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Note that not all members of G''k'' and R''k'' are superparticular. In particular, G(3''k'') is throdd-particular, and R(4''k'') is odd-particular. Such ratios will be excluded from consideration in this chart, though they will appear on companion no-twos and no-threes pages.
Note that not all members of G''k'' and R''k'' are superparticular. In particular, G(3''k'') is throdd-particular, and R(4''k'') is odd-particular. Such ratios will be excluded from consideration in this chart, though they will appear on companion no-twos and no-threes pages.


This list eventually aims to be complete to the 29-add-one-limit, i.e. the class of subgroups with at most one prime greater than 29, which is a superset of the 31-limit.
This list eventually aims to be complete to the 17-add-two-limit and the 29-add-one-limit, i.e. the union of the class of subgroups with at most one prime greater than 29, which is a superset of the 31-limit, and the class of subgroups with at most two primes greater than 17, which is a superset of the 23-limit.


== 2- and 3-prime subgroups (2.p and 2.3.p) ==
== 2- and 3-prime subgroups (2.p, 2.3.p, and 2.5.p) ==
Note that the following lists are ''complete'' and the insertion of higher primes will add no new inclusions to them.
Note that the following lists are ''complete'' and the insertion of higher primes will add no new inclusions to them.


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|-
|-
| G4 = R3
| G4 = R3
| '''2.5'''
| 2.5
| [[5/4]]
| [[5/4]]
| {{monzo| -2 1 }}
| {{monzo| -2 1 }}
|-
|-
| G5
| G5
| '''2.7'''
| 2.7
| [[8/7]]
| [[8/7]]
| {{monzo| 3 -1 }}
| {{monzo| 3 -1 }}
Line 78: Line 78:
|}
|}


== 3-prime no-threes and 4-prime subgroups ==
=== 3-prime subgroups (2.5.p) ===
{| class="wikitable center-1 center-2"
|-
! rowspan="2" | Superparticular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
|-
| R6
| 2.5.7
| [[50/49]]
| {{monzo| 1 2 -2 }}
|-
| G14
| 2.5.13
| [[65/64]]
| {{monzo| -6 1 1 }}
|}
 
== 4-prime subgroups with threes ==
Note that the following lists are ''complete'' and the insertion of higher primes will add no new inclusions to them.
 
=== 5-add-one-limit (L5.p) ===
{| class="wikitable center-1 center-2 center-5 center-6"
|-
! rowspan="2" | Third-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
! rowspan="2" | Semiparticular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
! Ratio
! Smonzo
|-
| G8 = T6
| [[7-limit|L7]]
| [[21/20]]
| {{monzo| -2 1 -1 1 }}
|
|
|
|
|-
| G26 = S15
| [[7-limit|L7]]
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| R26
| [[7-limit|L7]]
| [[4375/4374]]
| {{monzo| -1 -7 4 1 }}
|-
| G11
| L5.13
| [[40/39]]
| {{monzo| 3 -1 1 -1 }}
| R11 = T25
| L5.13
| [[325/324]]
| {{monzo| -2 -4 2 1 }}
|-
|
|
|
|
| R14 = S26
| L5.13
| [[676/675]]
| {{monzo| 2 -3 -2 2 }}
|-
| G17
| L5.19
| [[96/95]]
| {{monzo| 5 1 -1 -1 }}
| R17
| L5.19
| [[1216/1215]]
| {{monzo| 6 -5 -1 1 }}
|}
 
=== 2.3.13.p subgroups ===
{| class="wikitable center-1 center-2 center-5 center-6"
|-
! rowspan="2" | Third-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
! rowspan="2" | Semiparticular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
! Ratio
! Smonzo
|-
| G25
| 2.3.13.23
| [[208/207]]
| {{monzo| 4 -2 1 -1 }}
| R25
| 2.3.13.23
| [[3888/3887]]
| {{monzo| 4 5 -2 -1 }}
|}
 
== 4-prime no-threes subgroups and 5-prime subgroups ==
In the tables that follow, no-threes subgroups will be indicated in '''bold'''.
 
=== 5-add-two-limit (L5.p.q) ===
 
==== 7-add-one-limit (L7.p) ====
{| class="wikitable center-1 center-2 center-5 center-6"
|-
! rowspan="2" | Third-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
! rowspan="2" | Semiparticular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
! Ratio
! Smonzo
|-
|
|
|
|
| R9 = G23
| '''L11(-3)'''
| [[176/175]]
| {{monzo| 4 -2 -1 1 }}
|-
| G13
| '''L11(-3)'''
| [[56/55]]
| {{monzo| 3 -1 1 -1 }}
| R13
| [[11-limit|L11]]
| [[540/539]]
| {{monzo| 2 3 1 -2 -1 }}
|-
| G23 = R9
| '''L11(-3)'''
| [[176/175]]
| {{monzo| 4 -2 -1 1 }}
| R23 = S55
| [[11-limit|L11]]
| [[3025/3024]]
| {{monzo| -4 -3 2 -1 2 }}
|-
| G34
| [[11-limit|L11]]
| [[385/384]]
| {{monzo| -7 -1 1 1 1 }}
| R34 = S99
| [[11-limit|L11]]
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
|-
| G16
| L7.17
| [[85/84]]
| {{monzo| -2 -1 1 -1 1 }}
|
|
|
|
|-
| G19 = T15
| L7.17
| [[120/119]]
| {{monzo| 3 1 1 -1 -1 }}
| R19
| L7.17
| [[1701/1700]]
| {{monzo| -2 5 -2 1 -1 }}
|-
| G22
| '''2.5.7.23'''
| [[161/160]]
| {{monzo| -5 -1 1 1 }}
| R22
| L7.23
| [[2646/2645]]
| {{monzo| 1 3 -1 2 -2 }}
|-
| G47
| L7.23
| [[736/735]]
| {{monzo| 5 -1 -1 -2 1 }}
| R47 = S161
| L7.23
| [[25921/25920]]
| {{monzo| -6 -4 -1 2 2 }}
|-
| G29
| L7.31
| [[280/279]]
| {{monzo| 3 -2 1 1 -1 }}
| R29
| L7.31
| [[6076/6075]]
| {{monzo| 2 -2 -5 2 1 }}
|-
| G62
| L7.61
| [[1281/1280]]
| {{monzo| -8 1 -1 1 1 }}
| R62 = S244
| L7.61
| [[59536/59535]]
| {{monzo| 4 -5 -1 -2 2 }}
|-
| G82
| L7.83
| [[2241/2240]]
| {{monzo| -6 3 -1 -1 1 }}
| R82
| L7.83
| [[137781/137780]]
| {{monzo| -2 9 -1 1 -2 }}
|}
 
==== L5.11.p subgroups ====
{| class="wikitable center-1 center-2 center-5 center-6"
|-
! rowspan="2" | Third-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
! rowspan="2" | Semiparticular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
! Ratio
! Smonzo
|-
| G31
| '''2.5.11.29'''
| [[320/319]]
| {{monzo| 6 1 -1 -1 }}
| R31
| L5.11.29
| [[7425/7424]]
| {{monzo| -8 3 2 1 -1 }}
|-
| G46
| L5.11.47
| [[705/704]]
| {{monzo| -6 1 1 -1 1 }}
| R46
| L5.11.47
| [[24300/24299]]
| {{monzo| 2 5 2 -1 -2 }}
|-
| G98
| L5.11.97
| [[3201/3200]]
| {{monzo| -7 1 -2 1 1 }}
| R98 = S485
| L5.11.97
| [[235225/235224]]
| {{monzo| -3 -5 2 -2 2 }}
|-
| G241
| L5.11.239
| [[19360/19359]]
| {{monzo| 5 -4 1 2 -1 }}
| R241
| L5.11.239
| [[3499200/3499199]]
| {{monzo| 6 7 2 -4 -1 }}
|}
 
==== L5.13.p subgroups ====
{| class="wikitable center-1 center-2 center-5 center-6"
|-
! rowspan="2" | Third-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
! rowspan="2" | Semiparticular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
! Ratio
! Smonzo
|-
| G28
| L5.13.29
| [[261/260]]
| {{monzo| -2 2 -1 -1 1 }}
|
|
|
|
|-
| G38
| L5.13.37
| [[481/480]]
| {{monzo| -5 -1 -1 1 1 }}
| R38
| L5.13.37
| [[13690/13689]]
| {{monzo| 1 -4 1 -2 2 }}
|}
 
==== L5.17.p subgroups ====
{| class="wikitable center-1 center-2 center-5 center-6"
|-
! rowspan="2" | Third-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
! rowspan="2" | Semiparticular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
! Ratio
! Smonzo
|-
|
|
|
|
| R18
| '''2.5.17.19'''
| [[1445/1444]]
| {{monzo| -2 1 2 -2 }}
|-
| G49
| '''2.5.17.47'''
| [[800/799]]
| {{monzo| 5 2 -1 -1 }}
| R49
| L5.17.47
| [[29376/29375]]
| {{monzo| 6 3 -4 1 -1 }}
|-
| G52
| L5.17.53
| [[901/900]]
| {{monzo| -2 -2 -2 1 1 }}
|
|
|
|
|}
 
==== Higher-prime subgroups ====
{| class="wikitable center-1 center-2 center-5 center-6"
|-
! rowspan="2" | Third-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
! rowspan="2" | Semiparticular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
! Ratio
! Smonzo
|-
| G74
| L5.19.73
| [[1825/1824]]
| {{monzo| -3 -2 2 -1 1 }}
| R74
| L5.19.73
| [[101251/101250]]
| {{monzo| -1 -4 -4 1 2 }}
|-
| G73
| L5.37.71
| [[1776/1775]]
| {{monzo| 4 1 -2 1 -1 }}
| R73
| L5.37.71
| [[97200/97199]]
| {{monzo| 4 5 2 -2 -1 }}
|-
| G161
| L5.53.163
| [[8640/8639]]
| {{monzo| 6 3 1 -1 -1 }}
| R161
| L5.53.163
| [[1043200/1043199]]
| {{monzo| 8 -9 2 -1 1 }}
|-
| G242
| L5.61.241
| [[19521/19520]]
| {{monzo| -6 4 -1 -1 1 }}
| R242
| L5.61.241
| [[3542941/3542940]]
| {{monzo| -2 -11 -1 1 2 }}
|}
 
=== No-fives subgroups ===
 
==== 7-add-two-limit (2.3.7.p.q) ====


=== 5-add-one-limit (2.5.p and L5.p) ===
==== Higher primes ====


== See also ==
== See also ==
* [[User:Lériendil/Square_and_triangle_superparticulars_by_prime_subgroup|Square and triangle superparticulars by prime subgroup]]
* [[User:Lériendil/Square_and_triangle_superparticulars_by_prime_subgroup|Square and triangle superparticulars by prime subgroup]]
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