Some shorthand notation used here:
- Sk stands for k^2/[(k-1)(k+1)] by standard convention (the kth square superparticular).
- U(9k+4) = S(9k+4)/S(9k+5) stands for the (9k+4)th ultraparticular.
- Gk stands for S(k-1)*Sk*S(k+1) (the kth third-particular).
- R(3k) stands for S(3k-1)/S(3k+1) (the 3kth semiparticular).
- Hk = Gk * G(k+3) stands for the kth sixth-particular.
- Wp refers to the no-threes p-limit, i.e. the subgroup of primes less than or equal to p.
- Wp(-q) refers to the no-threes p limit with the prime q omitted: e.g. W17(-11) refers to the 2.5.7.13.17 subgroup; these omissions can be stacked so that W23(-5.17) refers to the group 2.7.11.13.19.23.
Both Gk and Hk can be superparticular or throdd-particular: specifically, G(3k) and H(3k) are throdd-particular, while G(9k+4) and G(9k+5), and H(9k+7) and H(9k+8), are superparticular but still no-threes. To each throdd-particular G(3k) corresponds a no-threes semiparticular R(3k). Because of this difference, superparticular and throdd-particular commas are listed in separate tables.
Note that not all members of R(3k) are superparticular. In specific, R(12k) is odd-particular. Members of this set, lacking both twos and threes, are included in this sheet. No-twos-or-threes subgroups will be labeled in bold.
This list eventually aims to be complete to the no-threes 19-add-two-limit and 31-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 37-limit, and the class of subgroups with at most two primes greater than 19, which is a superset of the 29-limit.
2- and 3-prime subgroups
Note that the following lists are complete and the insertion of higher primes will add no new inclusions to them.
2-prime subgroups (2.p)
| Third-particular
|
Subgroup
|
Comma
|
| Ratio
|
Smonzo
|
| G4 = R3
|
2.5
|
5/4
|
[-2 1⟩
|
| G5
|
2.7
|
8/7
|
[3 -1⟩
|
| Throdd-particular
|
Subgroup
|
Comma
|
Semiparticular
|
Subgroup
|
Comma
|
| Ratio
|
Smonzo
|
Ratio
|
Smonzo
|
| G3
|
2.5
|
8/5
|
[3 -1⟩
|
R3 = G4
|
2.5
|
5/4
|
[-2 1⟩
|
| Sixth-particular
|
Subgroup
|
Comma
|
| Ratio
|
Smonzo
|
| H3
|
2.7
|
7/4
|
[-2 1⟩
|
| Ultraparticular
|
Subgroup
|
Comma
|
| Ratio
|
Smonzo
|
| U4
|
2.5
|
128/125
|
[7 -3⟩
|
3-prime subgroups
| Third-particular
|
Subgroup
|
Comma
|
| Ratio
|
Smonzo
|
| G14
|
2.5.13
|
65/64
|
[-6 1 1⟩
|
| Throdd-particular
|
Subgroup
|
Comma
|
Semiparticular
|
Subgroup
|
Comma
|
| Ratio
|
Smonzo
|
Ratio
|
Smonzo
|
| G6
|
2.5.7
|
35/32
|
[-5 1 1⟩
|
R6 = H16
|
2.5.7
|
50/49
|
[1 2 -2⟩
|
| Sixth-particular
|
Subgroup
|
Comma
|
| Ratio
|
Smonzo
|
| H6
|
2.5.11
|
25/22
|
[-1 2 -1⟩
|
| H9
|
2.7.13
|
52/49
|
[2 -2 1⟩
|
| Sixth-particular
|
Subgroup
|
Comma
|
| Ratio
|
Smonzo
|
| H16 = R6
|
2.5.7
|
50/49
|
[1 2 -2⟩
|
| H7
|
2.5.11
|
11/10
|
[1 1 -1⟩
|
| H26
|
2.5.31
|
125/124
|
[-2 3 -1⟩
|
| H8
|
2.7.13
|
14/13
|
[-1 -1 1⟩
|
| H44
|
2.7.43
|
344/343
|
[3 -3 1⟩
|
4-prime subgroups
5-add-two-limit (2.5.p.q)
7-add-one-limit (2.5.7.p)
| Third-particular
|
Subgroup
|
Comma
|
| Ratio
|
Smonzo
|
| G13
|
W11
|
56/55
|
[3 -1 1 -1⟩
|
| G23 = R9
|
W11
|
176/175
|
[4 -2 -1 1⟩
|
| G22
|
2.5.7.23
|
161/160
|
[-5 -1 1 1⟩
|
| Throdd-particular
|
Subgroup
|
Comma
|
Semiparticular
|
Subgroup
|
Comma
|
| Ratio
|
Smonzo
|
Ratio
|
Smonzo
|
| G9
|
W11
|
80/77
|
[4 1 -1 -1⟩
|
R9 = G23
|
W11
|
176/175
|
[4 -2 -1 1⟩
|
Higher-prime subgroups
| Third-particular
|
Subgroup
|
Comma
|
| Ratio
|
Smonzo
|
| G31
|
2.5.11.29
|
320/319
|
[6 1 -1 -1⟩
|
| G49
|
2.5.17.47
|
800/799
|
[5 2 -1 -1⟩
|
| Throdd-particular
|
Subgroup
|
Comma
|
Semiparticular
|
Subgroup
|
Comma
|
| Ratio
|
Smonzo
|
Ratio
|
Smonzo
|
| G18
|
2.5.17.19
|
323/320
|
[-6 -1 1 1⟩
|
R18
|
2.5.17.19
|
1445/1444
|
[-2 1 2 -2⟩
|
No-fives subgroups
7-add-two-limit (2.7.p.q)
| Third-particular
|
Subgroup
|
Comma
|
| Ratio
|
Smonzo
|
| G50 = R15
|
2.7.13.17
|
833/832
|
[-6 2 -1 1⟩
|
| Throdd-particular
|
Subgroup
|
Comma
|
Semiparticular
|
Subgroup
|
Comma
|
| Ratio
|
Smonzo
|
Ratio
|
Smonzo
|
| G15
|
2.7.13.17
|
224/221
|
[5 1 -1 -1⟩
|
R15 = G50
|
2.7.13.17
|
833/832
|
[-6 2 -1 1⟩
|
| G30
|
2.7.29.31
|
899/896
|
[-7 -1 1 1⟩
|
R30
|
2.7.29.31
|
6728/6727
|
[3 -1 2 -2⟩
|
See also