Some shorthand notation used here:
- Sk stands for k^2/[(k-1)(k+1)] by standard convention (the kth square superparticular).
- Tk = Sk * S(k+1) stands for [k(k+1)/2]/[(k-1)(k+2)/2] (the kth triangle superparticular).
- Uk = Sk/S(k+1) stands for the kth ultraparticular, which has the same subgroup as Tk except in the case where k is congruent to 4 (mod 9), in which case the subgroup of Uk lacks prime 3 from that of Tk.
- Lp refers to the p-limit, i.e. the subgroup of primes less than or equal to p.
- Lp(-q) refers to the p limit with the prime q omitted: e.g. L17(-11) refers to the 2.3.5.7.13.17 subgroup; these omissions can be stacked so that L23(-5.17) refers to the group 2.3.7.11.13.19.23.
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.
2- and 3-prime subgroups (2.3 and 2.3.p)
Note that the following lists are complete and the insertion of higher primes will add no new inclusions to them.
2-prime subgroup (2.3)
| Square-particular
|
Subgroup
|
Comma
|
| Ratio
|
Smonzo
|
| S2
|
2.3
|
4/3
|
[2 -1⟩
|
| S3
|
2.3
|
9/8
|
[-3 2⟩
|
| Triangle-particular
|
Subgroup
|
Comma
|
| Ratio
|
Smonzo
|
| T2
|
2.3
|
3/2
|
[-1 1⟩
|
3-prime subgroups (2.3.p)
| Triangle-particular
|
Subgroup
|
Comma
|
| Ratio
|
Smonzo
|
| T3
|
L5
|
6/5
|
[1 1 -1⟩
|
| T4
|
L5
|
10/9
|
[1 -2 1⟩
|
| T7
|
2.3.7
|
28/27
|
[2 -3 1⟩
|
4-prime subgroups
Semi-5-limit (L5.p)
| Triangle-particular
|
Subgroup
|
Comma
|
| Ratio
|
Smonzo
|
| T5
|
L7
|
15/14
|
[-1 1 1 -1⟩
|
| T6
|
L7
|
21/20
|
[-2 1 -1 1⟩
|
| T8
|
L7
|
21/20
|
[-2 1 -1 1⟩
|
| T9
|
L5.11
|
45/44
|
[-2 2 1 -1⟩
|
| T10
|
L5.11
|
55/54
|
[-1 -3 1 1⟩
|
| T25
|
L5.13
|
325/324
|
[-2 -4 2 1⟩
|