Finite Group DatabasebetaFinite Group DatabasebetaOrder 23079.
40 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C72×C157 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | #15693074 | — | 7 | 8 | no | maximal |
| 3 | C7×C157 | ΓL(1,𝔽8) | 1 | 8 | yes | complemented |
| 4 | C157⋊C3 | — | 49 | 1 | no | Hall |
| 5 | C157 | #3861 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 6 | #3861 | — | 157 | 1 | no | maximal, Hall |
| 7 | C72 | C157⋊C3 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 8 | ΓL(1,𝔽8) | — | 1099 | 8 | no | — |
| 9 | C7 | #15693074 | 1 | 8 | yes | complemented |
| 10 | C3 | — | 7693 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C72×C157 | C3 |
| LCS | G ▷ C72×C157 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C72×C157 | C3 | semidirect |
| 3. C7×C157 | ΓL(1,𝔽8) | semidirect |
| 5. C157 | #3861 | semidirect |
| 7. C72 | C157⋊C3 | semidirect |
| 9. C7 | #15693074 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C72×C157 | 3 |
| #15693074 | 7 |
| #3861 | 157 |