Finite Group DatabasebetaFinite Group DatabasebetaOrder 23961.
12 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C49×C163 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | #16811961 | — | 7 | 1 | no | maximal |
| 3 | C7×C163 | ΓL(1,𝔽8) | 1 | 1 | yes | characteristic |
| 4 | C163⋊C3 | — | 49 | 1 | no | Hall |
| 5 | C163 | C49⋊C3 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 6 | C49⋊C3 | — | 163 | 1 | no | maximal, Hall |
| 7 | C49 | C163⋊C3 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 8 | ΓL(1,𝔽8) | — | 1141 | 1 | no | — |
| 9 | C7 | #16811961 | 1 | 1 | yes | characteristic |
| 10 | C3 | — | 7987 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C49×C163 | C3 |
| Frattini | C7 | #16811961 |
| socle | C7×C163 | ΓL(1,𝔽8) |
| LCS | G ▷ C49×C163 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C49×C163 | C3 | semidirect |
| 5. C163 | C49⋊C3 | semidirect |
| 7. C49 | C163⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C49×C163 | 3 |
| #16811961 | 7 |
| C49⋊C3 | 163 |