Finite Group DatabasebetaFinite Group DatabasebetaOrder 7712.
12 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #16838773 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #14515349 | C4 | 1 | 1 | yes | characteristic |
| 3 | C4×C241 | C8 | 1 | 1 | yes | characteristic |
| 4 | C2×C241 | C16 | 1 | 1 | yes | characteristic |
| 5 | C241 | C32 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 6 | C32 | — | 241 | 1 | no | maximal, Sylow |
| 7 | C16 | — | 241 | 1 | no | — |
| 8 | C8 | — | 241 | 1 | no | — |
| 9 | C4 | #14515358 | 1 | 1 | yes | characteristic |
| 10 | C2 | #16838782 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | #14515358 |
| commutator, hypocenter | C241 | C32 |
| Fitting | C4×C241 | C8 |
| socle | C2×C241 | C16 |
| LCS | G ▷ C241 | |
| UCS | C1 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. C241 | C32 | semidirect |