Finite Group DatabasebetaFinite Group DatabasebetaOrder 7712.
40 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#14515356 | C2 | 1 | 2 | yes | maximal |
| 2 | C4×Dic241 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #14515356 | C4 | 1 | 4 | yes | complemented |
| 4 | C2×Dic241 | C4 | 1 | 1 | yes | characteristic |
| 5 | C2×Dic241 | C22 | 1 | 1 | yes | characteristic |
| 6 | C2×C4×C241 | C4 | 1 | 1 | yes | characteristic |
| 7 | C22×C241 | C2×C4 | 1 | 1 | yes | characteristic |
| 8 | C4×C241 | — | 2 | 1 | no | — |
| 9 | Dic241 | — | 2 | 1 | no | — |
| 10 | Dic241 | C2×C4 | 1 | 2 | yes | — |
| 11 | C2×C241 | C42 | 1 | 1 | yes | characteristic |
| 12 | C2×C241 | M16 | 1 | 1 | yes | characteristic |
| 13 | C2×C241 | M16 | 1 | 1 | yes | characteristic |
| 14 | C241 | #97 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 15 | #97 | — | 241 | 1 | no | maximal, Sylow |
| 16 | C2×C8 | — | 241 | 2 | no | — |
| 17 | C42 | — | 241 | 1 | no | — |
| 18 | C2×C4 | — | 241 | 1 | no | — |
| 19 | C2×C4 | — | 241 | 1 | no | — |
| 20 | C2×C4 | #11757210 | 1 | 1 | yes | characteristic |
| 21 | C8 | — | 241 | 4 | no | — |
| 22 | C22 | C2×#11757210 | 1 | 1 | yes | characteristic |
| 23 | C4 | — | 2 | 1 | no | — |
| 24 | C4 | — | 482 | 1 | no | — |
| 25 | C4 | — | 241 | 2 | no | — |
| 26 | C2 | C4×#11757210 | 1 | 1 | yes | characteristic |
| 27 | C2 | #16838779 | 1 | 1 | yes | characteristic |
| 28 | C2 | #16838780 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×#11757210 |
| commutator | C2×C241 | C42 |
| Fitting | C2×C4×C241 | C4 |
| socle | C22×C241 | C2×C4 |
| hypercenter | C2×C4 | #11757210 |
| hypocenter | C241 | #97 |
| LCS | G ▷ C2×C241 ▷ C241 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #14515356 | C4 | semidirect |
| 14. C241 | #97 | semidirect |