Finite Group DatabasebetaFinite Group DatabasebetaOrder 7232.
14 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #16813528 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C16×C113 | C4 | 1 | 1 | yes | characteristic |
| 3 | C8×C113 | C8 | 1 | 1 | yes | characteristic |
| 4 | C4×C113 | C16 | 1 | 1 | yes | characteristic |
| 5 | C2×C113 | C32 | 1 | 1 | yes | characteristic |
| 6 | C113 | C64 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 7 | C64 | — | 113 | 1 | no | maximal, Sylow |
| 8 | C32 | — | 113 | 1 | no | — |
| 9 | C16 | #90673 | 1 | 1 | yes | characteristic |
| 10 | C8 | #11744753 | 1 | 1 | yes | characteristic |
| 11 | C4 | #14268650 | 1 | 1 | yes | characteristic |
| 12 | C2 | #16813672 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C16 | #90673 |
| commutator, hypocenter | C113 | C64 |
| Fitting | C16×C113 | C4 |
| socle | C2×C113 | C32 |
| LCS | G ▷ C113 | |
| UCS | C1 ◁ C16 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 6. C113 | C64 | semidirect |