Finite Group DatabasebetaFinite Group DatabasebetaOrder 7232.
84 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #16813696 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | C2×#14268650 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #16813629 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #14268650 | C22 | 1 | 1 | yes | characteristic |
| 5 | #14268650 | C22 | 1 | 3 | yes | — |
| 6 | #14268631 | C4 | 1 | 3 | yes | — |
| 7 | C2×#11744751 | C22 | 1 | 3 | yes | — |
| 8 | #41×C113 | C4 | 1 | 1 | yes | characteristic |
| 9 | Q8×C113 | C8 | 1 | 1 | yes | characteristic |
| 10 | D4×C113 | C8 | 1 | 3 | yes | — |
| 11 | C2×C4×C113 | C2×C4 | 1 | 3 | yes | — |
| 12 | #11744751 | C2×C4 | 1 | 3 | yes | — |
| 13 | #11744751 | C23 | 1 | 1 | yes | characteristic |
| 14 | C22×C113 | C2×C8 | 1 | 3 | yes | — |
| 15 | C4×C113 | C2×C8 | 1 | 3 | yes | — |
| 16 | C4×C113 | C22×C4 | 1 | 1 | yes | characteristic |
| 17 | C2×C113 | — | 2 | 3 | no | — |
| 18 | C2×C113 | C22×C8 | 1 | 1 | yes | characteristic |
| 19 | C113 | #504 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 20 | #504 | — | 113 | 1 | no | maximal, Sylow |
| 21 | #131 | — | 113 | 1 | no | — |
| 22 | M32 | — | 113 | 3 | no | — |
| 23 | C2×C16 | — | 113 | 3 | no | — |
| 24 | #41 | #90673 | 1 | 1 | yes | characteristic |
| 25 | M16 | — | 113 | 3 | no | — |
| 26 | C2×C8 | — | 113 | 3 | no | — |
| 27 | C16 | — | 113 | 1 | no | — |
| 28 | C16 | — | 113 | 3 | no | — |
| 29 | Q8 | #11744753 | 1 | 1 | yes | characteristic |
| 30 | D4 | #11744753 | 1 | 3 | yes | — |
| 31 | C2×C4 | C2×#90673 | 1 | 3 | yes | — |
| 32 | C8 | — | 113 | 1 | no | — |
| 33 | C8 | — | 113 | 3 | no | — |
| 34 | C22 | C2×#11744753 | 1 | 3 | yes | — |
| 35 | C4 | C2×#11744753 | 1 | 3 | yes | — |
| 36 | C4 | C22×#90673 | 1 | 1 | yes | characteristic |
| 37 | C2 | — | 2 | 3 | no | — |
| 38 | C2 | C22×#11744753 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C22×#90673 |
| commutator, socle | C2×C113 | C22×C8 |
| Fitting | #41×C113 | C4 |
| hypercenter | #41 | #90673 |
| hypocenter | C113 | #504 |
| LCS | G ▷ C2×C113 ▷ C113 | |
| UCS | C1 ◁ C4 ◁ #41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16813696 | C2 | semidirect |
| 2. C2×#14268650 | C2 | semidirect |
| 19. C113 | #504 | semidirect |