Finite Group DatabasebetaFinite Group DatabasebetaOrder 7232.
292 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D113 |
| commutator, socle | C22×C113 | C24 |
| Fitting | #123×C113 | C2 |
| hypercenter | #123 | D113 |
| hypocenter | C113 | #547 |
| LCS | G ▷ C22×C113 ▷ C113 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#14268647 | C2 | semidirect |
| 2. C2×#14268646 | C2 | semidirect |
| 3. #16813621 | C2 | semidirect |
| 4. #16813607 | C2 | semidirect |
| 5. #16813606 | C2 | semidirect |
| 6. #16813604 | C2 | semidirect |
| 7. #16813600 | C2 | semidirect |
| 8. #16813599 | C2 | semidirect |
| 9. #16813595 | C2 | semidirect |
| 10. #16813594 | C2 | semidirect |
| 11. #16813582 | C2 | semidirect |
| 12. #16813582 | C2 | semidirect |
| 13. C2×#14268626 | C2 | semidirect |
| 14. C4⋊C4×D113 | C2 | semidirect |
| 15. #123×C113 | C2 | semidirect |
| 67. C22⋊D113 | D4 | semidirect |
| 107. Dic113 | #41 | semidirect |
| 115. C113 | #547 | semidirect |
| 123. #123 | D113 | semidirect |