Finite Group DatabasebetaFinite Group DatabasebetaOrder 7232.
334 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 103 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 | #120×C113 | C2 |
| hypercenter | #120 | D113 |
| hypocenter | C113 | #535 |
| LCS | G ▷ C22×C113 ▷ C113 | |
| UCS | C1 ◁ C22 ◁ #120 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#14268646 | C2 | semidirect |
| 2. C22×C22⋊D113 | C2 | semidirect |
| 3. #16813618 | C2 | semidirect |
| 4. #16813608 | C2 | semidirect |
| 5. #16813595 | C2 | semidirect |
| 6. #16813594 | C2 | semidirect |
| 7. #16813592 | C2 | semidirect |
| 8. #16813591 | C2 | semidirect |
| 9. #16813590 | C2 | semidirect |
| 10. #16813578 | C2 | semidirect |
| 11. #120×C113 | C2 | semidirect |
| 14. C2×Dic226 | C22 | semidirect |
| 15. C2×D452 | C22 | semidirect |
| 20. C2×C22⋊D113 | C22 | semidirect |
| 21. C2×C22⋊D113 | C22 | semidirect |
| 22. C2×C22⋊D113 | C22 | semidirect |
| 24. C22×Dic113 | C22 | semidirect |
| 25. C2×C4×D113 | C22 | semidirect |
| 26. #14268626 | C22 | semidirect |
| 28. C22⋊Dic113 | C22 | semidirect |
| 29. #14268624 | C22 | semidirect |
| 30. #14268624 | C22 | semidirect |
| 31. C4×Dic113 | C22 | semidirect |
| 34. C2×D4×C113 | C22 | semidirect |
| 36. C22⋊C4×C113 | C22 | semidirect |
| 57. C2×Dic113 | D4 | semidirect |
| 93. C113 | #535 | semidirect |
| 104. #120 | D113 | semidirect |
| 119. C2×D4 | C2×D113 | semidirect |
| 129. C22⋊C4 | C2×D113 | semidirect |