Finite Group DatabasebetaFinite Group DatabasebetaOrder 7232.
334 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 107 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 | C22⋊Q8×C113 | C2 |
| hypercenter | C22⋊Q8 | D113 |
| hypocenter | C113 | #546 |
| LCS | G ▷ C22×C113 ▷ C113 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#14268648 | C2 | semidirect |
| 2. C22×D452 | C2 | semidirect |
| 3. #16813611 | C2 | semidirect |
| 4. #16813607 | C2 | semidirect |
| 5. #16813597 | C2 | semidirect |
| 6. #16813597 | C2 | semidirect |
| 7. #16813592 | C2 | semidirect |
| 8. #16813584 | C2 | semidirect |
| 9. C4×C22⋊D113 | C2 | semidirect |
| 10. C22⋊C4×D113 | C2 | semidirect |
| 11. C22⋊Q8×C113 | C2 | semidirect |
| 18. C2×D452 | C22 | semidirect |
| 19. C2×D452 | C22 | semidirect |
| 21. C2×C4×D113 | C22 | semidirect |
| 22. C2×C4×D113 | C22 | semidirect |
| 23. C2×C4×D113 | C22 | semidirect |
| 24. #14268626 | C22 | semidirect |
| 26. #14268624 | C22 | semidirect |
| 27. #14268624 | C22 | semidirect |
| 28. #14268624 | C22 | semidirect |
| 29. #14268624 | C22 | semidirect |
| 30. #14268624 | C22 | semidirect |
| 31. C4×Dic113 | C22 | semidirect |
| 32. C2×Q8×C113 | C22 | semidirect |
| 34. C4⋊C4×C113 | C22 | semidirect |
| 35. C4⋊C4×C113 | C22 | semidirect |
| 58. D452 | D4 | semidirect |
| 64. C22×C113 | #41 | semidirect |
| 90. C113 | #546 | semidirect |
| 96. C22⋊Q8 | D113 | semidirect |
| 105. C2×Q8 | C2×D113 | semidirect |
| 118. C4⋊C4 | C2×D113 | semidirect |
| 119. C4⋊C4 | C2×D113 | semidirect |
| 165. C22 | #14268648 | semidirect |