Finite Group DatabasebetaFinite Group DatabasebetaOrder 7232.
258 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 258 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 | C2 | C22×C22⋊D113 |
| commutator | C4×C113 | C24 |
| Frattini | C4 | C23×D113 |
| Fitting | 2−1+4×C113 | C2 |
| socle | C2×C113 | C22×D4 |
| hypercenter | 2−1+4 | D113 |
| hypocenter | C113 | #577 |
| LCS | G ▷ C4×C113 ▷ C2×C113 ▷ C113 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ 2−1+4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16813668 | C2 | semidirect |
| 2. #16813653 | C2 | semidirect |
| 3. #16813647 | C2 | semidirect |
| 4. #16813642 | C2 | semidirect |
| 5. C2×#14268634 | C2 | semidirect |
| 6. #16813629 | C2 | semidirect |
| 7. 2−1+4×C113 | C2 | semidirect |
| 12. #14268637 | C22 | semidirect |
| 13. #14268635 | C22 | semidirect |
| 14. #14268634 | C22 | semidirect |
| 15. #14268634 | C22 | semidirect |
| 16. #14268632 | C22 | semidirect |
| 17. #14268631 | C22 | semidirect |
| 18. C2×#11744751 | C22 | semidirect |
| 21. #41×C113 | C22 | semidirect |
| 23. C2×Q8×C113 | C22 | semidirect |
| 53. C113 | #577 | semidirect |
| 55. 2−1+4 | D113 | semidirect |
| 66. #41 | C2×D113 | semidirect |
| 68. C2×Q8 | C2×D113 | semidirect |