Finite Group DatabasebetaFinite Group DatabasebetaOrder 7232.
240 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 93 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 | #540 |
| 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. #16813618 | C2 | semidirect |
| 2. #16813612 | C2 | semidirect |
| 3. #16813607 | C2 | semidirect |
| 4. #16813606 | C2 | semidirect |
| 5. #16813604 | C2 | semidirect |
| 6. #16813597 | C2 | semidirect |
| 7. #16813596 | C2 | semidirect |
| 8. #16813595 | C2 | semidirect |
| 9. #16813595 | C2 | semidirect |
| 10. #16813595 | C2 | semidirect |
| 11. #16813594 | C2 | semidirect |
| 12. C4×C22⋊D113 | C2 | semidirect |
| 13. #16813582 | C2 | semidirect |
| 14. #16813580 | C2 | semidirect |
| 15. #123×C113 | C2 | semidirect |
| 115. C113 | #540 | semidirect |
| 122. #123 | D113 | semidirect |