Finite Group DatabasebetaFinite Group DatabasebetaOrder 32576.
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×D509 |
| commutator, socle, cosocle | C22×C509 | C24 |
| Fitting | #123×C509 | C2 |
| hypercenter | #123 | D509 |
| hypocenter | C509 | #540 |
| LCS | G ▷ C22×C509 ▷ C509 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #30654211 | C2 | semidirect |
| 2. #30654205 | C2 | semidirect |
| 3. #30654200 | C2 | semidirect |
| 4. #30654199 | C2 | semidirect |
| 5. #30654197 | C2 | semidirect |
| 6. #30654190 | C2 | semidirect |
| 7. #30654189 | C2 | semidirect |
| 8. #30654188 | C2 | semidirect |
| 9. #30654188 | C2 | semidirect |
| 10. #30654188 | C2 | semidirect |
| 11. #30654187 | C2 | semidirect |
| 12. C4×C22⋊D509 | C2 | semidirect |
| 13. #30654175 | C2 | semidirect |
| 14. #30654173 | C2 | semidirect |
| 15. #123×C509 | C2 | semidirect |
| 115. C509 | #540 | semidirect |
| 122. #123 | D509 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #30654211 | 2 |
| #30654205 | 2 |
| #30654200 | 2 |
| #30654199 | 2 |
| #30654197 | 2 |
| #30654190 | 2 |
| #30654189 | 2 |
| #30654188 | 2 |
| #30654187 | 2 |
| C4×C22⋊D509 | 2 |
| #30654175 | 2 |
| #30654173 | 2 |
| #123×C509 | 2 |
| #540 | 509 |