Finite Group DatabasebetaFinite Group DatabasebetaOrder 32576.
192 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 192 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 | #126×C509 | C2 |
| prime radical | C23×C509 | C23 |
| hypercenter | #126 | D509 |
| hypocenter | C509 | #541 |
| LCS | G ▷ C22×C509 ▷ C509 | |
| UCS | C1 ◁ C22 ◁ #126 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #30654214 | C2 | semidirect |
| 5. #30654207 | C2 | semidirect |
| 6. #30654206 | C2 | semidirect |
| 7. #30654205 | C2 | semidirect |
| 8. #30654205 | C2 | semidirect |
| 9. #30654205 | C2 | semidirect |
| 12. C4×Dic1018 | C2 | semidirect |
| 13. #30654180 | C2 | semidirect |
| 91. C509 | #541 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #30654214 | 2 |
| #30654208 | 2 |
| #30654207 | 2 |
| #30654206 | 2 |
| #30654205 | 2 |
| #30654192 | 2 |
| C4×Dic1018 | 2 |
| #30654180 | 2 |
| #30654171 | 2 |
| #126×C509 | 2 |
| #541 | 509 |