Finite Group DatabasebetaFinite Group DatabasebetaOrder 32576.
218 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 218 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 | #30654188 | C2 |
| hypercenter | #126 | D509 |
| hypocenter | C509 | #539 |
| 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. #30654210 | C2 | semidirect |
| 2. #30654207 | C2 | semidirect |
| 3. #30654206 | C2 | semidirect |
| 4. #30654200 | C2 | semidirect |
| 5. #30654199 | C2 | semidirect |
| 6. #30654197 | C2 | semidirect |
| 7. #30654197 | C2 | semidirect |
| 8. #30654195 | C2 | semidirect |
| 9. #30654193 | C2 | semidirect |
| 10. #30654192 | C2 | semidirect |
| 12. C4×D2036 | C2 | semidirect |
| 13. #30654175 | C2 | semidirect |
| 14. C4⋊C4×D509 | C2 | semidirect |
| 15. #126×C509 | C2 | semidirect |
| 103. C509 | #539 | semidirect |
| 106. #126 | D509 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #30654210 | 2 |
| #30654207 | 2 |
| #30654206 | 2 |
| #30654200 | 2 |
| #30654199 | 2 |
| #30654197 | 2 |
| #30654195 | 2 |
| #30654193 | 2 |
| #30654192 | 2 |
| #30654188 | 2 |
| C4×D2036 | 2 |
| #30654175 | 2 |
| C4⋊C4×D509 | 2 |
| #126×C509 | 2 |
| #539 | 509 |