Finite Group DatabasebetaFinite Group DatabasebetaOrder 32576.
250 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 95 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 | C2×C22⋊C4×C509 | C2 |
| hypercenter | C2×C22⋊C4 | D509 |
| hypocenter | C509 | #538 |
| LCS | G ▷ C22×C509 ▷ C509 | |
| UCS | C1 ◁ C22 ◁ C2×C22⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #30654208 | C2 | semidirect |
| 2. #30654201 | C2 | semidirect |
| 3. #30654197 | C2 | semidirect |
| 4. #30654194 | C2 | semidirect |
| 5. #30654189 | C2 | semidirect |
| 6. #30654188 | C2 | semidirect |
| 7. #30654184 | C2 | semidirect |
| 8. C4×C22⋊D509 | C2 | semidirect |
| 9. C2×C22⋊C4×C509 | C2 | semidirect |
| 10. C2×Dic1018 | C22 | semidirect |
| 11. C2×D2036 | C22 | semidirect |
| 15. C2×C4×D509 | C22 | semidirect |
| 16. C4⋊Dic509 | C22 | semidirect |
| 17. C4⋊Dic509 | C22 | semidirect |
| 18. #22550840 | C22 | semidirect |
| 19. #22550840 | C22 | semidirect |
| 23. #22550838 | C22 | semidirect |
| 24. #22550838 | C22 | semidirect |
| 25. C4×Dic509 | C22 | semidirect |
| 74. C509 | #538 | semidirect |
| 84. C2×C22⋊C4 | D509 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #30654208 | 2 |
| #30654201 | 2 |
| #30654197 | 2 |
| #30654194 | 2 |
| #30654189 | 2 |
| #30654188 | 2 |
| #30654184 | 2 |
| C4×C22⋊D509 | 2 |
| C2×C22⋊C4×C509 | 2 |
| #538 | 509 |