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 | #121×C509 | C2 |
| hypercenter | #121 | D509 |
| hypocenter | C509 | #538 |
| LCS | G ▷ C22×C509 ▷ C509 | |
| UCS | C1 ◁ C22 ◁ #121 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #30654210 | C2 | semidirect |
| 2. #30654201 | C2 | semidirect |
| 3. #30654199 | C2 | semidirect |
| 4. #30654194 | C2 | semidirect |
| 5. #30654187 | C2 | semidirect |
| 6. #30654183 | C2 | semidirect |
| 7. D4×Dic509 | C2 | semidirect |
| 8. C2×#22550838 | C2 | semidirect |
| 9. #121×C509 | C2 | semidirect |
| 11. C2×Dic1018 | C22 | semidirect |
| 15. C22×Dic509 | C22 | semidirect |
| 16. C4⋊Dic509 | C22 | semidirect |
| 17. C4⋊Dic509 | C22 | semidirect |
| 18. #22550840 | C22 | semidirect |
| 20. C22⋊Dic509 | C22 | semidirect |
| 25. C4×Dic509 | C22 | semidirect |
| 27. C2×D4×C509 | C22 | semidirect |
| 29. C4⋊C4×C509 | C22 | semidirect |
| 30. C22⋊C4×C509 | C22 | semidirect |
| 74. C509 | #538 | semidirect |
| 81. #121 | D509 | semidirect |
| 90. C2×D4 | C2×D509 | semidirect |
| 97. C4⋊C4 | C2×D509 | semidirect |
| 104. C22⋊C4 | C2×D509 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #30654210 | 2 |
| #30654201 | 2 |
| #30654199 | 2 |
| #30654194 | 2 |
| #30654187 | 2 |
| #30654183 | 2 |
| D4×Dic509 | 2 |
| C2×#22550838 | 2 |
| #121×C509 | 2 |
| #538 | 509 |