Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
5796 conjugacy classes of subgroups; only the 3201 normal subgroups are shown.
This group has 5796 conjugacy classes of subgroups — too many to draw. Only its 3201 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C25×D251 |
| commutator, socle, cosocle | C2×C251 | C26 |
| Fitting | C2×2−1+4×C251 | C2 |
| hypercenter | C2×2−1+4 | D251 |
| hypocenter | C251 | 2−1+6 |
| LCS | G ▷ C2×C251 ▷ C251 | |
| UCS | C1 ◁ C2 ◁ C2×2−1+4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #30632524 | C2 | semidirect |
| 2. #30632523 | C2 | semidirect |
| 3. C2×#22548763 | C2 | semidirect |
| 4. 2−1+4×D251 | C2 | semidirect |
| 5. C2×2−1+4×C251 | C2 | semidirect |
| 6. #22548763 | C22 | semidirect |
| 7. #22548763 | C22 | semidirect |
| 8. #22548762 | C22 | semidirect |
| 9. #22548761 | C22 | semidirect |
| 10. #22548760 | C22 | semidirect |
| 11. C2×#16860811 | C22 | semidirect |
| 12. C2×#16860810 | C22 | semidirect |
| 13. #41×D251 | C22 | semidirect |
| 14. C2×Q8×D251 | C22 | semidirect |
| 15. 2−1+4×C251 | C22 | semidirect |
| 16. C2×#41×C251 | C22 | semidirect |
| 17. C22×Q8×C251 | C22 | semidirect |
| 56. C251 | 2−1+6 | semidirect |
| 62. C2×2−1+4 | D251 | semidirect |
| 66. 2−1+4 | C2×D251 | semidirect |
| 72. C2×#41 | C2×D251 | semidirect |
| 74. C22×Q8 | C2×D251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #30632524 | 2 |
| #30632523 | 2 |
| C2×#22548763 | 2 |
| 2−1+4×D251 | 2 |
| C2×2−1+4×C251 | 2 |
| 2−1+6 | 251 |