Finite Group DatabasebetaFinite Group DatabasebetaOrder 25664.
234 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 97 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×D401 |
| commutator, socle, cosocle | C22×C401 | C24 |
| Fitting | #126×C401 | C2 |
| prime radical | #27165710 | C2 |
| hypercenter | #126 | D401 |
| hypocenter | C401 | #529 |
| LCS | G ▷ C22×C401 ▷ C401 | |
| UCS | C1 ◁ C22 ◁ #126 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #27165731 | C2 | semidirect |
| 2. #27165727 | C2 | semidirect |
| 3. #27165723 | C2 | semidirect |
| 4. #27165722 | C2 | semidirect |
| 5. #27165719 | C2 | semidirect |
| 6. #27165715 | C2 | semidirect |
| 8. #27165702 | C2 | semidirect |
| 9. #27165699 | C2 | semidirect |
| 10. #27165697 | C2 | semidirect |
| 11. #27165697 | C2 | semidirect |
| 12. #27165695 | C2 | semidirect |
| 13. #27165693 | C2 | semidirect |
| 14. C42×D401 | C2 | semidirect |
| 15. #126×C401 | C2 | semidirect |
| 103. C401 | #529 | semidirect |
| 106. #126 | D401 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #27165731 | 2 |
| #27165727 | 2 |
| #27165723 | 2 |
| #27165722 | 2 |
| #27165719 | 2 |
| #27165715 | 2 |
| #27165710 | 2 |
| #27165702 | 2 |
| #27165699 | 2 |
| #27165697 | 2 |
| #27165695 | 2 |
| #27165693 | 2 |
| C42×D401 | 2 |
| #126×C401 | 2 |
| #529 | 401 |