Finite Group DatabasebetaFinite Group DatabasebetaOrder 25408.
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×D397 |
| commutator, socle, cosocle | C22×C397 | C24 |
| Fitting | #126×C397 | C2 |
| prime radical | #27163069 | C2 |
| hypercenter | #126 | D397 |
| hypocenter | C397 | #529 |
| LCS | G ▷ C22×C397 ▷ C397 | |
| UCS | C1 ◁ C22 ◁ #126 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #27163090 | C2 | semidirect |
| 2. #27163086 | C2 | semidirect |
| 3. #27163082 | C2 | semidirect |
| 4. #27163081 | C2 | semidirect |
| 5. #27163078 | C2 | semidirect |
| 6. #27163074 | C2 | semidirect |
| 8. #27163061 | C2 | semidirect |
| 9. #27163058 | C2 | semidirect |
| 10. #27163056 | C2 | semidirect |
| 11. #27163056 | C2 | semidirect |
| 12. #27163054 | C2 | semidirect |
| 13. #27163052 | C2 | semidirect |
| 14. C42×D397 | C2 | semidirect |
| 15. #126×C397 | C2 | semidirect |
| 103. C397 | #529 | semidirect |
| 106. #126 | D397 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #27163090 | 2 |
| #27163086 | 2 |
| #27163082 | 2 |
| #27163081 | 2 |
| #27163078 | 2 |
| #27163074 | 2 |
| #27163069 | 2 |
| #27163061 | 2 |
| #27163058 | 2 |
| #27163056 | 2 |
| #27163054 | 2 |
| #27163052 | 2 |
| C42×D397 | 2 |
| #126×C397 | 2 |
| #529 | 397 |