Finite Group DatabasebetaFinite Group DatabasebetaOrder 25216.
996 conjugacy classes of subgroups; only the 509 normal subgroups are shown.
This group has 996 conjugacy classes of subgroups — too many to draw. Only its 509 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 | C24×D197 |
| commutator, socle, cosocle | C22×C197 | C25 |
| Fitting | #543×C197 | C2 |
| prime radical | C24×C197 | C23 |
| hypercenter | #543 | D197 |
| hypocenter | C197 | #3458 |
| LCS | G ▷ C22×C197 ▷ C197 | |
| UCS | C1 ◁ C22 ◁ #543 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #27161708 | C2 | semidirect |
| 3. #27161631 | C2 | semidirect |
| 7. #20263325 | C22 | semidirect |
| 8. #20263324 | C22 | semidirect |
| 9. #20263316 | C22 | semidirect |
| 12. Q8×Dic197 | C22 | semidirect |
| 13. #20263291 | C22 | semidirect |
| 22. C22×Dic197 | Q8 | semidirect |
| 61. C197 | #3458 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #27161708 | 2 |
| #27161705 | 2 |
| #27161631 | 2 |
| C2×#20263282 | 2 |
| #543×C197 | 2 |
| #3458 | 197 |