Finite Group DatabasebetaFinite Group DatabasebetaOrder 25216.
568 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 568 conjugacy classes of subgroups — too many to draw. Only its 147 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D197 |
| commutator, socle, cosocle | C23×C197 | C24 |
| Fitting | #393×C197 | C2 |
| prime radical | C22×C22⋊D197 | C22 |
| hypercenter | #393 | D197 |
| hypocenter | C197 | #2398 |
| LCS | G ▷ C23×C197 ▷ C197 | |
| UCS | C1 ◁ C23 ◁ #393 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×Q8×Dic197 | C2 | semidirect |
| 3. #27161021 | C2 | semidirect |
| 4. #27161006 | C2 | semidirect |
| 5. #27161004 | C2 | semidirect |
| 7. #27160990 | C2 | semidirect |
| 8. C22⋊C4×Dic197 | C2 | semidirect |
| 9. #27160926 | C2 | semidirect |
| 10. #27160924 | C2 | semidirect |
| 11. #393×C197 | C2 | semidirect |
| 138. C197 | #2398 | semidirect |
| 147. #393 | D197 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q8×Dic197 | 2 |
| C2×#20263286 | 2 |
| #27161021 | 2 |
| #27161006 | 2 |
| #27161004 | 2 |
| #27160995 | 2 |
| #27160990 | 2 |
| C22⋊C4×Dic197 | 2 |
| #27160926 | 2 |
| #27160924 | 2 |
| #393×C197 | 2 |
| #2398 | 197 |