Finite Group DatabasebetaFinite Group DatabasebetaOrder 25216.
1212 conjugacy classes of subgroups; only the 509 normal subgroups are shown.
This group has 1212 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 | C42 | C22×D197 |
| commutator, socle, cosocle | C22×C197 | C25 |
| Frattini | C22 | C24×D197 |
| Fitting | C22×C42×C197 | C2 |
| hypercenter | C22×C42 | D197 |
| hypocenter | C197 | #3444 |
| LCS | G ▷ C22×C197 ▷ C197 | |
| UCS | C1 ◁ C42 ◁ C22×C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #27161636 | C2 | semidirect |
| 2. #27161629 | C2 | semidirect |
| 3. #27161613 | C2 | semidirect |
| 4. #27161590 | C2 | semidirect |
| 5. C4×#15691358 | C2 | semidirect |
| 6. C22×C42×C197 | C2 | semidirect |
| 8. #20263326 | C22 | semidirect |
| 9. #20263323 | C22 | semidirect |
| 10. #20263320 | C22 | semidirect |
| 11. #20263318 | C22 | semidirect |
| 12. #20263314 | C22 | semidirect |
| 17. C4×Dic394 | C22 | semidirect |
| 18. C4×D788 | C22 | semidirect |
| 21. #20263283 | C22 | semidirect |
| 22. C42×D197 | C22 | semidirect |
| 60. C2×C4×C197 | #41 | semidirect |
| 76. C197 | #3444 | semidirect |
| 83. C22×C42 | D197 | semidirect |
| 137. C2×C4 | #15691358 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #27161636 | 2 |
| #27161629 | 2 |
| #27161613 | 2 |
| #27161590 | 2 |
| C4×#15691358 | 2 |
| C22×C42×C197 | 2 |
| #3444 | 197 |