Finite Group DatabasebetaFinite Group DatabasebetaOrder 12608.
238 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 95 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×D197 |
| commutator, socle, cosocle | C22×C197 | C24 |
| Fitting | #126×C197 | C2 |
| hypercenter | #126 | D197 |
| hypocenter | C197 | #553 |
| LCS | G ▷ C22×C197 ▷ C197 | |
| UCS | C1 ◁ C22 ◁ #126 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20263321 | C2 | semidirect |
| 2. #20263316 | C2 | semidirect |
| 3. #20263311 | C2 | semidirect |
| 4. #20263310 | C2 | semidirect |
| 5. #20263301 | C2 | semidirect |
| 6. #20263299 | C2 | semidirect |
| 7. #20263299 | C2 | semidirect |
| 8. #20263299 | C2 | semidirect |
| 9. C4×D788 | C2 | semidirect |
| 10. #20263288 | C2 | semidirect |
| 11. #20263286 | C2 | semidirect |
| 12. #20263286 | C2 | semidirect |
| 13. #20263283 | C2 | semidirect |
| 14. C4⋊C4×D197 | C2 | semidirect |
| 15. #126×C197 | C2 | semidirect |
| 109. C197 | #553 | semidirect |
| 112. #126 | D197 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20263321 | 2 |
| #20263316 | 2 |
| #20263311 | 2 |
| #20263310 | 2 |
| #20263301 | 2 |
| #20263299 | 2 |
| C4×D788 | 2 |
| #20263288 | 2 |
| #20263286 | 2 |
| #20263283 | 2 |
| C4⋊C4×D197 | 2 |
| #126×C197 | 2 |
| #553 | 197 |