Finite Group DatabasebetaFinite Group DatabasebetaOrder 12608.
196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 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 |
| prime radical | C2×C22⋊D197 | C22 |
| hypercenter | #126 | D197 |
| hypocenter | C197 | #563 |
| 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. #20263326 | C2 | semidirect |
| 2. #20263325 | C2 | semidirect |
| 3. #20263321 | C2 | semidirect |
| 4. #20263320 | C2 | semidirect |
| 5. #20263319 | C2 | semidirect |
| 6. #20263317 | C2 | semidirect |
| 7. #20263316 | C2 | semidirect |
| 11. #20263306 | C2 | semidirect |
| 12. #20263304 | C2 | semidirect |
| 13. #20263303 | C2 | semidirect |
| 14. #20263303 | C2 | semidirect |
| 15. #126×C197 | C2 | semidirect |
| 97. C197 | #563 | semidirect |
| 104. #126 | D197 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20263326 | 2 |
| #20263325 | 2 |
| #20263321 | 2 |
| #20263320 | 2 |
| #20263319 | 2 |
| #20263317 | 2 |
| #20263316 | 2 |
| #20263312 | 2 |
| #20263310 | 2 |
| #20263308 | 2 |
| #20263306 | 2 |
| #20263304 | 2 |
| #20263303 | 2 |
| #126×C197 | 2 |
| #563 | 197 |