Finite Group DatabasebetaFinite Group DatabasebetaOrder 12608.
236 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 236 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 | #123×C197 | C2 |
| prime radical | #20263322 | C2 |
| hypercenter | #123 | D197 |
| hypocenter | C197 | #559 |
| LCS | G ▷ C22×C197 ▷ C197 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #20263312 | C2 | semidirect |
| 3. #20263312 | C2 | semidirect |
| 4. #20263311 | C2 | semidirect |
| 5. #20263307 | C2 | semidirect |
| 6. #20263291 | C2 | semidirect |
| 7. C4×C22⋊D197 | C2 | semidirect |
| 8. #20263286 | C2 | semidirect |
| 9. #123×C197 | C2 | semidirect |
| 67. C197 | #559 | semidirect |
| 74. #123 | D197 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20263322 | 2 |
| #20263312 | 2 |
| #20263311 | 2 |
| #20263307 | 2 |
| #20263291 | 2 |
| C4×C22⋊D197 | 2 |
| #20263286 | 2 |
| #123×C197 | 2 |
| #559 | 197 |