Finite Group DatabasebetaFinite Group DatabasebetaOrder 12736.
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×D199 |
| commutator, socle, cosocle | C22×C199 | C24 |
| Fitting | #123×C199 | C2 |
| prime radical | #20263951 | C2 |
| hypercenter | #123 | D199 |
| hypocenter | C199 | #559 |
| LCS | G ▷ C22×C199 ▷ C199 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #20263941 | C2 | semidirect |
| 3. #20263941 | C2 | semidirect |
| 4. #20263940 | C2 | semidirect |
| 5. #20263936 | C2 | semidirect |
| 6. #20263920 | C2 | semidirect |
| 7. C4×C22⋊D199 | C2 | semidirect |
| 8. (C22⋊D199)⋊C4 | C2 | semidirect |
| 9. #123×C199 | C2 | semidirect |
| 67. C199 | #559 | semidirect |
| 74. #123 | D199 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20263951 | 2 |
| #20263941 | 2 |
| #20263940 | 2 |
| #20263936 | 2 |
| #20263920 | 2 |
| C4×C22⋊D199 | 2 |
| (C22⋊D199)⋊C4 | 2 |
| #123×C199 | 2 |
| #559 | 199 |