Finite Group DatabasebetaFinite Group DatabasebetaOrder 12736.
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×D199 |
| commutator, socle, cosocle | C22×C199 | C24 |
| Fitting | #123×C199 | C2 |
| prime radical | C2×D4×C199 | C22 |
| hypercenter | #123 | D199 |
| hypocenter | C199 | #563 |
| LCS | G ▷ C22×C199 ▷ C199 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20263954 | C2 | semidirect |
| 2. #20263948 | C2 | semidirect |
| 3. #20263948 | C2 | semidirect |
| 4. #20263945 | C2 | semidirect |
| 7. #20263934 | C2 | semidirect |
| 8. #20263932 | C2 | semidirect |
| 61. C199 | #563 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20263954 | 2 |
| #20263948 | 2 |
| #20263945 | 2 |
| #20263942 | 2 |
| #20263936 | 2 |
| #20263934 | 2 |
| #20263932 | 2 |
| #123×C199 | 2 |
| #563 | 199 |