Finite Group DatabasebetaFinite Group DatabasebetaOrder 12736.
212 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 97 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 | #124×C199 | C2 |
| prime radical | C2×D4×C199 | C22 |
| hypercenter | #124 | D199 |
| hypocenter | C199 | #556 |
| LCS | G ▷ C22×C199 ▷ C199 | |
| UCS | C1 ◁ C22 ◁ #124 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20263953 | C2 | semidirect |
| 2. #20263948 | C2 | semidirect |
| 3. #20263948 | C2 | semidirect |
| 5. #20263932 | C2 | semidirect |
| 6. Q8×Dic199 | C2 | semidirect |
| 7. C4×Dic398 | C2 | semidirect |
| 8. C4×Dic398 | C2 | semidirect |
| 10. #20263911 | C2 | semidirect |
| 69. C199 | #556 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20263953 | 2 |
| #20263948 | 2 |
| #20263942 | 2 |
| #20263932 | 2 |
| Q8×Dic199 | 2 |
| C4×Dic398 | 2 |
| D4×Dic199 | 2 |
| #20263911 | 2 |
| #124×C199 | 2 |
| #556 | 199 |