Finite Group DatabasebetaFinite Group DatabasebetaOrder 12736.
216 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 93 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 | #126×C199 | C2 |
| prime radical | #20263928 | C2 |
| hypercenter | #126 | D199 |
| hypocenter | C199 | #542 |
| LCS | G ▷ C22×C199 ▷ C199 | |
| UCS | C1 ◁ C22 ◁ #126 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20263954 | C2 | semidirect |
| 2. #20263950 | C2 | semidirect |
| 3. #20263943 | C2 | semidirect |
| 4. #20263941 | C2 | semidirect |
| 5. #20263941 | C2 | semidirect |
| 6. #20263940 | C2 | semidirect |
| 7. #20263939 | C2 | semidirect |
| 8. #20263935 | C2 | semidirect |
| 9. #20263933 | C2 | semidirect |
| 10. #20263932 | C2 | semidirect |
| 12. C4×Dic398 | C2 | semidirect |
| 13. (C22⋊D199)⋊C4 | C2 | semidirect |
| 14. #20263913 | C2 | semidirect |
| 15. #126×C199 | C2 | semidirect |
| 103. C199 | #542 | semidirect |
| 107. #126 | D199 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20263954 | 2 |
| #20263950 | 2 |
| #20263943 | 2 |
| #20263941 | 2 |
| #20263940 | 2 |
| #20263939 | 2 |
| #20263935 | 2 |
| #20263933 | 2 |
| #20263932 | 2 |
| #20263928 | 2 |
| C4×Dic398 | 2 |
| (C22⋊D199)⋊C4 | 2 |
| #20263913 | 2 |
| #126×C199 | 2 |
| #542 | 199 |