Finite Group DatabasebetaFinite Group DatabasebetaOrder 12736.
330 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 111 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 | #120×C199 | C2 |
| hypercenter | #120 | D199 |
| hypocenter | C199 | #525 |
| LCS | G ▷ C22×C199 ▷ C199 | |
| UCS | C1 ◁ C22 ◁ #120 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#15691885 | C2 | semidirect |
| 2. C23×Dic199 | C2 | semidirect |
| 3. #20263939 | C2 | semidirect |
| 4. #20263936 | C2 | semidirect |
| 5. #20263932 | C2 | semidirect |
| 6. #20263927 | C2 | semidirect |
| 7. #20263924 | C2 | semidirect |
| 8. D4×Dic199 | C2 | semidirect |
| 9. (C22⋊D199)⋊C4 | C2 | semidirect |
| 10. #20263911 | C2 | semidirect |
| 11. #120×C199 | C2 | semidirect |
| 13. C2×Dic398 | C22 | semidirect |
| 16. C2×C22⋊D199 | C22 | semidirect |
| 22. C22×Dic199 | C22 | semidirect |
| 23. C2×C4×D199 | C22 | semidirect |
| 24. C4⋊Dic199 | C22 | semidirect |
| 25. Dic199⋊C4 | C22 | semidirect |
| 26. Dic199⋊C4 | C22 | semidirect |
| 29. C22⋊Dic199 | C22 | semidirect |
| 30. (C2×D199)⋊C4 | C22 | semidirect |
| 31. C4×Dic199 | C22 | semidirect |
| 46. C2×Dic199 | D4 | semidirect |
| 69. C22×C199 | #41 | semidirect |
| 86. C199 | #525 | semidirect |
| 95. #120 | D199 | semidirect |
| 158. C22 | #15691885 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#15691885 | 2 |
| C23×Dic199 | 2 |
| #20263939 | 2 |
| #20263936 | 2 |
| #20263932 | 2 |
| #20263927 | 2 |
| #20263924 | 2 |
| D4×Dic199 | 2 |
| (C22⋊D199)⋊C4 | 2 |
| #20263911 | 2 |
| #120×C199 | 2 |
| #525 | 199 |