Finite Group DatabasebetaFinite Group DatabasebetaOrder 12736.
248 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 248 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 | #20263923 | C2 |
| hypercenter | #123 | D199 |
| hypocenter | C199 | #551 |
| 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. #20263948 | C2 | semidirect |
| 2. #20263942 | C2 | semidirect |
| 3. #20263937 | C2 | semidirect |
| 4. #20263933 | C2 | semidirect |
| 6. C4×C22⋊D199 | C2 | semidirect |
| 7. #20263913 | C2 | semidirect |
| 8. C22⋊C4×D199 | C2 | semidirect |
| 9. #123×C199 | C2 | semidirect |
| 11. C2×Dic398 | C22 | semidirect |
| 15. C4⋊Dic199 | C22 | semidirect |
| 16. Dic199⋊C4 | C22 | semidirect |
| 17. Dic199⋊C4 | C22 | semidirect |
| 20. C22⋊Dic199 | C22 | semidirect |
| 24. C4×Dic199 | C22 | semidirect |
| 25. C4×Dic199 | C22 | semidirect |
| 27. C22×C4×C199 | C22 | semidirect |
| 28. C4⋊C4×C199 | C22 | semidirect |
| 30. C22⋊C4×C199 | C22 | semidirect |
| 68. C199 | #551 | semidirect |
| 73. #123 | D199 | semidirect |
| 85. C22×C4 | C2×D199 | semidirect |
| 89. C4⋊C4 | C2×D199 | semidirect |
| 97. C22⋊C4 | C2×D199 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20263948 | 2 |
| #20263942 | 2 |
| #20263937 | 2 |
| #20263933 | 2 |
| #20263923 | 2 |
| C4×C22⋊D199 | 2 |
| #20263913 | 2 |
| C22⋊C4×D199 | 2 |
| #123×C199 | 2 |
| #551 | 199 |