Finite Group DatabasebetaFinite Group DatabasebetaOrder 12736.
292 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 105 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 |
| hypercenter | #123 | D199 |
| hypocenter | C199 | #547 |
| 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. C2×#15691886 | C2 | semidirect |
| 2. C2×#15691885 | C2 | semidirect |
| 3. #20263954 | C2 | semidirect |
| 4. #20263940 | C2 | semidirect |
| 5. #20263939 | C2 | semidirect |
| 6. #20263937 | C2 | semidirect |
| 7. #20263933 | C2 | semidirect |
| 8. #20263932 | C2 | semidirect |
| 9. #20263928 | C2 | semidirect |
| 10. #20263927 | C2 | semidirect |
| 11. (C22⋊D199)⋊C4 | C2 | semidirect |
| 12. (C22⋊D199)⋊C4 | C2 | semidirect |
| 13. C2×Dic199⋊C4 | C2 | semidirect |
| 14. C4⋊C4×D199 | C2 | semidirect |
| 15. #123×C199 | C2 | semidirect |
| 61. C22⋊D199 | D4 | semidirect |
| 107. Dic199 | #41 | semidirect |
| 115. C199 | #547 | semidirect |
| 123. #123 | D199 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#15691886 | 2 |
| C2×#15691885 | 2 |
| #20263954 | 2 |
| #20263940 | 2 |
| #20263939 | 2 |
| #20263937 | 2 |
| #20263933 | 2 |
| #20263932 | 2 |
| #20263928 | 2 |
| #20263927 | 2 |
| (C22⋊D199)⋊C4 | 2 |
| C2×Dic199⋊C4 | 2 |
| C4⋊C4×D199 | 2 |
| #123×C199 | 2 |
| #547 | 199 |