Finite Group DatabasebetaFinite Group DatabasebetaOrder 12736.
334 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 103 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 | #127×C199 | C2 |
| hypercenter | #127 | D199 |
| hypocenter | C199 | #535 |
| LCS | G ▷ C22×C199 ▷ C199 | |
| UCS | C1 ◁ C22 ◁ #127 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#15691885 | C2 | semidirect |
| 2. C2×D4×D199 | C2 | semidirect |
| 3. #20263951 | C2 | semidirect |
| 4. #20263943 | C2 | semidirect |
| 5. #20263942 | C2 | semidirect |
| 6. #20263927 | C2 | semidirect |
| 7. #20263923 | C2 | semidirect |
| 8. #20263912 | C2 | semidirect |
| 9. #127×C199 | C2 | semidirect |
| 12. C2×Dic398 | C22 | semidirect |
| 16. C2×C22⋊D199 | C22 | semidirect |
| 18. C22×Dic199 | C22 | semidirect |
| 20. Dic199⋊C4 | C22 | semidirect |
| 21. C22⋊Dic199 | C22 | semidirect |
| 23. (C2×D199)⋊C4 | C22 | semidirect |
| 24. C4×Dic199 | C22 | semidirect |
| 26. C2×D4×C199 | C22 | semidirect |
| 27. C2×D4×C199 | C22 | semidirect |
| 28. C42×C199 | C22 | semidirect |
| 42. C4×D199 | D4 | semidirect |
| 71. C199 | #535 | semidirect |
| 76. #127 | D199 | semidirect |
| 91. C2×D4 | C2×D199 | semidirect |
| 92. C2×D4 | C2×D199 | semidirect |
| 100. C42 | C2×D199 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#15691885 | 2 |
| C2×D4×D199 | 2 |
| #20263951 | 2 |
| #20263943 | 2 |
| #20263942 | 2 |
| #20263927 | 2 |
| #20263923 | 2 |
| #20263912 | 2 |
| #127×C199 | 2 |
| #535 | 199 |