Finite Group DatabasebetaFinite Group DatabasebetaOrder 12736.
234 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 99 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 | #124×C199 | C2 |
| prime radical | #20263926 | C2 |
| hypercenter | #124 | D199 |
| hypocenter | C199 | #529 |
| LCS | G ▷ C22×C199 ▷ C199 | |
| UCS | C1 ◁ C22 ◁ #124 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20263948 | C2 | semidirect |
| 2. #20263947 | C2 | semidirect |
| 3. #20263937 | C2 | semidirect |
| 4. #20263931 | C2 | semidirect |
| 6. Q8×Dic199 | C2 | semidirect |
| 7. D4×Dic199 | C2 | semidirect |
| 8. (C22⋊D199)⋊C4 | C2 | semidirect |
| 9. #20263911 | C2 | semidirect |
| 10. C42×D199 | C2 | semidirect |
| 11. #124×C199 | C2 | semidirect |
| 74. C199 | #529 | semidirect |
| 78. #124 | D199 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20263948 | 2 |
| #20263947 | 2 |
| #20263937 | 2 |
| #20263931 | 2 |
| #20263926 | 2 |
| Q8×Dic199 | 2 |
| D4×Dic199 | 2 |
| (C22⋊D199)⋊C4 | 2 |
| #20263911 | 2 |
| C42×D199 | 2 |
| #124×C199 | 2 |
| #529 | 199 |