Finite Group DatabasebetaFinite Group DatabasebetaOrder 12736.
186 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 77 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D199 |
| commutator | C22×C199 | C22×C4 |
| Fitting | #95×C199 | C2 |
| socle, prime radical, cosocle | C23×C199 | C23 |
| hypercenter | #95 | D199 |
| hypocenter | C199 | #391 |
| LCS | G ▷ C22×C199 ▷ C199 | |
| UCS | C1 ◁ C23 ◁ #95 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 10. C2×Dic398 | C4 | semidirect |
| 32. C2×Dic199 | Q8 | semidirect |
| 42. C2×C4×C199 | Q8 | semidirect |
| 64. C199 | #391 | semidirect |
| 107. C2×C4 | Dic398 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Dic398 | 2 |
| C2×Dic199⋊C4 | 2 |
| C2×C4×Dic199 | 2 |
| C199⋊#95 | 2 |
| #95×C199 | 2 |
| #391 | 199 |