Finite Group DatabasebetaFinite Group DatabasebetaOrder 704.
196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 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×D11 |
| commutator, socle, cosocle | C22×C11 | C24 |
| Fitting | C11×#123 | C2 |
| prime radical | C2×D4×C11 | C22 |
| hypercenter | #123 | D11 |
| hypocenter | C11 | #563 |
| LCS | G ▷ C22×C11 ▷ C11 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #66977 | C2 | semidirect |
| 5. #66942 | C2 | semidirect |
| 6. #66941 | C2 | semidirect |
| 7. #66932 | C2 | semidirect |
| 8. #66932 | C2 | semidirect |
| 9. #66931 | C2 | semidirect |
| 92. C11 | #563 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C11×#123 | 2 |
| #66989 | 2 |
| #66988 | 2 |
| #66977 | 2 |
| #66942 | 2 |
| #66941 | 2 |
| #66932 | 2 |
| #66931 | 2 |
| #563 | 11 |