Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
448 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 448 conjugacy classes of subgroups — too many to draw. Only its 165 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 | C23×D7 |
| commutator, socle, prime radical, cosocle | C23×C7 | C24 |
| Fitting | C7×#384 | C2 |
| hypercenter | #384 | D7 |
| hypocenter | C7 | #2660 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ #384 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 44. C42×C7 | Q8 | semidirect |
| 52. C4×Dic7 | Q8 | semidirect |
| 140. C42 | Dic14 | semidirect |
| 169. C7 | #2660 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#6453 | 2 |
| C2×#6451 | 2 |
| C7×#384 | 2 |
| #89756 | 2 |
| #89751 | 2 |
| #89749 | 2 |
| #89748 | 2 |
| #89708 | 2 |
| #89704 | 2 |
| #2660 | 7 |