Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
564 conjugacy classes of subgroups; only the 177 normal subgroups are shown.
This group has 564 conjugacy classes of subgroups — too many to draw. Only its 177 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, cosocle | C23×C7 | C24 |
| Fitting | C7×#380 | C2 |
| prime radical | C24×C7 | C23 |
| hypercenter | #380 | D7 |
| hypocenter | C7 | #2855 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ #380 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #89757 | C2 | semidirect |
| 5. #89746 | C2 | semidirect |
| 8. #89425 | C2 | semidirect |
| 38. C22×Dic7 | Q8 | semidirect |
| 181. C7 | #2855 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#6487 | 2 |
| C22×C4×Dic7 | 2 |
| C7×#380 | 2 |
| #89757 | 2 |
| #89746 | 2 |
| #89722 | 2 |
| #89715 | 2 |
| #89425 | 2 |
| #2855 | 7 |