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 | C2×C7×C22⋊Q8 | C2 |
| prime radical | C24×C7 | C23 |
| hypercenter | C2×C22⋊Q8 | D7 |
| hypocenter | C7 | #2855 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C2×C22⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #90002 | C2 | semidirect |
| 5. #89750 | C2 | semidirect |
| 6. #89745 | C2 | semidirect |
| 41. 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×C7×C22⋊Q8 | 2 |
| C22×C4×Dic7 | 2 |
| #90002 | 2 |
| #89984 | 2 |
| #89750 | 2 |
| #89745 | 2 |
| #89718 | 2 |
| #89715 | 2 |
| #2855 | 7 |