Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
444 conjugacy classes of subgroups; only the 229 normal subgroups are shown.
This group has 444 conjugacy classes of subgroups — too many to draw. Only its 229 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 | C22×C7 | C23×C4 |
| Fitting | C2×C4×C7×Q8 | C2 |
| socle, prime radical, cosocle | C23×C7 | C24 |
| hypercenter | C2×C4×Q8 | D7 |
| hypocenter | C7 | #2320 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 10. C4×C7×Q8 | C4 | semidirect |
| 77. C4×Q8 | Dic7 | semidirect |
| 152. C7 | #2320 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×C7×Q8 | 2 |
| #90002 | 2 |
| #89757 | 2 |
| #89753 | 2 |
| #89709 | 2 |
| #89708 | 2 |
| C4×C4⋊Dic7 | 2 |
| #89704 | 2 |
| #2320 | 7 |