Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
496 conjugacy classes of subgroups; only the 209 normal subgroups are shown.
This group has 496 conjugacy classes of subgroups — too many to draw. Only its 209 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 | C7×#391 | C2 |
| socle, prime radical, cosocle | C23×C7 | C24 |
| hypercenter | #391 | D7 |
| hypocenter | C7 | #2369 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ #391 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 14. #6507 | C4 | semidirect |
| 55. Dic7⋊C4 | Q8 | semidirect |
| 199. C7 | #2369 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#6507 | 2 |
| C7×#391 | 2 |
| #90002 | 2 |
| #89744 | 2 |
| C4×Dic7⋊C4 | 2 |
| #89421 | 2 |
| #89419 | 2 |
| #89418 | 2 |
| #2369 | 7 |