Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
548 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 548 conjugacy classes of subgroups — too many to draw. Only its 149 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, prime radical | C2×C7×#121 | C2 |
| hypercenter | C2×#121 | D7 |
| hypocenter | C7 | #2624 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C2×#121 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#6489 | C2 | semidirect |
| 3. #89994 | C2 | semidirect |
| 4. #89993 | C2 | semidirect |
| 5. #89984 | C2 | semidirect |
| 6. #89745 | C2 | semidirect |
| 7. #89725 | C2 | semidirect |
| 8. #89721 | C2 | semidirect |
| 190. C7 | #2624 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×#121 | 2 |
| C2×#6489 | 2 |
| #89994 | 2 |
| #89993 | 2 |
| #89984 | 2 |
| #89745 | 2 |
| #89725 | 2 |
| #89721 | 2 |
| #2624 | 7 |