Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
696 conjugacy classes of subgroups; only the 189 normal subgroups are shown.
This group has 696 conjugacy classes of subgroups — too many to draw. Only its 189 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 | C22×C42×C7 | C2 |
| prime radical | C2×#6493 | C2 |
| hypercenter | C22×C42 | D7 |
| hypocenter | C7 | #2850 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C22×C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C42×C7 | C2 | semidirect |
| 3. C2×#6487 | C2 | semidirect |
| 4. C2×#6438 | C2 | semidirect |
| 5. #89985 | C2 | semidirect |
| 6. #89712 | C2 | semidirect |
| 7. #89711 | C2 | semidirect |
| 8. #89708 | C2 | semidirect |
| 9. #89701 | C2 | semidirect |
| 57. C42×C7 | D4 | semidirect |
| 68. C22×C42 | D7 | semidirect |
| 134. C22×C7 | #124 | semidirect |
| 184. C42 | C22⋊D7 | semidirect |
| 227. C7 | #2850 | semidirect |
| 237. C22 | #6438 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C42×C7 | 2 |
| C2×#6493 | 2 |
| C2×#6487 | 2 |
| C2×#6438 | 2 |
| #89985 | 2 |
| #89712 | 2 |
| #89711 | 2 |
| #89708 | 2 |
| #89701 | 2 |
| #2850 | 7 |