Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
712 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 712 conjugacy classes of subgroups — too many to draw. Only its 133 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×#120 | C2 |
| hypercenter | C2×#120 | D7 |
| hypocenter | C7 | #2812 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C2×#120 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C7×#120 | C2 | semidirect |
| 2. C2×#6516 | C2 | semidirect |
| 3. #90019 | C2 | semidirect |
| 4. #89995 | C2 | semidirect |
| 5. #89994 | C2 | semidirect |
| 6. #89731 | C2 | semidirect |
| 7. #89724 | C2 | semidirect |
| 11. C22×C22⋊D7 | C22 | semidirect |
| 18. C2×C22⋊Dic7 | C22 | semidirect |
| 19. C2×(C2×D7)⋊C4 | C22 | semidirect |
| 20. C2×Dic7⋊C4 | C22 | semidirect |
| 21. C7⋊#95 | C22 | semidirect |
| 22. C7⋊#95 | C22 | semidirect |
| 26. C24×C7 | D4 | semidirect |
| 49. C2×#120 | D7 | semidirect |
| 124. C24 | C22⋊D7 | semidirect |
| 181. C7 | #2812 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×#120 | 2 |
| C2×#6516 | 2 |
| #90019 | 2 |
| #89995 | 2 |
| #89994 | 2 |
| #89731 | 2 |
| #89724 | 2 |
| #2812 | 7 |