Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
712 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 712 conjugacy classes of subgroups — too many to draw. Only its 125 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 | C7×#397 | C2 |
| hypercenter | #397 | D7 |
| hypocenter | C7 | #2812 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ #397 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#6445 | C2 | semidirect |
| 2. C7×#397 | C2 | semidirect |
| 3. #89763 | C2 | semidirect |
| 4. #89732 | C2 | semidirect |
| 5. #89731 | C2 | semidirect |
| 6. #89727 | C2 | semidirect |
| 7. #89447 | C2 | semidirect |
| 8. #89444 | C2 | semidirect |
| 9. #89443 | C2 | semidirect |
| 14. C22×D28 | C22 | semidirect |
| 15. C2×C7×C4⋊C4 | C22 | semidirect |
| 23. C2×(C2×D7)⋊C4 | C22 | semidirect |
| 24. C2×(C2×D7)⋊C4 | C22 | semidirect |
| 25. C2×(C2×D7)⋊C4 | C22 | semidirect |
| 26. C2×Dic7⋊C4 | C22 | semidirect |
| 28. C7×#95 | C22 | semidirect |
| 29. C7×#95 | C22 | semidirect |
| 30. C7⋊#95 | C22 | semidirect |
| 31. C7⋊#95 | C22 | semidirect |
| 38. C23×D7 | D4 | semidirect |
| 72. #397 | D7 | semidirect |
| 131. C2×C4⋊C4 | C2×D7 | semidirect |
| 144. #95 | C2×D7 | semidirect |
| 145. #95 | C2×D7 | semidirect |
| 267. C7 | #2812 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#6445 | 2 |
| C7×#397 | 2 |
| #89763 | 2 |
| #89732 | 2 |
| #89731 | 2 |
| #89727 | 2 |
| #89447 | 2 |
| #89444 | 2 |
| #89443 | 2 |
| #2812 | 7 |