Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
688 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 688 conjugacy classes of subgroups — too many to draw. Only its 173 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×#379 | C2 |
| socle, cosocle | C23×C7 | C24 |
| prime radical | C2×#6516 | C2 |
| hypercenter | #379 | D7 |
| hypocenter | C7 | #2375 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ #379 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×#379 | C2 | semidirect |
| 3. #90019 | C2 | semidirect |
| 4. #89729 | C2 | semidirect |
| 5. #89728 | C2 | semidirect |
| 6. #89717 | C2 | semidirect |
| 7. #89716 | C2 | semidirect |
| 10. C22×C4×D7 | C22 | semidirect |
| 13. #6516 | C4 | semidirect |
| 17. C2×(C2×D7)⋊C4 | C22 | semidirect |
| 18. C2×Dic7⋊C4 | C22 | semidirect |
| 19. C2×C4×Dic7 | C22 | semidirect |
| 20. C7⋊#95 | C22 | semidirect |
| 21. C7⋊#95 | C22 | semidirect |
| 25. C24×C7 | C2×C4 | semidirect |
| 41. C22⋊Dic7 | C2×C4 | semidirect |
| 51. #379 | D7 | semidirect |
| 115. C24 | C4×D7 | semidirect |
| 166. C7 | #2375 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#6516 | 2 |
| C7×#379 | 2 |
| #90019 | 2 |
| #89729 | 2 |
| #89728 | 2 |
| #89717 | 2 |
| #89716 | 2 |
| #2375 | 7 |