Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
1088 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 1088 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, socle, cosocle | C23×C7 | C24 |
| Fitting | C7×#390 | C2 |
| prime radical | C22×D4×D7 | C2 |
| hypercenter | #390 | D7 |
| hypocenter | C7 | #2435 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ #390 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#6499 | C2 | semidirect |
| 3. C2×#6460 | C2 | semidirect |
| 4. C2×#6438 | C2 | semidirect |
| 5. C7×#390 | C2 | semidirect |
| 6. #89760 | C2 | semidirect |
| 7. #89733 | C2 | semidirect |
| 8. #89727 | C2 | semidirect |
| 9. C4×(C2×D7)⋊C4 | C2 | semidirect |
| 18. C22×Dic14 | C22 | semidirect |
| 19. C2×C7×C4⋊C4 | C22 | semidirect |
| 20. C2×C7×C22⋊C4 | C22 | semidirect |
| 21. C2×C42×C7 | C22 | semidirect |
| 22. C2×C22⋊Dic7 | C22 | semidirect |
| 26. C2×(C2×D7)⋊C4 | C22 | semidirect |
| 27. C2×(C2×D7)⋊C4 | C22 | semidirect |
| 28. C2×C4⋊Dic7 | C22 | semidirect |
| 29. C2×C4×Dic7 | C22 | semidirect |
| 32. C7⋊#95 | C22 | semidirect |
| 80. #390 | D7 | semidirect |
| 139. C2×C4⋊C4 | C2×D7 | semidirect |
| 145. C2×C22⋊C4 | C2×D7 | semidirect |
| 147. C2×C42 | C2×D7 | semidirect |
| 169. C2×D7 | #124 | semidirect |
| 267. C7 | #2435 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×D4×D7 | 2 |
| C2×#6499 | 2 |
| C2×#6460 | 2 |
| C2×#6438 | 2 |
| C7×#390 | 2 |
| #89760 | 2 |
| #89733 | 2 |
| #89727 | 2 |
| C4×(C2×D7)⋊C4 | 2 |
| #2435 | 7 |