Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
1492 conjugacy classes of subgroups; only the 553 normal subgroups are shown.
This group has 1492 conjugacy classes of subgroups — too many to draw. Only its 553 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D7 |
| commutator, socle, cosocle | C22×C7 | C25 |
| Fitting | C7×#519 | C2 |
| hypercenter | #519 | D7 |
| hypocenter | C7 | #3486 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #519 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#6551 | C2 | semidirect |
| 2. C22×Q8×D7 | C2 | semidirect |
| 3. C7×#519 | C2 | semidirect |
| 4. #90267 | C2 | semidirect |
| 5. Q8×D28 | C2 | semidirect |
| 6. #90217 | C2 | semidirect |
| 7. #90215 | C2 | semidirect |
| 10. C2×#1161 | C22 | semidirect |
| 13. C2×Q8×D7 | C22 | semidirect |
| 17. C4×C7×Q8 | C22 | semidirect |
| 20. #6460 | C22 | semidirect |
| 24. #6438 | C22 | semidirect |
| 26. #6432 | C22 | semidirect |
| 29. C2×C7×Q8 | D4 | semidirect |
| 58. #519 | D7 | semidirect |
| 94. C4×Q8 | C2×D7 | semidirect |
| 105. C2×D7 | 2−1+4 | semidirect |
| 118. C2×Q8 | D28 | semidirect |
| 158. C7 | #3486 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#6551 | 2 |
| C22×Q8×D7 | 2 |
| C7×#519 | 2 |
| #90267 | 2 |
| Q8×D28 | 2 |
| #90217 | 2 |
| #90215 | 2 |
| #3486 | 7 |