Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
526 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 526 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 | C22 | C2×D4×D7 |
| commutator, cosocle | C2×C4×C7 | C24 |
| Frattini | C2×C4 | C23×D7 |
| Fitting | C7×#418 | C2 |
| socle | C22×C7 | C22×D4 |
| hypercenter | #418 | D7 |
| hypocenter | C7 | #3041 |
| LCS | G ▷ C2×C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #418 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #90496 | C2 | semidirect |
| 2. C22×#1141 | C2 | semidirect |
| 3. C2×#6471 | C2 | semidirect |
| 4. #90212 | C2 | semidirect |
| 5. C7×#418 | C2 | semidirect |
| 6. #89891 | C2 | semidirect |
| 7. #89774 | C2 | semidirect |
| 8. #89559 | C2 | semidirect |
| 9. #89558 | C2 | semidirect |
| 16. #6503 | C22 | semidirect |
| 18. #6499 | C22 | semidirect |
| 24. C2×D56 | C22 | semidirect |
| 25. C2×#1132 | C22 | semidirect |
| 26. #6456 | C22 | semidirect |
| 29. C7×#102 | C22 | semidirect |
| 30. Dic7⋊C8 | C22 | semidirect |
| 31. #6384 | C22 | semidirect |
| 32. #6383 | C22 | semidirect |
| 47. C2×D28 | D4 | semidirect |
| 75. #418 | D7 | semidirect |
| 130. #102 | C2×D7 | semidirect |
| 219. C7 | #3041 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #90496 | 2 |
| C22×#1141 | 2 |
| C2×#6471 | 2 |
| #90212 | 2 |
| C7×#418 | 2 |
| #89891 | 2 |
| #89774 | 2 |
| #89559 | 2 |
| #89558 | 2 |
| #3041 | 7 |