Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
560 conjugacy classes of subgroups; only the 176 normal subgroups are shown.
This group has 560 conjugacy classes of subgroups — too many to draw. Only its 176 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C7 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C7 |
| prime radical | C25×C7 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#397 | C2 | semidirect |
| 2. C7×#393 | C2 | semidirect |
| 3. C7×#388 | C2 | semidirect |
| 6. C22×C7×Q8 | C22 | semidirect |
| 7. C2×C7×C4⋊C4 | C22 | semidirect |
| 12. C2×C42×C7 | C22 | semidirect |
| 13. C7×#95 | C22 | semidirect |
| 14. C7×#95 | C22 | semidirect |
| 15. C7×#95 | C22 | semidirect |
| 16. #2736 | C7 | direct (N × H) |
| 31. #397 | C2×C7 | semidirect |
| 32. #393 | C2×C7 | semidirect |
| 33. #388 | C2×C7 | semidirect |
| 53. C22×Q8 | C22×C7 | semidirect |
| 54. C2×C4⋊C4 | C22×C7 | semidirect |
| 59. C2×C42 | C22×C7 | semidirect |
| 60. #95 | C22×C7 | semidirect |
| 61. #95 | C22×C7 | semidirect |
| 62. #95 | C22×C7 | semidirect |
| 110. C7 | #2736 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#397 | 2 |
| C7×#393 | 2 |
| C7×#388 | 2 |
| C7×#379 | 2 |
| #2736 | 7 |