Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
788 conjugacy classes of subgroups; only the 248 normal subgroups are shown.
This group has 788 conjugacy classes of subgroups — too many to draw. Only its 248 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 |
| 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. C2×C7×#121 | C2 | semidirect |
| 2. C2×C4×C7×D4 | C2 | semidirect |
| 3. C7×#394 | C2 | semidirect |
| 4. C7×#384 | C2 | semidirect |
| 5. C7×#380 | C2 | semidirect |
| 18. #2537 | C7 | direct (N × H) |
| 24. C2×C7×D4 | D4 | semidirect |
| 37. C42×C7 | D4 | semidirect |
| 38. C2×#121 | C2×C7 | semidirect |
| 39. C2×C4×D4 | C2×C7 | semidirect |
| 40. #394 | C2×C7 | semidirect |
| 41. #384 | C2×C7 | semidirect |
| 42. #380 | C2×C7 | semidirect |
| 92. C4×C7 | #121 | semidirect |
| 98. C2×D4 | C7×D4 | semidirect |
| 111. C42 | C7×D4 | semidirect |
| 140. C7 | #2537 | direct (N × H) |
| 156. C4 | C7×#121 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×#121 | 2 |
| C2×C4×C7×D4 | 2 |
| C7×#394 | 2 |
| C7×#384 | 2 |
| C7×#380 | 2 |
| #2537 | 7 |