Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
232 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #102 |
| commutator | C8 | C22×C4×C7 |
| Frattini, cosocle | C2×C8 | C23×C7 |
| prime radical | C2×C7×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C22×C7×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C7×M32 | C2 | semidirect |
| 3. C2×C7×#106 | C2 | semidirect |
| 4. C7×#360 | C2 | direct (N × H) |
| 8. C7×#135 | C4 | semidirect |
| 10. C2×C7×Q16 | C4 | semidirect |
| 12. C2×C7×D8 | C4 | semidirect |
| 16. C7×M32 | C22 | semidirect |
| 19. C7×#106 | C22 | semidirect |
| 20. C2×#360 | C7 | direct (N × H) |
| 32. C7×Q16 | C2×C4 | semidirect |
| 36. C7×D8 | C2×C4 | semidirect |
| 47. C2×M32 | C2×C7 | semidirect |
| 48. C2×#106 | C2×C7 | semidirect |
| 49. #360 | C2×C7 | direct (N × H) |
| 76. #135 | C4×C7 | semidirect |
| 78. C2×Q16 | C4×C7 | semidirect |
| 80. C2×D8 | C4×C7 | semidirect |
| 84. M32 | C22×C7 | semidirect |
| 87. #106 | C22×C7 | semidirect |
| 111. Q16 | C2×C4×C7 | semidirect |
| 115. D8 | C2×C4×C7 | semidirect |
| 128. C2×C7 | #360 | direct (N × H) |
| 153. C7 | C2×#360 | direct (N × H) |
| 169. C2 | C7×#360 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×#135 | 2 |
| C2×C7×M32 | 2 |
| C2×C7×#106 | 2 |
| C7×#360 | 2 |
| C2×#360 | 7 |