Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
178 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 178 conjugacy classes of subgroups — too many to draw. Only its 102 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 | #357 |
| commutator | C16 | C42×C7 |
| Frattini, cosocle | C22×C16 | C22×C7 |
| prime radical | C23×C7 | D16 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C22×C7×C8 ◁ C22×C7×C16 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 8. C7×M64 | C4 | semidirect |
| 12. C7×#367 | C4 | semidirect |
| 13. C7×#366 | C4 | semidirect |
| 14. #7477 | C7 | direct (N × H) |
| 53. M64 | C4×C7 | semidirect |
| 57. #367 | C4×C7 | semidirect |
| 58. #366 | C4×C7 | semidirect |
| 132. C7 | #7477 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×M64 | 2 |
| C7×#2158 | 2 |
| C2×C7×#367 | 2 |
| #7477 | 7 |