Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
224 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 82 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 | #466 |
| commutator, Frattini, cosocle | C4×C8 | C23×C7 |
| prime radical | C2×C7×D4 | D8 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C42×C7 ◁ C7×C8×D4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#2241 | C2 | semidirect |
| 5. C7×#1372 | C2 | semidirect |
| 6. C7×#1364 | C2 | semidirect |
| 7. C7×#1362 | C2 | semidirect |
| 23. #12470 | C7 | direct (N × H) |
| 45. #2241 | C2×C7 | semidirect |
| 49. #1372 | C2×C7 | semidirect |
| 50. #1364 | C2×C7 | semidirect |
| 51. #1362 | C2×C7 | semidirect |
| 196. C7 | #12470 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#2241 | 2 |
| C7×D4×C16 | 2 |
| C7×#1673 | 2 |
| C7×#1669 | 2 |
| C7×#1372 | 2 |
| C7×#1364 | 2 |
| C7×#1362 | 2 |
| #12470 | 7 |