Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
498 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 498 conjugacy classes of subgroups — too many to draw. Only its 142 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 | #381 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| prime radical | C22×C7×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C7×#1303 | C2 | semidirect |
| 5. C7×#1286 | C2 | semidirect |
| 19. #8765 | C7 | direct (N × H) |
| 49. #1303 | C2×C7 | semidirect |
| 50. #1286 | C2×C7 | semidirect |
| 243. C7 | #8765 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#2349 | 2 |
| C7×#1800 | 2 |
| C7×#1460 | 2 |
| C7×#1303 | 2 |
| C7×#1286 | 2 |
| #8765 | 7 |