Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
626 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 626 conjugacy classes of subgroups — too many to draw. Only its 150 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 | #386 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| prime radical | C7×#2535 | C2 |
| 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 |
|---|---|---|
| 1. C7×#2972 | C2 | semidirect |
| 3. C7×#1800 | C2 | semidirect |
| 4. C7×#1486 | C2 | semidirect |
| 5. C7×#1286 | C2 | semidirect |
| 21. #9485 | C7 | direct (N × H) |
| 49. C4×C7×C8 | D4 | semidirect |
| 53. #2972 | C2×C7 | semidirect |
| 55. #1800 | C2×C7 | semidirect |
| 56. #1486 | C2×C7 | semidirect |
| 57. #1286 | C2×C7 | semidirect |
| 181. C4×C8 | C7×D4 | semidirect |
| 298. C7 | #9485 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#2972 | 2 |
| C7×#2535 | 2 |
| C7×#1800 | 2 |
| C7×#1486 | 2 |
| C7×#1286 | 2 |
| #9485 | 7 |