Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
456 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 118 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 | C42 | C22×C4×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| prime radical | C7×#532 | C22 |
| LCS | G ▷ C42 ▷ 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 |
|---|---|---|
| 2. C7×#2371 | C2 | semidirect |
| 3. C7×#1534 | C2 | semidirect |
| 5. C7×#1300 | C2 | semidirect |
| 8. C7×#491 | C4 | semidirect |
| 9. C7×#490 | C4 | semidirect |
| 21. #9442 | C7 | direct (N × H) |
| 51. #2371 | C2×C7 | semidirect |
| 52. #1534 | C2×C7 | semidirect |
| 54. #1300 | C2×C7 | semidirect |
| 83. #491 | C4×C7 | semidirect |
| 84. #490 | C4×C7 | semidirect |
| 217. C7 | #9442 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3142 | 2 |
| C7×#2371 | 2 |
| C7×#1534 | 2 |
| C7×#1505 | 2 |
| C7×#1300 | 2 |
| #9442 | 7 |