Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
556 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 556 conjugacy classes of subgroups — too many to draw. Only its 126 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×#3151 | C2 |
| 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×#2979 | C2 | semidirect |
| 3. C7×#1560 | C2 | semidirect |
| 4. C7×#1545 | C2 | semidirect |
| 5. C7×#1299 | C2 | semidirect |
| 22. #9437 | C7 | direct (N × H) |
| 53. #2979 | C2×C7 | semidirect |
| 54. #1560 | C2×C7 | semidirect |
| 55. #1545 | C2×C7 | semidirect |
| 56. #1299 | C2×C7 | semidirect |
| 252. C7 | #9437 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3151 | 2 |
| C7×#2979 | 2 |
| C7×#1560 | 2 |
| C7×#1545 | 2 |
| C7×#1299 | 2 |
| #9437 | 7 |