Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
500 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 500 conjugacy classes of subgroups — too many to draw. Only its 94 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 | #394 |
| commutator, Frattini, cosocle | C23×C4 | C23×C7 |
| prime radical | C7×#3186 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×#1900 | C2 | semidirect |
| 3. C7×#1851 | C2 | semidirect |
| 4. C7×#1507 | C2 | semidirect |
| 5. C7×#1293 | C2 | semidirect |
| 21. #10557 | C7 | direct (N × H) |
| 53. #1900 | C2×C7 | semidirect |
| 54. #1851 | C2×C7 | semidirect |
| 55. #1507 | C2×C7 | semidirect |
| 56. #1293 | C2×C7 | semidirect |
| 284. C7 | #10557 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3186 | 2 |
| C7×#1900 | 2 |
| C7×#1851 | 2 |
| C7×#1507 | 2 |
| C7×#1293 | 2 |
| #10557 | 7 |