Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
204 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 204 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 | C2×C4×C7 | C2×D8 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C4×C8 | C23×C7 |
| socle | C22×C7 | C2×#107 |
| prime radical | C23×C7 | #107 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ C2×C4×C7×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C7×#1372 | C2 | semidirect |
| 5. C7×#1371 | C2 | semidirect |
| 17. #12034 | C7 | direct (N × H) |
| 36. #1372 | C2×C7 | semidirect |
| 37. #1371 | C2×C7 | semidirect |
| 147. C7 | #12034 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#2162 | 2 |
| C7×#1568 | 2 |
| C7×#1564 | 2 |
| C7×#1372 | 2 |
| C7×#1371 | 2 |
| #12034 | 7 |