Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
402 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 402 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #481 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C2×C7 | #1853 |
| prime radical | C24×C7 | SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C22×C7×M16 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×#2155 | C2 | semidirect |
| 5. C7×#1389 | C2 | semidirect |
| 16. C7×#349 | C4 | semidirect |
| 18. #12932 | C7 | direct (N × H) |
| 36. C7×M32 | C2×C4 | semidirect |
| 46. #2155 | C2×C7 | semidirect |
| 49. #1389 | C2×C7 | semidirect |
| 92. #349 | C4×C7 | semidirect |
| 136. M32 | C2×C4×C7 | semidirect |
| 221. C7 | #12932 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#2911 | 2 |
| C7×#2155 | 2 |
| C2×C7×#349 | 2 |
| C7×#1811 | 2 |
| C7×#1389 | 2 |
| #12932 | 7 |