Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
416 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 416 conjugacy classes of subgroups — too many to draw. Only its 162 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 | #381 |
| commutator | C23 | C22×C7×C8 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C2×C7 | #1817 |
| prime radical | C24×C7 | C2×C8 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C7×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C7×#1378 | C2 | semidirect |
| 14. #12851 | C7 | direct (N × H) |
| 41. #1378 | C2×C7 | semidirect |
| 197. C7 | #12851 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#2873 | 2 |
| C2×C7×#349 | 2 |
| C7×#1378 | 2 |
| #12851 | 7 |