Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
370 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 370 conjugacy classes of subgroups — too many to draw. Only its 138 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 | #1824 |
| prime radical | C24×C7 | Q16 |
| 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 |
|---|---|---|
| 4. C7×#1388 | C2 | semidirect |
| 5. C7×#1385 | C2 | semidirect |
| 18. #12876 | C7 | direct (N × H) |
| 45. #1388 | C2×C7 | semidirect |
| 46. #1385 | C2×C7 | semidirect |
| 208. C7 | #12876 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×#430 | 2 |
| C2×C7×#349 | 2 |
| C7×#1811 | 2 |
| C7×#1388 | 2 |
| C7×#1385 | 2 |
| #12876 | 7 |