Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
274 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 274 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1533 |
| commutator | C42 | C22×C4×C7 |
| Frattini, cosocle | #97 | C23×C7 |
| prime radical | C23×C7 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C4×C7×M16 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C7×#1356 | C2 | semidirect |
| 14. #11916 | C7 | direct (N × H) |
| 35. #1356 | C2×C7 | semidirect |
| 148. C7 | #11916 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3139 | 2 |
| C7×#2165 | 2 |
| C7×#1356 | 2 |
| #11916 | 7 |