Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
1266 conjugacy classes of subgroups; only the 446 normal subgroups are shown.
This group has 1266 conjugacy classes of subgroups — too many to draw. Only its 446 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C7 |
| Frattini, cosocle | C24 | C24×C7 |
| prime radical | C24×C7 | C24 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C7×Q8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 7. C2×C4×C7×Q8 | C4 | semidirect |
| 14. #20691 | C7 | direct (N × H) |
| 60. C2×C4×Q8 | C4×C7 | semidirect |
| 152. C7 | #20691 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×#519 | 2 |
| C7×#2310 | 2 |
| C7×#1820 | 2 |
| C7×#1740 | 2 |
| #20691 | 7 |