Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
784 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 784 conjugacy classes of subgroups — too many to draw. Only its 142 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 | B(4,𝔽2) |
| commutator | C2×Q8 | C22×C4×C7 |
| Frattini, cosocle | C22×Q8 | C23×C7 |
| prime radical | C23×C7 | #120 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C7 ◁ C22×C7×Q8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. C7×Q82 | C4 | semidirect |
| 14. #12777 | C7 | direct (N × H) |
| 67. Q82 | C4×C7 | semidirect |
| 173. C7 | #12777 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×Q82 | 2 |
| C2×C7×#356 | 2 |
| C7×#1886 | 2 |
| #12777 | 7 |