Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
416 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 416 conjugacy classes of subgroups — too many to draw. Only its 98 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, Frattini, cosocle | C22×Q8 | C23×C7 |
| prime radical | C23×C7 | #120 |
| LCS | G ▷ C22×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 |
|---|---|---|
| 11. #13464 | C7 | direct (N × H) |
| 25. C7×#101 | Q8 | semidirect |
| 90. #101 | C7×Q8 | semidirect |
| 148. C7 | #13464 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#2811 | 2 |
| C7×#1857 | 2 |
| C7×#1394 | 2 |
| #13464 | 7 |