Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
198 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C7 | #466 |
| commutator, Frattini, cosocle | C4×C8 | C23×C7 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C42×C7 ◁ C7×C8×Q8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 23. #12459 | C7 | direct (N × H) |
| 70. C7×C16 | Q16 | semidirect |
| 153. C16 | C7×Q16 | semidirect |
| 175. C7 | #12459 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#2254 | 2 |
| C7×Q8×C16 | 2 |
| C7×#1676 | 2 |
| C7×#1674 | 2 |
| C7×#1372 | 2 |
| C7×#1365 | 2 |
| C7×#1364 | 2 |
| #12459 | 7 |