Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
198 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 154 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C7×C16 | C2×D4 |
| commutator | C4 | C22×C7×C16 |
| Frattini, cosocle | C2×C16 | C23×C7 |
| socle | C2×C7 | C2×C4⋊C16 |
| prime radical | C23×C7 | M32 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C7×C16 ◁ C22×C7×C16 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C7×#1422 | C2 | semidirect |
| 12. #13630 | C7 | direct (N × H) |
| 27. #1422 | C2×C7 | semidirect |
| 106. C7 | #13630 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×M64 | 2 |
| C7×#2108 | 2 |
| C7×#1422 | 2 |
| #13630 | 7 |