Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
212 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C7×C8 | C2×D4 |
| commutator | C4 | C22×C7×C16 |
| Frattini, cosocle | C4×C8 | C23×C7 |
| socle, prime radical | C22×C7 | C8×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7×C8 ◁ C4×C7×C16 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 16. #12059 | C7 | direct (N × H) |
| 36. C7×Q16 | C16 | semidirect |
| 95. Q16 | C7×C16 | semidirect |
| 125. C7 | #12059 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×Q8×C16 | 2 |
| C7×C8×Q16 | 2 |
| C7×#1367 | 2 |
| C7×Q8⋊C16 | 2 |
| C7×#1312 | 2 |
| #12059 | 7 |