Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
302 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 302 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, prime radical | C23×C7 | D16 |
| commutator | C16 | C22×C4×C7 |
| Frattini, cosocle | C2×C16 | C23×C7 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ C22×C4×C7 ◁ C22×C7×C8 ◁ C22×C7×C16 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C7×#1416 | C2 | direct (N × H) |
| 8. C2×C7×Q32 | C4 | semidirect |
| 16. C2×#1416 | C7 | direct (N × H) |
| 26. C7×Q32 | C2×C4 | semidirect |
| 37. #1416 | C2×C7 | direct (N × H) |
| 60. C2×Q32 | C4×C7 | semidirect |
| 92. Q32 | C2×C4×C7 | semidirect |
| 128. C2×C7 | #1416 | direct (N × H) |
| 147. C7 | C2×#1416 | direct (N × H) |
| 157. C2 | C7×#1416 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C7×Q32 | 2 |
| C22×C7×C32 | 2 |
| C2×C7×#366 | 2 |
| C7×#1416 | 2 |
| C2×#1416 | 7 |