Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
280 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1514 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C2×M16 | C23×C7 |
| prime radical | C23×C7 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C4×C7×M16 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 7. C7×#498 | C4 | semidirect |
| 8. C7×#494 | C4 | semidirect |
| 13. C7×#441 | C4 | semidirect |
| 22. #11838 | C7 | direct (N × H) |
| 83. #498 | C4×C7 | semidirect |
| 84. #494 | C4×C7 | semidirect |
| 89. #441 | C4×C7 | semidirect |
| 204. C7 | #11838 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3185 | 2 |
| C7×#2165 | 2 |
| C7×#1841 | 2 |
| C7×#1342 | 2 |
| C7×#1340 | 2 |
| #11838 | 7 |