Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
318 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 318 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #498 |
| commutator, Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C2×C7 | #2058 |
| prime radical | C23×C7 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C2×C7×#97 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 17. #13459 | C7 | direct (N × H) |
| 30. C2×C7×C16 | Q8 | semidirect |
| 37. C7×#106 | Q8 | semidirect |
| 108. C2×C16 | C7×Q8 | semidirect |
| 115. #106 | C7×Q8 | semidirect |
| 173. C7 | #13459 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3162 | 2 |
| C7×#2155 | 2 |
| C7×#1946 | 2 |
| C7×#1384 | 2 |
| C7×#1375 | 2 |
| #13459 | 7 |