Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
542 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 542 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 | C22×C7 | #393 |
| commutator, Frattini, cosocle | C25 | C23×C7 |
| prime radical | C25×C7 | C23 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C25×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 9. #10391 | C7 | direct (N × H) |
| 12. C23×C4×C7 | Q8 | semidirect |
| 64. C23×C4 | C7×Q8 | semidirect |
| 124. C7 | #10391 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#2840 | 2 |
| C7×#1819 | 2 |
| C7×#1304 | 2 |
| #10391 | 7 |