Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
3998 conjugacy classes of subgroups; only the 702 normal subgroups are shown.
This group has 3998 conjugacy classes of subgroups — too many to draw. Only its 702 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C7 | C24 |
| commutator | C23 | C23×C4×C7 |
| Frattini, cosocle | C24 | C24×C7 |
| prime radical | C25×C7 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C7×#1446 | C2 | semidirect |
| 5. C23×C7×Q8 | C4 | semidirect |
| 11. #14336 | C7 | direct (N × H) |
| 26. #1446 | C2×C7 | semidirect |
| 44. C23×Q8 | C4×C7 | semidirect |
| 119. C7 | #14336 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C7×Q8 | 2 |
| C2×C7×#381 | 2 |
| C7×#1446 | 2 |
| #14336 | 7 |