Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
786 conjugacy classes of subgroups; only the 222 normal subgroups are shown.
This group has 786 conjugacy classes of subgroups — too many to draw. Only its 222 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 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C7 |
| prime radical | C23×C7 | #120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C7×#1663 | C2 | semidirect |
| 28. #19008 | C7 | direct (N × H) |
| 73. #1663 | C2×C7 | semidirect |
| 252. C7 | #19008 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×#564 | 2 |
| C7×#3236 | 2 |
| C7×#3166 | 2 |
| C7×#1663 | 2 |
| C7×#1534 | 2 |
| C7×#1528 | 2 |
| #19008 | 7 |