Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
542 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 542 conjugacy classes of subgroups — too many to draw. Only its 122 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 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23×C7 |
| prime radical | C2×C7×#127 | C22 |
| LCS | G ▷ C2×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 |
|---|---|---|
| 2. C7×#3162 | C2 | semidirect |
| 4. C7×#1518 | C2 | semidirect |
| 5. C7×#1281 | C2 | semidirect |
| 20. #10220 | C7 | direct (N × H) |
| 49. #3162 | C2×C7 | semidirect |
| 51. #1518 | C2×C7 | semidirect |
| 52. #1281 | C2×C7 | semidirect |
| 70. C2×C7×C8 | SD16 | semidirect |
| 180. C2×C8 | C7×SD16 | semidirect |
| 222. C7 | #10220 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3208 | 2 |
| C7×#3162 | 2 |
| C7×#1961 | 2 |
| C7×#1518 | 2 |
| C7×#1281 | 2 |
| #10220 | 7 |