Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
500 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 500 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C7 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C23×C7 |
| prime radical | C2×C7×#127 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ C2×C42×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#2571 | C2 | semidirect |
| 4. C7×#1296 | C2 | semidirect |
| 5. C7×#1279 | C2 | semidirect |
| 17. #10351 | C7 | direct (N × H) |
| 41. #2571 | C2×C7 | semidirect |
| 44. #1296 | C2×C7 | semidirect |
| 45. #1279 | C2×C7 | semidirect |
| 239. C7 | #10351 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#2571 | 2 |
| C7×#1982 | 2 |
| C7×#1957 | 2 |
| C7×#1296 | 2 |
| C7×#1279 | 2 |
| #10351 | 7 |