Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
1066 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 1066 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 | C23×C4 | C23×C7 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#3186 | C2 | semidirect |
| 2. C7×#2847 | C2 | semidirect |
| 3. C7×#1779 | C2 | semidirect |
| 4. C7×#1519 | C2 | semidirect |
| 5. C7×#1289 | C2 | semidirect |
| 21. #10320 | C7 | direct (N × H) |
| 25. C22×C7×D4 | D4 | semidirect |
| 35. C22×C7×C8 | D4 | semidirect |
| 44. C2×C7×C4⋊C4 | D4 | semidirect |
| 55. C7×#95 | D4 | semidirect |
| 56. #3186 | C2×C7 | semidirect |
| 57. #2847 | C2×C7 | semidirect |
| 58. #1779 | C2×C7 | semidirect |
| 59. #1519 | C2×C7 | semidirect |
| 60. #1289 | C2×C7 | semidirect |
| 68. C24×C7 | D8 | semidirect |
| 193. C22×D4 | C7×D4 | semidirect |
| 203. C22×C8 | C7×D4 | semidirect |
| 212. C2×C4⋊C4 | C7×D4 | semidirect |
| 223. #95 | C7×D4 | semidirect |
| 265. C24 | C7×D8 | semidirect |
| 381. C7 | #10320 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3186 | 2 |
| C7×#2847 | 2 |
| C7×#1779 | 2 |
| C7×#1519 | 2 |
| C7×#1289 | 2 |
| #10320 | 7 |