Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
736 conjugacy classes of subgroups; only the 230 normal subgroups are shown.
This group has 736 conjugacy classes of subgroups — too many to draw. Only its 230 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×#127 |
| commutator, Frattini, cosocle | C42 | C24×C7 |
| prime radical | C2×C7×#127 | C22 |
| 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 |
|---|---|---|
| 1. C7×#3185 | C2 | semidirect |
| 3. C7×#1720 | C2 | semidirect |
| 4. C7×#1711 | C2 | semidirect |
| 5. C7×#1455 | C2 | semidirect |
| 8. C7×#498 | C22 | semidirect |
| 9. C7×#494 | C22 | semidirect |
| 11. C7×#487 | C22 | semidirect |
| 18. C7×#322 | C22 | semidirect |
| 19. C7×C82 | C22 | semidirect |
| 20. #19753 | C7 | direct (N × H) |
| 39. #3185 | C2×C7 | semidirect |
| 41. #1720 | C2×C7 | semidirect |
| 42. #1711 | C2×C7 | semidirect |
| 43. #1455 | C2×C7 | semidirect |
| 63. #498 | C22×C7 | semidirect |
| 64. #494 | C22×C7 | semidirect |
| 66. #487 | C22×C7 | semidirect |
| 73. #322 | C22×C7 | semidirect |
| 74. C82 | C22×C7 | semidirect |
| 144. C7 | #19753 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3185 | 2 |
| C7×#3138 | 2 |
| C7×#1720 | 2 |
| C7×#1711 | 2 |
| C7×#1455 | 2 |
| #19753 | 7 |