Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
994 conjugacy classes of subgroups; only the 254 normal subgroups are shown.
This group has 994 conjugacy classes of subgroups — too many to draw. Only its 254 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 |
| 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×#3184 | C2 | semidirect |
| 2. C7×#3151 | C2 | semidirect |
| 3. C7×#1721 | C2 | semidirect |
| 4. C7×#1715 | C2 | semidirect |
| 5. C7×#1712 | C2 | semidirect |
| 6. C7×#1705 | C2 | semidirect |
| 7. C7×#1455 | C2 | semidirect |
| 11. C7×#498 | C22 | semidirect |
| 15. C7×#493 | C22 | semidirect |
| 18. C7×#486 | C22 | semidirect |
| 22. C7×#322 | C22 | semidirect |
| 23. C7×C82 | C22 | semidirect |
| 24. #19745 | C7 | direct (N × H) |
| 48. #3184 | C2×C7 | semidirect |
| 49. #3151 | C2×C7 | semidirect |
| 50. #1721 | C2×C7 | semidirect |
| 51. #1715 | C2×C7 | semidirect |
| 52. #1712 | C2×C7 | semidirect |
| 53. #1705 | C2×C7 | semidirect |
| 54. #1455 | C2×C7 | semidirect |
| 82. #498 | C22×C7 | semidirect |
| 86. #493 | C22×C7 | semidirect |
| 89. #486 | C22×C7 | semidirect |
| 93. #322 | C22×C7 | semidirect |
| 94. C82 | C22×C7 | semidirect |
| 113. C7×C8 | AGL(1,ℤ/8) | semidirect |
| 194. C8 | C7×AGL(1,ℤ/8) | semidirect |
| 195. C7 | #19745 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3184 | 2 |
| C7×#3151 | 2 |
| C7×#1721 | 2 |
| C7×#1715 | 2 |
| C7×#1712 | 2 |
| C7×#1705 | 2 |
| C7×#1455 | 2 |
| #19745 | 7 |