Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
814 conjugacy classes of subgroups; only the 230 normal subgroups are shown.
This group has 814 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 |
| 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×#3207 | C2 | semidirect |
| 2. C7×#3205 | C2 | semidirect |
| 3. C7×#1721 | C2 | semidirect |
| 4. C7×#1717 | C2 | semidirect |
| 5. C7×#1712 | C2 | semidirect |
| 6. C7×#1711 | C2 | semidirect |
| 7. C7×#1454 | C2 | semidirect |
| 12. C7×#493 | C22 | semidirect |
| 13. C7×#492 | C22 | semidirect |
| 14. C7×#487 | C22 | semidirect |
| 17. C7×#486 | C22 | semidirect |
| 23. C7×#322 | C22 | semidirect |
| 24. C7×C82 | C22 | semidirect |
| 25. #19743 | C7 | direct (N × H) |
| 51. #3207 | C2×C7 | semidirect |
| 52. #3205 | C2×C7 | semidirect |
| 53. #1721 | C2×C7 | semidirect |
| 54. #1717 | C2×C7 | semidirect |
| 55. #1712 | C2×C7 | semidirect |
| 56. #1711 | C2×C7 | semidirect |
| 57. #1454 | C2×C7 | semidirect |
| 87. #493 | C22×C7 | semidirect |
| 88. #492 | C22×C7 | semidirect |
| 89. #487 | C22×C7 | semidirect |
| 92. #486 | C22×C7 | semidirect |
| 98. #322 | C22×C7 | semidirect |
| 99. C82 | C22×C7 | semidirect |
| 206. C7 | #19743 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3207 | 2 |
| C7×#3205 | 2 |
| C7×#1721 | 2 |
| C7×#1717 | 2 |
| C7×#1712 | 2 |
| C7×#1711 | 2 |
| C7×#1454 | 2 |
| #19743 | 7 |