Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
822 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 822 conjugacy classes of subgroups — too many to draw. Only its 98 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×#3006 | C2 | semidirect |
| 2. C7×#2815 | C2 | semidirect |
| 3. C7×#1782 | C2 | semidirect |
| 4. C7×#1510 | C2 | semidirect |
| 5. C7×#1306 | C2 | semidirect |
| 21. #9980 | C7 | direct (N × H) |
| 25. C22×C7×Q8 | D4 | semidirect |
| 30. C22×C7×D4 | D4 | semidirect |
| 50. C7×C22⋊C8 | D4 | semidirect |
| 54. #3006 | C2×C7 | semidirect |
| 55. #2815 | C2×C7 | semidirect |
| 56. #1782 | C2×C7 | semidirect |
| 57. #1510 | C2×C7 | semidirect |
| 58. #1306 | C2×C7 | semidirect |
| 165. C22×Q8 | C7×D4 | semidirect |
| 170. C22×D4 | C7×D4 | semidirect |
| 190. C22⋊C8 | C7×D4 | semidirect |
| 312. C7 | #9980 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3006 | 2 |
| C7×#2815 | 2 |
| C7×#1782 | 2 |
| C7×#1510 | 2 |
| C7×#1306 | 2 |
| #9980 | 7 |