Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
294 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 294 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #2184 |
| commutator, Frattini, cosocle | C2×C16 | C23×C7 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C22×C7×C8 ◁ C2×C7×M32 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#3414 | C2 | semidirect |
| 2. C7×#2263 | C2 | semidirect |
| 3. C7×#2218 | C2 | semidirect |
| 4. C7×#1418 | C2 | semidirect |
| 5. C7×#1400 | C2 | semidirect |
| 11. C2×C7×D16 | C22 | semidirect |
| 16. C7×M64 | C22 | semidirect |
| 17. C7×#367 | C22 | semidirect |
| 19. #13667 | C7 | direct (N × H) |
| 32. C7×Q32 | D4 | semidirect |
| 37. C7×D16 | D4 | semidirect |
| 44. #3414 | C2×C7 | semidirect |
| 45. #2263 | C2×C7 | semidirect |
| 46. #2218 | C2×C7 | semidirect |
| 47. #1418 | C2×C7 | semidirect |
| 48. #1400 | C2×C7 | semidirect |
| 81. C2×D16 | C22×C7 | semidirect |
| 86. M64 | C22×C7 | semidirect |
| 87. #367 | C22×C7 | semidirect |
| 124. Q32 | C7×D4 | semidirect |
| 129. D16 | C7×D4 | semidirect |
| 204. C7 | #13667 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3414 | 2 |
| C7×#2263 | 2 |
| C7×#2218 | 2 |
| C7×#1418 | 2 |
| C7×#1400 | 2 |
| #13667 | 7 |