Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
468 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 468 conjugacy classes of subgroups — too many to draw. Only its 94 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 | #353 |
| commutator | C42 | C22×C4×C7 |
| Frattini, cosocle | #124 | C23×C7 |
| prime radical | C2×C7×#496 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C7 ◁ C22×C4×C7 ◁ C2×C7×#124 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C7×#355 | C2 | semidirect |
| 3. C2×C7×#354 | C2 | semidirect |
| 4. C7×#1406 | C2 | direct (N × H) |
| 7. C2×C7×#127 | C4 | semidirect |
| 11. C7×#496 | C4 | semidirect |
| 15. C2×C7×#97 | C4 | semidirect |
| 17. C7×#355 | C22 | semidirect |
| 19. C7×#354 | C22 | semidirect |
| 20. C2×#1406 | C7 | direct (N × H) |
| 37. C7×#127 | C2×C4 | semidirect |
| 51. C7×#97 | C2×C4 | semidirect |
| 53. C2×#355 | C2×C7 | semidirect |
| 54. C2×#354 | C2×C7 | semidirect |
| 55. #1406 | C2×C7 | direct (N × H) |
| 102. C2×#127 | C4×C7 | semidirect |
| 106. #496 | C4×C7 | semidirect |
| 110. C2×#97 | C4×C7 | semidirect |
| 112. #355 | C22×C7 | semidirect |
| 114. #354 | C22×C7 | semidirect |
| 168. #127 | C2×C4×C7 | semidirect |
| 182. #97 | C2×C4×C7 | semidirect |
| 253. C2×C7 | #1406 | direct (N × H) |
| 292. C7 | C2×#1406 | direct (N × H) |
| 319. C2 | C7×#1406 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×#496 | 2 |
| C2×C7×#355 | 2 |
| C2×C7×#354 | 2 |
| C7×#1406 | 2 |
| C2×#1406 | 7 |