Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
2398 conjugacy classes of subgroups.
The subgroup lattice has 264 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4×C7 | D4 |
| commutator | C4 | C22×C42×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C24×C7 | C2×D4 |
| prime radical | C25×C7 | D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4×C7 ◁ C24×C4×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C7×#1735 | C2 | direct (N × H) |
| 4. C2×C7×#340 | C2 | semidirect |
| 7. C23×C7×C8 | C4 | semidirect |
| 12. C2×C7×#117 | C4 | semidirect |
| 17. C7×#340 | C22 | semidirect |
| 18. C2×#1735 | C7 | direct (N × H) |
| 33. C22×C7×C8 | C2×C4 | semidirect |
| 37. C7×#117 | C2×C4 | semidirect |
| 48. #1735 | C2×C7 | direct (N × H) |
| 49. C2×#340 | C2×C7 | semidirect |
| 87. C23×C8 | C4×C7 | semidirect |
| 92. C2×#117 | C4×C7 | semidirect |
| 97. #340 | C22×C7 | semidirect |
| 142. C22×C8 | C2×C4×C7 | semidirect |
| 146. #117 | C2×C4×C7 | semidirect |
| 207. C2×C7 | #1735 | direct (N × H) |
| 241. C7 | C2×#1735 | direct (N × H) |
| 259. C2 | C7×#1735 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C7×C8 | 2 |
| C22×C7×#117 | 2 |
| C7×#1735 | 2 |
| C2×C7×#340 | 2 |
| C2×#1735 | 7 |