Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
11230 conjugacy classes of subgroups.
The subgroup lattice has 144 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C7 | C24 |
| commutator | C2 | C25×C4×C7 |
| Frattini, cosocle | C22 | C26×C7 |
| prime radical | C25×C7 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C7×#519 | C2 | direct (N × H) |
| 3. C22×C4×C7×Q8 | C2 | semidirect |
| 5. C23×C7×Q8 | C4 | semidirect |
| 9. C7×#519 | C22 | direct (N × H) |
| 10. C2×C4×C7×Q8 | C22 | semidirect |
| 15. C22×#519 | C7 | direct (N × H) |
| 18. C22×C7×Q8 | C2×C4 | semidirect |
| 25. C4×C7×Q8 | C23 | semidirect |
| 31. C2×#519 | C2×C7 | direct (N × H) |
| 32. C22×C4×Q8 | C2×C7 | semidirect |
| 39. C2×C7×Q8 | C22×C4 | semidirect |
| 50. C23×Q8 | C4×C7 | semidirect |
| 54. #519 | C22×C7 | direct (N × H) |
| 55. C2×C4×Q8 | C22×C7 | semidirect |
| 75. C22×Q8 | C2×C4×C7 | semidirect |
| 82. C4×Q8 | C23×C7 | semidirect |
| 89. C22×C7 | #519 | direct (N × H) |
| 101. C2×Q8 | C22×C4×C7 | semidirect |
| 113. C2×C7 | C2×#519 | direct (N × H) |
| 129. C7 | C22×#519 | direct (N × H) |
| 132. C22 | C7×#519 | direct (N × H) |
| 140. C2 | C2×C7×#519 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C7×Q8 | 2 |
| C2×C7×#519 | 2 |
| C22×C4×C7×Q8 | 2 |
| C22×C7×#117 | 2 |
| C22×#519 | 7 |