Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
658 conjugacy classes of subgroups.
The subgroup lattice has 160 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C22×D8 |
| commutator | C8 | C23×C4×C7 |
| Frattini, cosocle | C2×C8 | C24×C7 |
| prime radical | C2×C7×#41 | D4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C7×#41 ◁ C2×C7×#131 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C7×#2159 | C2 | semidirect |
| 4. C7×#2158 | C2 | semidirect |
| 5. C2×C7×#366 | C2 | semidirect |
| 8. C7×#504 | C4 | semidirect |
| 18. #33410 | C7 | direct (N × H) |
| 28. C7×M32 | C2×C4 | semidirect |
| 29. C2×C7×C16 | C2×C4 | semidirect |
| 38. #2159 | C2×C7 | semidirect |
| 39. #2158 | C2×C7 | semidirect |
| 40. C2×#366 | C2×C7 | semidirect |
| 61. #504 | C4×C7 | semidirect |
| 93. M32 | C2×C4×C7 | semidirect |
| 94. C2×C16 | C2×C4×C7 | semidirect |
| 145. C7 | #33410 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×#504 | 2 |
| C7×#2913 | 2 |
| C7×#2159 | 2 |
| C7×#2158 | 2 |
| C2×C7×#366 | 2 |
| #33410 | 7 |