Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
202 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 202 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C7×C16 | C23 |
| commutator | C22 | C22×C7×C16 |
| Frattini, cosocle | C2×C16 | C23×C7 |
| socle | C22×C7 | C22×C16 |
| prime radical | C2×C7×D4 | C16 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C7×C16 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C7×M64 | C2 | semidirect |
| 3. C7×C4⋊C32 | C2 | semidirect |
| 4. C7×C22⋊C32 | C2 | semidirect |
| 5. C4×C7×C32 | C2 | semidirect |
| 15. #13646 | C7 | direct (N × H) |
| 28. C7×C32 | D4 | semidirect |
| 29. C2×M64 | C2×C7 | semidirect |
| 31. C4⋊C32 | C2×C7 | semidirect |
| 32. C22⋊C32 | C2×C7 | semidirect |
| 33. C4×C32 | C2×C7 | semidirect |
| 81. C32 | C7×D4 | semidirect |
| 88. C4×C7 | M64 | semidirect |
| 121. C7 | #13646 | direct (N × H) |
| 128. C4 | C7×M64 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×M64 | 2 |
| C7×D4×C16 | 2 |
| C7×C4⋊C32 | 2 |
| C7×C22⋊C32 | 2 |
| C4×C7×C32 | 2 |
| #13646 | 7 |