Finite Group DatabasebetaFinite Group DatabasebetaOrder 1680.
160 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×C4×C5×C7 | C2 | semidirect |
| 2. C7×#1220 | C2 | semidirect |
| 3. C7×D60 | C2 | semidirect |
| 4. C4×C7×D15 | C2 | semidirect |
| 5. C7×Dic30 | C2 | semidirect |
| 13. C2×C4×C5×C7 | D3 | semidirect |
| 18. #6817 | C7 | direct (N × H) |
| 22. C2×C3×C4×C7 | D5 | semidirect |
| 32. C2×C3×C4×C5 | C2×C7 | semidirect |
| 33. #1220 | C2×C7 | semidirect |
| 34. D60 | C2×C7 | semidirect |
| 35. C4×D15 | C2×C7 | semidirect |
| 36. Dic30 | C2×C7 | semidirect |
| 38. C3×C5×C7 | #41 | semidirect |
| 57. C2×C4×C7 | D15 | semidirect |
| 62. C2×C4×C5 | D3×C7 | semidirect |
| 67. C5×C7 | #235 | semidirect |
| 76. C2×C3×C4 | C7×D5 | semidirect |
| 81. C3×C7 | #709 | semidirect |
| 88. C3×C5 | C7×#41 | semidirect |
| 104. C2×C4 | C7×D15 | semidirect |
| 105. C7 | #6817 | direct (N × H) |
| 109. C5 | C7×#235 | semidirect |
| 115. C3 | C7×#709 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×C4×C5×C7 | 2 |
| C7×#1220 | 2 |
| C7×D60 | 2 |
| C4×C7×D15 | 2 |
| C7×Dic30 | 2 |
| C7×#709 | 3 |
| C7×#235 | 5 |
| #6817 | 7 |