Finite Group DatabasebetaFinite Group DatabasebetaOrder 1680.
160 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 92 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. C3×C5×C7×D4 | C2 | semidirect |
| 2. C5×C7×#67 | C2 | semidirect |
| 3. C2×C5×C7×Dic3 | C2 | semidirect |
| 4. C4×C5×D3×C7 | C2 | semidirect |
| 5. C5×C7×Dic6 | C2 | semidirect |
| 12. C7×#237 | C5 | direct (N × H) |
| 15. C5×C7×D4 | D3 | semidirect |
| 18. C5×#237 | C7 | direct (N × H) |
| 22. C3×C7×D4 | C2×C5 | semidirect |
| 23. C7×#67 | C2×C5 | semidirect |
| 24. C2×C7×Dic3 | C2×C5 | semidirect |
| 25. C4×D3×C7 | C2×C5 | semidirect |
| 26. C7×Dic6 | C2×C5 | semidirect |
| 32. C3×C5×D4 | C2×C7 | semidirect |
| 33. C5×#67 | C2×C7 | semidirect |
| 34. C2×C5×Dic3 | C2×C7 | semidirect |
| 35. C4×C5×D3 | C2×C7 | semidirect |
| 36. C5×Dic6 | C2×C7 | semidirect |
| 38. C3×C5×C7 | #41 | semidirect |
| 55. C7×D4 | C5×D3 | semidirect |
| 58. #237 | C5×C7 | direct (N × H) |
| 64. C5×D4 | D3×C7 | semidirect |
| 67. C5×C7 | #237 | direct (N × H) |
| 76. C3×D4 | C2×C5×C7 | semidirect |
| 77. #67 | C2×C5×C7 | semidirect |
| 78. C2×Dic3 | C2×C5×C7 | semidirect |
| 79. C4×D3 | C2×C5×C7 | semidirect |
| 80. Dic6 | C2×C5×C7 | semidirect |
| 81. C3×C7 | C5×#41 | semidirect |
| 88. C3×C5 | C7×#41 | semidirect |
| 102. D4 | C5×D3×C7 | semidirect |
| 105. C7 | C5×#237 | direct (N × H) |
| 109. C5 | C7×#237 | direct (N × H) |
| 115. C3 | C5×C7×#41 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C5×C7×D4 | 2 |
| C5×C7×#67 | 2 |
| C2×C5×C7×Dic3 | 2 |
| C4×C5×D3×C7 | 2 |
| C5×C7×Dic6 | 2 |
| C5×C7×#41 | 3 |
| C7×#237 | 5 |
| C5×#237 | 7 |