Finite Group DatabasebetaFinite Group DatabasebetaOrder 1680.
216 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 108 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. C23×C3×C5×C7 | C2 | semidirect |
| 2. C22×C5×D3×C7 | C2 | semidirect |
| 3. C5×C7×#67 | C2 | direct (N × H) |
| 4. C2×C5×C7×Dic3 | C2 | semidirect |
| 8. C22×C3×C5×C7 | C22 | semidirect |
| 10. C2×C5×D3×C7 | C22 | semidirect |
| 11. C5×C7×Dic3 | C22 | semidirect |
| 12. C2×C7×#67 | C5 | direct (N × H) |
| 14. C23×C5×C7 | D3 | semidirect |
| 17. C2×C5×#67 | C7 | direct (N × H) |
| 19. C2×C3×C5×C7 | D4 | semidirect |
| 22. C23×C3×C7 | C2×C5 | semidirect |
| 23. C22×D3×C7 | C2×C5 | semidirect |
| 24. C7×#67 | C2×C5 | direct (N × H) |
| 25. C2×C7×Dic3 | C2×C5 | semidirect |
| 30. C22×C5×C7 | C2×D3 | semidirect |
| 32. C23×C3×C5 | C2×C7 | semidirect |
| 33. C22×C5×D3 | C2×C7 | semidirect |
| 34. C5×#67 | C2×C7 | direct (N × H) |
| 35. C2×C5×Dic3 | C2×C7 | semidirect |
| 37. C3×C5×C7 | C2×D4 | semidirect |
| 40. C22×C3×C7 | C22×C5 | semidirect |
| 42. C2×D3×C7 | C22×C5 | semidirect |
| 43. C7×Dic3 | C22×C5 | semidirect |
| 47. C2×C5×C7 | #67 | direct (N × H) |
| 51. C22×C3×C5 | C22×C7 | semidirect |
| 53. C2×C5×D3 | C22×C7 | semidirect |
| 54. C5×Dic3 | C22×C7 | semidirect |
| 56. C23×C7 | C5×D3 | semidirect |
| 59. C2×#67 | C5×C7 | direct (N × H) |
| 61. C2×C3×C7 | C5×D4 | semidirect |
| 65. C23×C5 | D3×C7 | semidirect |
| 68. C5×C7 | C2×#67 | direct (N × H) |
| 70. C2×C3×C5 | C7×D4 | semidirect |
| 77. C22×C7 | C2×C5×D3 | semidirect |
| 79. C23×C3 | C2×C5×C7 | semidirect |
| 80. C22×D3 | C2×C5×C7 | semidirect |
| 81. #67 | C2×C5×C7 | direct (N × H) |
| 82. C2×Dic3 | C2×C5×C7 | semidirect |
| 83. C3×C7 | C2×C5×D4 | semidirect |
| 88. C22×C5 | C2×D3×C7 | semidirect |
| 91. C3×C5 | C2×C7×D4 | semidirect |
| 94. C2×C7 | C5×#67 | direct (N × H) |
| 98. C22×C3 | C22×C5×C7 | semidirect |
| 100. C2×D3 | C22×C5×C7 | semidirect |
| 101. Dic3 | C22×C5×C7 | semidirect |
| 104. C2×C5 | C7×#67 | direct (N × H) |
| 107. C23 | C5×D3×C7 | semidirect |
| 110. C7 | C2×C5×#67 | direct (N × H) |
| 112. C2×C3 | C5×C7×D4 | semidirect |
| 115. C5 | C2×C7×#67 | direct (N × H) |
| 120. C22 | C2×C5×D3×C7 | semidirect |
| 122. C3 | C2×C5×C7×D4 | semidirect |
| 125. C2 | C5×C7×#67 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×C5×C7 | 2 |
| C22×C5×D3×C7 | 2 |
| C5×C7×#67 | 2 |
| C2×C5×C7×Dic3 | 2 |
| C2×C5×C7×D4 | 3 |
| C2×C7×#67 | 5 |
| C2×C5×#67 | 7 |