Finite Group DatabasebetaFinite Group DatabasebetaOrder 1680.
216 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | D7×D15 |
| commutator, hypocenter | C3×C5×C7 | C22×C4 |
| Frattini | C2 | C2×D7×D15 |
| Fitting | C3×C4×C5×C7 | C22 |
| socle, cosocle | C2×C3×C5×C7 | C23 |
| prime radical | C2×D7×D15 | C2 |
| LCS | G ▷ C3×C5×C7 | |
| UCS | C1 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C4×D105 | C2 | semidirect |
| 3. C4×C7×D15 | C2 | semidirect |
| 4. C3×C4×C5×D7 | C2 | semidirect |
| 5. #11719756 | C2 | semidirect |
| 6. Dic7×D15 | C2 | semidirect |
| 7. D7×Dic15 | C2 | semidirect |
| 12. D7×D15 | C4 | direct (N × H) |
| 13. C3×C4×C5×C7 | C22 | semidirect |
| 14. Dic105 | C22 | semidirect |
| 15. C7×Dic15 | C22 | semidirect |
| 16. C3×C5×Dic7 | C22 | semidirect |
| 21. C4×C5×D7 | D3 | semidirect |
| 27. D105 | C2×C4 | semidirect |
| 28. C7×D15 | C2×C4 | semidirect |
| 29. C3×C5×D7 | C2×C4 | semidirect |
| 33. C3×C4×D7 | D5 | semidirect |
| 41. C4×C5×C7 | C2×D3 | semidirect |
| 43. C5×Dic7 | C2×D3 | semidirect |
| 51. C4×D15 | D7 | direct (N × H) |
| 53. C3×C5×C7 | C22×C4 | semidirect |
| 58. C3×C4×C7 | C2×D5 | semidirect |
| 60. C3×Dic7 | C2×D5 | semidirect |
| 65. C5×D7 | C4×D3 | semidirect |
| 72. C3×C4×C5 | C2×D7 | semidirect |
| 74. Dic15 | C2×D7 | semidirect |
| 81. C4×D7 | D15 | direct (N × H) |
| 85. C3×D7 | C4×D5 | semidirect |
| 94. C5×C7 | C2×C4×D3 | semidirect |
| 98. D15 | C4×D7 | direct (N × H) |
| 104. C4×C7 | C2×D15 | semidirect |
| 106. Dic7 | C2×D15 | semidirect |
| 114. C3×C7 | C2×C4×D5 | semidirect |
| 120. C4×C5 | D3×D7 | semidirect |
| 124. C3×C5 | C2×C4×D7 | semidirect |
| 128. D7 | C4×D15 | direct (N × H) |
| 134. C3×C4 | D5×D7 | semidirect |
| 148. C7 | C2×C4×D15 | semidirect |
| 153. C5 | C4×D3×D7 | semidirect |
| 161. C4 | D7×D15 | direct (N × H) |
| 162. C3 | C4×D5×D7 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D7×D15 | 2 |
| C4×D105 | 2 |
| C4×C7×D15 | 2 |
| C3×C4×C5×D7 | 2 |
| #11719756 | 2 |
| Dic7×D15 | 2 |
| D7×Dic15 | 2 |
| C4×D5×D7 | 3 |
| C4×D3×D7 | 5 |
| C2×C4×D15 | 7 |