Finite Group DatabasebetaFinite Group DatabasebetaOrder 1680.
160 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D7×D15 |
| commutator, socle, cosocle | C2×C3×C5×C7 | C23 |
| Fitting | C3×C4×C5×C7 | C22 |
| hypercenter | C4 | D7×D15 |
| hypocenter | C3×C5×C7 | #41 |
| LCS | G ▷ C2×C3×C5×C7 ▷ C3×C5×C7 | |
| UCS | C1 ◁ C2 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D420 | C2 | semidirect |
| 2. C7×Dic30 | C2 | semidirect |
| 3. C3×C4×C5×D7 | C2 | semidirect |
| 4. #11719758 | C2 | semidirect |
| 5. #11719756 | C2 | semidirect |
| 15. C4×C5×D7 | D3 | semidirect |
| 24. C3×C4×D7 | D5 | semidirect |
| 36. Dic30 | D7 | semidirect |
| 38. C3×C5×C7 | #41 | semidirect |
| 57. C4×D7 | D15 | semidirect |
| 67. C5×C7 | #235 | semidirect |
| 81. C3×C7 | #709 | semidirect |
| 88. C3×C5 | #1161 | semidirect |
| 105. C7 | #6817 | semidirect |
| 109. C5 | #66677 | semidirect |
| 115. C3 | #10587946 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D420 | 2 |
| C7×Dic30 | 2 |
| C3×C4×C5×D7 | 2 |
| #11719758 | 2 |
| #11719756 | 2 |
| #10587946 | 3 |
| #66677 | 5 |
| #6817 | 7 |