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, hypercenter | C4 | D7×D15 |
| commutator, socle, cosocle | C2×C3×C5×C7 | C23 |
| Frattini | C2 | C2×D7×D15 |
| Fitting | C3×C4×C5×C7 | C22 |
| hypocenter | C3×C5×C7 | #41 |
| LCS | G ▷ C2×C3×C5×C7 ▷ C3×C5×C7 | |
| UCS | C1 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C4×D105 | C2 | semidirect |
| 2. C4×C7×D15 | C2 | semidirect |
| 3. C3×C4×C5×D7 | C2 | semidirect |
| 4. #11719760 | C2 | semidirect |
| 5. #11719759 | C2 | semidirect |
| 6. #11719758 | C2 | semidirect |
| 7. #11719757 | C2 | semidirect |
| 19. C4×C5×D7 | D3 | semidirect |
| 31. C3×C4×D7 | D5 | semidirect |
| 48. C4×D15 | D7 | semidirect |
| 51. C3×C5×C7 | #41 | semidirect |
| 76. C4×D7 | D15 | semidirect |
| 90. C5×C7 | #235 | semidirect |
| 109. C3×C7 | #709 | semidirect |
| 118. C3×C5 | #1157 | semidirect |
| 141. C7 | #6817 | semidirect |
| 146. C5 | #66675 | semidirect |
| 154. C3 | #10587944 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×D105 | 2 |
| C4×C7×D15 | 2 |
| C3×C4×C5×D7 | 2 |
| #11719760 | 2 |
| #11719759 | 2 |
| #11719758 | 2 |
| #11719757 | 2 |
| #10587944 | 3 |
| #66675 | 5 |
| #6817 | 7 |