Finite Group DatabasebetaFinite Group DatabasebetaOrder 1680.
160 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 64 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×C5 | D3×D7 |
| commutator, cosocle | C2×C3×C7 | C23×C5 |
| Frattini | C2 | C2×C5×D3×D7 |
| Fitting | C3×C4×C5×C7 | C22 |
| socle | C2×C3×C5×C7 | C23 |
| hypocenter | C3×C7 | C5×#41 |
| LCS | G ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C4×C5 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C4×C5×D21 | C2 | semidirect |
| 2. C4×C5×D3×C7 | C2 | semidirect |
| 3. C3×C4×C5×D7 | C2 | semidirect |
| 4. C5×#4248 | C2 | semidirect |
| 5. C5×#4247 | C2 | semidirect |
| 6. C5×#4246 | C2 | semidirect |
| 7. C5×#4245 | C2 | semidirect |
| 16. #66675 | C5 | direct (N × H) |
| 22. C4×C5×D7 | D3 | semidirect |
| 29. C4×D21 | C2×C5 | semidirect |
| 30. C4×D3×C7 | C2×C5 | semidirect |
| 31. C3×C4×D7 | C2×C5 | semidirect |
| 32. #4248 | C2×C5 | semidirect |
| 33. #4247 | C2×C5 | semidirect |
| 34. #4246 | C2×C5 | semidirect |
| 35. #4245 | C2×C5 | semidirect |
| 48. C4×C5×D3 | D7 | semidirect |
| 51. C3×C5×C7 | #41 | semidirect |
| 76. C4×D7 | C5×D3 | semidirect |
| 90. C5×C7 | #235 | semidirect |
| 107. C4×D3 | C5×D7 | semidirect |
| 109. C3×C7 | C5×#41 | semidirect |
| 118. C3×C5 | #1157 | semidirect |
| 141. C7 | C5×#235 | semidirect |
| 146. C5 | #66675 | direct (N × H) |
| 154. C3 | C5×#1157 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×C5×D21 | 2 |
| C4×C5×D3×C7 | 2 |
| C3×C4×C5×D7 | 2 |
| C5×#4248 | 2 |
| C5×#4247 | 2 |
| C5×#4246 | 2 |
| C5×#4245 | 2 |
| C5×#1157 | 3 |
| #66675 | 5 |
| C5×#235 | 7 |