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 | C22×C3×C5×C7 | C22 |
| hypercenter | C22 | D7×D15 |
| hypocenter | C3×C5×C7 | #41 |
| LCS | G ▷ C2×C3×C5×C7 ▷ C3×C5×C7 | |
| UCS | C1 ◁ C2 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×Dic105 | C2 | semidirect |
| 2. C7×#1220 | C2 | semidirect |
| 3. C3×C5×C22⋊D7 | C2 | semidirect |
| 4. #11719760 | C2 | semidirect |
| 5. #11719757 | C2 | semidirect |
| 6. Dic7×D15 | C2 | semidirect |
| 7. D7×Dic15 | C2 | semidirect |
| 19. C5×C22⋊D7 | D3 | semidirect |
| 31. C3×C22⋊D7 | D5 | semidirect |
| 45. #1220 | D7 | semidirect |
| 51. C3×C5×C7 | #41 | semidirect |
| 73. C22⋊D7 | D15 | semidirect |
| 90. C5×C7 | #237 | semidirect |
| 109. C3×C7 | #711 | semidirect |
| 118. C3×C5 | #1159 | semidirect |
| 141. C7 | #6819 | semidirect |
| 146. C5 | #66684 | semidirect |
| 154. C3 | #10587953 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Dic105 | 2 |
| C7×#1220 | 2 |
| C3×C5×C22⋊D7 | 2 |
| #11719760 | 2 |
| #11719757 | 2 |
| Dic7×D15 | 2 |
| D7×Dic15 | 2 |
| #10587953 | 3 |
| #66684 | 5 |
| #6819 | 7 |