Finite Group DatabasebetaFinite Group DatabasebetaOrder 1680.
132 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 21 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C5×C7×A4 | C22 |
| Fitting, socle | C22×C5×C7 | C2×D3 |
| LCS | G ▷ C5×C7×A4 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. A4×D35 | C2 | semidirect |
| 2. C5⋊(C7⋊S4) | C2 | semidirect |
| 3. C5×C7×S4 | C2 | semidirect |
| 5. C5×C7×A4 | C22 | semidirect |
| 9. C22×D35 | D3 | semidirect |
| 21. C7×S4 | D5 | semidirect |
| 23. C22×C5×C7 | C2×D3 | semidirect |
| 33. C5×S4 | D7 | semidirect |
| 36. C7×A4 | C2×D5 | semidirect |
| 43. D35 | S4 | direct (N × H) |
| 44. C5×A4 | C2×D7 | semidirect |
| 64. C5×C7 | C2×S4 | semidirect |
| 69. C22×C7 | D3×D5 | semidirect |
| 79. S4 | D35 | direct (N × H) |
| 82. C22×C5 | D3×D7 | semidirect |
| 98. A4 | C2×D35 | semidirect |
| 111. C7 | D5×S4 | semidirect |
| 115. C5 | D7×S4 | semidirect |
| 122. C22 | D3×D35 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| A4×D35 | 2 |
| C5⋊(C7⋊S4) | 2 |
| C5×C7×S4 | 2 |
| D4×D35 | 3 |
| D3×D35 | 4 |
| D7×S4 | 5 |
| D5×S4 | 7 |