Finite Group DatabasebetaFinite Group DatabasebetaOrder 1680.
132 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | C7⋊S4 |
| commutator, cosocle, hypocenter | C7×A4 | C22×C5 |
| Frattini | C1 | G |
| Fitting, socle | C23×C5×C7 | D3 |
| LCS | G ▷ C7×A4 | |
| UCS | C1 ◁ C2×C5 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C5×C7×A4 | C2 | semidirect |
| 2. C5×C7⋊S4 | C2 | direct (N × H) |
| 5. C5×C7×A4 | C22 | semidirect |
| 6. C2×C7⋊S4 | C5 | direct (N × H) |
| 7. C23×C5×C7 | D3 | semidirect |
| 15. C2×C7×A4 | C2×C5 | semidirect |
| 16. C7⋊S4 | C2×C5 | direct (N × H) |
| 19. C22×C5×C7 | C2×D3 | semidirect |
| 24. C2×C5×A4 | D7 | semidirect |
| 29. C7×A4 | C22×C5 | semidirect |
| 33. C2×C5×C7 | S4 | semidirect |
| 36. C5×A4 | C2×D7 | semidirect |
| 37. C23×C7 | C5×D3 | semidirect |
| 46. C23×C5 | D21 | semidirect |
| 50. C5×C7 | C2×S4 | semidirect |
| 55. C22×C7 | C2×C5×D3 | semidirect |
| 60. C2×A4 | C5×D7 | semidirect |
| 68. C22×C5 | C2×D21 | semidirect |
| 74. C2×C7 | C5×S4 | semidirect |
| 77. A4 | C2×C5×D7 | semidirect |
| 81. C2×C5 | C7⋊S4 | direct (N × H) |
| 83. C23 | C5×D21 | semidirect |
| 87. C7 | C2×C5×S4 | semidirect |
| 90. C5 | C2×C7⋊S4 | direct (N × H) |
| 96. C22 | C2×C5×D21 | semidirect |
| 102. C2 | C5×C7⋊S4 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C5×C7×A4 | 2 |
| C5×C7⋊S4 | 2 |
| C2×C5×C22⋊D7 | 3 |
| C2×C5×D21 | 4 |
| C2×C7⋊S4 | 5 |
| C2×C5×S4 | 7 |