Finite Group DatabasebetaFinite Group DatabasebetaOrder 1680.
156 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 38 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 | D7⋊A4 |
| commutator, cosocle, hypocenter | C22×C7 | C22×C3×C5 |
| Frattini | C1 | G |
| Fitting, socle | C23×C5×C7 | C2×C3 |
| LCS | G ▷ C22×C7 | |
| UCS | C1 ◁ C2×C5 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C5×C22⋊ΓL(1,𝔽8) | C2 | semidirect |
| 2. C5×D7⋊A4 | C2 | direct (N × H) |
| 3. C23×C5×D7 | C3 | semidirect |
| 4. C5×C22⋊ΓL(1,𝔽8) | C22 | semidirect |
| 6. C2×D7⋊A4 | C5 | direct (N × H) |
| 7. C23×C5×C7 | C2×C3 | semidirect |
| 11. C22×C5×D7 | C2×C3 | semidirect |
| 15. C2×C22⋊ΓL(1,𝔽8) | C2×C5 | semidirect |
| 16. D7⋊A4 | C2×C5 | direct (N × H) |
| 19. C22×C5×C7 | C22×C3 | semidirect |
| 25. C2×C5×D7 | A4 | semidirect |
| 28. C23×D7 | C3×C5 | semidirect |
| 30. C22⋊ΓL(1,𝔽8) | C22×C5 | semidirect |
| 35. C2×C5×C7 | C2×A4 | semidirect |
| 37. C5×D7 | C2×A4 | semidirect |
| 40. C23×C7 | C2×C3×C5 | semidirect |
| 44. C22×D7 | C2×C3×C5 | semidirect |
| 52. C23×C5 | AGL(1,𝔽7) | semidirect |
| 53. C5×C7 | C22×A4 | semidirect |
| 58. C22×C7 | C22×C3×C5 | semidirect |
| 64. C2×D7 | C5×A4 | semidirect |
| 76. C22×C5 | C2×AGL(1,𝔽7) | semidirect |
| 81. C2×C7 | C2×C5×A4 | semidirect |
| 83. D7 | C2×C5×A4 | semidirect |
| 90. C2×C5 | D7⋊A4 | direct (N × H) |
| 95. C23 | C5×AGL(1,𝔽7) | semidirect |
| 96. C7 | C22×C5×A4 | semidirect |
| 99. C5 | C2×D7⋊A4 | direct (N × H) |
| 108. C22 | C2×C5×AGL(1,𝔽7) | semidirect |
| 114. C2 | C5×D7⋊A4 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C5×C22⋊ΓL(1,𝔽8) | 2 |
| C5×D7⋊A4 | 2 |
| C23×C5×D7 | 3 |
| C2×C5×AGL(1,𝔽7) | 4 |
| C2×D7⋊A4 | 5 |
| C22×C5×A4 | 7 |