Finite Group DatabasebetaFinite Group DatabasebetaOrder 1680.
216 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×ΓL(1,𝔽8)⋊Dic5 | C2 | semidirect |
| 3. C2×Dic5×ΓL(1,𝔽8) | C2 | semidirect |
| 4. Dic5×AGL(1,𝔽7) | C2 | direct (N × H) |
| 5. C2×D7×Dic5 | C3 | semidirect |
| 7. C2×C5×AGL(1,𝔽7) | C4 | semidirect |
| 9. ΓL(1,𝔽8)⋊Dic5 | C22 | semidirect |
| 10. Dic5×ΓL(1,𝔽8) | C22 | semidirect |
| 13. C2×Dic35 | C2×C3 | semidirect |
| 14. C2×C7×Dic5 | C2×C3 | semidirect |
| 15. D7×Dic5 | C2×C3 | semidirect |
| 17. C2×C5×ΓL(1,𝔽8) | C2×C4 | semidirect |
| 19. C5×AGL(1,𝔽7) | C2×C4 | semidirect |
| 25. C2×C5×D7 | C3×C4 | semidirect |
| 27. Dic35 | C22×C3 | semidirect |
| 28. C7×Dic5 | C22×C3 | semidirect |
| 34. C5×ΓL(1,𝔽8) | C22×C4 | semidirect |
| 36. C2×AGL(1,𝔽7) | Dic5 | direct (N × H) |
| 41. C2×C5×C7 | C2×C3×C4 | semidirect |
| 43. C5×D7 | C2×C3×C4 | semidirect |
| 54. C2×ΓL(1,𝔽8) | C2×Dic5 | semidirect |
| 56. AGL(1,𝔽7) | C2×Dic5 | direct (N × H) |
| 60. C2×Dic5 | AGL(1,𝔽7) | direct (N × H) |
| 61. C5×C7 | C22×C3×C4 | semidirect |
| 66. C2×D7 | C3×Dic5 | semidirect |
| 74. ΓL(1,𝔽8) | C22×Dic5 | semidirect |
| 79. Dic5 | C2×AGL(1,𝔽7) | direct (N × H) |
| 82. C2×C7 | C2×C3×Dic5 | semidirect |
| 84. D7 | C2×C3×Dic5 | semidirect |
| 91. C2×C5 | C4×AGL(1,𝔽7) | semidirect |
| 97. C7 | C22×C3×Dic5 | semidirect |
| 101. C5 | C2×C4×AGL(1,𝔽7) | semidirect |
| 109. C2 | Dic5×AGL(1,𝔽7) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C5×AGL(1,𝔽7) | 2 |
| C2×ΓL(1,𝔽8)⋊Dic5 | 2 |
| C2×Dic5×ΓL(1,𝔽8) | 2 |
| Dic5×AGL(1,𝔽7) | 2 |
| C2×D7×Dic5 | 3 |
| C2×C4×AGL(1,𝔽7) | 5 |
| C22×C3×Dic5 | 7 |