Finite Group DatabasebetaFinite Group DatabasebetaOrder 1680.
216 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 91 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×C5×ΓL(1,𝔽8)⋊C4 | C2 | semidirect |
| 4. D5×ΓL(1,𝔽8)⋊C4 | C2 | direct (N × H) |
| 5. C2×D5×Dic7 | C3 | semidirect |
| 7. C2×D5×ΓL(1,𝔽8) | C4 | semidirect |
| 9. ΓL(1,𝔽8)⋊Dic5 | C22 | semidirect |
| 10. C5×ΓL(1,𝔽8)⋊C4 | C22 | semidirect |
| 13. C2×Dic35 | C2×C3 | semidirect |
| 14. C2×C5×Dic7 | C2×C3 | semidirect |
| 15. D5×Dic7 | C2×C3 | semidirect |
| 17. C2×C5×ΓL(1,𝔽8) | C2×C4 | semidirect |
| 19. D5×ΓL(1,𝔽8) | C2×C4 | semidirect |
| 23. C2×ΓL(1,𝔽8)⋊C4 | D5 | direct (N × H) |
| 25. C2×C7×D5 | C3×C4 | semidirect |
| 27. Dic35 | C22×C3 | semidirect |
| 28. C5×Dic7 | C22×C3 | semidirect |
| 34. C5×ΓL(1,𝔽8) | C22×C4 | semidirect |
| 39. ΓL(1,𝔽8)⋊C4 | C2×D5 | direct (N × H) |
| 41. C2×C5×C7 | C2×C3×C4 | semidirect |
| 43. C7×D5 | C2×C3×C4 | semidirect |
| 52. C2×Dic7 | C3×D5 | semidirect |
| 55. C2×ΓL(1,𝔽8) | C4×D5 | semidirect |
| 61. C5×C7 | C22×C3×C4 | semidirect |
| 69. Dic7 | C2×C3×D5 | semidirect |
| 74. ΓL(1,𝔽8) | C2×C4×D5 | semidirect |
| 76. C2×D5 | ΓL(1,𝔽8)⋊C4 | direct (N × H) |
| 83. C2×C7 | C3×C4×D5 | semidirect |
| 90. C2×C5 | C2×ΓL(1,𝔽8)⋊C4 | semidirect |
| 92. D5 | C2×ΓL(1,𝔽8)⋊C4 | direct (N × H) |
| 97. C7 | C2×C3×C4×D5 | semidirect |
| 101. C5 | C22×ΓL(1,𝔽8)⋊C4 | semidirect |
| 109. C2 | D5×ΓL(1,𝔽8)⋊C4 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×D5×ΓL(1,𝔽8) | 2 |
| C2×ΓL(1,𝔽8)⋊Dic5 | 2 |
| C2×C5×ΓL(1,𝔽8)⋊C4 | 2 |
| D5×ΓL(1,𝔽8)⋊C4 | 2 |
| C2×D5×Dic7 | 3 |
| C22×ΓL(1,𝔽8)⋊C4 | 5 |
| C2×C3×C4×D5 | 7 |