Finite Group DatabasebetaFinite Group DatabasebetaOrder 1680.
160 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D5×ΓL(1,𝔽8) |
| commutator, socle, cosocle | C2×C5×C7 | C23×C3 |
| Fitting | C5×C7×D4 | C2×C3 |
| hypercenter | D4 | D5×ΓL(1,𝔽8) |
| hypocenter | C5×C7 | C3×#41 |
| LCS | G ▷ C2×C5×C7 ▷ C5×C7 | |
| UCS | C1 ◁ C2 ◁ D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×D4×ΓL(1,𝔽8) | C2 | semidirect |
| 2. ΓL(1,𝔽8)×C22⋊D5 | C2 | semidirect |
| 3. C2×Dic5×ΓL(1,𝔽8) | C2 | semidirect |
| 4. C4×D5×ΓL(1,𝔽8) | C2 | semidirect |
| 5. ΓL(1,𝔽8)×Dic10 | C2 | semidirect |
| 6. C7×#711 | C3 | semidirect |
| 13. C5×C7×D4 | C2×C3 | semidirect |
| 14. C7×C22⋊D5 | C2×C3 | semidirect |
| 15. C2×C7×Dic5 | C2×C3 | semidirect |
| 16. C4×C7×D5 | C2×C3 | semidirect |
| 17. C7×Dic10 | C2×C3 | semidirect |
| 24. D4×ΓL(1,𝔽8) | D5 | semidirect |
| 38. C5×ΓL(1,𝔽8) | #41 | semidirect |
| 44. #711 | ΓL(1,𝔽8) | direct (N × H) |
| 55. C7×D4 | C3×D5 | semidirect |
| 62. C5×D4 | C2×ΓL(1,𝔽8) | semidirect |
| 63. C22⋊D5 | C2×ΓL(1,𝔽8) | semidirect |
| 64. C2×Dic5 | C2×ΓL(1,𝔽8) | semidirect |
| 65. C4×D5 | C2×ΓL(1,𝔽8) | semidirect |
| 66. Dic10 | C2×ΓL(1,𝔽8) | semidirect |
| 67. C5×C7 | C3×#41 | semidirect |
| 81. ΓL(1,𝔽8) | #711 | direct (N × H) |
| 102. D4 | D5×ΓL(1,𝔽8) | semidirect |
| 105. C7 | C3×#711 | semidirect |
| 109. C5 | #41×ΓL(1,𝔽8) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×D4×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×C22⋊D5 | 2 |
| C2×Dic5×ΓL(1,𝔽8) | 2 |
| C4×D5×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×Dic10 | 2 |
| C7×#711 | 3 |
| #41×ΓL(1,𝔽8) | 5 |
| C3×#711 | 7 |