Finite Group DatabasebetaFinite Group DatabasebetaOrder 1680.
160 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 55 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 | C2×D5×AGL(1,𝔽7) |
| commutator, socle, cosocle | C2×C5×C7 | C23×C3 |
| Fitting | C4×C5×C7 | C22×C3 |
| hypercenter | C4 | D5×AGL(1,𝔽7) |
| hypocenter | C5×C7 | C3×#41 |
| LCS | G ▷ C2×C5×C7 ▷ C5×C7 | |
| UCS | C1 ◁ C2 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C4×C5⋊AGL(1,𝔽7) | C2 | semidirect |
| 2. ΓL(1,𝔽8)×D20 | C2 | semidirect |
| 3. C5×ΓL(1,𝔽8)⋊Q8 | C2 | semidirect |
| 4. #63307⋊C3 | C2 | semidirect |
| 5. D5×ΓL(1,𝔽8)⋊C4 | C2 | semidirect |
| 6. #10587941 | C3 | semidirect |
| 13. C4×D35 | C2×C3 | semidirect |
| 14. C7×D20 | C2×C3 | semidirect |
| 15. C5×Dic14 | C2×C3 | semidirect |
| 16. #63307 | C2×C3 | semidirect |
| 17. D5×Dic7 | C2×C3 | semidirect |
| 26. ΓL(1,𝔽8)⋊Q8 | D5 | semidirect |
| 38. C5×ΓL(1,𝔽8) | #41 | semidirect |
| 57. Dic14 | C3×D5 | semidirect |
| 64. D20 | AGL(1,𝔽7) | semidirect |
| 67. C5×C7 | C3×#41 | semidirect |
| 81. ΓL(1,𝔽8) | #713 | semidirect |
| 105. C7 | C3×#713 | semidirect |
| 109. C5 | #1159⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×C5⋊AGL(1,𝔽7) | 2 |
| ΓL(1,𝔽8)×D20 | 2 |
| C5×ΓL(1,𝔽8)⋊Q8 | 2 |
| #63307⋊C3 | 2 |
| D5×ΓL(1,𝔽8)⋊C4 | 2 |
| #10587941 | 3 |
| #1159⋊C3 | 5 |
| C3×#713 | 7 |