Finite Group DatabasebetaFinite Group DatabasebetaOrder 1680.
120 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D7×ΓL(1,𝔽16) |
| commutator, hypocenter | C3×C5×C7 | C42 |
| Fitting, socle | C2×C3×C5×C7 | C2×C4 |
| prime radical, cosocle | C2×C3×C7×D5 | C22 |
| LCS | G ▷ C3×C5×C7 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 6. C3⋊(C7⋊AGL(1,𝔽5)) | C4 | semidirect |
| 7. C7×ΓL(1,𝔽16) | C4 | semidirect |
| 8. C3×Dic35 | C4 | semidirect |
| 9. C3×C5×Dic7 | C4 | semidirect |
| 24. Dic35 | Dic3 | semidirect |
| 25. C5×Dic7 | Dic3 | semidirect |
| 30. C3×C5×C7 | C42 | semidirect |
| 34. C3×Dic7 | AGL(1,𝔽5) | semidirect |
| 42. ΓL(1,𝔽16) | Dic7 | direct (N × H) |
| 55. C5×C7 | C4×Dic3 | semidirect |
| 63. Dic7 | ΓL(1,𝔽16) | direct (N × H) |
| 67. C3×C7 | C4×AGL(1,𝔽5) | semidirect |
| 74. C3×C5 | C4×Dic7 | semidirect |
| 89. C7 | C4×ΓL(1,𝔽16) | semidirect |
| 93. C5 | Dic3×Dic7 | semidirect |
| 99. C3 | AGL(1,𝔽5)×Dic7 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3⋊(C7⋊AGL(1,𝔽5)) | 2 |
| C2×C7×ΓL(1,𝔽16) | 2 |
| C3×D5×Dic7 | 2 |
| AGL(1,𝔽5)×Dic7 | 3 |
| Dic3×Dic7 | 5 |
| C4×ΓL(1,𝔽16) | 7 |