Finite Group DatabasebetaFinite Group DatabasebetaOrder 1680.
216 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D7×ΓL(1,𝔽16) |
| commutator, hypocenter | C3×C5×C7 | C22×C4 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C5×C7 | C2×C4 |
| prime radical | C2×C3×D5×D7 | C2 |
| cosocle | C3×C7×D5 | C23 |
| LCS | G ▷ C3×C5×C7 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C3⋊(C7⋊AGL(1,𝔽5)) | C2 | semidirect |
| 3. C2×C7×ΓL(1,𝔽16) | C2 | semidirect |
| 4. D7×ΓL(1,𝔽16) | C2 | direct (N × H) |
| 6. C2×C3×D35 | C4 | semidirect |
| 8. C2×C3×C5×D7 | C4 | semidirect |
| 9. C3×D5×D7 | C4 | semidirect |
| 11. C3⋊(C7⋊AGL(1,𝔽5)) | C22 | semidirect |
| 12. C7×ΓL(1,𝔽16) | C22 | semidirect |
| 19. C2×C3×C5×C7 | C2×C4 | semidirect |
| 20. C3×D35 | C2×C4 | semidirect |
| 21. C3×C7×D5 | C2×C4 | semidirect |
| 23. C3×C5×D7 | C2×C4 | semidirect |
| 28. C2×D35 | Dic3 | semidirect |
| 30. C2×C5×D7 | Dic3 | semidirect |
| 31. D5×D7 | Dic3 | semidirect |
| 38. C2×ΓL(1,𝔽16) | D7 | direct (N × H) |
| 40. C3×C5×C7 | C22×C4 | semidirect |
| 45. C2×C3×D7 | AGL(1,𝔽5) | semidirect |
| 49. C2×C5×C7 | C2×Dic3 | semidirect |
| 50. D35 | C2×Dic3 | semidirect |
| 51. C5×D7 | C2×Dic3 | semidirect |
| 52. C7×D5 | C2×Dic3 | semidirect |
| 60. ΓL(1,𝔽16) | C2×D7 | direct (N × H) |
| 68. C2×C3×C7 | C2×AGL(1,𝔽5) | semidirect |
| 70. C3×D7 | C2×AGL(1,𝔽5) | semidirect |
| 75. C5×C7 | C22×Dic3 | semidirect |
| 77. C2×C3×C5 | C4×D7 | semidirect |
| 79. C3×D5 | C4×D7 | semidirect |
| 85. C2×D7 | ΓL(1,𝔽16) | direct (N × H) |
| 92. C3×C7 | C22×AGL(1,𝔽5) | semidirect |
| 101. C3×C5 | C2×C4×D7 | semidirect |
| 104. C2×C7 | C2×ΓL(1,𝔽16) | semidirect |
| 106. D7 | C2×ΓL(1,𝔽16) | direct (N × H) |
| 115. C2×C5 | Dic3×D7 | semidirect |
| 117. D5 | Dic3×D7 | semidirect |
| 123. C7 | C22×ΓL(1,𝔽16) | semidirect |
| 128. C2×C3 | D7×AGL(1,𝔽5) | semidirect |
| 129. C5 | C2×Dic3×D7 | semidirect |
| 137. C3 | C2×D7×AGL(1,𝔽5) | semidirect |
| 142. C2 | D7×ΓL(1,𝔽16) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×D5×D7 | 2 |
| C2×C3⋊(C7⋊AGL(1,𝔽5)) | 2 |
| C2×C7×ΓL(1,𝔽16) | 2 |
| D7×ΓL(1,𝔽16) | 2 |
| C2×D7×AGL(1,𝔽5) | 3 |
| C2×Dic3×D7 | 5 |
| C22×ΓL(1,𝔽16) | 7 |