Finite Group DatabasebetaFinite Group DatabasebetaOrder 1680.
216 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 62 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×C5⋊AGL(1,𝔽7) |
| commutator, socle, cosocle | C2×C5×C7 | C23×C3 |
| Fitting | C5×C7×D4 | C2×C3 |
| hypercenter | D4 | C5⋊AGL(1,𝔽7) |
| hypocenter | C5×C7 | C2×C3×D4 |
| 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. C22×C5⋊AGL(1,𝔽7) | C2 | semidirect |
| 2. C5×D4×ΓL(1,𝔽8) | C2 | semidirect |
| 3. (C22⋊D35)⋊C3 | C2 | semidirect |
| 4. C4⋊(C5⋊AGL(1,𝔽7)) | C2 | semidirect |
| 5. C4×C5⋊AGL(1,𝔽7) | C2 | semidirect |
| 6. D4×D35 | C3 | semidirect |
| 7. C22×C5×ΓL(1,𝔽8) | C22 | semidirect |
| 10. C2×C5⋊AGL(1,𝔽7) | C22 | semidirect |
| 11. C4×C5×ΓL(1,𝔽8) | C22 | semidirect |
| 12. ΓL(1,𝔽8)⋊Dic5 | C22 | semidirect |
| 14. C22×D35 | C2×C3 | semidirect |
| 15. C5×C7×D4 | C2×C3 | semidirect |
| 16. C22⋊D35 | C2×C3 | semidirect |
| 17. D140 | C2×C3 | semidirect |
| 18. C4×D35 | C2×C3 | semidirect |
| 23. C5⋊AGL(1,𝔽7) | D4 | direct (N × H) |
| 25. D4×ΓL(1,𝔽8) | D5 | semidirect |
| 29. C22×C5×C7 | C22×C3 | semidirect |
| 32. C2×D35 | C22×C3 | semidirect |
| 33. C4×C5×C7 | C22×C3 | semidirect |
| 34. Dic35 | C22×C3 | semidirect |
| 41. C5×ΓL(1,𝔽8) | C2×D4 | semidirect |
| 42. C22×ΓL(1,𝔽8) | C2×D5 | semidirect |
| 46. C4×ΓL(1,𝔽8) | C2×D5 | semidirect |
| 52. D35 | C3×D4 | semidirect |
| 60. C7×D4 | C3×D5 | semidirect |
| 70. C5×D4 | AGL(1,𝔽7) | semidirect |
| 74. C5×C7 | C2×C3×D4 | semidirect |
| 79. C22×C7 | C2×C3×D5 | semidirect |
| 83. C4×C7 | C2×C3×D5 | semidirect |
| 90. ΓL(1,𝔽8) | D4×D5 | semidirect |
| 91. C22×C5 | C2×AGL(1,𝔽7) | semidirect |
| 95. C4×C5 | C2×AGL(1,𝔽7) | semidirect |
| 116. D4 | C5⋊AGL(1,𝔽7) | direct (N × H) |
| 118. C7 | C3×D4×D5 | semidirect |
| 123. C5 | D4×AGL(1,𝔽7) | semidirect |
| 127. C22 | C2×C5⋊AGL(1,𝔽7) | semidirect |
| 129. C4 | C2×C5⋊AGL(1,𝔽7) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C5⋊AGL(1,𝔽7) | 2 |
| C5×D4×ΓL(1,𝔽8) | 2 |
| (C22⋊D35)⋊C3 | 2 |
| C4⋊(C5⋊AGL(1,𝔽7)) | 2 |
| C4×C5⋊AGL(1,𝔽7) | 2 |
| D4×D35 | 3 |
| D4×AGL(1,𝔽7) | 5 |
| C3×D4×D5 | 7 |