Finite Group DatabasebetaFinite Group Databasebeta← C7×C3⋊(C5⋊AGL(1,𝔽7)) #17984385
Order 4410.
76 conjugacy classes of subgroups.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7 | C3⋊(C5⋊AGL(1,𝔽7)) |
| commutator, cosocle, hypocenter | C3×C5×C7 | C2×C3×C7 |
| Frattini | C1 | G |
| Fitting, socle | C3×C5×C72 | C2×C3 |
| LCS | G ▷ C3×C5×C7 | |
| UCS | C1 ◁ C7 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×C5×C7×ΓL(1,𝔽8) | C2 | semidirect |
| 2. C7×D105 | C3 | semidirect |
| 5. C3×C5×C72 | C2×C3 | semidirect |
| 7. C5×C7×ΓL(1,𝔽8) | D3 | semidirect |
| 9. C3⋊(C5⋊AGL(1,𝔽7)) | C7 | direct (N × H) |
| 11. C3×C7×ΓL(1,𝔽8) | D5 | semidirect |
| 13. C3×C5×ΓL(1,𝔽8) | C2×C7 | semidirect |
| 16. C5×C72 | C3×D3 | semidirect |
| 17. D105 | C3×C7 | semidirect |
| 21. C3×C72 | C3×D5 | semidirect |
| 23. C7×ΓL(1,𝔽8) | D15 | semidirect |
| 29. C3×C5×C7 | AGL(1,𝔽7) | semidirect |
| 30. C3×C5×C7 | C2×C3×C7 | semidirect |
| 32. C5×ΓL(1,𝔽8) | D3×C7 | semidirect |
| 38. C3×ΓL(1,𝔽8) | C7×D5 | semidirect |
| 39. C72 | C3×D15 | semidirect |
| 46. C5×C7 | C3⋊AGL(1,𝔽7) | semidirect |
| 47. C5×C7 | C3×D3×C7 | semidirect |
| 53. C3×C7 | C5⋊AGL(1,𝔽7) | semidirect |
| 54. C3×C7 | C3×C7×D5 | semidirect |
| 56. ΓL(1,𝔽8) | C7×D15 | semidirect |
| 60. C3×C5 | C7×AGL(1,𝔽7) | semidirect |
| 66. C7 | C3⋊(C5⋊AGL(1,𝔽7)) | direct (N × H) |
| 67. C7 | C3×C7×D15 | semidirect |
| 70. C5 | C7×C3⋊AGL(1,𝔽7) | semidirect |
| 73. C3 | C7×C5⋊AGL(1,𝔽7) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C5×C7×ΓL(1,𝔽8) | 2 |
| C7×D105 | 3 |
| C7×C5⋊AGL(1,𝔽7) | 3 |
| C7×C3⋊AGL(1,𝔽7) | 5 |
| C3×C7×D15 | 7 |
| C3⋊(C5⋊AGL(1,𝔽7)) | 7 |