Finite Group DatabasebetaFinite Group Databasebeta← AGL(1,𝔽7)×AΣL(1,𝔽16) #72276464
Order 2688.
876 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 876 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #100×AGL(1,𝔽7) | C2 | semidirect |
| 3. #99×AGL(1,𝔽7) | C2 | semidirect |
| 4. ΓL(1,𝔽8)×AΣL(1,𝔽16) | C2 | semidirect |
| 5. #89331⋊C3 | C2 | semidirect |
| 6. #89270⋊C3 | C2 | semidirect |
| 7. #89266⋊C3 | C2 | semidirect |
| 8. D7×AΣL(1,𝔽16) | C3 | semidirect |
| 9. C24×AGL(1,𝔽7) | C4 | semidirect |
| 13. #10620789 | C4 | semidirect |
| 18. #6445⋊C3 | C4 | semidirect |
| 21. C22⋊C4×AGL(1,𝔽7) | C4 | semidirect |
| 22. ΓL(1,𝔽8)×#100 | C22 | semidirect |
| 23. ΓL(1,𝔽8)×#99 | C22 | semidirect |
| 24. #6408⋊C3 | C22 | semidirect |
| 25. #6407⋊C3 | C22 | semidirect |
| 29. D7×#100 | C2×C3 | semidirect |
| 30. D7×#99 | C2×C3 | semidirect |
| 31. C7×AΣL(1,𝔽16) | C2×C3 | semidirect |
| 32. #89331 | C2×C3 | semidirect |
| 33. #89270 | C2×C3 | semidirect |
| 34. #89266 | C2×C3 | semidirect |
| 36. C24×ΓL(1,𝔽8) | C2×C4 | semidirect |
| 43. C23×AGL(1,𝔽7) | C2×C4 | semidirect |
| 62. C22⋊C4×ΓL(1,𝔽8) | C2×C4 | semidirect |
| 65. (C22⋊Dic7)⋊C3 | C2×C4 | semidirect |
| 73. C24×D7 | C3×C4 | semidirect |
| 77. #6516 | C3×C4 | semidirect |
| 82. #6445 | C3×C4 | semidirect |
| 85. D7×C22⋊C4 | C3×C4 | semidirect |
| 86. C7×#100 | C22×C3 | semidirect |
| 87. C7×#99 | C22×C3 | semidirect |
| 88. #6408 | C22×C3 | semidirect |
| 89. #6407 | C22×C3 | semidirect |
| 148. C24×C7 | C2×C3×C4 | semidirect |
| 155. C23×D7 | C2×C3×C4 | semidirect |
| 174. C7×C22⋊C4 | C2×C3×C4 | semidirect |
| 177. C22⋊Dic7 | C2×C3×C4 | semidirect |
| 245. AΣL(1,𝔽16) | AGL(1,𝔽7) | direct (N × H) |
| 340. AGL(1,𝔽7) | AΣL(1,𝔽16) | direct (N × H) |
| 356. #100 | C2×AGL(1,𝔽7) | semidirect |
| 359. #99 | C2×AGL(1,𝔽7) | semidirect |
| 443. ΓL(1,𝔽8) | C2×AΣL(1,𝔽16) | semidirect |
| 452. C24 | C4×AGL(1,𝔽7) | semidirect |
| 480. C22⋊C4 | C4×AGL(1,𝔽7) | semidirect |
| 490. D7 | C3×AΣL(1,𝔽16) | semidirect |
| 574. C7 | C2×C3×AΣL(1,𝔽16) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #120×AGL(1,𝔽7) | 2 |
| #100×AGL(1,𝔽7) | 2 |
| #99×AGL(1,𝔽7) | 2 |
| ΓL(1,𝔽8)×AΣL(1,𝔽16) | 2 |
| #89331⋊C3 | 2 |
| #89270⋊C3 | 2 |
| #89266⋊C3 | 2 |
| D7×AΣL(1,𝔽16) | 3 |
| C2×C3×AΣL(1,𝔽16) | 7 |