Finite Group DatabasebetaFinite Group Databasebeta← C22×Dic3×AGL(1,𝔽5) #11756855
Order 960.
996 conjugacy classes of subgroups; only the 410 normal subgroups are shown.
This group has 996 conjugacy classes of subgroups — too many to draw. Only its 410 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | D3×AGL(1,𝔽5) |
| commutator, hypocenter | C3×C5 | C22×C42 |
| Frattini | C2 | C22×D3×AGL(1,𝔽5) |
| Fitting, socle | C23×C3×C5 | C2×C4 |
| prime radical | C23×C3×D5 | C22 |
| cosocle | C2×C3×D5 | C24 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C2×Dic3×AGL(1,𝔽5) | C2 | direct (N × H) |
| 7. C22×ΓL(1,𝔽16) | C4 | semidirect |
| 9. C22×C3×AGL(1,𝔽5) | C4 | semidirect |
| 11. C22×Dic15 | C4 | semidirect |
| 12. C22×C5×Dic3 | C4 | semidirect |
| 13. C2×D5×Dic3 | C4 | semidirect |
| 15. Dic3×AGL(1,𝔽5) | C22 | direct (N × H) |
| 26. C2×ΓL(1,𝔽16) | C2×C4 | semidirect |
| 28. C2×C3×AGL(1,𝔽5) | C2×C4 | semidirect |
| 30. C2×Dic15 | C2×C4 | semidirect |
| 31. C2×C5×Dic3 | C2×C4 | semidirect |
| 32. D5×Dic3 | C2×C4 | semidirect |
| 41. C22×AGL(1,𝔽5) | Dic3 | direct (N × H) |
| 49. C22×C3×C5 | C42 | semidirect |
| 51. C2×C3×D5 | C42 | semidirect |
| 56. ΓL(1,𝔽16) | C22×C4 | semidirect |
| 57. C3×AGL(1,𝔽5) | C22×C4 | semidirect |
| 58. Dic15 | C22×C4 | semidirect |
| 59. C5×Dic3 | C22×C4 | semidirect |
| 68. C22×Dic3 | AGL(1,𝔽5) | direct (N × H) |
| 77. C2×AGL(1,𝔽5) | C2×Dic3 | direct (N × H) |
| 87. C2×C3×C5 | C2×C42 | semidirect |
| 89. C3×D5 | C2×C42 | semidirect |
| 104. C2×Dic3 | C2×AGL(1,𝔽5) | direct (N × H) |
| 105. C22×C5 | C4×Dic3 | semidirect |
| 107. C2×D5 | C4×Dic3 | semidirect |
| 113. AGL(1,𝔽5) | C22×Dic3 | direct (N × H) |
| 126. C3×C5 | C22×C42 | semidirect |
| 132. C22×C3 | C4×AGL(1,𝔽5) | semidirect |
| 137. Dic3 | C22×AGL(1,𝔽5) | direct (N × H) |
| 138. C2×C5 | C2×C4×Dic3 | semidirect |
| 140. D5 | C2×C4×Dic3 | semidirect |
| 159. C2×C3 | C2×C4×AGL(1,𝔽5) | semidirect |
| 161. C5 | C22×C4×Dic3 | semidirect |
| 167. C22 | Dic3×AGL(1,𝔽5) | direct (N × H) |
| 173. C3 | C22×C4×AGL(1,𝔽5) | semidirect |
| 177. C2 | C2×Dic3×AGL(1,𝔽5) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×ΓL(1,𝔽16) | 2 |
| C23×C3×AGL(1,𝔽5) | 2 |
| C22×D5×Dic3 | 2 |
| C2×Dic3×AGL(1,𝔽5) | 2 |
| C22×C4×AGL(1,𝔽5) | 3 |
| C22×C4×Dic3 | 5 |