Finite Group DatabasebetaFinite Group Databasebeta← C22×Dic5×AGL(1,𝔽5) #13107637
Order 1600.
1023 conjugacy classes of subgroups; only the 410 normal subgroups are shown.
This group has 1023 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 | D5×AGL(1,𝔽5) |
| commutator, hypocenter | C52 | C22×C42 |
| Frattini | C2 | C22×D5×AGL(1,𝔽5) |
| Fitting, socle | C23×C52 | C2×C4 |
| prime radical | C23×C5×D5 | C22 |
| cosocle | C2×C5×D5 | C24 |
| LCS | G ▷ C52 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C2×Dic5×AGL(1,𝔽5) | C2 | direct (N × H) |
| 6. C22×#1042 | C4 | semidirect |
| 8. C22×C5×AGL(1,𝔽5) | C4 | semidirect |
| 10. C22×#1039 | C4 | semidirect |
| 11. C22×C5×Dic5 | C4 | semidirect |
| 12. C2×D5×Dic5 | C4 | semidirect |
| 14. Dic5×AGL(1,𝔽5) | C22 | direct (N × H) |
| 22. C2×#1042 | C2×C4 | semidirect |
| 24. C2×C5×AGL(1,𝔽5) | C2×C4 | semidirect |
| 26. C2×#1039 | C2×C4 | semidirect |
| 27. C2×C5×Dic5 | C2×C4 | semidirect |
| 28. D5×Dic5 | C2×C4 | semidirect |
| 38. C22×C52 | C42 | semidirect |
| 40. C2×C5×D5 | C42 | semidirect |
| 45. #1042 | C22×C4 | semidirect |
| 46. C5×AGL(1,𝔽5) | C22×C4 | semidirect |
| 47. #1039 | C22×C4 | semidirect |
| 48. C5×Dic5 | C22×C4 | semidirect |
| 53. C22×AGL(1,𝔽5) | Dic5 | direct (N × H) |
| 65. C22×Dic5 | AGL(1,𝔽5) | direct (N × H) |
| 71. C2×C52 | C2×C42 | semidirect |
| 73. C5×D5 | C2×C42 | semidirect |
| 89. C2×AGL(1,𝔽5) | C2×Dic5 | direct (N × H) |
| 101. C2×Dic5 | C2×AGL(1,𝔽5) | direct (N × H) |
| 108. C52 | C22×C42 | semidirect |
| 116. C22×C5 | C4×Dic5 | semidirect |
| 117. C22×C5 | C4×AGL(1,𝔽5) | semidirect |
| 120. C2×D5 | C4×Dic5 | semidirect |
| 126. AGL(1,𝔽5) | C22×Dic5 | direct (N × H) |
| 133. Dic5 | C22×AGL(1,𝔽5) | direct (N × H) |
| 149. C2×C5 | C2×C4×Dic5 | semidirect |
| 150. C2×C5 | C2×C4×AGL(1,𝔽5) | semidirect |
| 153. D5 | C2×C4×Dic5 | semidirect |
| 170. C5 | C22×C4×Dic5 | semidirect |
| 171. C5 | C22×C4×AGL(1,𝔽5) | semidirect |
| 177. C22 | Dic5×AGL(1,𝔽5) | direct (N × H) |
| 186. C2 | C2×Dic5×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×#1042 | 2 |
| C23×C5×AGL(1,𝔽5) | 2 |
| C22×D5×Dic5 | 2 |
| C2×Dic5×AGL(1,𝔽5) | 2 |
| C22×C4×AGL(1,𝔽5) | 5 |
| C22×C4×Dic5 | 5 |