Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
2791 conjugacy classes of subgroups; only the 882 normal subgroups are shown.
This group has 2791 conjugacy classes of subgroups — too many to draw. Only its 882 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 | C24×C4 |
| Frattini | C1 | G |
| Fitting, socle | C23×C52 | C2×C4 |
| prime radical | C23×D52 | C2 |
| cosocle | C5×D5 | C25 |
| LCS | G ▷ C52 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C23×#1042 | C2 | semidirect |
| 3. C23×C5×AGL(1,𝔽5) | C2 | semidirect |
| 4. C22×D5×AGL(1,𝔽5) | C2 | direct (N × H) |
| 5. C23×C5⋊D5 | C4 | semidirect |
| 6. C23×C5×D5 | C4 | semidirect |
| 8. C22×D52 | C4 | semidirect |
| 10. C22×#1042 | C22 | semidirect |
| 11. C22×C5×AGL(1,𝔽5) | C22 | semidirect |
| 12. C2×D5×AGL(1,𝔽5) | C22 | direct (N × H) |
| 15. C23×C52 | C2×C4 | semidirect |
| 16. C22×C5⋊D5 | C2×C4 | semidirect |
| 17. C22×C5×D5 | C2×C4 | semidirect |
| 18. C22×C5×D5 | C2×C4 | semidirect |
| 20. C2×D52 | C2×C4 | semidirect |
| 22. C2×#1042 | C23 | semidirect |
| 23. C2×C5×AGL(1,𝔽5) | C23 | semidirect |
| 24. D5×AGL(1,𝔽5) | C23 | direct (N × H) |
| 29. C23×AGL(1,𝔽5) | D5 | direct (N × H) |
| 33. C22×C52 | C22×C4 | semidirect |
| 34. C2×C5⋊D5 | C22×C4 | semidirect |
| 35. C2×C5×D5 | C22×C4 | semidirect |
| 36. C2×C5×D5 | C22×C4 | semidirect |
| 38. D52 | C22×C4 | semidirect |
| 40. #1042 | C24 | semidirect |
| 41. C5×AGL(1,𝔽5) | C24 | semidirect |
| 51. C23×D5 | AGL(1,𝔽5) | direct (N × H) |
| 55. C22×AGL(1,𝔽5) | C2×D5 | direct (N × H) |
| 60. C2×C52 | C23×C4 | semidirect |
| 61. C5⋊D5 | C23×C4 | semidirect |
| 62. C5×D5 | C23×C4 | semidirect |
| 63. C5×D5 | C23×C4 | semidirect |
| 69. C23×C5 | C4×D5 | semidirect |
| 70. C23×C5 | C2×AGL(1,𝔽5) | semidirect |
| 78. C22×D5 | C4×D5 | semidirect |
| 79. C22×D5 | C2×AGL(1,𝔽5) | direct (N × H) |
| 83. C2×AGL(1,𝔽5) | C22×D5 | direct (N × H) |
| 91. C52 | C24×C4 | semidirect |
| 96. C22×C5 | C2×C4×D5 | semidirect |
| 97. C22×C5 | C22×AGL(1,𝔽5) | semidirect |
| 105. C2×D5 | C2×C4×D5 | semidirect |
| 106. C2×D5 | C22×AGL(1,𝔽5) | direct (N × H) |
| 109. AGL(1,𝔽5) | C23×D5 | direct (N × H) |
| 124. C2×C5 | C22×C4×D5 | semidirect |
| 125. C2×C5 | C23×AGL(1,𝔽5) | semidirect |
| 129. D5 | C22×C4×D5 | semidirect |
| 130. D5 | C23×AGL(1,𝔽5) | direct (N × H) |
| 138. C23 | D5×AGL(1,𝔽5) | direct (N × H) |
| 143. C5 | C23×C4×D5 | semidirect |
| 144. C5 | C24×AGL(1,𝔽5) | semidirect |
| 151. C22 | C2×D5×AGL(1,𝔽5) | direct (N × H) |
| 158. C2 | C22×D5×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×D52 | 2 |
| C23×#1042 | 2 |
| C23×C5×AGL(1,𝔽5) | 2 |
| C22×D5×AGL(1,𝔽5) | 2 |
| C24×AGL(1,𝔽5) | 5 |
| C23×C4×D5 | 5 |