Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
21104 conjugacy classes of subgroups; only the 6398 normal subgroups are shown.
This group has 21104 conjugacy classes of subgroups — too many to draw. Only its 6398 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24 | D3×AGL(1,𝔽5) |
| commutator, hypocenter | C3×C5 | C25×C4 |
| Frattini | C1 | G |
| Fitting, socle | C24×C3×C5 | C2×C4 |
| prime radical | C24×D3×D5 | C2 |
| cosocle | C3×D5 | C26 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C24×ΓL(1,𝔽16) | C2 | semidirect |
| 3. C24×C3×AGL(1,𝔽5) | C2 | semidirect |
| 4. C23×D3×AGL(1,𝔽5) | C2 | direct (N × H) |
| 6. C24×D15 | C4 | semidirect |
| 7. C24×C5×D3 | C4 | semidirect |
| 9. C23×D3×D5 | C4 | semidirect |
| 11. C23×ΓL(1,𝔽16) | C22 | semidirect |
| 12. C23×C3×AGL(1,𝔽5) | C22 | semidirect |
| 13. C22×D3×AGL(1,𝔽5) | C22 | direct (N × H) |
| 18. C24×AGL(1,𝔽5) | D3 | direct (N × H) |
| 19. C24×C3×C5 | C2×C4 | semidirect |
| 20. C23×D15 | C2×C4 | semidirect |
| 21. C23×C5×D3 | C2×C4 | semidirect |
| 22. C23×C3×D5 | C2×C4 | semidirect |
| 24. C22×D3×D5 | C2×C4 | semidirect |
| 26. C22×ΓL(1,𝔽16) | C23 | semidirect |
| 27. C22×C3×AGL(1,𝔽5) | C23 | semidirect |
| 28. C2×D3×AGL(1,𝔽5) | C23 | direct (N × H) |
| 40. C23×AGL(1,𝔽5) | C2×D3 | direct (N × H) |
| 42. C23×C3×C5 | C22×C4 | semidirect |
| 43. C22×D15 | C22×C4 | semidirect |
| 44. C22×C5×D3 | C22×C4 | semidirect |
| 45. C22×C3×D5 | C22×C4 | semidirect |
| 47. C2×D3×D5 | C22×C4 | semidirect |
| 49. C2×ΓL(1,𝔽16) | C24 | semidirect |
| 50. C2×C3×AGL(1,𝔽5) | C24 | semidirect |
| 51. D3×AGL(1,𝔽5) | C24 | direct (N × H) |
| 56. C24×D3 | AGL(1,𝔽5) | direct (N × H) |
| 61. C24×C5 | C4×D3 | semidirect |
| 65. C23×D5 | C4×D3 | semidirect |
| 69. C22×AGL(1,𝔽5) | C22×D3 | direct (N × H) |
| 74. C22×C3×C5 | C23×C4 | semidirect |
| 75. C2×D15 | C23×C4 | semidirect |
| 76. C2×C5×D3 | C23×C4 | semidirect |
| 77. C2×C3×D5 | C23×C4 | semidirect |
| 79. D3×D5 | C23×C4 | semidirect |
| 81. ΓL(1,𝔽16) | C25 | semidirect |
| 82. C3×AGL(1,𝔽5) | C25 | semidirect |
| 85. C24×C3 | C2×AGL(1,𝔽5) | semidirect |
| 89. C23×D3 | C2×AGL(1,𝔽5) | direct (N × H) |
| 94. C23×C5 | C2×C4×D3 | semidirect |
| 98. C22×D5 | C2×C4×D3 | semidirect |
| 102. C2×AGL(1,𝔽5) | C23×D3 | direct (N × H) |
| 111. C2×C3×C5 | C24×C4 | semidirect |
| 112. D15 | C24×C4 | semidirect |
| 113. C3×D5 | C24×C4 | semidirect |
| 115. C5×D3 | C24×C4 | semidirect |
| 118. C23×C3 | C22×AGL(1,𝔽5) | semidirect |
| 122. C22×D3 | C22×AGL(1,𝔽5) | direct (N × H) |
| 127. C22×C5 | C22×C4×D3 | semidirect |
| 131. C2×D5 | C22×C4×D3 | semidirect |
| 134. AGL(1,𝔽5) | C24×D3 | direct (N × H) |
| 141. C24 | D3×AGL(1,𝔽5) | direct (N × H) |
| 145. C3×C5 | C25×C4 | semidirect |
| 148. C22×C3 | C23×AGL(1,𝔽5) | semidirect |
| 152. C2×D3 | C23×AGL(1,𝔽5) | direct (N × H) |
| 156. C2×C5 | C23×C4×D3 | semidirect |
| 158. D5 | C23×C4×D3 | semidirect |
| 166. C23 | C2×D3×AGL(1,𝔽5) | direct (N × H) |
| 172. C2×C3 | C24×AGL(1,𝔽5) | semidirect |
| 174. D3 | C24×AGL(1,𝔽5) | direct (N × H) |
| 175. C5 | C24×C4×D3 | semidirect |
| 182. C22 | C22×D3×AGL(1,𝔽5) | direct (N × H) |
| 185. C3 | C25×AGL(1,𝔽5) | semidirect |
| 190. C2 | C23×D3×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 |
|---|---|
| C24×D3×D5 | 2 |
| C24×ΓL(1,𝔽16) | 2 |
| C24×C3×AGL(1,𝔽5) | 2 |
| C23×D3×AGL(1,𝔽5) | 2 |
| C25×AGL(1,𝔽5) | 3 |
| C24×C4×D3 | 5 |