Finite Group DatabasebetaFinite Group DatabasebetaOrder 36900.
212 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | D15×D123 |
| commutator, cosocle, hypocenter | C32×C5×C41 | C22×C5 |
| Frattini | C1 | G |
| Fitting, socle | C32×C52×C41 | C22 |
| LCS | G ▷ C32×C5×C41 | |
| UCS | C1 ◁ C5 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×C52×D123 | C2 | semidirect |
| 2. C3×C5×D15×C41 | C2 | semidirect |
| 3. C5×C3⋊D615 | C2 | semidirect |
| 6. C32×C52×C41 | C22 | semidirect |
| 7. D15×D123 | C5 | direct (N × H) |
| 12. C5×D15×C41 | D3 | semidirect |
| 13. C52×D123 | D3 | semidirect |
| 17. C3⋊D615 | C2×C5 | semidirect |
| 19. C3×D15×C41 | C2×C5 | semidirect |
| 21. C3×C5×D123 | D5 | semidirect |
| 22. C3×C5×D123 | C2×C5 | semidirect |
| 25. C3×C52×C41 | C2×D3 | semidirect |
| 26. C3×C52×C41 | C2×D3 | semidirect |
| 35. C32×C5×C41 | C2×D5 | semidirect |
| 36. C32×C5×C41 | C22×C5 | semidirect |
| 42. D15×C41 | C5×D3 | semidirect |
| 47. C5×D123 | C5×D3 | semidirect |
| 48. C5×D123 | D15 | direct (N × H) |
| 54. C52×C41 | D32 | semidirect |
| 59. C3×D123 | C5×D5 | semidirect |
| 66. C3×C5×C41 | D3×D5 | semidirect |
| 67. C3×C5×C41 | C2×C5×D3 | semidirect |
| 68. C3×C5×C41 | C2×C5×D3 | semidirect |
| 69. C3×C5×C41 | C2×D15 | semidirect |
| 73. C3×C5×D15 | D41 | semidirect |
| 82. C32×C41 | C2×C5×D5 | semidirect |
| 89. D123 | C5×D15 | direct (N × H) |
| 92. C32×C52 | C2×D41 | semidirect |
| 94. C5×C41 | C5×D32 | semidirect |
| 95. C5×C41 | D3×D15 | semidirect |
| 103. C5×D15 | D123 | direct (N × H) |
| 107. C3×C41 | C5×D3×D5 | semidirect |
| 108. C3×C41 | C2×C5×D15 | semidirect |
| 112. C3×D15 | C5×D41 | semidirect |
| 121. C3×C52 | D3×D41 | semidirect |
| 122. C3×C52 | C2×D123 | semidirect |
| 131. C32×C5 | D5×D41 | semidirect |
| 132. C32×C5 | C2×C5×D41 | semidirect |
| 133. C41 | C5×D3×D15 | semidirect |
| 142. D15 | C5×D123 | direct (N × H) |
| 151. C52 | D3×D123 | semidirect |
| 162. C3×C5 | C5×D3×D41 | semidirect |
| 163. C3×C5 | D15×D41 | semidirect |
| 164. C3×C5 | D5×D123 | semidirect |
| 165. C3×C5 | C2×C5×D123 | semidirect |
| 175. C32 | C5×D5×D41 | semidirect |
| 184. C5 | C5×D3×D123 | semidirect |
| 185. C5 | D15×D123 | direct (N × H) |
| 188. C3 | C5×D15×D41 | semidirect |
| 189. C3 | C5×D5×D123 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C52×D123 | 2 |
| C3×C5×D15×C41 | 2 |
| C5×C3⋊D615 | 2 |
| C5×D5×D123 | 3 |
| C5×D15×D41 | 3 |
| D15×D123 | 5 |
| C5×D3×D123 | 5 |
| C5×D3×D15 | 41 |