Finite Group DatabasebetaFinite Group DatabasebetaOrder 1800.
149 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 149 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C32×C52 | C2×C4 |
| prime radical | C3×D5×D15 | C2 |
| cosocle | C32×C5×D5 | C22 |
| LCS | G ▷ C32×C52 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3⋊(C3⋊#1042) | C2 | semidirect |
| 3. C3×C5×ΓL(1,𝔽16) | C2 | semidirect |
| 6. C3×C3⋊(C5⋊D5) | C4 | semidirect |
| 7. C3×C5×D15 | C4 | semidirect |
| 16. C5×ΓL(1,𝔽16) | D3 | semidirect |
| 18. C32×C52 | C2×C4 | semidirect |
| 23. C3×ΓL(1,𝔽16) | D5 | semidirect |
| 26. C3⋊(C5⋊D5) | Dic3 | semidirect |
| 27. C5×D15 | Dic3 | semidirect |
| 44. C3×D15 | AGL(1,𝔽5) | semidirect |
| 48. C3×C52 | C4×D3 | semidirect |
| 49. C3×C52 | C2×Dic3 | semidirect |
| 58. ΓL(1,𝔽16) | D15 | direct (N × H) |
| 69. C32×C5 | C4×D5 | semidirect |
| 70. C32×C5 | C2×AGL(1,𝔽5) | semidirect |
| 83. D15 | ΓL(1,𝔽16) | direct (N × H) |
| 92. C52 | D3×Dic3 | semidirect |
| 109. C3×C5 | D5×Dic3 | semidirect |
| 110. C3×C5 | C4×D15 | semidirect |
| 111. C3×C5 | D3×AGL(1,𝔽5) | semidirect |
| 112. C3×C5 | C2×ΓL(1,𝔽16) | semidirect |
| 127. C32 | D5×AGL(1,𝔽5) | semidirect |
| 137. C5 | Dic3×D15 | semidirect |
| 138. C5 | D3×ΓL(1,𝔽16) | semidirect |
| 143. C3 | AGL(1,𝔽5)×D15 | semidirect |
| 144. C3 | D5×ΓL(1,𝔽16) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×D5×D15 | 2 |
| C3⋊(C3⋊#1042) | 2 |
| C3×C5×ΓL(1,𝔽16) | 2 |
| D5×ΓL(1,𝔽16) | 3 |
| AGL(1,𝔽5)×D15 | 3 |
| D3×ΓL(1,𝔽16) | 5 |
| Dic3×D15 | 5 |