Finite Group DatabasebetaFinite Group DatabasebetaOrder 2160.
540 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 540 conjugacy classes of subgroups — too many to draw. Only its 77 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 | C33×C5 | C22×C4 |
| prime radical | D5×#1113 | C2 |
| cosocle | C33×D5 | C23 |
| LCS | G ▷ C33×C5 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3⋊D3×ΓL(1,𝔽16) | C2 | semidirect |
| 3. C3×ΓL(1,𝔽16)⋊D3 | C2 | semidirect |
| 5. C5⋊#1113 | C4 | semidirect |
| 6. C5×#1113 | C4 | semidirect |
| 8. C3⋊(C3⋊ΓL(1,𝔽16)) | C22 | semidirect |
| 9. C32×ΓL(1,𝔽16) | C22 | semidirect |
| 12. ΓL(1,𝔽16)⋊D3 | D3 | semidirect |
| 17. C3×C3⋊D15 | C2×C4 | semidirect |
| 18. C3×C5×C3⋊D3 | C2×C4 | semidirect |
| 34. C3×ΓL(1,𝔽16) | C2×D3 | semidirect |
| 37. C33×C5 | C22×C4 | semidirect |
| 46. #1113 | AGL(1,𝔽5) | semidirect |
| 49. C3⋊D15 | C4×D3 | semidirect |
| 50. C5×C3⋊D3 | C4×D3 | semidirect |
| 74. ΓL(1,𝔽16) | D32 | semidirect |
| 79. C3×C3⋊D3 | C2×AGL(1,𝔽5) | semidirect |
| 83. C32×C5 | C2×C4×D3 | semidirect |
| 108. C33 | C22×AGL(1,𝔽5) | semidirect |
| 124. C3⋊D3 | D3×AGL(1,𝔽5) | semidirect |
| 132. C3×C5 | C4×D32 | semidirect |
| 150. C32 | C2×D3×AGL(1,𝔽5) | semidirect |
| 163. C5 | #88949 | semidirect |
| 171. C3 | D32×AGL(1,𝔽5) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D5×#1113 | 2 |
| C3⋊D3×ΓL(1,𝔽16) | 2 |
| C3×ΓL(1,𝔽16)⋊D3 | 2 |
| D32×AGL(1,𝔽5) | 3 |
| #88949 | 5 |