Finite Group DatabasebetaFinite Group DatabasebetaOrder 2160.
180 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | D5×PSU(3,𝔽2) |
| commutator, cosocle | C5×C3⋊D3 | C23×C3 |
| Frattini | C1 | G |
| Fitting, socle | C33×C5 | C2×Q8 |
| prime radical | C3×D5×C3⋊D3 | C22 |
| hypocenter | C32×C5 | C2×C3×Q8 |
| LCS | G ▷ C5×C3⋊D3 ▷ C32×C5 | |
| UCS | C1 ◁ C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×C5⋊PSU(3,𝔽2) | C2 | semidirect |
| 2. C3×C5×PSU(3,𝔽2) | C2 | semidirect |
| 4. D5×PSU(3,𝔽2) | C3 | direct (N × H) |
| 9. C5⋊PSU(3,𝔽2) | C2×C3 | semidirect |
| 10. C5×PSU(3,𝔽2) | C2×C3 | semidirect |
| 12. C3×C3⋊D15 | Q8 | semidirect |
| 14. C33×D5 | Q8 | semidirect |
| 18. C3×PSU(3,𝔽2) | D5 | direct (N × H) |
| 24. C33×C5 | C2×Q8 | semidirect |
| 31. C3⋊D15 | C3×Q8 | semidirect |
| 37. C32×D5 | C3×Q8 | semidirect |
| 41. PSU(3,𝔽2) | C3×D5 | direct (N × H) |
| 52. C32×C5 | C2×C3×Q8 | semidirect |
| 65. C3×D5 | PSU(3,𝔽2) | direct (N × H) |
| 67. C33 | Q8×D5 | semidirect |
| 84. C3×C5 | C2×PSU(3,𝔽2) | semidirect |
| 91. D5 | C3×PSU(3,𝔽2) | direct (N × H) |
| 94. C32 | C3×Q8×D5 | semidirect |
| 105. C5 | C2×C3×PSU(3,𝔽2) | semidirect |
| 111. C3 | D5×PSU(3,𝔽2) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C5⋊PSU(3,𝔽2) | 2 |
| C3×C5×PSU(3,𝔽2) | 2 |
| C3×D5×#157 | 2 |
| D5×PSU(3,𝔽2) | 3 |
| C2×C3×PSU(3,𝔽2) | 5 |
| C3×Q8×D5 | 9 |