Finite Group DatabasebetaFinite Group DatabasebetaOrder 720.
248 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3 | ΓL(1,𝔽16) |
| commutator, hypocenter | C3×C5 | C22×C3×C4 |
| Frattini | C1 | G |
| Fitting, socle | C22×C32×C5 | C4 |
| prime radical | C22×C32×D5 | C2 |
| cosocle | C3×D5 | C23×C3 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C22×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C3×ΓL(1,𝔽16) | C2 | direct (N × H) |
| 3. C22×ΓL(1,𝔽16) | C3 | direct (N × H) |
| 5. C22×C32×C5 | C4 | semidirect |
| 6. C2×C32×D5 | C4 | semidirect |
| 8. C3×ΓL(1,𝔽16) | C22 | direct (N × H) |
| 13. C2×ΓL(1,𝔽16) | C2×C3 | direct (N × H) |
| 15. C2×C32×C5 | C2×C4 | semidirect |
| 16. C32×D5 | C2×C4 | semidirect |
| 22. C22×C3×C5 | Dic3 | semidirect |
| 23. C22×C3×C5 | C3×C4 | semidirect |
| 26. C2×C3×D5 | Dic3 | semidirect |
| 27. C2×C3×D5 | C3×C4 | semidirect |
| 30. ΓL(1,𝔽16) | C22×C3 | direct (N × H) |
| 34. C32×C5 | C22×C4 | semidirect |
| 39. C22×C32 | AGL(1,𝔽5) | semidirect |
| 42. C2×C3×C5 | C2×Dic3 | semidirect |
| 43. C2×C3×C5 | C2×C3×C4 | semidirect |
| 46. C3×D5 | C2×Dic3 | semidirect |
| 47. C3×D5 | C2×C3×C4 | semidirect |
| 55. C22×C5 | C3×Dic3 | semidirect |
| 56. C2×D5 | C3×Dic3 | semidirect |
| 61. C2×C32 | C2×AGL(1,𝔽5) | semidirect |
| 64. C3×C5 | C22×Dic3 | semidirect |
| 65. C3×C5 | C22×C3×C4 | semidirect |
| 73. C22×C3 | C3×AGL(1,𝔽5) | semidirect |
| 74. C22×C3 | ΓL(1,𝔽16) | direct (N × H) |
| 77. C2×C5 | C2×C3×Dic3 | semidirect |
| 78. D5 | C2×C3×Dic3 | semidirect |
| 80. C32 | C22×AGL(1,𝔽5) | semidirect |
| 90. C2×C3 | C2×C3×AGL(1,𝔽5) | semidirect |
| 91. C2×C3 | C2×ΓL(1,𝔽16) | direct (N × H) |
| 92. C5 | C22×C3×Dic3 | semidirect |
| 95. C22 | C3×ΓL(1,𝔽16) | direct (N × H) |
| 98. C3 | C22×C3×AGL(1,𝔽5) | semidirect |
| 99. C3 | C22×ΓL(1,𝔽16) | direct (N × H) |
| 102. C2 | C2×C3×ΓL(1,𝔽16) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C32×D5 | 2 |
| C2×C3×ΓL(1,𝔽16) | 2 |
| C22×ΓL(1,𝔽16) | 3 |
| C22×C3×AGL(1,𝔽5) | 3 |
| C22×C3×Dic3 | 5 |