Finite Group DatabasebetaFinite Group DatabasebetaOrder 16380.
176 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 176 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 | C3 | D21×D65 |
| commutator, cosocle, hypocenter | C3×C5×C7×C13 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C32×C5×C7×C13 | C22 |
| LCS | G ▷ C3×C5×C7×C13 | |
| UCS | C1 ◁ C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×D1365 | C2 | semidirect |
| 2. C3×C5×C13×D21 | C2 | semidirect |
| 3. C32×C7×D65 | C2 | semidirect |
| 4. D21×D65 | C3 | direct (N × H) |
| 6. C32×C5×C7×C13 | C22 | semidirect |
| 8. D1365 | C2×C3 | semidirect |
| 9. C5×C13×D21 | C2×C3 | semidirect |
| 12. C3×C7×D65 | D3 | semidirect |
| 13. C3×C7×D65 | C2×C3 | semidirect |
| 18. C3×C13×D21 | D5 | semidirect |
| 21. C3×C5×C7×C13 | C2×D3 | semidirect |
| 22. C3×C5×C7×C13 | C22×C3 | semidirect |
| 26. C32×D65 | D7 | semidirect |
| 30. C7×D65 | C3×D3 | semidirect |
| 32. C32×C7×C13 | C2×D5 | semidirect |
| 36. C3×C5×D21 | D13 | semidirect |
| 38. C32×C5×C13 | C2×D7 | semidirect |
| 40. C13×D21 | C3×D5 | semidirect |
| 47. C5×C7×C13 | C2×C3×D3 | semidirect |
| 55. C3×D65 | C3×D7 | semidirect |
| 56. C3×D65 | D21 | direct (N × H) |
| 58. C32×C5×C7 | C2×D13 | semidirect |
| 60. C3×C7×C13 | D3×D5 | semidirect |
| 61. C3×C7×C13 | C2×C3×D5 | semidirect |
| 68. C5×D21 | C3×D13 | semidirect |
| 75. C3×C5×C13 | C2×C3×D7 | semidirect |
| 76. C3×C5×C13 | C2×D21 | semidirect |
| 86. D65 | C3×D21 | direct (N × H) |
| 89. C3×D21 | D65 | direct (N × H) |
| 90. C32×C13 | D5×D7 | semidirect |
| 92. C3×C5×C7 | D3×D13 | semidirect |
| 93. C3×C5×C7 | C2×C3×D13 | semidirect |
| 94. C7×C13 | C3×D3×D5 | semidirect |
| 110. C5×C13 | C2×C3×D21 | semidirect |
| 111. C32×C7 | C2×D65 | semidirect |
| 115. C32×C5 | D7×D13 | semidirect |
| 120. D21 | C3×D65 | direct (N × H) |
| 124. C3×C13 | C3×D5×D7 | semidirect |
| 125. C3×C13 | D5×D21 | semidirect |
| 127. C5×C7 | C3×D3×D13 | semidirect |
| 140. C3×C7 | D3×D65 | semidirect |
| 141. C3×C7 | C2×C3×D65 | semidirect |
| 147. C3×C5 | C3×D7×D13 | semidirect |
| 148. C3×C5 | D13×D21 | semidirect |
| 152. C13 | C3×D5×D21 | semidirect |
| 158. C32 | D7×D65 | semidirect |
| 159. C7 | C3×D3×D65 | semidirect |
| 167. C5 | C3×D13×D21 | semidirect |
| 170. C3 | C3×D7×D65 | semidirect |
| 171. C3 | D21×D65 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×D1365 | 2 |
| C3×C5×C13×D21 | 2 |
| C32×C7×D65 | 2 |
| D21×D65 | 3 |
| C3×D7×D65 | 3 |
| C3×D13×D21 | 5 |
| C3×D3×D65 | 7 |
| C3×D5×D21 | 13 |