Finite Group DatabasebetaFinite Group DatabasebetaOrder 2016.
780 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 780 conjugacy classes of subgroups — too many to draw. Only its 152 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D7×C3⋊D3 |
| commutator, socle, cosocle | C22×C32×C7 | C23 |
| Fitting | C23×C32×C7 | C22 |
| hypercenter | C23 | D7×C3⋊D3 |
| hypocenter | C32×C7 | #120 |
| LCS | G ▷ C22×C32×C7 ▷ C32×C7 | |
| UCS | C1 ◁ C22 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×C32×D7 | C2 | semidirect |
| 2. C2×#92726 | C2 | semidirect |
| 3. C2×#92677 | C2 | semidirect |
| 4. C7×#3757 | C2 | semidirect |
| 5. (C2×D7)⋊(C3⋊Dic3) | C2 | semidirect |
| 8. C22×C3⋊D21 | C22 | semidirect |
| 13. C2×C3⋊Dic21 | C22 | semidirect |
| 14. C2×C7×C3⋊Dic3 | C22 | semidirect |
| 16. C23×C3×D7 | D3 | semidirect |
| 23. C22×C32×C7 | D4 | semidirect |
| 30. C2×C32×D7 | D4 | semidirect |
| 47. #3757 | D7 | semidirect |
| 53. C23×D7 | C3⋊D3 | semidirect |
| 60. C22×C3×C7 | #67 | semidirect |
| 67. C2×C3×D7 | #67 | semidirect |
| 78. C2×C3⋊Dic3 | C2×D7 | semidirect |
| 79. C32×C7 | #120 | semidirect |
| 104. C22×C32 | D28 | semidirect |
| 112. C22×C7 | #638 | semidirect |
| 119. C2×D7 | #638 | semidirect |
| 131. C3×C7 | #953 | semidirect |
| 152. C22×C3 | #4246 | semidirect |
| 158. C32 | #6445 | semidirect |
| 168. C7 | #64162 | semidirect |
| 181. C22 | #92677 | semidirect |
| 185. C3 | #10621126 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C32×D7 | 2 |
| C2×#92726 | 2 |
| C2×#92677 | 2 |
| C7×#3757 | 2 |
| (C2×D7)⋊(C3⋊Dic3) | 2 |
| #10621126 | 3 |
| #64162 | 7 |