Finite Group DatabasebetaFinite Group DatabasebetaOrder 20288.
426 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 426 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×D317 |
| commutator, cosocle | C22×C317 | C24 |
| Frattini | C22 | C23×D317 |
| Fitting | C23×C4×C317 | C2 |
| socle | C23×C317 | C23 |
| hypercenter | C23×C4 | D317 |
| hypocenter | C317 | C2×#121 |
| LCS | G ▷ C22×C317 ▷ C317 | |
| UCS | C1 ◁ C23 ◁ C23×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×D1268 | C2 | semidirect |
| 2. C22×C22⋊D317 | C2 | semidirect |
| 3. #23778754 | C2 | direct (N × H) |
| 4. C2×C4⋊Dic317 | C2 | semidirect |
| 5. C2×#19099551 | C2 | semidirect |
| 6. C23×C4×C317 | C2 | semidirect |
| 9. C2×D1268 | C22 | semidirect |
| 11. C2×C22⋊D317 | C22 | semidirect |
| 13. C4⋊Dic317 | C22 | semidirect |
| 14. #19099551 | C22 | semidirect |
| 19. C22×C4×C317 | C22 | semidirect |
| 30. C23×C317 | D4 | semidirect |
| 35. C2×C4×C317 | D4 | semidirect |
| 43. C22×C317 | C2×D4 | semidirect |
| 51. C4×C317 | C2×D4 | semidirect |
| 54. C2×C317 | #121 | semidirect |
| 59. C317 | C2×#121 | semidirect |
| 63. C23×C4 | D317 | semidirect |
| 77. C22×C4 | C2×D317 | semidirect |
| 86. C23 | D1268 | semidirect |
| 95. C2×C4 | C22⋊D317 | semidirect |
| 107. C22 | C2×D1268 | semidirect |
| 112. C4 | C2×C22⋊D317 | semidirect |
| 115. C2 | #23778754 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×D1268 | 2 |
| C22×C22⋊D317 | 2 |
| #23778754 | 2 |
| C2×C4⋊Dic317 | 2 |
| C2×#19099551 | 2 |
| C23×C4×C317 | 2 |
| C2×#121 | 317 |