Finite Group DatabasebetaFinite Group DatabasebetaOrder 20224.
65470 conjugacy classes of subgroups; only the 32651 normal subgroups are shown.
This group has 65470 conjugacy classes of subgroups — too many to draw. Only its 32651 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C25 | C22×D79 |
| commutator, cosocle | C2×C79 | C27 |
| Frattini | C2 | C26×D79 |
| Fitting | C24×D4×C79 | C2 |
| socle | C25×C79 | C23 |
| hypercenter | C24×D4 | D79 |
| hypocenter | C79 | C24×#41 |
| LCS | G ▷ C2×C79 ▷ C79 | |
| UCS | C1 ◁ C25 ◁ C24×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×#11929623 | C2 | direct (N × H) |
| 2. C24×Dic158 | C2 | semidirect |
| 3. C24×C22⋊D79 | C2 | semidirect |
| 4. C25×Dic79 | C2 | semidirect |
| 5. C24×C4×D79 | C2 | semidirect |
| 6. C24×D4×C79 | C2 | semidirect |
| 8. C22×#11929623 | C22 | direct (N × H) |
| 9. C23×Dic158 | C22 | semidirect |
| 10. C23×C22⋊D79 | C22 | semidirect |
| 13. C24×Dic79 | C22 | semidirect |
| 14. C23×C4×D79 | C22 | semidirect |
| 16. C23×D4×C79 | C22 | semidirect |
| 20. C2×#11929623 | C23 | direct (N × H) |
| 21. C22×Dic158 | C23 | semidirect |
| 22. C22×C22⋊D79 | C23 | semidirect |
| 25. C23×Dic79 | C23 | semidirect |
| 26. C22×C4×D79 | C23 | semidirect |
| 30. C22×D4×C79 | C23 | semidirect |
| 34. #11929623 | C24 | direct (N × H) |
| 35. C2×Dic158 | C24 | semidirect |
| 36. C2×C22⋊D79 | C24 | semidirect |
| 39. C22×Dic79 | C24 | semidirect |
| 40. C2×C4×D79 | C24 | semidirect |
| 42. C24×C79 | #41 | semidirect |
| 45. C2×D4×C79 | C24 | semidirect |
| 48. C23×C79 | C2×#41 | semidirect |
| 53. D4×C79 | C25 | semidirect |
| 55. C22⋊D79 | C25 | semidirect |
| 58. C2×Dic79 | C25 | semidirect |
| 59. C4×D79 | C25 | semidirect |
| 60. Dic158 | C25 | semidirect |
| 62. C22×C79 | C22×#41 | semidirect |
| 72. C2×C79 | C23×#41 | semidirect |
| 78. C24×D4 | D79 | semidirect |
| 81. C79 | C24×#41 | semidirect |
| 87. C23×D4 | C2×D79 | semidirect |
| 101. C22×D4 | C22×D79 | semidirect |
| 110. C24 | #11929623 | direct (N × H) |
| 116. C2×D4 | C23×D79 | semidirect |
| 125. C23 | C2×#11929623 | direct (N × H) |
| 130. D4 | C24×D79 | semidirect |
| 139. C22 | C22×#11929623 | direct (N × H) |
| 147. C2 | C23×#11929623 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#11929623 | 2 |
| C24×Dic158 | 2 |
| C24×C22⋊D79 | 2 |
| C25×Dic79 | 2 |
| C24×C4×D79 | 2 |
| C24×D4×C79 | 2 |
| C24×#41 | 79 |