Finite Group DatabasebetaFinite Group DatabasebetaOrder 320.
334 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 105 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 | C23×D5 |
| commutator, socle, cosocle | C22×C5 | C24 |
| Fitting | C5×#121 | C2 |
| hypercenter | #121 | D5 |
| hypocenter | C5 | #546 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ #121 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#711 | C2 | semidirect |
| 2. C2×D4×D5 | C2 | semidirect |
| 3. C5×#121 | C2 | semidirect |
| 4. #4085 | C2 | semidirect |
| 5. #4083 | C2 | semidirect |
| 6. #4083 | C2 | semidirect |
| 7. #4081 | C2 | semidirect |
| 8. C4×C22⋊D5 | C2 | semidirect |
| 9. C2×#685 | C2 | semidirect |
| 10. #4042 | C2 | semidirect |
| 11. #4040 | C2 | semidirect |
| 12. #4031 | C2 | semidirect |
| 13. #4030 | C2 | semidirect |
| 14. #4027 | C2 | semidirect |
| 15. D5×C22⋊C4 | C2 | semidirect |
| 19. C2×C5×D4 | C22 | semidirect |
| 20. C2×C5×D4 | C22 | semidirect |
| 21. C22×C4×C5 | C22 | semidirect |
| 25. C2×C22⋊D5 | C22 | semidirect |
| 26. C2×C22⋊D5 | C22 | semidirect |
| 27. C22×Dic5 | C22 | semidirect |
| 28. C22×Dic5 | C22 | semidirect |
| 33. C2×C4×D5 | C22 | semidirect |
| 34. C2×C4×D5 | C22 | semidirect |
| 35. C2×Dic10 | C22 | semidirect |
| 36. C5×C4⋊C4 | C22 | semidirect |
| 37. C5×C22⋊C4 | C22 | semidirect |
| 38. C5×C22⋊C4 | C22 | semidirect |
| 39. C22⋊Dic5 | C22 | semidirect |
| 40. C22⋊Dic5 | C22 | semidirect |
| 41. C22⋊Dic5 | C22 | semidirect |
| 45. #685 | C22 | semidirect |
| 46. #685 | C22 | semidirect |
| 47. #685 | C22 | semidirect |
| 48. #685 | C22 | semidirect |
| 49. #683 | C22 | semidirect |
| 50. #683 | C22 | semidirect |
| 51. #683 | C22 | semidirect |
| 52. C4×Dic5 | C22 | semidirect |
| 78. C22⋊D5 | D4 | semidirect |
| 99. #121 | D5 | semidirect |
| 126. C2×D5 | #41 | semidirect |
| 148. C2×D4 | C2×D5 | semidirect |
| 149. C2×D4 | C2×D5 | semidirect |
| 155. C22×C4 | C2×D5 | semidirect |
| 159. C4⋊C4 | C2×D5 | semidirect |
| 170. C22⋊C4 | C2×D5 | semidirect |
| 171. C22⋊C4 | C2×D5 | semidirect |
| 219. C5 | #546 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#711 | 2 |
| C2×D4×D5 | 2 |
| C5×#121 | 2 |
| #4085 | 2 |
| #4083 | 2 |
| #4081 | 2 |
| C4×C22⋊D5 | 2 |
| C2×#685 | 2 |
| #4042 | 2 |
| #4040 | 2 |
| #4031 | 2 |
| #4030 | 2 |
| #4027 | 2 |
| D5×C22⋊C4 | 2 |
| #546 | 5 |