Finite Group DatabasebetaFinite Group DatabasebetaOrder 320.
250 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 95 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×#120 | C2 |
| hypercenter | #120 | D5 |
| hypocenter | C5 | #538 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ #120 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×#120 | C2 | semidirect |
| 2. #4099 | C2 | semidirect |
| 3. C2×C22⋊Dic5 | C2 | semidirect |
| 4. #4085 | C2 | semidirect |
| 5. #4085 | C2 | semidirect |
| 6. #4084 | C2 | semidirect |
| 7. #4082 | C2 | semidirect |
| 8. #4081 | C2 | semidirect |
| 9. D4×Dic5 | C2 | semidirect |
| 10. #4031 | C2 | semidirect |
| 11. #4029 | C2 | semidirect |
| 12. #4027 | C2 | semidirect |
| 13. #4025 | C2 | semidirect |
| 14. #4025 | C2 | semidirect |
| 15. #4024 | C2 | semidirect |
| 23. C2×C22⋊D5 | C22 | semidirect |
| 26. C22×Dic5 | C22 | semidirect |
| 27. C2×C4×D5 | C22 | semidirect |
| 28. C2×Dic10 | C22 | semidirect |
| 39. C22⋊Dic5 | C22 | semidirect |
| 40. C22⋊Dic5 | C22 | semidirect |
| 41. #685 | C22 | semidirect |
| 42. #685 | C22 | semidirect |
| 43. C4⋊Dic5 | C22 | semidirect |
| 44. C4⋊Dic5 | C22 | semidirect |
| 45. #683 | C22 | semidirect |
| 46. #683 | C22 | semidirect |
| 47. #683 | C22 | semidirect |
| 48. #683 | C22 | semidirect |
| 49. C4×Dic5 | C22 | semidirect |
| 50. C4×Dic5 | C22 | semidirect |
| 93. #120 | D5 | semidirect |
| 197. C5 | #538 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×#120 | 2 |
| #4099 | 2 |
| C2×C22⋊Dic5 | 2 |
| #4085 | 2 |
| #4084 | 2 |
| #4082 | 2 |
| #4081 | 2 |
| D4×Dic5 | 2 |
| #4031 | 2 |
| #4029 | 2 |
| #4027 | 2 |
| #4025 | 2 |
| #4024 | 2 |
| #538 | 5 |