Finite Group DatabasebetaFinite Group DatabasebetaOrder 320.
248 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 248 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×#126 | C2 |
| prime radical | #4028 | C2 |
| hypercenter | #126 | D5 |
| hypocenter | C5 | #551 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ #126 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×#126 | C2 | semidirect |
| 2. #4043 | C2 | semidirect |
| 3. #4042 | C2 | semidirect |
| 4. #4040 | C2 | semidirect |
| 5. #4040 | C2 | semidirect |
| 6. #4039 | C2 | semidirect |
| 7. #4038 | C2 | semidirect |
| 8. #4031 | C2 | semidirect |
| 9. #4029 | C2 | semidirect |
| 11. D5×C22⋊C4 | C2 | semidirect |
| 12. D5×C22⋊C4 | C2 | semidirect |
| 13. #4025 | C2 | semidirect |
| 14. C4×D20 | C2 | semidirect |
| 15. #4018 | C2 | semidirect |
| 24. C2×C4×D5 | C22 | semidirect |
| 25. C2×Dic10 | C22 | semidirect |
| 26. C5×C4⋊C4 | C22 | semidirect |
| 27. C5×C4⋊C4 | C22 | semidirect |
| 28. C5×C4⋊C4 | C22 | semidirect |
| 32. C42×C5 | C22 | semidirect |
| 41. #685 | C22 | semidirect |
| 42. #685 | C22 | semidirect |
| 43. #685 | C22 | semidirect |
| 44. C4⋊Dic5 | C22 | semidirect |
| 45. C4⋊Dic5 | C22 | semidirect |
| 46. #683 | C22 | semidirect |
| 47. #683 | C22 | semidirect |
| 48. #683 | C22 | semidirect |
| 49. C4×Dic5 | C22 | semidirect |
| 50. C4×Dic5 | C22 | semidirect |
| 80. #126 | D5 | semidirect |
| 134. C4⋊C4 | C2×D5 | semidirect |
| 135. C4⋊C4 | C2×D5 | semidirect |
| 136. C4⋊C4 | C2×D5 | semidirect |
| 153. C42 | C2×D5 | semidirect |
| 188. C5 | #551 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×#126 | 2 |
| #4043 | 2 |
| #4042 | 2 |
| #4040 | 2 |
| #4039 | 2 |
| #4038 | 2 |
| #4031 | 2 |
| #4029 | 2 |
| #4028 | 2 |
| D5×C22⋊C4 | 2 |
| #4025 | 2 |
| C4×D20 | 2 |
| #4018 | 2 |
| #551 | 5 |