Finite Group DatabasebetaFinite Group DatabasebetaOrder 320.
322 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 322 conjugacy classes of subgroups — too many to draw. Only its 121 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×C22⋊Q8 | C2 |
| prime radical | C23×D5 | C22 |
| hypercenter | C22⋊Q8 | D5 |
| hypocenter | C5 | C2×C22⋊Q8 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×Q8×D5 | C2 | semidirect |
| 3. C5×C22⋊Q8 | C2 | semidirect |
| 4. #4092 | C2 | semidirect |
| 5. #4070 | C2 | semidirect |
| 6. #4043 | C2 | semidirect |
| 7. #4042 | C2 | semidirect |
| 8. D5×C4⋊C4 | C2 | semidirect |
| 9. D5×C4⋊C4 | C2 | semidirect |
| 11. #4024 | C2 | semidirect |
| 13. C2×C5×Q8 | C22 | semidirect |
| 24. C2×Dic10 | C22 | semidirect |
| 25. C2×Dic10 | C22 | semidirect |
| 26. C5×C4⋊C4 | C22 | semidirect |
| 27. C5×C4⋊C4 | C22 | semidirect |
| 32. C4⋊Dic5 | C22 | semidirect |
| 33. C4⋊Dic5 | C22 | semidirect |
| 34. #683 | C22 | semidirect |
| 35. #683 | C22 | semidirect |
| 36. #683 | C22 | semidirect |
| 41. C22×D5 | Q8 | semidirect |
| 70. C22⋊Q8 | D5 | direct (N × H) |
| 76. C22×C5 | C2×Q8 | semidirect |
| 83. C2×D5 | C2×Q8 | semidirect |
| 97. C2×Q8 | C2×D5 | semidirect |
| 112. C4⋊C4 | C2×D5 | semidirect |
| 113. C4⋊C4 | C2×D5 | semidirect |
| 123. D5 | C22⋊Q8 | direct (N × H) |
| 149. C5 | C2×C22⋊Q8 | semidirect |
| 159. C22 | Q8×D5 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q8×D5 | 2 |
| C22×C4×D5 | 2 |
| C5×C22⋊Q8 | 2 |
| #4092 | 2 |
| #4070 | 2 |
| #4043 | 2 |
| #4042 | 2 |
| D5×C4⋊C4 | 2 |
| D5×C22⋊C4 | 2 |
| #4024 | 2 |
| C2×C22⋊Q8 | 5 |