Finite Group DatabasebetaFinite Group DatabasebetaOrder 320.
216 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 93 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 | #4030 | C2 |
| hypercenter | C22⋊Q8 | D5 |
| hypocenter | C5 | #542 |
| 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. C5×C22⋊Q8 | C2 | semidirect |
| 2. #4093 | C2 | semidirect |
| 3. #4090 | C2 | semidirect |
| 4. #4075 | C2 | semidirect |
| 5. C4×C22⋊D5 | C2 | semidirect |
| 6. #4044 | C2 | semidirect |
| 7. #4042 | C2 | semidirect |
| 8. #4042 | C2 | semidirect |
| 9. #4040 | C2 | semidirect |
| 10. #4038 | C2 | semidirect |
| 11. #4033 | C2 | semidirect |
| 12. #4031 | C2 | semidirect |
| 13. #4031 | C2 | semidirect |
| 15. #4025 | C2 | semidirect |
| 86. C22⋊Q8 | D5 | semidirect |
| 178. C5 | #542 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×C22⋊Q8 | 2 |
| #4093 | 2 |
| #4090 | 2 |
| #4075 | 2 |
| C4×C22⋊D5 | 2 |
| #4044 | 2 |
| #4042 | 2 |
| #4040 | 2 |
| #4038 | 2 |
| #4033 | 2 |
| #4031 | 2 |
| #4030 | 2 |
| #4025 | 2 |
| #542 | 5 |