Finite Group DatabasebetaFinite Group DatabasebetaOrder 320.
196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 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×C4⋊Q8 | C2 |
| prime radical | C2×D20 | C22 |
| hypercenter | C4⋊Q8 | D5 |
| hypocenter | C5 | #563 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C4⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×C4⋊Q8 | C2 | semidirect |
| 3. #4092 | C2 | semidirect |
| 4. #4090 | C2 | semidirect |
| 5. #4044 | C2 | semidirect |
| 6. #4042 | C2 | semidirect |
| 8. #4035 | C2 | semidirect |
| 9. #4022 | C2 | semidirect |
| 47. C4⋊Q8 | D5 | semidirect |
| 104. C5 | #563 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×C4⋊Q8 | 2 |
| #4093 | 2 |
| #4092 | 2 |
| #4090 | 2 |
| #4044 | 2 |
| #4042 | 2 |
| #4040 | 2 |
| #4035 | 2 |
| #4022 | 2 |
| #563 | 5 |