Finite Group DatabasebetaFinite Group DatabasebetaOrder 320.
186 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 64 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 | C22×AGL(1,𝔽5) |
| commutator, socle | C22×C5 | C22×C4 |
| Fitting | C2×C5×Q8 | C4 |
| prime radical, cosocle | C22×D5 | C23 |
| hypercenter | C2×Q8 | AGL(1,𝔽5) |
| hypocenter | C5 | #391 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 7. C2×C5×Q8 | C4 | semidirect |
| 12. C2×Dic10 | C4 | semidirect |
| 13. C2×Dic10 | C4 | semidirect |
| 22. C2×AGL(1,𝔽5) | Q8 | semidirect |
| 53. C2×Q8 | AGL(1,𝔽5) | semidirect |
| 82. C5 | #391 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q8×D5 | 2 |
| C2×C4⋊AGL(1,𝔽5) | 2 |
| C2×C4×AGL(1,𝔽5) | 2 |
| (C2×AGL(1,𝔽5))⋊C4 | 2 |
| #391 | 5 |