Finite Group DatabasebetaFinite Group DatabasebetaOrder 80.
26 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×AGL(1,𝔽5) | C2 | 1 | 2 | yes | maximal |
| 2 | C4×D5 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×D5 | C22 | 1 | 1 | yes | characteristic |
| 4 | AGL(1,𝔽5) | — | 2 | 2 | no | — |
| 5 | C4×C5 | C4 | 1 | 1 | yes | characteristic, complemented |
| 6 | Dic5 | C4 | 1 | 1 | yes | characteristic, complemented |
| 7 | C4⋊C4 | — | 5 | 1 | no | maximal, Sylow |
| 8 | C2×C5 | C2×C4 | 1 | 1 | yes | characteristic |
| 9 | D5 | D4 | 1 | 1 | yes | characteristic |
| 10 | D5 | Q8 | 1 | 1 | yes | characteristic |
| 11 | C2×C4 | — | 5 | 1 | no | — |
| 12 | C2×C4 | — | 5 | 2 | no | — |
| 13 | C5 | C4⋊C4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 14 | C22 | — | 5 | 1 | no | — |
| 15 | C4 | — | 5 | 1 | no | — |
| 16 | C4 | — | 10 | 2 | no | — |
| 17 | C4 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 18 | C2 | — | 5 | 1 | no | — |
| 19 | C2 | — | 5 | 1 | no | — |
| 20 | C2 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×AGL(1,𝔽5) |
| commutator, socle | C2×C5 | C2×C4 |
| Fitting | C4×C5 | C4 |
| prime radical, cosocle | C2×D5 | C22 |
| hypercenter | C4 | AGL(1,𝔽5) |
| hypocenter | C5 | C4⋊C4 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. C4×C5 | C4 | semidirect |
| 6. Dic5 | C4 | semidirect |
| 13. C5 | C4⋊C4 | semidirect |
| 17. C4 | AGL(1,𝔽5) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×AGL(1,𝔽5) | 2 |
| C4×D5 | 2 |
| C4⋊C4 | 5 |