Finite Group DatabasebetaFinite Group DatabasebetaOrder 500.
192 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C5⋊(C5⋊D5) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C53 | C4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 3 | #1043 | — | 5 | 31 | no | maximal |
| 4 | C5⋊D5 | — | 5 | 31 | no | — |
| 5 | C52 | AGL(1,𝔽5) | 1 | 31 | yes | complemented |
| 6 | AGL(1,𝔽5) | — | 25 | 31 | no | — |
| 7 | D5 | — | 25 | 31 | no | — |
| 8 | C5 | #1043 | 1 | 31 | yes | complemented |
| 9 | C4 | — | 125 | 1 | no | Sylow |
| 10 | C2 | — | 125 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C53 | C4 |
| prime radical, cosocle | C5⋊(C5⋊D5) | C2 |
| LCS | G ▷ C53 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53 | C4 | semidirect |
| 5. C52 | AGL(1,𝔽5) | semidirect |
| 8. C5 | #1043 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5⋊(C5⋊D5) | 2 |
| #1043 | 5 |