Finite Group DatabasebetaFinite Group DatabasebetaOrder 500.
72 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C5×C5⋊D5 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | C53 | C22 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 3 | D52 | — | 5 | 3 | no | maximal |
| 4 | C5⋊D5 | D5 | 1 | 3 | yes | complemented |
| 5 | C5×D5 | — | 5 | 6 | no | — |
| 6 | C5×D5 | — | 10 | 6 | no | — |
| 7 | C52 | — | 2 | 6 | no | — |
| 8 | C52 | — | 4 | 4 | no | — |
| 9 | C52 | C2×D5 | 1 | 3 | yes | complemented |
| 10 | C2×D5 | — | 25 | 3 | no | — |
| 11 | C2×C5 | — | 25 | 3 | no | — |
| 12 | D5 | — | 5 | 6 | no | — |
| 13 | D5 | — | 10 | 6 | no | — |
| 14 | C5 | — | 2 | 6 | no | — |
| 15 | C5 | — | 4 | 4 | no | — |
| 16 | C5 | D52 | 1 | 3 | yes | complemented |
| 17 | C22 | — | 125 | 1 | no | Sylow |
| 18 | C2 | — | 25 | 3 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C53 | C22 |
| LCS | G ▷ C53 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×C5⋊D5 | C2 | semidirect |
| 2. C53 | C22 | semidirect |
| 4. C5⋊D5 | D5 | semidirect |
| 9. C52 | C2×D5 | semidirect |
| 16. C5 | D52 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×C5⋊D5 | 2 |
| D52 | 5 |