Finite Group DatabasebetaFinite Group DatabasebetaOrder 500.
29 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×H(3,𝔽5) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | H(3,𝔽5) | C4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 3 | #1039 | C5 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C5×Dic5 | — | 5 | 1 | no | maximal |
| 5 | C2×C52 | — | 2 | 2 | no | — |
| 6 | C2×C52 | D5 | 1 | 1 | yes | characteristic |
| 7 | C2×C52 | C2×C5 | 1 | 1 | yes | characteristic |
| 8 | C52 | — | 2 | 2 | no | — |
| 9 | C52 | Dic5 | 1 | 1 | yes | characteristic, complemented |
| 10 | C52 | C4×C5 | 1 | 1 | yes | characteristic, complemented |
| 11 | C4×C5 | — | 25 | 1 | no | — |
| 12 | Dic5 | — | 5 | 1 | no | — |
| 13 | Dic5 | — | 25 | 1 | no | — |
| 14 | C2×C5 | — | 5 | 1 | no | — |
| 15 | C2×C5 | — | 5 | 1 | no | — |
| 16 | C2×C5 | — | 10 | 2 | no | — |
| 17 | C2×C5 | C5×D5 | 1 | 1 | yes | characteristic |
| 18 | C5 | — | 5 | 1 | no | — |
| 19 | C5 | — | 5 | 1 | no | — |
| 20 | C5 | — | 10 | 2 | no | — |
| 21 | C5 | C5×Dic5 | 1 | 1 | yes | characteristic |
| 22 | C4 | — | 25 | 1 | no | Sylow |
| 23 | C2 | (C5⋊D5)⋊C5 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | (C5⋊D5)⋊C5 |
| commutator, hypocenter | C52 | C4×C5 |
| Frattini, socle | C2×C5 | C5×D5 |
| Fitting, prime radical | C2×H(3,𝔽5) | C2 |
| cosocle | C2×C52 | C2×C5 |
| LCS | G ▷ C52 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. H(3,𝔽5) | C4 | semidirect |
| 3. #1039 | C5 | semidirect |
| 9. C52 | Dic5 | semidirect |
| 10. C52 | C4×C5 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×H(3,𝔽5) | 2 |
| #1039 | 5 |
| C5×Dic5 | 5 |