Finite Group DatabasebetaFinite Group DatabasebetaOrder 500.
75 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×H(3,𝔽5) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #6953 | C2 | 1 | 2 | yes | maximal, direct product factor |
| 3 | H(3,𝔽5) | C22 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 4 | C2×C5×D5 | — | 5 | 6 | no | maximal |
| 5 | C2×C52 | D5 | 1 | 6 | yes | complemented |
| 6 | C5×D5 | — | 5 | 12 | no | — |
| 7 | C52 | C2×D5 | 1 | 6 | yes | complemented |
| 8 | C22×C5 | — | 25 | 1 | no | — |
| 9 | C2×D5 | — | 5 | 6 | no | — |
| 10 | C2×C5 | — | 5 | 6 | no | — |
| 11 | C2×C5 | — | 25 | 2 | no | — |
| 12 | C2×C5 | C5⋊D5 | 1 | 1 | yes | characteristic |
| 13 | D5 | — | 5 | 12 | no | — |
| 14 | C5 | — | 5 | 6 | no | — |
| 15 | C5 | C2×C5⋊D5 | 1 | 1 | yes | characteristic |
| 16 | C22 | — | 25 | 1 | no | Sylow |
| 17 | C2 | — | 25 | 2 | no | — |
| 18 | C2 | #6953 | 1 | 1 | yes | characteristic, direct product factor |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C2×C5 | C5⋊D5 |
| commutator, cosocle, hypocenter | H(3,𝔽5) | C22 |
| Frattini | C5 | C2×C5⋊D5 |
| Fitting | C2×H(3,𝔽5) | C2 |
| LCS | G ▷ H(3,𝔽5) | |
| UCS | C1 ◁ C2×C5 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×H(3,𝔽5) | C2 | semidirect |
| 2. #6953 | C2 | direct (N × H) |
| 3. H(3,𝔽5) | C22 | semidirect |
| 5. C2×C52 | D5 | semidirect |
| 7. C52 | C2×D5 | semidirect |
| 18. C2 | #6953 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×H(3,𝔽5) | 2 |
| #6953 | 2 |
| C2×C5×D5 | 5 |