long-range order v.s. symmetry-breaking

Question

Long-range order characterizes physical systems in which remote portions of the same sample exhibit correlated behavior. This can be expressed as a correlation function, namely the correlation function of some local operators $O(x)$:

$$G(x,x')=\langle O(x) O(x')\rangle.$$

Does long-range order necessarily imply the symmetry-breaking? And does the symmetry-breaking necessarily imply the long-range order?