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A group 
$G$ is said to be an 
$FC$-group if each element of 
$G$ has only finitely many conjugates, and 
$G$ is minimal non
$FC$ if all its proper subgroups have the property 
$FC$ but 
$G$ is not an 
$FC$-group. It is an open question whether there exists a group of infinite rank which is minimal non
$FC$. We consider here groups of infinite rank in which all proper subgroups of infinite rank are 
$FC$, and prove that in most cases such groups are either 
$FC$-groups or minimal non
$FC$.
