  It is proved that the following conditions are equivalent for an infinite
non-associative commutative Moufang loop $Q$: 1) $Q$ satisfies the minimum
condition for subloops; 2) if the loop $Q$ contains a centrally solvable
subloop of class $s$, then it satisfies the minimum condition for centrally
solvable subloops of class $s$; 3) if the loop $Q$ contains a centrally
nilpotent subloop of class $n$, then it satisfies the minimum condition for
centrally nilpotent subloops of class $n$; 4) $Q$ satisfies the minimum
condition for non-invatiant associative subloops. The structure of the
commutative Moufang loops, whose infinite non-associative subloops are normal,
is examined.
