Alternative definition for carmichael function

Question

As my textbook describes it, the Carmichael function is the minimum integer m such that, for all a coprime with n, $$a^m \equiv 1 \pmod{n}$$

So, as far as I understand it, m is the minimum number that:

1. divides $\phi(n)$ (because it is the order of a subgroup and for Lagrange theorem divides the order of the group).
2. is divided by the order of every subgroup generated by any a s.t. $(a,n) = 1$

So my question is: wouldn't this make m the least common multiple of the order of every subgroup generated by any a with $(a,n) = 1$?