Typically, large cardinals with stronger consistency strength are larger, but there is an important general exception:

If a large cardinal axiom is $Σ_{k+1}$, then typically the least example is larger than the least example for a $Σ_k$ axiom regardless of the consistency strength. This is because $Σ_{k+1}$ large cardinal axioms typically imply that the cardinal is a $Σ_k$ elementary substructure of $V$ (and thus larger than the least example satisfying a $Σ_k$ axiom). However, a consistency-wise stronger $Σ_k$ axiom typically implies existence of $V_κ$ that satisfies a consistency-wise weaker $Σ_{k+1}$ axiom.

For example, the least strong cardinal ($Σ^V_3$ existence axiom) is larger than the least Woodin cardinal ($Σ^V_2$ existence axiom), which in turn is larger than the least cardinal strong up to an inaccessible ($Σ^V_2$ existence axiom), which in turn implies existence of $V_κ$ satisfying "ZFC + there is a strong cardinal".

There are also ad hoc exceptions (such as, at least in some generic extensions, strongly compact cardinals), but their relative infrequency is a testament to the coherency of the large cardinal hierarchy.

consistency and implication strengthorders with each other. It could be of your interest as well. $\endgroup$ – Morteza Azad Aug 29 '17 at 10:18