Abstract. Let be the ordinal notation system from Buchholz-Schtte (1988). [The order type of the countable segment is - by Rathjen (1988) - the proof-theoretic ordinal the proof-theoretic ordinal of .] In particular let denote the enumeration function of the infinite cardinals and let denote the partial collapsing operation on which maps ordinals of into the countable segment of . Assume that the (fast growing) extended Grzegorczyk hierarchy and the slow growing hierarchy are defined with respect to the natural system of distinguished fundamental sequences of Buchholz and Schtte (1988) in the following way: where is a countable limit ordinal (term) and is the distinguished fundamental sequence assigned to . For each ordinal (term) in and each natural number let be the formal term which results from the ordinal term by successively replacing every occurence of by where is considered as a defined function symbol, namely (Note that ) In this article it is shown that for each ordinal term in there exists a natural number such that and holds for all This hierarchy comparison theorem yields a plethora of new results on nontrivial lower bounds for the slow growing ordinals - i.e. ordinals for which the slow growing hierarchy yields a classification of the provably total functions of the theory in question - of various theories of iterated inductive definitions (and subsystems of ) and on the number and size of the subrecursively inaccessible ordinals - i.e. ordinals at which the extended Grzegorczyk hierarchy and the slow growing hierarchy catch up - below the proof-theoretic ordinal of . In particular these subrecursively inaccessibles ordinals are necessarily of the form