The following question is "ideologically related" to the one I recently asked here.

For a prime $p$, let $M_p$ denotes the least common multiple of the orders modulo $p$ of all odd prime divisors of $p-1$: $$ M_p := {\rm lcm}\{{\rm ord}_p(q)\colon q\mid p-1,\ q\ \text{is an odd prime}\}. $$ I am interested in the primes $p\equiv5\pmod 8$, and I want to show that normally, $M_p>\sqrt p$ holds for these primes. In the range $5\le p<100,000,000$, there are only three exceptions (primes $p\equiv 5\pmod 8$ with $M_p<\sqrt p$): namely, $5$, $13$, and $148,997$. Are there any more such exceptional primes and if so, is the set of all these primes finite?

Notice that allowing $p\equiv 1\pmod 8$ would make every Fermat prime a bold exception.