The following question is "ideologically related" to this one.

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", $M_p>\sqrt p$ for these primes. Computations show that in the range $5\le p<50,000,000$, there are only three exceptional primes (that is, primes $p\equiv 5\pmod 8$ with $M_p<\sqrt p$): namely, $p=5$, $p=13$, and $p=148,997$. Are there any more such exceptional primes and if so, is the set of all these primes finite?