  Let $T$ be a tournament of odd order $n\ge 5,$ $c_{m}(T)$ be the number of
its $m$-cycles, and $s_{m}(T)$ be the number of its strongly connected
$m$-subtournaments. Due to work of L.W. Beineke and F. Harary, it is well known
that $s_{m}(T)\le s_{m}(RLT_{n}),$ where $RLT_{n}$ is the regular locally
transitive tournament of order $n.$ For $m=3$ and $m=4,$ $c_{m}(T)$ equals
$s_{m}(T),$ but it is not so for $m\ge 5.$ As J.W. Moon pointed out in his note
in 1966, the problem of determining the maximum of $c_{m}(T)$ seems very
difficult in general (i.e. for $m\ge 5$). In the present paper, based on the
Komarov-Mackey formula for $c_{5}(T)$ obtained recently, we prove that
$c_{5}(T)\le (n+1)n(n-1)(n-2)(n-3)/160$ with equality holding iff $T$ is doubly
regular. A formula for $s_{5}(T)$ is also deduced. With the use of it, we show
that $s_{5}(T)\le (n+1)n(n-1)(n-3)(11n-47)/1920$ with equality holding iff
$T=RLT_{n}$ or $n=7$ and $T$ is regular or $n=5$ and $T$ is strong. It is also
proved that for a regular tournament $T$ of (odd) order $n\ge 9,$ a lower bound
$(n+1)n(n-1)(n-3)(17n-59)/3840\le s_{5}(T)$ holds with equality iff $T$ is
doubly regular. These results are compared with the ones recently obtained by
the author for $c_{5}(T).$
