Let $n$ be a positive integer and $\zeta_n$ a primitive root of unity. Computation of the class group $C_n$ of the maximal real subfield $\Bbb{Q} (\zeta_n + \zeta_n^{-1} )$ is a difficult matter. \par In [Math. Comp. {\bf 72} (2003), no.~242, 913--937; MR1954975 (2004f:11116)], R. Schoof introduced a new method for computing certain invariants of $C_n$ for prime $n$. More precisely, let $B_n$ be the quotient group of the unit group $O_n^{*}$ modulo the subgroup of cyclotomic units. It is known that the orders of $B_n$ and $C_n$ are the same. Schoof refined this relation by considering the Jordan-Hölder factors of the dual of $B_n$ as a Galois module. \par In this paper under review, the author extends Schoof's method to the case where $n$ is a product of two distinct primes $p$ and $q$. She avoids using the group of full cyclotomic units but does use a certain cyclic subgroup that has finite index in the unit group, and considers the quotient group $B_n$. This choice enables her to extend Schoof's idea directly to the composite conductor case. To determine the structure of the Jordan-Hölder filtration of the dual of $B_n$, she uses the Gröbner basis because she has to deal with two variable polynomials, in contrast to one variable in Schoof's paper. \par The algorithm obtained is used to compute the $l$-parts of $C_{pq}$ for ${2 < l < 10000}$, $pq < 2000$. \par John C. Miller recently succeeded in computing the exact order of $C_n$ for several prime and composite $n$'s [see Acta Arith. {\bf 164} (2014), no.~4, 381--398; 3244941 ; LMS J. Comput. Math. {\bf 17} (2014), suppl. A, 404--417; 3240817 ]. \par REVISED (July, 2017) \prevrevtext Let $n$ be a positive integer and $\zeta_n$ a primitive root of unity. Computation of the class group $C_n$ of the maximal real subfield $\Bbb{Q} (\zeta_n + \zeta_n^{-1} )$ is a difficult matter. \par In [Math. Comp. {\bf 72} (2003), no.~242, 913--937; MR1954975 (2004f:11116)], R. Schoof introduced a new method for computing certain invariants of $C_n$ for prime $n$. More precisely, let $B_n$ be the quotient group of the unit group $O_n^{*}$ modulo the subgroup of cyclotomic units. It is known that the orders of $B_n$ and $C_n$ are the same. Schoof refined this relation by considering the Jordan-Hölder factors of the dual of $B_n$ as a Galois module. \par In this paper under review, the author extends Schoof's method to the case where $n$ is a product of two distinct primes $p$ and $q$. She avoids using the group of full cyclotomic units but does use a certain cyclic subgroup that has finite index in the unit group, and considers the quotient group $B_n$. This choice enables her to extend Schoof's idea directly to the composite conductor case. To determine the structure of the Jordan-Hölder filtration of the dual of $B_n$, she uses the Gröbner basis because she has to deal with two variable polynomials, in contrast to one variable in Schoof's paper. \par The algorithm obtained is used to compute (a part of) the $l$-parts of $C_{pq}$ for ${2 < l < 10000}$, $pq < 2000$. \par John C. Miller recently succeeded in computing the exact order of $C_n$ for several prime and composite $n$'s [see Acta Arith. {\bf 164} (2014), no.~4, 381--398; 3244941 ; LMS J. Comput. Math. {\bf 17} (2014), suppl. A, 404--417; 3240817 ]. \prevrevr Masanari\ Kida \endprevrevtext