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\section{The groups that have to be considered}\label{sec:groups}
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Every finite subgroup of $E(K)$ is isomorphic to $\tg{m}{n}$ with $m\mid n$. The set $\mathcal G_{10}$ of the $69$ groups
of \Cref{thm:main} is closed under passing to subgroups, and by \Cref{sec:occur,sec:sporadic} it is contained in
$\Phi(10)$. Recall that $S(10)$ is the set of primes up to $37$ \cite[Theorem 1.1]{Sd}.

\begin{lemma}\label{lem:minimal}
Let $\mathcal M_{10}$ be the set of groups $\tg mn$ with $m\mid n$ such that $\varphi(m)\mid 10$, every prime dividing
$n$ lies in $S(10)$, $\tg mn\notin\mathcal G_{10}$, and every proper subgroup of $\tg mn$ lies in $\mathcal G_{10}$.
Then $\mathcal M_{10}$ consists of the following $110$ groups:
\[
\begin{array}{l}
\torz{n}\ \text{ for }\ n\in\{44,46,48,49,51,52,54,55,56,57,58,60,62,63,64,65,66,68,69,70,\\
\qquad\quad 72,74,75,76,77,78,80,81,84,85,87,90,91,93,95,99,100,105,111,115,117,\\
\qquad\quad 119,121,125,133,135,143,145,155,161,169,175,185,187,203,209,217,221,\\
\qquad\quad 247,253,259,289,299,319,323,341,361,377,391,403,407,437,481,493,527,\\
\qquad\quad 529,551,589,629,667,703,713,841,851,899,961,1073,1147,1369\},\\[2pt]
\tg{2}{2n}\ \text{ for }\ n\in\{16,17,18,19,20,21,25\},\\[2pt]
\tg{3}{3n}\ \text{ for }\ n\in\{7,8,9,10,11,12,13,15\},\\[2pt]
\tg{4}{4n}\ \text{ for }\ n\in\{4,5,6,7\},\\[2pt]
\tg{6}{18},\quad \tg{11}{11}.
\end{array}
\]
Every group in $\Phi(10)\setminus\mathcal G_{10}$ contains a member of $\mathcal M_{10}$.
\end{lemma}

\begin{proof}
By the Weil pairing and by \cite[Theorem 1.1]{Sd}, every group in $\Phi(10)$ is of the form $\tg mn$ with
$\varphi(m)\mid 10$, that is, $m\in\{1,2,3,4,6,11,22\}$, and with all the primes dividing $n$ in $S(10)$. As
$\mathcal G_{10}$ is closed under passing to subgroups, a group in $\Phi(10)\setminus\mathcal G_{10}$ contains a subgroup
of minimal order which is not in $\mathcal G_{10}$, and this subgroup lies in $\mathcal M_{10}$. If
$\torz n\in\mathcal M_{10}$, then $\Z/(n/p)\Z\in\mathcal G_{10}$ for every prime $p\mid n$, so $n\leq37\cdot50$, and if
$\tg mn\in\mathcal M_{10}$ with $m>1$, then $\torz n\in\mathcal G_{10}$, so $n\leq 50$. It remains to run through these
finitely many groups, which we \gitlink{candidates.py}{do} by computer.
\end{proof}

Note that $\tg{22}{22}$ does not occur in $\mathcal M_{10}$, as it contains $\tg{11}{11}$.

\begin{remark}\label{rem:exactdegree}
Let $\tg mn\in\mathcal M_{10}$ with $m\leq 2$, and suppose that $n$ is divisible by a prime $p\geq23$. As
$S(1)\cup S(2)\cup S(5)$ consists of the primes up to $19$ \cite{mazur,kamienny,DKSS}, a point
$x\in Y_1(m,n)$ defined over a number field of degree $10$ has $[\Q(x):\Q]=10$, so its trace divisor is a closed point of
degree $10$.
\end{remark}

To rule out a group $\tg mn\in\mathcal M_{10}$ we show that no effective $\Q(\zeta_m)$-rational divisor of degree
$d_m$ on $X_1(m,n)$ with non-cuspidal support is the trace divisor (\Cref{subsec:trace}) of a point
$x\in Y_1(m,n)(K)$ with $[K:\Q]=10$.

Several members of $\mathcal M_{10}$ are reduced to others by an isogeny.

\begin{lemma}\label{lem:isogeny}
Let $m\mid n$, let $K$ be a number field containing $\mu_m$, and let $E/K$ be an elliptic curve with a point of order
$mn$ over $K$. Then there is an elliptic curve $E'$ over $K$, isogenous to $E$ over $K$, with $\tg mn\subseteq E'(K)$.
In particular, if $\tg{2}{2k}$ is not a subgroup of the torsion of any elliptic curve over a number field $K$, then
neither is $\torz{4k}$.
\end{lemma}

\begin{proof}
Let $P\in E(K)$ have order $mn$, and let $\phi:E\to E'=E/\dm{nP}$, which is defined over $K$. The kernel of the dual
isogeny $\widehat\phi$ is $\phi(E[m])$, which is cyclic of order $m$ and, by the Weil pairing, isomorphic to $\mu_m$
as a Galois module, as $\dm{nP}\simeq\torz m$. As $\mu_m\subseteq K$, all the points of $\ker\widehat\phi$ are defined
over $K$. The point $Q=\phi(P)\in E'(K)$ has order $n$, and $\widehat\phi(Q)=mP$ also has order $n$, so
$\dm Q\cap\ker\widehat\phi=0$. Hence $\ker\widehat\phi+\dm Q\simeq\tg mn$ is a subgroup of $E'(K)$. For the last
statement take $m=2$ and $n=2k$.
\end{proof}

By \Cref{lem:isogeny}, the exclusion of $\tg{2}{2k}$ for $2k\in\{32,34,36,38,40,42,50\}$ in \Cref{sec:2x2n} also
excludes $\torz{4k}$ for $4k\in\{64,68,72,76,80,84,100\}$. It does not apply to $\torz{52}$, as $\tg{2}{26}$ lies in
$\Phi(10)$, and we treat this group in \Cref{prop:52}. \Cref{sec:proof} lists where each of the groups in
$\mathcal M_{10}$ is excluded.