The Prüfer group $\mathbb{Z}(p^\infty)$, for $p$ prime, has the interesting property that it is infinite, and every proper subgroup is finite, but can be arbitrarily large, so there is no proper subgroup of maximum cardinality.

For any group $G$, let $\text{Sub}(G)$ be the collection of subgroups of $G$. Is there an example of an uncountable group $G$ such that the set $$\{\text{card}(S):S\in \text{Sub}(G)\setminus\{G\}\}$$ has no largest member?

The Prüfer group $\mathbb{Z}(p^\infty)$, for $p$ prime, has the interesting property that it is infinite, and every proper subgroup is finite, but can be arbitrarily large, so there is no proper subgroup of maximum cardinality.

For any group $G$, let $\text{Sub}(G)$ be the collection of subgroups of $G$. Is there an example of an uncountable group $G$ such that the set of cardinals $$\{\text{card}(S):S\in \text{Sub}(G)\setminus\{G\}\}$$ has no largest member?

The Prüfer group $\mathbb{Z}(p^\infty)$, for $p$ prime, has the interesting property that it is infinite, and every proper subgroup is finite, but can be arbitrarily large, so there is no proper subgroup of maximum cardinality.

For any group $G$, let $\text{Sub}(G)$ be the collection of subgroups of $G$. Is there an example of a group $G$ such that the set $$\{\text{card}(S):S\in \text{Sub}(G)\setminus\{G\}\}$$ contains $\aleph_0$ as an element, but has no largest member?

The Prüfer group $\mathbb{Z}(p^\infty)$, for $p$ prime, has the interesting property that it is infinite, and every proper subgroup is finite, but can be arbitrarily large, so there is no proper subgroup of maximum cardinality.

For any group $G$, let $\text{Sub}(G)$ be the collection of subgroups of $G$. Is there an example of an uncountable group $G$ such that the set $$\{\text{card}(S):S\in \text{Sub}(G)\setminus\{G\}\}$$ has no largest member?

The Prüfer group $\mathbb{Z}(p^\infty)$, for $p$ prime, has the interesting property that it is infinite, and every proper subgroup is finite, but can be arbitrarily large, so there is no proper subgroup of maximum cardinality.

For any group $G$, let $\text{Sub}(G)$ be the collection of subgroups of $G$. Is there an example of a group $G$ such that the set $$\{|S|:S\in \text{Sub}(G)\setminus\{G\}\}$$ contains $\aleph_0$ as an element, but has no largest member?

The Prüfer group $\mathbb{Z}(p^\infty)$, for $p$ prime, has the interesting property that it is infinite, and every proper subgroup is finite, but can be arbitrarily large, so there is no proper subgroup of maximum cardinality.

For any group $G$, let $\text{Sub}(G)$ be the collection of subgroups of $G$. Is there an example of a group $G$ such that the set $$\{\text{card}(S):S\in \text{Sub}(G)\setminus\{G\}\}$$ contains $\aleph_0$ as an element, but has no largest member?