Finite groups all of whose subgroups are CLT

abstract-algebrafinite-groupsgroup-theorysolvable-groups

A $CLT$-group is a finite group with the property that for every divisor of the order of the group,
there is a subgroup of that order (converse lagrange theorem).

I know that:

  • There is a $CLT$-group with a non-$CLT$ subgroup (e.g., $S_4$).

  • Every subgroup of a supersolvable group is a $CLT$-group (because every
    supersolvable group is $CLT$ and it subgroups are all supersolvable).

Now, my question:

(Q1) Is there a name (or an easy criterion) for groups all of whose subgroups are $CLT$?

(Q2) Does anybody know an example of such groups that is not supersolvable?

Thanks in advance

Best Answer

Since every subgroup of a supersolvable group is supersolvable, and every supersolvable group is CLT, it follows that if $G$ is supersolvable, then every subgroup of $G$ is CLT. W. Deskins (see A Characterization of Finite Supersolvable Groups, Am. Math. Monthly Vol. 75, No. 2 (1968)) proved that $G$ is supersolvable if and only if every subgroup of $G$ is CLT. With this, I think you can answer Q2 yourself...

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