# Subgroup structure of groups of order 64

## Contents

This article gives specific information, namely, subgroup structure, about a family of groups, namely: groups of order 64.
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## Abelian subgroups

### Counts of abelian subgroups and abelian normal subgroups

Note the following:

The upshot is that all counts in the table below are odd.

• Index two implies normal, so all the abelian subgroups of order 32 are normal. Thus the count for abelian subgroups of order 8 is the same as the count for abelian normal subgroups of order 32.
• For the abelian groups: note that abelian implies every subgroup is normal and also that subgroup lattice and quotient lattice of finite abelian group are isomorphic. Thus, when the whole group is abelian, we have: number of abelian subgroups of order 2 = number of abelian normal subgroups of order 2 = number of abelian subgroups of order 32 = number of abelian normal subgroups of order 32. Separately, we have number of abelian subgroups of order 4 = number of abelian normal subgroups of order 4 = number of abelian subgroups of order 16 = number of abelian normal subgroups of order 16. Finally, we also have number of abelian subgroups of order 8 = number of abelian normal subgroups of order 8.
• The "number of abelian normal subgroups" columns depend only on the Hall-Senior genus, i.e., two groups with the same Hall-Senior genus have the same "number of abelian normal subgroups" of each order. The Hall-Senior genus is the part of the Hall-Senior symbol excluding the very final subscript, so for instance $64\Gamma_2a_1$ and $64\Gamma_2a_2$ both belong to the Hall-Senior genus $64\Gamma_2a$.