Subgroup structure of groups of prime-cube order

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This article gives specific information, namely, subgroup structure, about a family of groups, namely: groups of prime-cube order.
View subgroup structure of group families | View other specific information about groups of prime-cube order

Note that the tables here work correctly only for groups of order p^3 where p is an odd prime. The case p = 2 behaves differently. For that case, see subgroup structure of groups of order 8.

Numerical information on counts of subgroups by isomorphism type

FACTS TO CHECK AGAINST FOR SUBGROUP STRUCTURE: (group of prime power order)
Lagrange's theorem (order of subgroup times index of subgroup equals order of whole group, so all subgroups have prime power orders)|order of quotient group divides order of group (and equals index of corresponding normal subgroup, so all quotients have prime power orders)
prime power order implies not centerless | prime power order implies nilpotent | prime power order implies center is normality-large
size of conjugacy class of subgroups divides index of center
congruence condition on number of subgroups of given prime power order: The total number of subgroups of any fixed prime power order is congruent to 1 mod the prime.

Number of subgroups per isomorphism type

The number in each column is the number of subgroups in the given group (by row) of the isomorphism type of the (column) group:

Group Second part of GAP ID Nilpotency class Minimum size of generating set group of prime order cyclic group of prime-square order elementary abelian group of prime-square order Total (row sum + 2, for trivial group and whole group)
cyclic group of prime-cube order 1 1 1 1 1 0 4
direct product of ... 2 1 2 p + 1 p 1 2p + 4
prime-cube order group:U(3,p) 3 2 2 p^2 + p + 1 0 p + 1 p^2 + 2p + 4
semidirect product of ... 4 2 2 p + 1 p 1 2p + 4
elementary abelian group of prime-cube order 5 1 3 p^2 + p + 1 0 p^2 + p + 1 2p^2 + 2p + 4

Number of conjugacy classes of subgroups per isomorphism type

The number in each column is the number of conjugacy classes of subgroups in the given group (by row) of the isomorphism type of the (column) group:

Group Second part of GAP ID nilpotency class minimum size of generating set group of prime order cyclic group of prime-square order elementary abelian group of prime-square order Total (row sum + 2, for trivial group and whole group)
cyclic group of prime-cube order 1 1 1 1 1 0 4
direct product of ... 2 1 2 p + 1 p 1 2p + 4
prime-cube order group:U(3,p) 3 2 2 p + 2 0 p + 1 2p + 5
semidirect product of ... 4 2 2 2 p 1 p + 5
elementary abelian group of prime-cube order 5 1 3 p^2 + p + 1 0 p^2 + p + 1 2p^2 + 2p + 4

Number of normal subgroups per isomorphism type

Group Second part of GAP ID nilpotency class minimum size of generating set group of prime order cyclic group of prime-square order elementary abelian group of prime-square order Total (row sum + 2, for trivial group and whole group)
cyclic group of prime-cube order 1 1 1 1 1 0 4
direct product of ... 2 1 2 p + 1 p 1 2p + 4
prime-cube order group:U(3,p) 3 2 2 1 0 p + 1 p + 4
semidirect product of ... 4 2 2 1 p 1 p + 4
elementary abelian group of prime-cube order 5 1 3 p^2 + p + 1 0 p^2 + p + 1 2p^2 + 2p + 4

Number of automorphism classes of subgroups per isomorphism type

The number in each column is the number of automorphism classes of subgroups in the given group (by row) of the isomorphism type of the (column) group:

Group Second part of GAP ID nilpotency class minimum size of generating set group of prime order cyclic group of prime-square order elementary abelian group of prime-square order Total (row sum + 2, for trivial group and whole group)
cyclic group of prime-cube order 1 1 1 1 1 0 4
direct product of ... 2 1 2 2 1 1 6
prime-cube order group:U(3,p) 3 2 2 2 0 1 5
semidirect product of ... 4 2 2 2 1 1 6
elementary abelian group of prime-cube order 5 1 3 1 0 1 4

Number of characteristic subgroups per isomorphism type

Group Second part of GAP ID nilpotency class minimum size of generating set group of prime order cyclic group of prime-square order elementary abelian group of prime-square order Total (row sum + 2, for trivial group and whole group)
cyclic group of prime-cube order 1 1 1 1 1 0 4
direct product of ... 2 1 2 1 0 1 4
prime-cube order group:U(3,p) 3 2 1 0 0 3
semidirect product of ... 4 2 2 1 0 1 4
elementary abelian group of prime-cube order 5 1 3 0 0 0 2

Numerical information on counts of subgroups by order

Number of subgroups per order

We have the following:

Group Second part of GAP ID number of subgroups of order p number of normal subgroups of order p number of subgroups of order p^2 number of normal subgroups of order p^2
cyclic group of prime-cube order 1 1 1 1 1
direct product of ... 2 p + 1 p + 1 p + 1 p + 1
prime-cube order group:U(3,p) 3 p^2 + p + 1 1 p + 1 p + 1
semidirect product of ... 4 p + 1 1 p + 1 p + 1
elementary abelian group of prime-cube order 5 p^2 + p + 1 p^2 + p + 1 p^2 + p + 1 p^2 + p + 1