Subgroup structure of groups of order 20

This article gives specific information, namely, subgroup structure, about a family of groups, namely: groups of order 20.
View subgroup structure of group families | View subgroup structure of groups of a particular order |View other specific information about groups of order 20

List of groups

The groups of order 20 are:

Group	Second part of GAP ID (GAP ID is (20,second part))	abelian?
dicyclic group:Dic20	1	No
cyclic group:Z20	2	Yes
general affine group:GA(1,5)	3	No
dihedral group:D20	4	No
direct product of Z10 and Z2	5	Yes

Number of subgroups

The groups of order 20 are not all able to be distinguished by their number of subgroups. dicyclic group:Dic20 and direct product of Z10 and Z2 both have the same number of subgroups, 10. These two groups can be distinguished via other means, such as abelianness.

Group	number of subgroups
dicyclic group:Dic20	10
cyclic group:Z20	6
general affine group:GA(1,5)	14
dihedral group:D20	22
direct product of Z10 and Z2	10

Number of normal subgroups

The groups, however, all have distinct numbers of normal subgroups:

Group	number of normal subgroups
dicyclic group:Dic20	5
cyclic group:Z20	6
general affine group:GA(1,5)	4
dihedral group:D20	7
direct product of Z10 and Z2	10

Sylow subgroups

Group	Sylow 2-subgroup isomorphism class	Sylow 5-subgroup isomorphism class
dicyclic group:Dic20	cyclic group:Z4	cyclic group:Z5
cyclic group:Z20	cyclic group:Z4	cyclic group:Z5
general affine group:GA(1,5)	cyclic group:Z4	cyclic group:Z5
dihedral group:D20	Klein four-group	cyclic group:Z5
direct product of Z10 and Z2	Klein four-group	cyclic group:Z5

Other information

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