Subgroup structure of nontrivial semidirect product of Z4 and Z4

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This article gives specific information, namely, subgroup structure, about a particular group, namely: nontrivial semidirect product of Z4 and Z4.
View subgroup structure of particular groups | View other specific information about nontrivial semidirect product of Z4 and Z4

This article describes the subgroup structure of nontrivial semidirect product of Z4 and Z4, given explicitly by the following presentation, where e denotes the identity element:

\langle x,y \mid x^4 = y^4 = e, yxy^{-1} = x^3 \rangle

The elements are:

\{ e,x,x^2,x^3,y,xy,x^2y,x^3y,y^2,xy^2,x^2y^2,x^3y^2,y^3,xy^3,x^2y^3,x^3y^3 \}

Tables for quick information

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.

Table classifying subgroups up to automorphisms


In case a single equivalence class of subgroups under automorphisms comprises multiple conjugacy classes of subgroups, outer curly braces are used to bucket the conjugacy classes.

Automorphism class of subgroups List of subgroups Isomorphism class Order of subgroups Index of subgroups Number of conjugacy classes (=1 iff automorph-conjugate subgroup) Size of each conjugacy class (=1 iff normal subgroup) Total number of subgroups (=1 iff characteristic subgroup) Isomorphism class of quotient (if exists) Subnormal depth Nilpotency class
trivial subgroup \{ e \} trivial group 1 16 1 1 1 nontrivial semidirect product of Z4 and Z4 1 0
derived subgroup of nontrivial semidirect product of Z4 and Z4 \{ e, x^2 \} cyclic group:Z2 2 8 1 1 1 direct product of Z4 and Z2 1 1
subgroup generated by a non-commutator square in nontrivial semidirect product of Z4 and Z4 \{ e, y^2 \} cyclic group:Z2 2 8 1 1 1 dihedral group:D8 1 1
central subgroup generated by a non-square in nontrivial semidirect product of Z4 and Z4 \{ e, x^2y^2 \} cyclic group:Z2 2 8 1 1 1 quaternion group 1 1
center of nontrivial semidirect product of Z4 and Z4 \{ e, x^2, y^2, x^2y^2 \} Klein four-group 4 4 1 1 1 Klein four-group 1 1
a bunch of cyclic subgroups of order four \{ \langle y \rangle, \langle x^2 y \rangle \}
\{ \langle xy \rangle, \langle x^3y \rangle \}
cyclic group:Z4 4 4 2 2 4 -- 2 1
another bunch of cyclic subgroups of order four \langle x \rangle, \langle xy^2 \rangle cyclic group:Z4 4 4 2 1 2 cyclic group:Z4 1 1
abelian maximal subgroups that are not characteristic \langle x^2, y \rangle, \langle x^2, xy \rangle direct product of Z4 and Z2 8 2 2 1 2 cyclic group:Z2 1 1
abelian maximal subgroup that is characteristic \langle x,y^2 \rangle direct product of Z4 and Z2 8 2 1 1 1 cyclic group:Z2 1 1
whole group \langle x,y \rangle nontrivial semidirect product of Z4 and Z4 16 1 1 1 1 trivial group 1 1
Total -- -- -- -- 13 -- 15 -- -- --