Finite Group Databasebeta

Subgroup lattice of Cp3 × Cp2 × (Cp)2

Valid at every prime. The subgroup lattice of this shape is one object: each count below is a polynomial in p, and substituting a prime gives the lattice of that group.

Orderp7
Subgroups2p5 + 10p4 + 12p3 + 10p2 + 6p + 8
Classes of subgroups40
Automorphismsp21 − 3p20 + 2p19 + 2p18 − 3p17 + p16
Classes with a complement10

Diagram

Each circle is one class of subgroups, labeled by the exponents of its shape — 3,2 is Cp3 × Cp2, and 2²,1 is (Cp2)2 × Cp. The badge is how many subgroups the class holds. Hover a row of the table to light its circle, or click a circle to jump to its row.

Classes of subgroups

SubgroupQuotientSubgroups
Cp (direct factor)Cp3 × Cp2 × Cpp3 + p2
CpCp3 × (Cp)3p
Cp (characteristic)(Cp2)2 × (Cp)21
Cp2 (direct factor)Cp3 × (Cp)2p4
Cp2(Cp2)2 × Cpp3 − p
Cp2Cp2 × (Cp)3p
(Cp)2 (direct factor)Cp3 × Cp2p4
(Cp)2Cp3 × (Cp)2p3 + p2
(Cp)2(Cp2)2 × Cpp2 + p
(Cp)2 (characteristic)Cp2 × (Cp)31
Cp3 (direct factor)Cp2 × (Cp)2p4
Cp2 × Cp (direct factor)Cp3 × Cpp5 + p4
Cp2 × Cp(Cp2)2p4 − p2
Cp2 × CpCp2 × (Cp)2p2 − 1
Cp2 × CpCp2 × (Cp)2p3
Cp2 × CpCp2 × (Cp)2p3 + p2
Cp2 × Cp (characteristic)(Cp)41
(Cp)3Cp3 × Cpp3
(Cp)3(Cp2)2p2
(Cp)3Cp2 × (Cp)2p + 1
Cp3 × Cp (direct factor)Cp2 × Cpp5 + p4
Cp3 × Cp(Cp)3p3
(Cp2)2Cp2 × Cpp4 − p2
(Cp2)2(Cp)3p2
Cp2 × (Cp)2 (direct factor)Cp3p4
Cp2 × (Cp)2Cp2 × Cpp2 − 1
Cp2 × (Cp)2Cp2 × Cpp3 + p2
Cp2 × (Cp)2Cp2 × Cpp3
Cp2 × (Cp)2(Cp)3p + 1
(Cp)4 (characteristic)Cp2 × Cp1
Cp3 × Cp2 (direct factor)(Cp)2p4
Cp3 × (Cp)2 (direct factor)Cp2p4
Cp3 × (Cp)2(Cp)2p3 + p2
(Cp2)2 × CpCp2p3 − p
(Cp2)2 × Cp(Cp)2p2 + p
Cp2 × (Cp)3Cp2p
Cp2 × (Cp)3 (characteristic)(Cp)21
Cp3 × Cp2 × Cp (direct factor)Cpp3 + p2
Cp3 × (Cp)3Cpp
(Cp2)2 × (Cp)2 (characteristic)Cp1

← All shapes