Subgroup Lattices of Finite Cyclic Groups

notation



n

〈

a

〉

n

20

subgroup lattice of



20

Explore the subgroup lattices of finite cyclic groups of order up to 1000. The cyclic group of order

n

can be represented as



n

(the integers mod

n

under addition) or as generated by an abstract element

a

. Mouse over a vertex of the lattice to see the order and index of the subgroup represented by that vertex; placing the cursor over an edge displays the index of the smaller subgroup in the larger subgroup. Mouse over a vertex label to see all the elements in the group that generate that subgroup.

If

G

is a cyclic group of order

n

, then every subgroup of

G

is likewise cyclic. The order of each subgroup of

G

is a divisor of

n

, and

G

has exactly one subgroup of order

k

for each positive divisor

k

of

n

.