# Template:Finite p-group subgroup structure facts to check against

From Groupprops

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.