Template:Finite solvable group subgroup structure facts to check against

FACTS TO CHECK AGAINST FOR SUBGROUP STRUCTURE: (finite solvable group)
Lagrange's theorem (order of subgroup times index of subgroup equals order of whole group, so both divide it), |order of quotient group divides order of group (and equals index of corresponding normal subgroup)
Sylow's theorem (Sylow subgroups exist, Sylow implies order-dominant, congruence condition on Sylow numbers|congruence condition on number of subgroups of prime power order
Hall subgroups exist in finite solvable|Hall implies order-dominating in finite solvable| Schur-Zassenhaus theorem (normal Hall implies permutably complemented + Hall retract implies order-conjugate
MINIMAL, MAXIMAL: minimal normal implies elementary abelian in finite solvable | maximal subgroup has prime power index in finite solvable group