Graphical output

Lattices of subgroups L₁ - L₄, displayed in 4 separate windows, can be browsed with a mouse; resizing of windows is possible. Four zooming ratios (1,2,3,4; 2 being standard) can be set in the upper-right corner input field.

After a mouse click on any node in the lattice a new window opens. Program shows a detailed information on subgroup P'(T', or H' ), resp. pair (P',T'), namely generators of the group P'( H' ), and a primitive basis of the translation group T'. Also, there is given order of P', and index of T' in T, |T:T'|, resp. the index |G:H'|=|P:P'||T:T'|. Last, but not least, minimal supergroups (resp. covering pairs) and maximal subgroups (resp. covered pairs) are given.

Normal subgroups can be highlighted with yellow/orange (button “2 color” - see Figure below), or their sublattice displayed by omitting the other groups (button “normal”) .

The lattice L₁ is a sublattice of the lattice L(P₁), consists of all subgroups of the point group P₁, L₁ L(P₁). The sublattice L₁ has the greatest and the least element , i.e. the point groups P₁ and P₂, respectively, and is called a quotient of lattice L(P₁).

Similarly, one speaks of quotients L₂ L(T₁), L₃ L((P₁ ,T₁)), and L₄ L(G). Five kinds of possible mutual relationships among these quotients can be distinguished:

1. T₁ = T₂ ~ equitranslational subgroups -> Quotient L₂ consists only of the group T₁, and quotients L₁, L₃ and L₄ are isomorphic, L₁ ~ L₃ ~ L₄ .

2. P₁ = P₂ ~ equiclass subgroups -> Quotient L₁ consists only of the group P₁, and quotients L₂ ~ L₃ ~ L₄ are isomorphic. Condition: with each translational subgroup T'  L₂ the space group G contains only one group H' ~ (P₁,T') .

3. L₃ ~ L₄ -> Condition: for each pair (P',T') L₃ the space group G contains only one group H' ~ (P',T').

4. L₁ ~ L₂ -> Conditions (sufficient but not necessary): P₂ is a normal subgroup of P₁, the factor group P₁/P₂ is Abelian and isomorphic to the factor group T₁/T_{2 .}

5. A general case.

EXAMPLES:

(ii)   L₂ ~ L₃ ~ L₄ :     L₂

                                                  L₃

                                                               L₄

(iii)   L₃ ~ L_{4 :}                           L₃

                                                               L₄

(iv)   L₁~ L₂ :     L₁

                                              L₂
 

(v) general case: G=Cmmm, H'=P-1, T=C_1,1,1: (a-b)/2,(a+b)/2,c; T'=P_1,1,1: a,b,c

                                                                      L₁

                           L₂

                                                                          L₃ = L₁ x L₂

                                                         L₄