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Sebastien Palcoux
Sebastien Palcoux
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Answer to Question 1:

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All other lattices of size at most 7 are known to be representable as the congruence lattice of a finite algebra. See my thesis and this MO question.

Answer to Question 2: The same lattice that answers question 1.

However, there may be other 7-element lattices that are not known to occur as intervals in subgroup lattices of finite groups.

It is know that every lattice of size at most 6 is "group representable" in the above sense. See Watatani (1996) MR1409040 and Aschbacher (2008) MR2393428.

Answer to Question 1:

enter image description here

All other lattices of size at most 7 are known to be representable as the congruence lattice of a finite algebra. See my thesis and this MO question.

Answer to Question 2: The same lattice that answers question 1.

However, there may be other 7-element lattices that are not known to occur as intervals in subgroup lattices of finite groups.

It is know that every lattice of size at most 6 is "group representable" in the above sense. See Watatani (1996) MR1409040 and Aschbacher (2008) MR2393428.

Answer to Question 1:

enter image description here

All other lattices of size at most 7 are known to be representable as the congruence lattice of a finite algebra. See my thesis and this MO question.

Answer to Question 2: The same lattice that answers question 1.

However, there may be other 7-element lattices that are not known to occur as intervals in subgroup lattices of finite groups.

It is know that every lattice of size at most 6 is "group representable" in the above sense. See Watatani (1996) MR1409040 and Aschbacher (2008) MR2393428.

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William DeMeo
William DeMeo
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  • 1
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Answer to Question 1:

enter image description here

All other lattices of size at most 7 are known to be representable as the congruence lattice of a finite algebra. See my thesis and this MO question.

Answer to Question 2: The same lattice that answers question 1, shown above.

However, there may be other 7-element lattices that are not known to occur as intervals in subgroup lattices of finite groups.

It is know that every lattice of size at most 6 is "group representable" in the above sense. See Watatani (1996) MR1409040 and Aschbacher (2008) MR2393428.

Answer to Question 1:

enter image description here

All other lattices of size at most 7 are known to be representable as the congruence lattice of a finite algebra. See my thesis and this MO question.

Answer to Question 2: The same lattice that answers question 1, shown above.

However, there may be other 7-element lattices that are not known to occur as intervals in subgroup lattices of finite groups.

It is know that every lattice of size at most 6 is "group representable" in the above sense. See Watatani (1996) MR1409040 and Aschbacher (2008) MR2393428.

Answer to Question 1:

enter image description here

All other lattices of size at most 7 are known to be representable as the congruence lattice of a finite algebra. See my thesis and this MO question.

Answer to Question 2: The same lattice that answers question 1.

However, there may be other 7-element lattices that are not known to occur as intervals in subgroup lattices of finite groups.

It is know that every lattice of size at most 6 is "group representable" in the above sense. See Watatani (1996) MR1409040 and Aschbacher (2008) MR2393428.

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William DeMeo
William DeMeo
  • 1.2k
  • 1
  • 12
  • 16

Answer to Question 1:

enter image description here

All other lattices of size at most 7 are known to be representable as the congruence lattice of a finite algebra. See my thesis and this MO question.

Answer to Question 2: The same lattice that answers question 1, shown above.

However, there may be other 7-element lattices that are not known to occur as intervals in subgroup lattices of finite groups.

It is know that every lattice of size at most 6 is "group representable" in the above sense. See Watatani (1996) MR1409040 and Aschbacher (2008) MR2393428.