Subgroup structure of symmetric group:S5
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This article gives specific information, namely, subgroup structure, about a particular group, namely: symmetric group:S5.
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The symmetric group of degree five has many subgroups. We'll take the five letters as . The group has order 120.
Note that since is a complete group, every automorphism of it is inner, so the classification of subgroups upto conjugacy is the same as the classification of subgroups upto automorphism. In other words, every subgroup is an automorph-conjugate subgroup.
Tables for quick information
FACTS TO CHECK AGAINST FOR SUBGROUP STRUCTURE: (finite 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 subgroups exist, Sylow implies order-dominating, congruence condition on Sylow numbers|congruence condition on number of subgroups of given prime power order
normal Hall implies permutably complemented, Hall retract implies order-conjugate
Quick summary
Item | Value |
---|---|
Number of subgroups | 156 Compared with : 1,2,6,30,156,1455,11300, 151221 |
Number of conjugacy classes of subgroups | 19 Compared with , : 1,2,4,11,19,56,96,296,554,1593 |
Number of automorphism classes of subgroups | 19 Compared with , : 1,2,4,11,19,37,96,296,554,1593 |
Isomorphism classes of Sylow subgroups and the corresponding Sylow numbers and fusion systems | 2-Sylow: dihedral group:D8 (order 8), Sylow number is 15, fusion system is non-inner non-simple fusion system for dihedral group:D8 3-Sylow: cyclic group:Z3, Sylow number is 10, fusion system is non-inner fusion system for cyclic group:Z3 5-Sylow: Z5 in S5, Sylow number is 6, fusion system is universal fusion system for cyclic group:Z5 |
Hall subgroups | -Hall subgroup: S4 in S5 (order 24) No -Hall subgroup or -Hall subgroup |
maximal subgroups | maximal subgroups have orders 12 (direct product of S3 and S2 in S5), 20 (GA(1,5) in S5), 24 (S4 in S5), 60 (A5 in S5) |
normal subgroups | There are three normal subgroups: the whole group, A5 in S5, and the trivial subgroup. |
Table classifying subgroups up to automorphisms
Note that the only normal subgroups are the trivial subgroup, the whole group, and A5 in S5, so we do not waste a column on specifying whether the subgroup is normal and on the quotient group.
TABLE SORTING AND INTERPRETATION: Note that the subgroups in the table below are sorted based on the powers of the prime divisors of the order, first covering the smallest prime in ascending order of powers, then powers of the next prime, then products of powers of the first two primes, then the third prime, then products of powers of the first and third, second and third, and all three primes. The rationale is to cluster together subgroups with similar prime powers in their order. The subgroups are not sorted by the magnitude of the order. To sort that way, click the sorting button for the order column. Similarly you can sort by index or by number of subgroups of the automorphism class.
Table classifying isomorphism types of subgroups
Group name | Order | Second part of GAP ID (first part is order) | Occurrences as subgroup | Conjugacy classes of occurrence as subgroup | Automorphism classes of occurrence as subgroup | Occurrences as normal subgroup | Occurrences as characteristic subgroup |
---|---|---|---|---|---|---|---|
trivial group | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
cyclic group:Z2 | 2 | 1 | 25 | 2 | 2 | 0 | 0 |
cyclic group:Z3 | 3 | 1 | 10 | 1 | 1 | 0 | 0 |
cyclic group:Z4 | 4 | 1 | 15 | 1 | 1 | 0 | 0 |
Klein four-group | 4 | 2 | 20 | 2 | 2 | 0 | 0 |
cyclic group:Z5 | 5 | 1 | 6 | 1 | 1 | 0 | 0 |
symmetric group:S3 | 6 | 1 | 20 | 2 | 2 | 0 | 0 |
cyclic group:Z6 | 6 | 2 | 10 | 1 | 1 | 0 | 0 |
dihedral group:D8 | 8 | 3 | 15 | 1 | 1 | 0 | 0 |
dihedral group:D10 | 10 | 1 | 6 | 1 | 1 | 0 | 0 |
alternating group:A4 | 12 | 3 | 5 | 1 | 1 | 0 | 0 |
direct product of S3 and Z2 | 12 | 4 | 10 | 1 | 1 | 0 | 0 |
general affine group:GA(1,5) | 20 | 3 | 6 | 1 | 1 | 0 | 0 |
symmetric group:S4 | 24 | 12 | 5 | 1 | 1 | 0 | 0 |
alternating group:A5 | 60 | 5 | 1 | 1 | 1 | 1 | 1 |
symmetric group:S5 | 120 | 34 | 1 | 1 | 1 | 1 | 1 |
Total | -- | -- | 156 | 19 | 19 | 3 | 3 |
Table listing number of subgroups by order
Note that these orders satisfy the congruence condition on number of subgroups of given prime power order: the number of subgroups of order is congruent to modulo .
Group order | Occurrences as subgroup | Conjugacy classes of occurrence as subgroup | Automorphism classes of occurrence as subgroup | Occurrences as normal subgroup | Occurrences as characteristic subgroup |
---|---|---|---|---|---|
1 | 1 | 1 | 1 | 1 | 1 |
2 | 25 | 2 | 2 | 0 | 0 |
3 | 10 | 1 | 1 | 0 | 0 |
4 | 35 | 3 | 3 | 0 | 0 |
5 | 6 | 1 | 1 | 0 | 0 |
6 | 30 | 3 | 3 | 0 | 0 |
8 | 15 | 1 | 1 | 0 | 0 |
10 | 6 | 1 | 1 | 0 | 0 |
12 | 15 | 2 | 2 | 0 | 0 |
20 | 6 | 1 | 1 | 0 | 0 |
24 | 5 | 1 | 1 | 0 | 0 |
60 | 1 | 1 | 1 | 1 | 1 |
120 | 1 | 1 | 1 | 1 | 1 |
Total | 156 | 19 | 19 | 3 | 3 |