Subgroups — C₂²×C₃²×C₁₃₁⋊C₅ #37556096

Order 23580.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₁₃₁⋊C₅	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₃×C₁₃₁⋊C₅	C₃	1	4	yes	maximal, direct product factor
3	C₃²×C₁₃₁⋊C₅	C₂²	1	1	yes	characteristic, direct product factor, Hall
4	C₂²×C₃²×C₁₃₁	C₅	1	1	yes	maximal, characteristic, complemented, Hall
5	C₂×C₃×C₁₃₁⋊C₅	C₂×C₃	1	12	yes	direct product factor
6	C₂²×C₁₃₁⋊C₅	C₃²	1	1	yes	characteristic, direct product factor, Hall
7	C₂×C₃²×C₁₃₁	C₂×C₅	1	3	yes	complemented
8	C₃×C₁₃₁⋊C₅	C₂²×C₃	1	4	yes	direct product factor
9	C₂²×C₃×C₁₃₁	C₃×C₅	1	4	yes	complemented
10	C₂×C₁₃₁⋊C₅	C₂×C₃²	1	3	yes	direct product factor
11	C₃²×C₁₃₁	C₂²×C₅	1	1	yes	characteristic, complemented, Hall
12	C₂×C₃×C₁₃₁	C₂×C₃×C₅	1	12	yes	complemented
13	C₁₃₁⋊C₅	C₂²×C₃²	1	1	yes	characteristic, direct product factor, Hall
14	C₂²×C₁₃₁	C₃²×C₅	1	1	yes	characteristic, complemented, Hall
15	C₃×C₁₃₁	C₂²×C₃×C₅	1	4	yes	complemented
16	C₂×C₁₃₁	C₂×C₃²×C₅	1	3	yes	complemented
17	C₂²×C₃²×C₅	—	131	1	no	maximal, Hall
18	C₁₃₁	C₂²×C₃²×C₅	1	1	yes	characteristic, complemented, Sylow
19	C₂×C₃²×C₅	—	131	3	no	—
20	C₂²×C₃×C₅	—	131	4	no	—
21	C₃²×C₅	—	131	1	no	Hall
22	C₂²×C₃²	C₁₃₁⋊C₅	1	1	yes	characteristic, direct product factor, Hall
23	C₂×C₃×C₅	—	131	12	no	—
24	C₂²×C₅	—	131	1	no	Hall
25	C₂×C₃²	C₂×C₁₃₁⋊C₅	1	3	yes	direct product factor
26	C₃×C₅	—	131	4	no	—
27	C₂²×C₃	C₃×C₁₃₁⋊C₅	1	4	yes	direct product factor
28	C₂×C₅	—	131	3	no	—
29	C₃²	C₂²×C₁₃₁⋊C₅	1	1	yes	characteristic, direct product factor, Sylow
30	C₂×C₃	C₂×C₃×C₁₃₁⋊C₅	1	12	yes	direct product factor
31	C₅	—	131	1	no	Sylow
32	C₂²	C₃²×C₁₃₁⋊C₅	1	1	yes	characteristic, direct product factor, Sylow
33	C₃	C₂²×C₃×C₁₃₁⋊C₅	1	4	yes	direct product factor
34	C₂	C₂×C₃²×C₁₃₁⋊C₅	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₃²	C₁₃₁⋊C₅
commutator, cosocle, hypocenter	C₁₃₁	C₂²×C₃²×C₅
Frattini	C₁	G
Fitting, socle	C₂²×C₃²×C₁₃₁	C₅
LCS	G ▷ C₁₃₁
UCS	C₁ ◁ C₂²×C₃²

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₃²×C₁₃₁⋊C₅	C₂	direct (N × H)
2. C₂²×C₃×C₁₃₁⋊C₅	C₃	direct (N × H)
3. C₃²×C₁₃₁⋊C₅	C₂²	direct (N × H)
4. C₂²×C₃²×C₁₃₁	C₅	semidirect
5. C₂×C₃×C₁₃₁⋊C₅	C₂×C₃	direct (N × H)
6. C₂²×C₁₃₁⋊C₅	C₃²	direct (N × H)
7. C₂×C₃²×C₁₃₁	C₂×C₅	semidirect
8. C₃×C₁₃₁⋊C₅	C₂²×C₃	direct (N × H)
9. C₂²×C₃×C₁₃₁	C₃×C₅	semidirect
10. C₂×C₁₃₁⋊C₅	C₂×C₃²	direct (N × H)
11. C₃²×C₁₃₁	C₂²×C₅	semidirect
12. C₂×C₃×C₁₃₁	C₂×C₃×C₅	semidirect
13. C₁₃₁⋊C₅	C₂²×C₃²	direct (N × H)
14. C₂²×C₁₃₁	C₃²×C₅	semidirect
15. C₃×C₁₃₁	C₂²×C₃×C₅	semidirect
16. C₂×C₁₃₁	C₂×C₃²×C₅	semidirect
18. C₁₃₁	C₂²×C₃²×C₅	semidirect
22. C₂²×C₃²	C₁₃₁⋊C₅	direct (N × H)
25. C₂×C₃²	C₂×C₁₃₁⋊C₅	direct (N × H)
27. C₂²×C₃	C₃×C₁₃₁⋊C₅	direct (N × H)
29. C₃²	C₂²×C₁₃₁⋊C₅	direct (N × H)
30. C₂×C₃	C₂×C₃×C₁₃₁⋊C₅	direct (N × H)
32. C₂²	C₃²×C₁₃₁⋊C₅	direct (N × H)
33. C₃	C₂²×C₃×C₁₃₁⋊C₅	direct (N × H)
34. C₂	C₂×C₃²×C₁₃₁⋊C₅	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃²×C₁₃₁⋊C₅	2
C₂²×C₃×C₁₃₁⋊C₅	3
C₂²×C₃²×C₁₃₁	5
C₂²×C₃²×C₅	131

Subgroups — C22×C32×C131⋊C5 #37556096

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃²×C₁₃₁⋊C₅ #37556096