Subgroups — C₃×Dic₁₅×C₁₃₁ #37556076

Order 23580.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₅×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	Dic₁₅×C₁₃₁	C₃	1	1	yes	maximal, characteristic, direct product factor
3	C₃×Dic₅×C₁₃₁	—	3	1	no	maximal
4	C₃²×C₅×C₁₃₁	C₄	1	1	yes	characteristic, complemented, Hall
5	C₃×Dic₃×C₁₃₁	—	5	1	no	maximal, Hall
6	C₂×C₃×C₅×C₁₃₁	—	2	1	no	—
7	C₂×C₃×C₅×C₁₃₁	D₃	1	1	yes	characteristic
8	C₂×C₃×C₅×C₁₃₁	C₂×C₃	1	1	yes	characteristic
9	Dic₅×C₁₃₁	—	3	1	no	Hall
10	C₂×C₃²×C₁₃₁	D₅	1	1	yes	characteristic
11	C₃×C₅×C₁₃₁	—	2	1	no	—
12	C₃×C₅×C₁₃₁	Dic₃	1	1	yes	characteristic, complemented
13	C₃×C₅×C₁₃₁	C₃×C₄	1	1	yes	characteristic, complemented
14	C₃×C₄×C₁₃₁	—	15	1	no	—
15	Dic₃×C₁₃₁	—	5	1	no	—
16	C₂×C₅×C₁₃₁	C₃×D₃	1	1	yes	characteristic
17	C₃²×C₁₃₁	Dic₅	1	1	yes	characteristic, complemented, Hall
18	C₂×C₃×C₁₃₁	—	2	1	no	—
19	C₂×C₃×C₁₃₁	C₃×D₅	1	1	yes	characteristic
20	C₂×C₃×C₁₃₁	D₁₅	1	1	yes	characteristic
21	C₅×C₁₃₁	C₃×Dic₃	1	1	yes	characteristic, complemented, Hall
22	C₄×C₁₃₁	—	15	1	no	Hall
23	C₃×C₁₃₁	—	2	1	no	—
24	C₃×C₁₃₁	C₃×Dic₅	1	1	yes	characteristic, complemented
25	C₃×C₁₃₁	Dic₁₅	1	1	yes	characteristic, direct product factor
26	C₂×C₁₃₁	C₃×D₁₅	1	1	yes	characteristic
27	C₃×Dic₁₅	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
28	C₁₃₁	C₃×Dic₁₅	1	1	yes	characteristic, direct product factor, Sylow
29	C₂×C₃²×C₅	C₂×C₁₃₁	1	1	yes	characteristic
30	Dic₁₅	C₃×C₁₃₁	1	1	yes	characteristic, direct product factor
31	C₃×Dic₅	—	3	1	no	—
32	C₃²×C₅	C₄×C₁₃₁	1	1	yes	characteristic, complemented, Hall
33	C₃×Dic₃	—	5	1	no	Hall
34	C₂×C₃×C₅	—	2	1	no	—
35	C₂×C₃×C₅	D₃×C₁₃₁	1	1	yes	characteristic
36	C₂×C₃×C₅	C₂×C₃×C₁₃₁	1	1	yes	characteristic
37	Dic₅	—	3	1	no	Hall
38	C₂×C₃²	D₅×C₁₃₁	1	1	yes	characteristic
39	C₃×C₅	—	2	1	no	—
40	C₃×C₅	Dic₃×C₁₃₁	1	1	yes	characteristic, complemented
41	C₃×C₅	C₃×C₄×C₁₃₁	1	1	yes	characteristic, complemented
42	C₃×C₄	—	15	1	no	—
43	Dic₃	—	5	1	no	—
44	C₂×C₅	C₃×D₃×C₁₃₁	1	1	yes	characteristic
45	C₃²	Dic₅×C₁₃₁	1	1	yes	characteristic, complemented, Sylow
46	C₂×C₃	—	2	1	no	—
47	C₂×C₃	C₃×D₅×C₁₃₁	1	1	yes	characteristic
48	C₂×C₃	D₁₅×C₁₃₁	1	1	yes	characteristic
49	C₅	C₃×Dic₃×C₁₃₁	1	1	yes	characteristic, complemented, Sylow
50	C₄	—	15	1	no	Sylow
51	C₃	—	2	1	no	—
52	C₃	C₃×Dic₅×C₁₃₁	1	1	yes	characteristic, complemented
53	C₃	Dic₁₅×C₁₃₁	1	1	yes	characteristic, direct product factor
54	C₂	C₃×D₁₅×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃×C₁₃₁	D₁₅
commutator, hypocenter	C₃×C₅	C₃×C₄×C₁₃₁
Frattini	C₂	C₃×D₁₅×C₁₃₁
Fitting, socle, prime radical	C₂×C₃²×C₅×C₁₃₁	C₂
cosocle	C₂×C₃×C₅	C₂×C₃×C₁₃₁
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₃×C₁₃₁

Complements

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

N	H ≅ G/N	type
2. Dic₁₅×C₁₃₁	C₃	direct (N × H)
4. C₃²×C₅×C₁₃₁	C₄	semidirect
12. C₃×C₅×C₁₃₁	Dic₃	semidirect
13. C₃×C₅×C₁₃₁	C₃×C₄	semidirect
17. C₃²×C₁₃₁	Dic₅	semidirect
21. C₅×C₁₃₁	C₃×Dic₃	semidirect
24. C₃×C₁₃₁	C₃×Dic₅	semidirect
25. C₃×C₁₃₁	Dic₁₅	direct (N × H)
27. C₃×Dic₁₅	C₁₃₁	direct (N × H)
28. C₁₃₁	C₃×Dic₁₅	direct (N × H)
30. Dic₁₅	C₃×C₁₃₁	direct (N × H)
32. C₃²×C₅	C₄×C₁₃₁	semidirect
40. C₃×C₅	Dic₃×C₁₃₁	semidirect
41. C₃×C₅	C₃×C₄×C₁₃₁	semidirect
45. C₃²	Dic₅×C₁₃₁	semidirect
49. C₅	C₃×Dic₃×C₁₃₁	semidirect
52. C₃	C₃×Dic₅×C₁₃₁	semidirect
53. C₃	Dic₁₅×C₁₃₁	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×C₅×C₁₃₁	2
Dic₁₅×C₁₃₁	3
C₃×Dic₅×C₁₃₁	3
C₃×Dic₃×C₁₃₁	5
C₃×Dic₁₅	131

Subgroups — C3×Dic15×C131 #37556076

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×Dic₁₅×C₁₃₁ #37556076