Subgroups — C₃⋊(C₃⋊#16861527) #49147180

Order 36900.

156 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅×D₅×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₃⋊#16861527	—	3	4	no	maximal
3	C₃²×C₅²×C₄₁	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃⋊(C₃⋊#11717963)	—	5	1	no	maximal
5	C₃⋊Dic₆₁₅	—	5	1	no	maximal
6	C₃×C₅×D₅×C₄₁	D₃	1	4	yes	—
7	#16861527	—	9	1	no	Hall
8	C₃²×D₅×C₄₁	D₅	1	1	yes	characteristic
9	C₂×C₃²×C₅×C₄₁	—	5	1	no	—
10	C₃×C₅²×C₄₁	Dic₃	1	4	yes	complemented
11	C₃⋊#11717963	—	15	4	no	—
12	Dic₆₁₅	—	15	4	no	—
13	C₅×D₅×C₄₁	C₃⋊D₃	1	1	yes	characteristic
14	C₃²×C₅×C₄₁	—	4	1	no	—
15	C₃²×C₅×C₄₁	Dic₅	1	1	yes	characteristic, complemented
16	C₃²×C₅×C₄₁	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
17	C₃⋊Dic₁₂₃	—	25	1	no	Hall
18	C₂×C₃×C₅×C₄₁	—	5	4	no	—
19	C₃×D₅×C₄₁	D₁₅	1	4	yes	—
20	C₅²×C₄₁	C₃⋊Dic₃	1	1	yes	characteristic, complemented, Hall
21	C₃⋊(C₃⋊#1042)	—	41	1	no	maximal, Hall
22	#11717963	—	45	1	no	—
23	Dic₂₀₅	—	45	1	no	—
24	C₂×C₃²×C₄₁	—	5	1	no	—
25	C₃×C₅×C₄₁	—	4	4	no	—
26	C₃×C₅×C₄₁	Dic₁₅	1	4	yes	complemented
27	C₃×C₅×C₄₁	ΓL(1,𝔽₁₆)	1	4	yes	complemented
28	Dic₁₂₃	—	75	4	no	—
29	C₃²×C₅×D₅	D₄₁	1	1	yes	characteristic
30	C₂×C₅×C₄₁	—	5	1	no	—
31	D₅×C₄₁	C₃⋊D₁₅	1	1	yes	characteristic
32	C₃²×C₄₁	#1042	1	1	yes	characteristic, complemented, Hall
33	C₃⋊#1042	—	123	4	no	—
34	C₂×C₃×C₄₁	—	5	4	no	—
35	C₃²×C₅²	Dic₄₁	1	1	yes	characteristic, complemented, Hall
36	C₅×C₄₁	—	4	1	no	—
37	C₅×C₄₁	C₃⋊Dic₁₅	1	1	yes	characteristic, complemented
38	C₅×C₄₁	C₃⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
39	C₃⋊ΓL(1,𝔽₁₆)	—	205	1	no	—
40	C₃⋊Dic₁₅	—	205	1	no	—
41	Dic₄₁	—	225	1	no	Hall
42	C₃×C₅×D₅	D₁₂₃	1	4	yes	—
43	C₃×C₄₁	C₃⋊#1042	1	4	yes	complemented
44	#1042	—	369	1	no	Hall
45	C₂×C₃²×C₅	—	5	1	no	—
46	C₃²×D₅	D₂₀₅	1	1	yes	characteristic
47	C₂×C₄₁	—	5	1	no	—
48	C₃×C₅²	Dic₁₂₃	1	4	yes	complemented
49	ΓL(1,𝔽₁₆)	—	615	4	no	—
50	Dic₁₅	—	615	4	no	—
51	C₅×D₅	C₃⋊D₁₂₃	1	1	yes	characteristic
52	C₃²×C₅	—	4	1	no	—
53	C₃²×C₅	Dic₂₀₅	1	1	yes	characteristic, complemented
54	C₃²×C₅	#11717963	1	1	yes	characteristic, complemented
55	C₄₁	C₃⋊(C₃⋊#1042)	1	1	yes	characteristic, complemented, Sylow
56	C₃⋊Dic₃	—	1025	1	no	Hall
57	C₂×C₃×C₅	—	5	4	no	—
58	C₃×D₅	D₆₁₅	1	4	yes	—
59	C₅²	C₃⋊Dic₁₂₃	1	1	yes	characteristic, complemented, Sylow
60	AGL(1,𝔽₅)	—	1845	1	no	—
61	Dic₅	—	1845	1	no	—
62	C₂×C₃²	—	5	1	no	—
63	C₃×C₅	—	4	4	no	—
64	C₃×C₅	Dic₆₁₅	1	4	yes	complemented
65	C₃×C₅	C₃⋊#11717963	1	4	yes	complemented
66	Dic₃	—	3075	4	no	—
67	C₂×C₅	—	5	1	no	—
68	D₅	C₃⋊D₆₁₅	1	1	yes	characteristic
69	C₃²	#16861527	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₃	—	5	4	no	—
71	C₅	—	4	1	no	—
72	C₅	C₃⋊Dic₆₁₅	1	1	yes	characteristic, complemented
73	C₅	C₃⋊(C₃⋊#11717963)	1	1	yes	characteristic, complemented
74	C₄	—	9225	1	no	Sylow
75	C₃	C₃⋊#16861527	1	4	yes	complemented
76	C₂	—	5	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃²×C₅²×C₄₁	C₄
prime radical, cosocle	C₃²×C₅×D₅×C₄₁	C₂
LCS	G ▷ C₃²×C₅²×C₄₁
UCS	C₁

Complements

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

N	H ≅ G/N	type
3. C₃²×C₅²×C₄₁	C₄	semidirect
10. C₃×C₅²×C₄₁	Dic₃	semidirect
15. C₃²×C₅×C₄₁	Dic₅	semidirect
16. C₃²×C₅×C₄₁	AGL(1,𝔽₅)	semidirect
20. C₅²×C₄₁	C₃⋊Dic₃	semidirect
26. C₃×C₅×C₄₁	Dic₁₅	semidirect
27. C₃×C₅×C₄₁	ΓL(1,𝔽₁₆)	semidirect
32. C₃²×C₄₁	#1042	semidirect
35. C₃²×C₅²	Dic₄₁	semidirect
37. C₅×C₄₁	C₃⋊Dic₁₅	semidirect
38. C₅×C₄₁	C₃⋊ΓL(1,𝔽₁₆)	semidirect
43. C₃×C₄₁	C₃⋊#1042	semidirect
48. C₃×C₅²	Dic₁₂₃	semidirect
53. C₃²×C₅	Dic₂₀₅	semidirect
54. C₃²×C₅	#11717963	semidirect
55. C₄₁	C₃⋊(C₃⋊#1042)	semidirect
59. C₅²	C₃⋊Dic₁₂₃	semidirect
64. C₃×C₅	Dic₆₁₅	semidirect
65. C₃×C₅	C₃⋊#11717963	semidirect
69. C₃²	#16861527	semidirect
72. C₅	C₃⋊Dic₆₁₅	semidirect
73. C₅	C₃⋊(C₃⋊#11717963)	semidirect
75. C₃	C₃⋊#16861527	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₅×D₅×C₄₁	2
C₃⋊#16861527	3
C₃⋊(C₃⋊#11717963)	5
C₃⋊Dic₆₁₅	5
C₃⋊(C₃⋊#1042)	41

Subgroups — C3⋊(C3⋊#16861527) #49147180

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃⋊(C₃⋊#16861527) #49147180