Subgroups — C₅×#20269083 #45686166

Order 33300.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₅⋊(#66526⋊C₂)	C₂	1	1	yes	maximal, characteristic
2	C₅×#10625192⋊C₃	C₃	1	1	yes	maximal, characteristic
3	C₅²×#66526	C₄	1	1	yes	characteristic, complemented, Hall
4	#20269083	C₅	1	1	yes	maximal, characteristic, direct product factor
5	C₅×AGL(1,𝔽₃₇)	—	5	1	no	maximal
6	C₅×C₅⋊(D₃₇⋊C₃)	C₂×C₃	1	1	yes	characteristic
7	C₅×#10625192	C₉	1	1	yes	characteristic, complemented, Hall
8	C₅⋊(#66526⋊C₂)	C₂×C₅	1	1	yes	characteristic
9	C₅×#66526⋊C₂	—	5	1	no	—
10	C₅²×C₃₇⋊C₃	C₃×C₄	1	1	yes	characteristic
11	#10625192⋊C₃	C₃×C₅	1	1	yes	characteristic
12	C₅×#3866⋊C₃	—	5	1	no	—
13	C₅×D₁₈₅	C₂×C₉	1	1	yes	characteristic
14	C₅×#66526	—	4	1	no	—
15	C₅×#66526	C₄×C₅	1	1	yes	characteristic, complemented
16	C₅×#66526	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
17	AGL(1,𝔽₃₇)	—	5	1	no	Hall
18	C₅⋊(D₃₇⋊C₃)	C₂×C₃×C₅	1	1	yes	characteristic
19	C₅×D₃₇⋊C₃	—	5	1	no	—
20	C₅²×C₃₇	C₄×C₉	1	1	yes	characteristic, complemented, Hall
21	C₅×C₉×AGL(1,𝔽₅)	—	37	1	no	maximal, Hall
22	#10625192	C₅×C₉	1	1	yes	characteristic, complemented
23	C₅×#3866	—	5	1	no	—
24	#66526⋊C₂	—	5	1	no	—
25	C₅×C₃₇⋊C₃	—	4	1	no	—
26	C₅×C₃₇⋊C₃	C₃×C₄×C₅	1	1	yes	characteristic
27	C₅×C₃₇⋊C₃	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
28	C₅×C₉×D₅	—	37	1	no	—
29	#3866⋊C₃	—	5	1	no	—
30	D₁₈₅	C₂×C₅×C₉	1	1	yes	characteristic
31	C₅×D₃₇	—	5	1	no	—
32	#66526	C₅×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
33	C₃×C₅×AGL(1,𝔽₅)	—	37	1	no	—
34	C₅²×C₉	—	37	1	no	Hall
35	D₃₇⋊C₃	—	5	1	no	—
36	C₅×C₃₇	—	4	1	no	—
37	C₅×C₃₇	C₄×C₅×C₉	1	1	yes	characteristic, complemented
38	C₅×C₃₇	C₉×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
39	C₉×AGL(1,𝔽₅)	—	37	1	no	—
40	C₄×C₅×C₉	—	185	1	no	—
41	C₃×C₅×D₅	—	37	1	no	—
42	#3866	—	5	1	no	Hall
43	C₃₇⋊C₃	C₃×C₅×AGL(1,𝔽₅)	1	1	yes	characteristic
44	C₅×AGL(1,𝔽₅)	—	37	1	no	Hall
45	C₂×C₅×C₉	—	185	1	no	—
46	C₉×D₅	—	37	1	no	—
47	C₃×C₅²	—	37	1	no	—
48	D₃₇	—	5	1	no	—
49	C₃×AGL(1,𝔽₅)	—	37	1	no	—
50	C₃×C₄×C₅	—	185	1	no	—
51	C₅×D₅	—	37	1	no	—
52	C₅×C₉	—	37	1	no	—
53	C₅×C₉	—	37	1	no	—
54	C₅×C₉	—	148	1	no	—
55	C₃₇	C₅×C₉×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
56	C₄×C₉	—	185	1	no	Hall
57	C₂×C₃×C₅	—	185	1	no	—
58	C₃×D₅	—	37	1	no	—
59	C₅²	AGL(1,𝔽₃₇)	1	1	yes	characteristic, complemented, Sylow
60	AGL(1,𝔽₅)	—	37	1	no	—
61	C₄×C₅	—	185	1	no	—
62	C₂×C₉	—	185	1	no	—
63	C₃×C₅	—	37	1	no	—
64	C₃×C₅	—	37	1	no	—
65	C₃×C₅	—	148	1	no	—
66	C₃×C₄	—	185	1	no	—
67	C₂×C₅	—	185	1	no	—
68	D₅	—	37	1	no	—
69	C₉	—	37	1	no	Sylow
70	C₂×C₃	—	185	1	no	—
71	C₅	—	4	1	no	—
72	C₅	C₅×AGL(1,𝔽₃₇)	1	1	yes	characteristic, complemented
73	C₅	#20269083	1	1	yes	characteristic, direct product factor
74	C₄	—	185	1	no	Sylow
75	C₃	—	37	1	no	—
76	C₂	—	185	1	no	—

	subgroup	G/H
center, hypercenter	C₅	#20269083
commutator, hypocenter	C₅×C₃₇	C₄×C₅×C₉
Frattini	C₁	G
Fitting, socle	C₅²×C₃₇	C₄×C₉
prime radical	C₅×C₅⋊(D₃₇⋊C₃)	C₂×C₃
cosocle	C₅⋊(D₃₇⋊C₃)	C₂×C₃×C₅
LCS	G ▷ C₅×C₃₇
UCS	C₁ ◁ C₅

Complements

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

N	H ≅ G/N	type
3. C₅²×#66526	C₄	semidirect
4. #20269083	C₅	direct (N × H)
7. C₅×#10625192	C₉	semidirect
15. C₅×#66526	C₄×C₅	semidirect
16. C₅×#66526	AGL(1,𝔽₅)	semidirect
20. C₅²×C₃₇	C₄×C₉	semidirect
22. #10625192	C₅×C₉	semidirect
32. #66526	C₅×AGL(1,𝔽₅)	semidirect
37. C₅×C₃₇	C₄×C₅×C₉	semidirect
38. C₅×C₃₇	C₉×AGL(1,𝔽₅)	semidirect
55. C₃₇	C₅×C₉×AGL(1,𝔽₅)	semidirect
59. C₅²	AGL(1,𝔽₃₇)	semidirect
72. C₅	C₅×AGL(1,𝔽₃₇)	semidirect
73. C₅	#20269083	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×C₅⋊(#66526⋊C₂)	2
C₅×#10625192⋊C₃	3
#20269083	5
C₅×AGL(1,𝔽₃₇)	5
C₅×C₉×AGL(1,𝔽₅)	37

Subgroups — C5×#20269083 #45686166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#20269083 #45686166