Group #49087926 — subgroups

Subgroups — C₅²×D₇×D₅₁ #49087926

Order 35700.

320 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅²×D₇×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅²×C₇×D₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅²×D₃₅₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅²×D₇×D₁₇	—	3	1	no	maximal, Hall
5	C₃×C₅²×C₇×C₁₇	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₅×D₇×D₅₁	C₅	1	6	yes	maximal, direct product factor
7	C₅²×D₁₁₉	—	3	1	no	—
8	C₅²×D₇×C₁₇	D₃	1	1	yes	characteristic, complemented
9	C₅²×C₇×D₁₇	—	3	1	no	—
10	C₂×C₅²×D₅₁	—	7	1	no	maximal, Hall
11	C₅×D₃₅₇	C₂×C₅	1	6	yes	complemented
12	C₅×C₇×D₅₁	C₂×C₅	1	6	yes	complemented
13	C₃×C₅×D₇×C₁₇	C₂×C₅	1	6	yes	complemented
14	C₅²×C₇×C₁₇	C₂×D₃	1	1	yes	characteristic, complemented, Hall
15	C₅²×D₅₁	—	7	1	no	—
16	C₅²×D₅₁	D₇	1	1	yes	characteristic, direct product factor
17	C₂×C₃×C₅²×C₁₇	—	7	1	no	—
18	C₅×D₇×D₁₇	—	3	6	no	—
19	C₅²×D₃×D₇	—	17	1	no	maximal, Hall
20	C₃×C₅×C₇×C₁₇	C₂²×C₅	1	6	yes	complemented
21	C₂×C₅²×D₁₇	—	21	1	no	Hall
22	D₇×D₅₁	C₅²	1	1	yes	characteristic, direct product factor, Hall
23	C₃×C₅²×C₁₇	C₂×D₇	1	1	yes	characteristic, complemented, Hall
24	C₅×D₁₁₉	—	3	6	no	—
25	C₅×D₇×C₁₇	C₅×D₃	1	6	yes	complemented
26	C₅×C₇×D₁₇	—	3	6	no	—
27	C₅²×D₂₁	—	17	1	no	—
28	C₅²×D₃×C₇	—	17	1	no	—
29	C₃×C₅²×D₇	D₁₇	1	1	yes	characteristic, complemented
30	C₂×C₅×D₅₁	—	7	6	no	—
31	C₂×C₅²×C₁₇	—	7	1	no	—
32	C₅²×D₁₇	—	3	1	no	—
33	C₅²×D₁₇	—	21	1	no	—
34	D₃₅₇	C₂×C₅²	1	1	yes	characteristic, complemented
35	C₇×D₅₁	C₂×C₅²	1	1	yes	characteristic, complemented
36	C₃×D₇×C₁₇	C₂×C₅²	1	1	yes	characteristic, complemented
37	C₂×C₅²×D₇	—	51	1	no	Hall
38	C₅×C₇×C₁₇	C₂×C₅×D₃	1	6	yes	complemented
39	C₃×C₅²×C₇	C₂×D₁₇	1	1	yes	characteristic, complemented, Hall
40	C₂×C₃×C₅×C₁₇	—	7	6	no	—
41	C₅×D₅₁	—	7	6	no	—
42	C₅×D₅₁	C₅×D₇	1	6	yes	direct product factor
43	D₇×D₁₇	—	3	1	no	Hall
44	C₅²×C₁₇	D₃×D₇	1	1	yes	characteristic, complemented, Hall
45	C₅×D₃×D₇	—	17	6	no	—
46	C₃×C₇×C₁₇	C₂²×C₅²	1	1	yes	characteristic, complemented, Hall
47	C₂×C₅²×C₇	—	51	1	no	—
48	C₅²×D₇	—	51	1	no	—
49	C₅²×D₇	D₅₁	1	1	yes	characteristic, direct product factor
50	C₂×C₅×D₁₇	—	21	6	no	—
51	C₂×C₅²×D₃	—	119	1	no	Hall
52	C₃×C₅×C₁₇	C₂×C₅×D₇	1	6	yes	complemented
53	D₁₁₉	—	3	1	no	—
54	C₇×D₁₇	—	3	1	no	—
55	D₇×C₁₇	C₅²×D₃	1	1	yes	characteristic, complemented
56	C₅×D₂₁	—	17	6	no	—
57	C₅×D₃×C₇	—	17	6	no	—
58	C₃×C₅×D₇	C₅×D₁₇	1	6	yes	complemented
59	C₂×D₅₁	—	7	1	no	Hall
60	C₅²×C₇	C₂×D₅₁	1	1	yes	characteristic, complemented, Hall
61	C₂×C₅×C₁₇	—	7	6	no	—
62	C₅×D₁₇	—	3	6	no	—
63	C₅×D₁₇	—	21	6	no	—
64	C₂×C₃×C₅²	—	7	1	no	—
65	C₅²×D₃	—	17	1	no	—
66	C₅²×D₃	—	119	1	no	—
67	C₂×C₅×D₇	—	51	6	no	—
68	C₇×C₁₇	C₂×C₅²×D₃	1	1	yes	characteristic, complemented, Hall
69	C₃×C₅×C₇	C₂×C₅×D₁₇	1	6	yes	complemented
70	C₂×C₃×C₁₇	—	7	1	no	—
71	D₅₁	—	7	1	no	—
72	D₅₁	C₅²×D₇	1	1	yes	characteristic, direct product factor
73	C₂²×C₅²	—	357	1	no	Hall
74	C₅×C₁₇	C₅×D₃×D₇	1	6	yes	complemented
75	D₃×D₇	—	17	1	no	Hall
76	C₃×C₅²	D₇×D₁₇	1	1	yes	characteristic, complemented, Hall
77	C₂×C₅×C₇	—	51	6	no	—
78	C₅×D₇	—	51	6	no	—
79	C₅×D₇	C₅×D₅₁	1	6	yes	direct product factor
80	C₂×D₁₇	—	21	1	no	Hall
81	C₂×C₅×D₃	—	119	6	no	—
82	C₃×C₁₇	C₂×C₅²×D₇	1	1	yes	characteristic, complemented, Hall
83	C₂×C₅²	—	7	1	no	—
84	C₂×C₅²	—	51	1	no	—
85	C₂×C₅²	—	357	1	no	—
86	D₂₁	—	17	1	no	—
87	C₃×D₇	C₅²×D₁₇	1	1	yes	characteristic, complemented
88	D₃×C₇	—	17	1	no	—
89	C₅×C₇	C₂×C₅×D₅₁	1	6	yes	complemented
90	C₂×C₁₇	—	7	1	no	—
91	D₁₇	—	3	1	no	—
92	D₁₇	—	21	1	no	—
93	C₂×C₃×C₅	—	7	6	no	—
94	C₅×D₃	—	17	6	no	—
95	C₅×D₃	—	119	6	no	—
96	C₂×D₇	—	51	1	no	Hall
97	C₅²	D₇×D₅₁	1	1	yes	characteristic, direct product factor, Sylow
98	C₃×C₇	C₂×C₅²×D₁₇	1	1	yes	characteristic, complemented, Hall
99	C₂²×C₅	—	357	6	no	—
100	C₁₇	C₅²×D₃×D₇	1	1	yes	characteristic, complemented, Sylow
101	C₃×C₅	C₅×D₇×D₁₇	1	6	yes	complemented
102	C₂×C₇	—	51	1	no	—
103	D₇	—	51	1	no	—
104	D₇	C₅²×D₅₁	1	1	yes	characteristic, direct product factor
105	C₂×D₃	—	119	1	no	Hall
106	C₂×C₅	—	7	6	no	—
107	C₂×C₅	—	51	6	no	—
108	C₂×C₅	—	357	6	no	—
109	C₇	C₂×C₅²×D₅₁	1	1	yes	characteristic, complemented, Sylow
110	C₂×C₃	—	7	1	no	—
111	D₃	—	17	1	no	—
112	D₃	—	119	1	no	—
113	C₅	C₅×D₇×D₅₁	1	6	yes	direct product factor
114	C₂²	—	357	1	no	Sylow
115	C₃	C₅²×D₇×D₁₇	1	1	yes	characteristic, complemented, Sylow
116	C₂	—	7	1	no	—
117	C₂	—	51	1	no	—
118	C₂	—	357	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₅²×D₇×C₁₇

C₂

yes

maximal, characteristic, complemented

C₅²×C₇×D₅₁

C₂

yes

maximal, characteristic, complemented

C₅²×D₃₅₇

C₂

yes

maximal, characteristic, complemented

C₅²×D₇×D₁₇

—

maximal, Hall

C₃×C₅²×C₇×C₁₇

C₂²

yes

characteristic, complemented, Hall

C₅×D₇×D₅₁

C₅

yes

maximal, direct product factor

C₅²×D₁₁₉

—

C₅²×D₇×C₁₇

D₃

yes

characteristic, complemented

C₅²×C₇×D₁₇

—

C₂×C₅²×D₅₁

—

maximal, Hall

C₅×D₃₅₇

C₂×C₅

yes

complemented

C₅×C₇×D₅₁

C₂×C₅

yes

complemented

C₃×C₅×D₇×C₁₇

C₂×C₅

yes

complemented

C₅²×C₇×C₁₇

C₂×D₃

yes

characteristic, complemented, Hall

C₅²×D₅₁

—

C₅²×D₅₁

D₇

yes

characteristic, direct product factor

C₂×C₃×C₅²×C₁₇

—

C₅×D₇×D₁₇

—

C₅²×D₃×D₇

—

maximal, Hall

C₃×C₅×C₇×C₁₇

C₂²×C₅

yes

complemented

C₂×C₅²×D₁₇

—

Hall

D₇×D₅₁

C₅²

yes

characteristic, direct product factor, Hall

C₃×C₅²×C₁₇

C₂×D₇

yes

characteristic, complemented, Hall

C₅×D₁₁₉

—

C₅×D₇×C₁₇

C₅×D₃

yes

complemented

C₅×C₇×D₁₇

—

C₅²×D₂₁

—

C₅²×D₃×C₇

—

C₃×C₅²×D₇

D₁₇

yes

characteristic, complemented

C₂×C₅×D₅₁

—

C₂×C₅²×C₁₇

—

C₅²×D₁₇

—

C₅²×D₁₇

—

D₃₅₇

C₂×C₅²

yes

characteristic, complemented

C₇×D₅₁

C₂×C₅²

yes

characteristic, complemented

C₃×D₇×C₁₇

C₂×C₅²

yes

characteristic, complemented

C₂×C₅²×D₇

—

Hall

C₅×C₇×C₁₇

C₂×C₅×D₃

yes

complemented

C₃×C₅²×C₇

C₂×D₁₇

yes

characteristic, complemented, Hall

C₂×C₃×C₅×C₁₇

—

C₅×D₅₁

—

C₅×D₅₁

C₅×D₇

yes

direct product factor

D₇×D₁₇

—

Hall

C₅²×C₁₇

D₃×D₇

yes

characteristic, complemented, Hall

C₅×D₃×D₇

—

C₃×C₇×C₁₇

C₂²×C₅²

yes

characteristic, complemented, Hall

C₂×C₅²×C₇

—

C₅²×D₇

—

C₅²×D₇

D₅₁

yes

characteristic, direct product factor

C₂×C₅×D₁₇

—

C₂×C₅²×D₃

—

119

Hall

C₃×C₅×C₁₇

C₂×C₅×D₇

yes

complemented

D₁₁₉

—

C₇×D₁₇

—

D₇×C₁₇

C₅²×D₃

yes

characteristic, complemented

C₅×D₂₁

—

C₅×D₃×C₇

—

C₃×C₅×D₇

C₅×D₁₇

yes

complemented

C₂×D₅₁

—

Hall

C₅²×C₇

C₂×D₅₁

yes

characteristic, complemented, Hall

C₂×C₅×C₁₇

—

C₅×D₁₇

—

C₅×D₁₇

—

C₂×C₃×C₅²

—

C₅²×D₃

—

C₅²×D₃

—

119

—

C₂×C₅×D₇

—

C₇×C₁₇

C₂×C₅²×D₃

yes

characteristic, complemented, Hall

C₃×C₅×C₇

C₂×C₅×D₁₇

yes

complemented

C₂×C₃×C₁₇

—

D₅₁

—

D₅₁

C₅²×D₇

yes

characteristic, direct product factor

C₂²×C₅²

—

357

Hall

C₅×C₁₇

C₅×D₃×D₇

yes

complemented

D₃×D₇

—

Hall

C₃×C₅²

D₇×D₁₇

yes

characteristic, complemented, Hall

C₂×C₅×C₇

—

C₅×D₇

—

C₅×D₇

C₅×D₅₁

yes

direct product factor

C₂×D₁₇

—

Hall

C₂×C₅×D₃

—

119

—

C₃×C₁₇

C₂×C₅²×D₇

yes

characteristic, complemented, Hall

C₂×C₅²

—

C₂×C₅²

—

C₂×C₅²

—

357

—

D₂₁

—

C₃×D₇

C₅²×D₁₇

yes

characteristic, complemented

D₃×C₇

—

C₅×C₇

C₂×C₅×D₅₁

yes

complemented

C₂×C₁₇

—

D₁₇

—

D₁₇

—

C₂×C₃×C₅

—

C₅×D₃

—

C₅×D₃

—

119

—

C₂×D₇

—

Hall

C₅²

D₇×D₅₁

yes

characteristic, direct product factor, Sylow

C₃×C₇

C₂×C₅²×D₁₇

yes

characteristic, complemented, Hall

C₂²×C₅

—

357

—

100

C₁₇

C₅²×D₃×D₇

yes

characteristic, complemented, Sylow

101

C₃×C₅

C₅×D₇×D₁₇

yes

complemented

102

C₂×C₇

—

103

D₇

—

104

D₇

C₅²×D₅₁

yes

characteristic, direct product factor

105

C₂×D₃

—

119

Hall

106

C₂×C₅

—

107

C₂×C₅

—

108

C₂×C₅

—

357

—

109

C₇

C₂×C₅²×D₅₁

yes

characteristic, complemented, Sylow

110

C₂×C₃

—

111

D₃

—

112

D₃

—

119

—

113

C₅

C₅×D₇×D₅₁

yes

direct product factor

114

C₂²

—

357

Sylow

115

C₃

C₅²×D₇×D₁₇

yes

characteristic, complemented, Sylow

116

C₂

—

117

C₂

—

118

C₂

—

357

—

	subgroup	G/H
center, hypercenter	C₅²	D₇×D₅₁
commutator, cosocle, hypocenter	C₃×C₇×C₁₇	C₂²×C₅²
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₇×C₁₇	C₂²
LCS	G ▷ C₃×C₇×C₁₇
UCS	C₁ ◁ C₅²

subgroup

G/H

center, hypercenter

C₅²

D₇×D₅₁

commutator, cosocle, hypocenter

C₃×C₇×C₁₇

C₂²×C₅²

Frattini

C₁

Fitting, socle

C₃×C₅²×C₇×C₁₇

C₂²

LCS

G ▷ C₃×C₇×C₁₇

UCS

C₁ ◁ C₅²

N	H ≅ G/N	type
1. C₃×C₅²×D₇×C₁₇	C₂	semidirect
2. C₅²×C₇×D₅₁	C₂	semidirect
3. C₅²×D₃₅₇	C₂	semidirect
5. C₃×C₅²×C₇×C₁₇	C₂²	semidirect
6. C₅×D₇×D₅₁	C₅	direct (N × H)
8. C₅²×D₇×C₁₇	D₃	semidirect
11. C₅×D₃₅₇	C₂×C₅	semidirect
12. C₅×C₇×D₅₁	C₂×C₅	semidirect
13. C₃×C₅×D₇×C₁₇	C₂×C₅	semidirect
14. C₅²×C₇×C₁₇	C₂×D₃	semidirect
16. C₅²×D₅₁	D₇	direct (N × H)
20. C₃×C₅×C₇×C₁₇	C₂²×C₅	semidirect
22. D₇×D₅₁	C₅²	direct (N × H)
23. C₃×C₅²×C₁₇	C₂×D₇	semidirect
25. C₅×D₇×C₁₇	C₅×D₃	semidirect
29. C₃×C₅²×D₇	D₁₇	semidirect
34. D₃₅₇	C₂×C₅²	semidirect
35. C₇×D₅₁	C₂×C₅²	semidirect
36. C₃×D₇×C₁₇	C₂×C₅²	semidirect
38. C₅×C₇×C₁₇	C₂×C₅×D₃	semidirect
39. C₃×C₅²×C₇	C₂×D₁₇	semidirect
42. C₅×D₅₁	C₅×D₇	direct (N × H)
44. C₅²×C₁₇	D₃×D₇	semidirect
46. C₃×C₇×C₁₇	C₂²×C₅²	semidirect
49. C₅²×D₇	D₅₁	direct (N × H)
52. C₃×C₅×C₁₇	C₂×C₅×D₇	semidirect
55. D₇×C₁₇	C₅²×D₃	semidirect
58. C₃×C₅×D₇	C₅×D₁₇	semidirect
60. C₅²×C₇	C₂×D₅₁	semidirect
68. C₇×C₁₇	C₂×C₅²×D₃	semidirect
69. C₃×C₅×C₇	C₂×C₅×D₁₇	semidirect
72. D₅₁	C₅²×D₇	direct (N × H)
74. C₅×C₁₇	C₅×D₃×D₇	semidirect
76. C₃×C₅²	D₇×D₁₇	semidirect
79. C₅×D₇	C₅×D₅₁	direct (N × H)
82. C₃×C₁₇	C₂×C₅²×D₇	semidirect
87. C₃×D₇	C₅²×D₁₇	semidirect
89. C₅×C₇	C₂×C₅×D₅₁	semidirect
97. C₅²	D₇×D₅₁	direct (N × H)
98. C₃×C₇	C₂×C₅²×D₁₇	semidirect
100. C₁₇	C₅²×D₃×D₇	semidirect
101. C₃×C₅	C₅×D₇×D₁₇	semidirect
104. D₇	C₅²×D₅₁	direct (N × H)
109. C₇	C₂×C₅²×D₅₁	semidirect
113. C₅	C₅×D₇×D₅₁	direct (N × H)
115. C₃	C₅²×D₇×D₁₇	semidirect

H ≅ G/N

type

1. C₃×C₅²×D₇×C₁₇

C₂

semidirect

2. C₅²×C₇×D₅₁

C₂

semidirect

3. C₅²×D₃₅₇

C₂

semidirect

5. C₃×C₅²×C₇×C₁₇

C₂²

semidirect

6. C₅×D₇×D₅₁

C₅

direct (N × H)

8. C₅²×D₇×C₁₇

D₃

semidirect

11. C₅×D₃₅₇

C₂×C₅

semidirect

12. C₅×C₇×D₅₁

C₂×C₅

semidirect

13. C₃×C₅×D₇×C₁₇

C₂×C₅

semidirect

14. C₅²×C₇×C₁₇

C₂×D₃

semidirect

16. C₅²×D₅₁

D₇

direct (N × H)

20. C₃×C₅×C₇×C₁₇

C₂²×C₅

semidirect

22. D₇×D₅₁

C₅²

direct (N × H)

23. C₃×C₅²×C₁₇

C₂×D₇

semidirect

25. C₅×D₇×C₁₇

C₅×D₃

semidirect

29. C₃×C₅²×D₇

D₁₇

semidirect

34. D₃₅₇

C₂×C₅²

semidirect

35. C₇×D₅₁

C₂×C₅²

semidirect

36. C₃×D₇×C₁₇

C₂×C₅²

semidirect

38. C₅×C₇×C₁₇

C₂×C₅×D₃

semidirect

39. C₃×C₅²×C₇

C₂×D₁₇

semidirect

42. C₅×D₅₁

C₅×D₇

direct (N × H)

44. C₅²×C₁₇

D₃×D₇

semidirect

46. C₃×C₇×C₁₇

C₂²×C₅²

semidirect

49. C₅²×D₇

D₅₁

direct (N × H)

52. C₃×C₅×C₁₇

C₂×C₅×D₇

semidirect

55. D₇×C₁₇

C₅²×D₃

semidirect

58. C₃×C₅×D₇

C₅×D₁₇

semidirect

60. C₅²×C₇

C₂×D₅₁

semidirect

68. C₇×C₁₇

C₂×C₅²×D₃

semidirect

69. C₃×C₅×C₇

C₂×C₅×D₁₇

semidirect

72. D₅₁

C₅²×D₇

direct (N × H)

74. C₅×C₁₇

C₅×D₃×D₇

semidirect

76. C₃×C₅²

D₇×D₁₇

semidirect

79. C₅×D₇

C₅×D₅₁

direct (N × H)

82. C₃×C₁₇

C₂×C₅²×D₇

semidirect

87. C₃×D₇

C₅²×D₁₇

semidirect

89. C₅×C₇

C₂×C₅×D₅₁

semidirect

97. C₅²

D₇×D₅₁

direct (N × H)

98. C₃×C₇

C₂×C₅²×D₁₇

semidirect

100. C₁₇

C₅²×D₃×D₇

semidirect

101. C₃×C₅

C₅×D₇×D₁₇

semidirect

104. D₇

C₅²×D₅₁

direct (N × H)

109. C₇

C₂×C₅²×D₅₁

semidirect

113. C₅

C₅×D₇×D₅₁

direct (N × H)

115. C₃

C₅²×D₇×D₁₇

semidirect

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

subgroup	index
C₃×C₅²×D₇×C₁₇	2
C₅²×C₇×D₅₁	2
C₅²×D₃₅₇	2
C₅²×D₇×D₁₇	3
C₅×D₇×D₅₁	5
C₂×C₅²×D₅₁	7
C₅²×D₃×D₇	17

subgroup

index

C₃×C₅²×D₇×C₁₇

C₅²×C₇×D₅₁

C₅²×D₃₅₇

C₅²×D₇×D₁₇

C₅×D₇×D₅₁

C₂×C₅²×D₅₁

C₅²×D₃×D₇

Subgroups — C₅²×D₇×D₅₁ #49087926

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C52×D7×D51 #49087926

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×D₇×D₅₁ #49087926