Group #44472607 — subgroups

Subgroups — D₇×D₁₀₇₁ #44472607

Order 29988.

156 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 40 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	C₇⋊D₁₀₇₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×D₁₀₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×C₉×D₇×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	D₇×D₃₅₇	—	3	1	no	maximal
5	C₇²×C₉×C₁₇	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₃⋊(C₇⋊D₁₁₉)	—	3	1	no	—
7	C₇×D₃₅₇	—	3	1	no	—
8	C₃×C₇×D₇×C₁₇	D₃	1	1	yes	characteristic
9	C₂×D₁₀₇₁	—	7	1	no	maximal
10	D₇×D₁₅₃	—	7	1	no	maximal
11	D₇×D₁₁₉	—	9	1	no	Hall
12	C₃×C₇²×C₁₇	C₂×D₃	1	1	yes	characteristic
13	D₁₀₇₁	—	7	1	no	—
14	D₁₀₇₁	—	7	1	no	—
15	D₁₀₇₁	—	14	3	no	—
16	D₁₀₇₁	D₇	1	1	yes	characteristic, direct product factor
17	C₇×D₁₅₃	—	7	1	no	—
18	C₉×D₇×C₁₇	D₇	1	1	yes	characteristic, complemented
19	C₂×C₇×C₉×C₁₇	—	7	1	no	—
20	D₇×D₆₃	—	17	1	no	maximal, Hall
21	C₇⋊D₁₁₉	—	9	1	no	—
22	C₇×D₁₁₉	—	9	1	no	—
23	C₇×D₇×C₁₇	D₉	1	1	yes	characteristic, complemented
24	C₂×D₃₅₇	—	21	1	no	—
25	D₇×D₅₁	—	21	1	no	—
26	C₇×C₉×C₁₇	—	2	3	no	—
27	C₇×C₉×C₁₇	C₂×D₇	1	1	yes	characteristic, complemented
28	C₇×C₉×C₁₇	C₂×D₇	1	1	yes	characteristic, complemented
29	C₇⋊D₆₃	—	17	1	no	—
30	C₇×D₆₃	—	17	1	no	—
31	C₇×C₉×D₇	D₁₇	1	1	yes	characteristic, complemented
32	C₇²×C₁₇	C₂×D₉	1	1	yes	characteristic, complemented, Hall
33	C₂×C₃×C₇×C₁₇	—	7	1	no	—
34	D₃₅₇	—	3	1	no	—
35	D₃₅₇	—	21	1	no	—
36	D₃₅₇	—	21	1	no	—
37	D₃₅₇	—	42	3	no	—
38	C₇×D₅₁	—	21	1	no	—
39	C₃×D₇×C₁₇	D₂₁	1	1	yes	characteristic
40	C₂×D₁₅₃	—	49	1	no	Hall
41	D₇×D₂₁	—	51	1	no	—
42	C₂×D₁₁₉	—	63	1	no	—
43	D₇×D₁₇	—	63	1	no	—
44	C₇²×C₉	C₂×D₁₇	1	1	yes	characteristic, complemented, Hall
45	C₃×C₇×C₁₇	—	2	3	no	—
46	C₃×C₇×C₁₇	D₃×D₇	1	1	yes	characteristic
47	C₃×C₇×C₁₇	C₂×D₂₁	1	1	yes	characteristic
48	C₂×C₉×C₁₇	—	7	1	no	—
49	D₁₅₃	—	7	1	no	—
50	D₁₅₃	—	49	1	no	—
51	C₃⋊(C₇⋊D₇)	—	51	1	no	—
52	C₇×D₂₁	—	51	1	no	—
53	C₃×C₇×D₇	D₅₁	1	1	yes	characteristic
54	C₂×D₆₃	—	119	1	no	—
55	D₇×D₉	—	119	1	no	—
56	C₂×C₇×C₁₇	—	7	1	no	—
57	D₁₁₉	—	9	1	no	—
58	D₁₁₉	—	63	1	no	—
59	D₁₁₉	—	63	1	no	—
60	D₁₁₉	—	126	3	no	—
61	C₇×D₁₇	—	63	1	no	—
62	D₇×C₁₇	D₆₃	1	1	yes	characteristic, complemented
63	C₂×D₅₁	—	147	1	no	—
64	D₇²	—	153	1	no	Hall
65	C₉×C₁₇	D₇²	1	1	yes	characteristic, complemented, Hall
66	C₃×C₇²	C₂×D₅₁	1	1	yes	characteristic
67	C₂×C₇×C₉	—	7	1	no	—
68	D₆₃	—	17	1	no	—
69	D₆₃	—	119	1	no	—
70	D₆₃	—	119	1	no	—
71	D₆₃	—	238	3	no	—
72	C₉×D₇	D₁₁₉	1	1	yes	characteristic, complemented
73	C₇×D₉	—	119	1	no	—
74	C₇×C₁₇	—	2	3	no	—
75	C₇×C₁₇	D₇×D₉	1	1	yes	characteristic, complemented
76	C₇×C₁₇	C₂×D₆₃	1	1	yes	characteristic, complemented
77	C₂×C₃×C₁₇	—	7	1	no	—
78	D₅₁	—	21	1	no	—
79	D₅₁	—	147	1	no	—
80	C₇⋊D₇	—	153	1	no	—
81	C₇×D₇	—	153	1	no	—
82	C₇×D₇	D₁₅₃	1	1	yes	characteristic, complemented
83	C₂×D₂₁	—	357	1	no	—
84	D₃×D₇	—	357	1	no	—
85	C₂×D₁₇	—	441	1	no	Hall
86	C₇×C₉	—	2	3	no	—
87	C₇×C₉	D₇×D₁₇	1	1	yes	characteristic, complemented
88	C₇×C₉	C₂×D₁₁₉	1	1	yes	characteristic, complemented
89	C₃×C₁₇	D₇×D₂₁	1	1	yes	characteristic
90	C₇²	C₂×D₁₅₃	1	1	yes	characteristic, complemented, Sylow
91	C₂×C₃×C₇	—	7	1	no	—
92	D₂₁	—	51	1	no	—
93	D₂₁	—	357	1	no	—
94	D₂₁	—	357	1	no	—
95	D₂₁	—	714	3	no	—
96	C₃×D₇	D₃₅₇	1	1	yes	characteristic
97	D₃×C₇	—	357	1	no	—
98	C₂×D₉	—	833	1	no	Hall
99	C₂×C₁₇	—	7	1	no	—
100	D₁₇	—	63	1	no	—
101	D₁₇	—	441	1	no	—
102	C₂×D₇	—	1071	1	no	—
103	C₂×D₇	—	1071	1	no	—
104	C₃×C₇	—	2	3	no	—
105	C₃×C₇	D₇×D₅₁	1	1	yes	characteristic
106	C₃×C₇	C₂×D₃₅₇	1	1	yes	characteristic
107	C₂×C₉	—	7	1	no	—
108	D₉	—	833	1	no	—
109	D₉	—	119	1	no	—
110	C₁₇	D₇×D₆₃	1	1	yes	characteristic, complemented, Sylow
111	C₂×C₇	—	7	1	no	—
112	C₂×C₇	—	1071	1	no	—
113	D₇	—	1071	1	no	—
114	D₇	—	1071	1	no	—
115	D₇	—	2142	3	no	—
116	D₇	—	153	1	no	—
117	D₇	D₁₀₇₁	1	1	yes	characteristic, direct product factor
118	C₂×D₃	—	2499	1	no	—
119	C₉	D₇×D₁₁₉	1	1	yes	characteristic, complemented, Sylow
120	C₇	—	2	3	no	—
121	C₇	D₇×D₁₅₃	1	1	yes	characteristic, complemented
122	C₇	C₂×D₁₀₇₁	1	1	yes	characteristic, complemented
123	C₂×C₃	—	7	1	no	—
124	D₃	—	357	1	no	—
125	D₃	—	2499	1	no	—
126	C₂²	—	7497	1	no	Sylow
127	C₃	D₇×D₃₅₇	1	1	yes	characteristic
128	C₂	—	7	1	no	—
129	C₂	—	1071	1	no	—
130	C₂	—	7497	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇⋊D₁₀₇₁

C₂

yes

maximal, characteristic, complemented

C₇×D₁₀₇₁

C₂

yes

maximal, characteristic, complemented

C₇×C₉×D₇×C₁₇

C₂

yes

maximal, characteristic, complemented

D₇×D₃₅₇

—

maximal

C₇²×C₉×C₁₇

C₂²

yes

characteristic, complemented, Hall

C₃⋊(C₇⋊D₁₁₉)

—

C₇×D₃₅₇

—

C₃×C₇×D₇×C₁₇

D₃

yes

characteristic

C₂×D₁₀₇₁

—

maximal

D₇×D₁₅₃

—

maximal

D₇×D₁₁₉

—

Hall

C₃×C₇²×C₁₇

C₂×D₃

yes

characteristic

D₁₀₇₁

—

D₁₀₇₁

—

D₁₀₇₁

—

D₁₀₇₁

D₇

yes

characteristic, direct product factor

C₇×D₁₅₃

—

C₉×D₇×C₁₇

D₇

yes

characteristic, complemented

C₂×C₇×C₉×C₁₇

—

D₇×D₆₃

—

maximal, Hall

C₇⋊D₁₁₉

—

C₇×D₁₁₉

—

C₇×D₇×C₁₇

D₉

yes

characteristic, complemented

C₂×D₃₅₇

—

D₇×D₅₁

—

C₇×C₉×C₁₇

—

C₇×C₉×C₁₇

C₂×D₇

yes

characteristic, complemented

C₇×C₉×C₁₇

C₂×D₇

yes

characteristic, complemented

C₇⋊D₆₃

—

C₇×D₆₃

—

C₇×C₉×D₇

D₁₇

yes

characteristic, complemented

C₇²×C₁₇

C₂×D₉

yes

characteristic, complemented, Hall

C₂×C₃×C₇×C₁₇

—

D₃₅₇

—

D₃₅₇

—

D₃₅₇

—

D₃₅₇

—

C₇×D₅₁

—

C₃×D₇×C₁₇

D₂₁

yes

characteristic

C₂×D₁₅₃

—

Hall

D₇×D₂₁

—

C₂×D₁₁₉

—

D₇×D₁₇

—

C₇²×C₉

C₂×D₁₇

yes

characteristic, complemented, Hall

C₃×C₇×C₁₇

—

C₃×C₇×C₁₇

D₃×D₇

yes

characteristic

C₃×C₇×C₁₇

C₂×D₂₁

yes

characteristic

C₂×C₉×C₁₇

—

D₁₅₃

—

D₁₅₃

—

C₃⋊(C₇⋊D₇)

—

C₇×D₂₁

—

C₃×C₇×D₇

D₅₁

yes

characteristic

C₂×D₆₃

—

119

—

D₇×D₉

—

119

—

C₂×C₇×C₁₇

—

D₁₁₉

—

D₁₁₉

—

D₁₁₉

—

D₁₁₉

—

126

—

C₇×D₁₇

—

D₇×C₁₇

D₆₃

yes

characteristic, complemented

C₂×D₅₁

—

147

—

D₇²

—

153

Hall

C₉×C₁₇

D₇²

yes

characteristic, complemented, Hall

C₃×C₇²

C₂×D₅₁

yes

characteristic

C₂×C₇×C₉

—

D₆₃

—

D₆₃

—

119

—

D₆₃

—

119

—

D₆₃

—

238

—

C₉×D₇

D₁₁₉

yes

characteristic, complemented

C₇×D₉

—

119

—

C₇×C₁₇

—

C₇×C₁₇

D₇×D₉

yes

characteristic, complemented

C₇×C₁₇

C₂×D₆₃

yes

characteristic, complemented

C₂×C₃×C₁₇

—

D₅₁

—

D₅₁

—

147

—

C₇⋊D₇

—

153

—

C₇×D₇

—

153

—

C₇×D₇

D₁₅₃

yes

characteristic, complemented

C₂×D₂₁

—

357

—

D₃×D₇

—

357

—

C₂×D₁₇

—

441

Hall

C₇×C₉

—

C₇×C₉

D₇×D₁₇

yes

characteristic, complemented

C₇×C₉

C₂×D₁₁₉

yes

characteristic, complemented

C₃×C₁₇

D₇×D₂₁

yes

characteristic

C₇²

C₂×D₁₅₃

yes

characteristic, complemented, Sylow

C₂×C₃×C₇

—

D₂₁

—

D₂₁

—

357

—

D₂₁

—

357

—

D₂₁

—

714

—

C₃×D₇

D₃₅₇

yes

characteristic

D₃×C₇

—

357

—

C₂×D₉

—

833

Hall

C₂×C₁₇

—

100

D₁₇

—

101

D₁₇

—

441

—

102

C₂×D₇

—

1071

—

103

C₂×D₇

—

1071

—

104

C₃×C₇

—

105

C₃×C₇

D₇×D₅₁

yes

characteristic

106

C₃×C₇

C₂×D₃₅₇

yes

characteristic

107

C₂×C₉

—

108

D₉

—

833

—

109

D₉

—

119

—

110

C₁₇

D₇×D₆₃

yes

characteristic, complemented, Sylow

111

C₂×C₇

—

112

C₂×C₇

—

1071

—

113

D₇

—

1071

—

114

D₇

—

1071

—

115

D₇

—

2142

—

116

D₇

—

153

—

117

D₇

D₁₀₇₁

yes

characteristic, direct product factor

118

C₂×D₃

—

2499

—

119

C₉

D₇×D₁₁₉

yes

characteristic, complemented, Sylow

120

C₇

—

121

C₇

D₇×D₁₅₃

yes

characteristic, complemented

122

C₇

C₂×D₁₀₇₁

yes

characteristic, complemented

123

C₂×C₃

—

124

D₃

—

357

—

125

D₃

—

2499

—

126

C₂²

—

7497

Sylow

127

C₃

D₇×D₃₅₇

yes

characteristic

128

C₂

—

129

C₂

—

1071

—

130

C₂

—

7497

—

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₇²×C₉×C₁₇	C₂²
Frattini	C₃	D₇×D₃₅₇
socle	C₃×C₇²×C₁₇	C₂×D₃
LCS	G ▷ C₇²×C₉×C₁₇
UCS	C₁

subgroup

G/H

center, hypercenter

C₁

commutator, Fitting, cosocle, hypocenter

C₇²×C₉×C₁₇

C₂²

Frattini

C₃

D₇×D₃₅₇

socle

C₃×C₇²×C₁₇

C₂×D₃

LCS

G ▷ C₇²×C₉×C₁₇

UCS

C₁

N	H ≅ G/N	type
1. C₇⋊D₁₀₇₁	C₂	semidirect
2. C₇×D₁₀₇₁	C₂	semidirect
3. C₇×C₉×D₇×C₁₇	C₂	semidirect
5. C₇²×C₉×C₁₇	C₂²	semidirect
16. D₁₀₇₁	D₇	direct (N × H)
18. C₉×D₇×C₁₇	D₇	semidirect
23. C₇×D₇×C₁₇	D₉	semidirect
27. C₇×C₉×C₁₇	C₂×D₇	semidirect
28. C₇×C₉×C₁₇	C₂×D₇	semidirect
31. C₇×C₉×D₇	D₁₇	semidirect
32. C₇²×C₁₇	C₂×D₉	semidirect
44. C₇²×C₉	C₂×D₁₇	semidirect
62. D₇×C₁₇	D₆₃	semidirect
65. C₉×C₁₇	D₇²	semidirect
72. C₉×D₇	D₁₁₉	semidirect
75. C₇×C₁₇	D₇×D₉	semidirect
76. C₇×C₁₇	C₂×D₆₃	semidirect
82. C₇×D₇	D₁₅₃	semidirect
87. C₇×C₉	D₇×D₁₇	semidirect
88. C₇×C₉	C₂×D₁₁₉	semidirect
90. C₇²	C₂×D₁₅₃	semidirect
110. C₁₇	D₇×D₆₃	semidirect
117. D₇	D₁₀₇₁	direct (N × H)
119. C₉	D₇×D₁₁₉	semidirect
121. C₇	D₇×D₁₅₃	semidirect
122. C₇	C₂×D₁₀₇₁	semidirect

H ≅ G/N

type

1. C₇⋊D₁₀₇₁

C₂

semidirect

2. C₇×D₁₀₇₁

C₂

semidirect

3. C₇×C₉×D₇×C₁₇

C₂

semidirect

5. C₇²×C₉×C₁₇

C₂²

semidirect

16. D₁₀₇₁

D₇

direct (N × H)

18. C₉×D₇×C₁₇

D₇

semidirect

23. C₇×D₇×C₁₇

D₉

semidirect

27. C₇×C₉×C₁₇

C₂×D₇

semidirect

28. C₇×C₉×C₁₇

C₂×D₇

semidirect

31. C₇×C₉×D₇

D₁₇

semidirect

32. C₇²×C₁₇

C₂×D₉

semidirect

44. C₇²×C₉

C₂×D₁₇

semidirect

62. D₇×C₁₇

D₆₃

semidirect

65. C₉×C₁₇

D₇²

semidirect

72. C₉×D₇

D₁₁₉

semidirect

75. C₇×C₁₇

D₇×D₉

semidirect

76. C₇×C₁₇

C₂×D₆₃

semidirect

82. C₇×D₇

D₁₅₃

semidirect

87. C₇×C₉

D₇×D₁₇

semidirect

88. C₇×C₉

C₂×D₁₁₉

semidirect

90. C₇²

C₂×D₁₅₃

semidirect

110. C₁₇

D₇×D₆₃

semidirect

117. D₇

D₁₀₇₁

direct (N × H)

119. C₉

D₇×D₁₁₉

semidirect

121. C₇

D₇×D₁₅₃

semidirect

122. C₇

C₂×D₁₀₇₁

semidirect

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

subgroup	index
C₇⋊D₁₀₇₁	2
C₇×D₁₀₇₁	2
C₇×C₉×D₇×C₁₇	2
D₇×D₃₅₇	3
C₂×D₁₀₇₁	7
D₇×D₁₅₃	7
D₇×D₆₃	17

subgroup

index

C₇⋊D₁₀₇₁

C₇×D₁₀₇₁

C₇×C₉×D₇×C₁₇

D₇×D₃₅₇

C₂×D₁₀₇₁

D₇×D₁₅₃

D₇×D₆₃

Subgroups — D₇×D₁₀₇₁ #44472607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D7×D1071 #44472607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×D₁₀₇₁ #44472607