Group #31834589 — subgroups

Subgroups — C₇₁×#13717 #31834589

Order 18176.

170 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 66 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	C₇₁×#2218	C₂	1	1	yes	maximal, characteristic
2	C₇₁×#2161	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₁×#1426	C₂	1	2	yes	maximal, complemented
4	C₇₁×#1419	C₂	1	2	yes	maximal, complemented
5	C₇₁×#1400	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₆×C₇₁	C₂²	1	1	yes	characteristic
7	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
8	#466×C₇₁	—	2	1	no	—
9	C₂×#108×C₇₁	—	2	1	no	—
10	M₆₄×C₇₁	C₂²	1	2	yes	—
11	#368×C₇₁	—	2	1	no	—
12	#368×C₇₁	C₂²	1	2	yes	—
13	#367×C₇₁	C₂²	1	1	yes	characteristic
14	#357×C₇₁	—	2	1	no	—
15	C₂×D₈×C₇₁	—	2	1	no	—
16	C₂×M₁₆×C₇₁	—	4	1	no	—
17	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
18	#121×C₇₁	—	4	1	no	—
19	D₁₆×C₇₁	—	2	2	no	—
20	M₃₂×C₇₁	—	2	2	no	—
21	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
23	#108×C₇₁	—	2	2	no	—
24	#108×C₇₁	—	4	1	no	—
25	#107×C₇₁	—	2	1	no	—
26	#102×C₇₁	—	4	1	no	—
27	C₃₂×C₇₁	—	2	2	no	—
28	C₂×D₄×C₇₁	—	4	1	no	—
29	C₂×D₄×C₇₁	—	8	1	no	—
30	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
31	D₈×C₇₁	—	4	1	no	—
32	D₈×C₇₁	—	4	1	no	—
33	M₁₆×C₇₁	—	4	2	no	—
34	M₁₆×C₇₁	—	8	1	no	—
35	C₂×C₈×C₇₁	—	2	1	no	—
36	C₂×C₈×C₇₁	—	2	1	no	—
37	C₂×C₈×C₇₁	—	8	1	no	—
38	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
39	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
40	C₄⋊C₄×C₇₁	—	4	1	no	—
41	C₂²⋊C₄×C₇₁	—	8	1	no	—
42	C₁₆×C₇₁	—	2	1	no	—
43	C₁₆×C₇₁	#41	1	2	yes	—
44	C₂³×C₇₁	—	8	1	no	—
45	C₂³×C₇₁	SD₃₂	1	1	yes	characteristic
46	D₄×C₇₁	—	8	1	no	—
47	D₄×C₇₁	—	8	1	no	—
48	D₄×C₇₁	—	16	2	no	—
49	C₂×C₄×C₇₁	—	2	1	no	—
50	C₂×C₄×C₇₁	—	2	1	no	—
51	C₂×C₄×C₇₁	—	8	1	no	—
52	C₂×C₄×C₇₁	SD₃₂	1	1	yes	characteristic
53	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
54	C₈×C₇₁	—	2	1	no	—
55	C₈×C₇₁	—	8	2	no	—
56	C₈×C₇₁	#123	1	1	yes	characteristic
57	C₈×C₇₁	#137	1	1	yes	characteristic
58	C₂²×C₇₁	—	2	1	no	—
59	C₂²×C₇₁	—	4	1	no	—
60	C₂²×C₇₁	—	16	1	no	—
61	C₂²×C₇₁	—	16	1	no	—
62	C₂²×C₇₁	—	16	1	no	—
63	C₂²×C₇₁	C₂×SD₃₂	1	1	yes	characteristic
64	C₄×C₇₁	—	2	1	no	—
65	C₄×C₇₁	—	16	1	no	—
66	C₄×C₇₁	#480	1	1	yes	characteristic
67	C₄×C₇₁	#509	1	1	yes	characteristic
68	#13717	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂×C₇₁	—	2	1	no	—
70	C₂×C₇₁	—	4	1	no	—
71	C₂×C₇₁	—	32	1	no	—
72	C₂×C₇₁	#2233	1	1	yes	characteristic
73	#2218	C₂×C₇₁	1	1	yes	characteristic
74	#2161	C₂×C₇₁	1	1	yes	characteristic, complemented
75	#1426	C₂×C₇₁	1	2	yes	complemented
76	#1419	C₂×C₇₁	1	2	yes	complemented
77	#1400	C₂×C₇₁	1	1	yes	characteristic, complemented
78	C₇₁	#13717	1	1	yes	characteristic, direct product factor, Sylow
79	C₂×D₁₆	C₂²×C₇₁	1	1	yes	characteristic
80	C₂×M₃₂	C₂²×C₇₁	1	1	yes	characteristic
81	#466	—	2	1	no	—
82	C₂×#108	—	2	1	no	—
83	M₆₄	C₂²×C₇₁	1	2	yes	—
84	#368	—	2	1	no	—
85	#368	C₂²×C₇₁	1	2	yes	—
86	#367	C₂²×C₇₁	1	1	yes	characteristic
87	#357	—	2	1	no	—
88	C₂×D₈	—	2	1	no	—
89	C₂×M₁₆	—	4	1	no	—
90	C₂²×C₈	D₄×C₇₁	1	1	yes	characteristic
91	#121	—	4	1	no	—
92	D₁₆	—	2	2	no	—
93	M₃₂	—	2	2	no	—
94	C₂×C₁₆	D₄×C₇₁	1	1	yes	characteristic
95	C₂×C₁₆	C₂³×C₇₁	1	1	yes	characteristic
96	#108	—	2	2	no	—
97	#108	—	4	1	no	—
98	#107	—	2	1	no	—
99	#102	—	4	1	no	—
100	C₃₂	—	2	2	no	—
101	C₂×D₄	—	4	1	no	—
102	C₂×D₄	—	8	1	no	—
103	C₂²×C₄	D₈×C₇₁	1	1	yes	characteristic
104	D₈	—	4	1	no	—
105	D₈	—	4	1	no	—
106	M₁₆	—	4	2	no	—
107	M₁₆	—	8	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	8	1	no	—
111	C₂×C₈	D₈×C₇₁	1	1	yes	characteristic
112	C₂×C₈	C₂×D₄×C₇₁	1	1	yes	characteristic
113	C₄⋊C₄	—	4	1	no	—
114	C₂²⋊C₄	—	8	1	no	—
115	C₁₆	—	2	1	no	—
116	C₁₆	#41×C₇₁	1	2	yes	—
117	C₂³	—	8	1	no	—
118	C₂³	SD₃₂×C₇₁	1	1	yes	characteristic
119	D₄	—	8	1	no	—
120	D₄	—	8	1	no	—
121	D₄	—	16	2	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	8	1	no	—
125	C₂×C₄	SD₃₂×C₇₁	1	1	yes	characteristic
126	C₂×C₄	C₂×D₈×C₇₁	1	1	yes	characteristic
127	C₈	—	2	1	no	—
128	C₈	—	8	2	no	—
129	C₈	#123×C₇₁	1	1	yes	characteristic
130	C₈	#137×C₇₁	1	1	yes	characteristic
131	C₂²	—	2	1	no	—
132	C₂²	—	4	1	no	—
133	C₂²	—	16	1	no	—
134	C₂²	—	16	1	no	—
135	C₂²	—	16	1	no	—
136	C₂²	C₂×SD₃₂×C₇₁	1	1	yes	characteristic
137	C₄	—	2	1	no	—
138	C₄	—	16	1	no	—
139	C₄	#480×C₇₁	1	1	yes	characteristic
140	C₄	#509×C₇₁	1	1	yes	characteristic
141	C₂	—	2	1	no	—
142	C₂	—	4	1	no	—
143	C₂	—	32	1	no	—
144	C₂	C₇₁×#2233	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₁×#2218

C₂

yes

maximal, characteristic

C₇₁×#2161

C₂

yes

maximal, characteristic, complemented

C₇₁×#1426

C₂

yes

maximal, complemented

C₇₁×#1419

C₂

yes

maximal, complemented

C₇₁×#1400

C₂

yes

maximal, characteristic, complemented

C₂×D₁₆×C₇₁

C₂²

yes

characteristic

C₂×M₃₂×C₇₁

C₂²

yes

characteristic

#466×C₇₁

—

C₂×#108×C₇₁

—

M₆₄×C₇₁

C₂²

yes

—

#368×C₇₁

—

#368×C₇₁

C₂²

yes

—

#367×C₇₁

C₂²

yes

characteristic

#357×C₇₁

—

C₂×D₈×C₇₁

—

C₂×M₁₆×C₇₁

—

C₂²×C₈×C₇₁

D₄

yes

characteristic

#121×C₇₁

—

D₁₆×C₇₁

—

M₃₂×C₇₁

—

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

C₂³

yes

characteristic

#108×C₇₁

—

#108×C₇₁

—

#107×C₇₁

—

#102×C₇₁

—

C₃₂×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂²×C₄×C₇₁

D₈

yes

characteristic

D₈×C₇₁

—

D₈×C₇₁

—

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

D₈

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₁₆×C₇₁

—

C₁₆×C₇₁

#41

yes

—

C₂³×C₇₁

—

C₂³×C₇₁

SD₃₂

yes

characteristic

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

SD₃₂

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

yes

C₈×C₇₁

yes

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×SD₃₂

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₇₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#2233

yes

characteristic

#2218

C₂×C₇₁

yes

characteristic

#2161

C₂×C₇₁

yes

characteristic, complemented

#1426

C₂×C₇₁

yes

complemented

#1419

C₂×C₇₁

yes

complemented

#1400

C₂×C₇₁

yes

characteristic, complemented

C₇₁

#13717

yes

characteristic, direct product factor, Sylow

C₂×D₁₆

C₂²×C₇₁

yes

characteristic

C₂×M₃₂

C₂²×C₇₁

yes

characteristic

#466

—

C₂×#108

—

M₆₄

C₂²×C₇₁

yes

—

#368

—

#368

C₂²×C₇₁

yes

—

#367

C₂²×C₇₁

yes

characteristic

#357

—

C₂×D₈

—

C₂×M₁₆

—

C₂²×C₈

D₄×C₇₁

yes

characteristic

#121

—

D₁₆

—

M₃₂

—

C₂×C₁₆

D₄×C₇₁

yes

characteristic

C₂×C₁₆

C₂³×C₇₁

yes

—

—

—

—

100

C₃₂

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂²×C₄

D₈×C₇₁

yes

characteristic

104

D₈

—

105

D₈

—

106

M₁₆

—

107

M₁₆

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

D₈×C₇₁

yes

characteristic

112

C₂×C₈

C₂×D₄×C₇₁

yes

characteristic

113

C₄⋊C₄

—

114

C₂²⋊C₄

—

115

C₁₆

—

116

C₁₆

#41×C₇₁

yes

—

117

C₂³

—

118

C₂³

SD₃₂×C₇₁

yes

characteristic

119

D₄

—

120

D₄

—

121

D₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

SD₃₂×C₇₁

yes

characteristic

126

C₂×C₄

C₂×D₈×C₇₁

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

#123×C₇₁

yes

characteristic

130

C₈

#137×C₇₁

yes

characteristic

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

C₂×SD₃₂×C₇₁

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

#480×C₇₁

yes

characteristic

140

C₄

#509×C₇₁

yes

characteristic

141

C₂

—

142

C₂

—

143

C₂

—

144

C₂

C₇₁×#2233

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₇₁	#2233
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₇₁
prime radical	C₇₁×#2218	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₁ ◁ C₂×C₄×C₇₁ ◁ C₂²×C₈×C₇₁ ◁ C₂×M₃₂×C₇₁ ◁ G

subgroup

G/H

center, socle

C₂×C₇₁

#2233

commutator, Frattini, cosocle

C₂×C₁₆

C₂³×C₇₁

prime radical

C₇₁×#2218

C₂

LCS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₇₁ ◁ C₂×C₄×C₇₁ ◁ C₂²×C₈×C₇₁ ◁ C₂×M₃₂×C₇₁ ◁ G

N	H ≅ G/N	type
2. C₇₁×#2161	C₂	semidirect
3. C₇₁×#1426	C₂	semidirect
4. C₇₁×#1419	C₂	semidirect
5. C₇₁×#1400	C₂	semidirect
68. #13717	C₇₁	direct (N × H)
74. #2161	C₂×C₇₁	semidirect
75. #1426	C₂×C₇₁	semidirect
76. #1419	C₂×C₇₁	semidirect
77. #1400	C₂×C₇₁	semidirect
78. C₇₁	#13717	direct (N × H)

H ≅ G/N

type

2. C₇₁×#2161

C₂

semidirect

3. C₇₁×#1426

C₂

semidirect

4. C₇₁×#1419

C₂

semidirect

5. C₇₁×#1400

C₂

semidirect

68. #13717

C₇₁

direct (N × H)

74. #2161

C₂×C₇₁

semidirect

75. #1426

C₂×C₇₁

semidirect

76. #1419

C₂×C₇₁

semidirect

77. #1400

C₂×C₇₁

semidirect

78. C₇₁

#13717

direct (N × H)

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

subgroup	index
C₇₁×#2218	2
C₇₁×#2161	2
C₇₁×#1426	2
C₇₁×#1419	2
C₇₁×#1400	2
#13717	71

subgroup

index

C₇₁×#2218

C₇₁×#2161

C₇₁×#1426

C₇₁×#1419

C₇₁×#1400

#13717

Subgroups — C₇₁×#13717 #31834589

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C71×#13717 #31834589

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#13717 #31834589