Group #31834383 — subgroups

Subgroups — C₇₁×#13511 #31834383

Order 18176.

250 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 110 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×#2155	C₂	1	2	yes	maximal
2	C₇₁×#1946	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#1849	C₂	1	1	yes	maximal, characteristic
4	C₇₁×#1385	C₂	1	2	yes	maximal
5	C₇₁×#1375	C₂	1	1	yes	maximal, characteristic
6	C₂×M₃₂×C₇₁	C₂²	1	2	yes	—
7	C₂×#108×C₇₁	—	2	1	no	—
8	C₂×#108×C₇₁	C₂²	1	2	yes	—
9	C₂×#108×C₇₁	C₂²	1	1	yes	characteristic
10	#427×C₇₁	C₂²	1	1	yes	characteristic
11	#424×C₇₁	—	2	1	no	—
12	#423×C₇₁	—	2	1	no	—
13	C₂×#97×C₇₁	C₂²	1	1	yes	characteristic
14	#365×C₇₁	—	2	4	no	—
15	#339×C₇₁	—	2	1	no	—
16	C₂×M₁₆×C₇₁	—	2	2	no	—
17	C₂×M₁₆×C₇₁	—	2	1	no	—
18	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
20	#117×C₇₁	—	2	1	no	—
21	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
22	M₃₂×C₇₁	—	2	2	no	—
23	M₃₂×C₇₁	—	2	2	no	—
24	C₂×C₁₆×C₇₁	Q₈	1	4	yes	—
25	#108×C₇₁	—	2	2	no	—
26	#108×C₇₁	—	2	2	no	—
27	#108×C₇₁	—	2	2	no	—
28	#108×C₇₁	—	4	2	no	—
29	#107×C₇₁	—	2	1	no	—
30	#106×C₇₁	D₄	1	2	yes	—
31	C₄⋊C₈×C₇₁	—	4	2	no	—
32	C₄⋊C₈×C₇₁	—	4	2	no	—
33	#97×C₇₁	—	2	2	no	—
34	C₂²×C₄×C₇₁	—	2	1	no	—
35	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
36	M₁₆×C₇₁	—	4	2	no	—
37	M₁₆×C₇₁	—	4	1	no	—
38	M₁₆×C₇₁	—	4	2	no	—
39	M₁₆×C₇₁	—	4	1	no	—
40	C₂×C₈×C₇₁	—	2	1	no	—
41	C₂×C₈×C₇₁	—	2	1	no	—
42	C₂×C₈×C₇₁	—	4	2	no	—
43	C₂×C₈×C₇₁	—	4	1	no	—
44	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
45	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
47	C₂×C₈×C₇₁	#41	1	2	yes	—
48	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
49	C₄⋊C₄×C₇₁	—	2	2	no	—
50	C₂²⋊C₄×C₇₁	—	4	1	no	—
51	C₄²×C₇₁	—	2	1	no	—
52	C₄²×C₇₁	—	4	1	no	—
53	C₄²×C₇₁	SD₁₆	1	2	yes	—
54	C₁₆×C₇₁	—	2	4	no	—
55	C₂³×C₇₁	#137	1	1	yes	characteristic
56	C₂×C₄×C₇₁	—	2	2	no	—
57	C₂×C₄×C₇₁	—	4	2	no	—
58	C₂×C₄×C₇₁	—	8	1	no	—
59	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
60	C₂×C₄×C₇₁	#123	1	1	yes	characteristic
61	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
64	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₈×C₇₁	—	2	2	no	—
66	C₈×C₇₁	—	8	2	no	—
67	C₈×C₇₁	—	8	1	no	—
68	C₈×C₇₁	C₂²⋊Q₈	1	2	yes	—
69	C₈×C₇₁	#125	1	2	yes	—
70	C₂²×C₇₁	—	4	1	no	—
71	C₂²×C₇₁	#461	1	1	yes	characteristic
72	C₂²×C₇₁	#483	1	1	yes	characteristic
73	C₂²×C₇₁	#501	1	1	yes	characteristic
74	C₄×C₇₁	—	4	2	no	—
75	C₄×C₇₁	—	8	2	no	—
76	C₄×C₇₁	#398	1	1	yes	characteristic
77	C₄×C₇₁	#460	1	1	yes	characteristic
78	C₄×C₇₁	#481	1	1	yes	characteristic
79	C₄×C₇₁	#498	1	1	yes	characteristic
80	#13511	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂×C₇₁	—	2	1	no	—
82	C₂×C₇₁	—	2	1	no	—
83	C₂×C₇₁	—	2	1	no	—
84	C₂×C₇₁	#2084	1	1	yes	characteristic
85	#2155	C₂×C₇₁	1	2	yes	—
86	#1946	C₂×C₇₁	1	1	yes	characteristic
87	#1849	C₂×C₇₁	1	1	yes	characteristic
88	#1385	C₂×C₇₁	1	2	yes	—
89	#1375	C₂×C₇₁	1	1	yes	characteristic
90	C₇₁	#13511	1	1	yes	characteristic, direct product factor, Sylow
91	C₂×M₃₂	C₂²×C₇₁	1	2	yes	—
92	C₂×#108	—	2	1	no	—
93	C₂×#108	C₂²×C₇₁	1	2	yes	—
94	C₂×#108	C₂²×C₇₁	1	1	yes	characteristic
95	#427	C₂²×C₇₁	1	1	yes	characteristic
96	#424	—	2	1	no	—
97	#423	—	2	1	no	—
98	C₂×#97	C₂²×C₇₁	1	1	yes	characteristic
99	#365	—	2	4	no	—
100	#339	—	2	1	no	—
101	C₂×M₁₆	—	2	1	no	—
102	C₂×M₁₆	—	2	2	no	—
103	C₂²×C₈	D₄×C₇₁	1	1	yes	characteristic
104	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
105	#117	—	2	1	no	—
106	C₂×C₄²	D₄×C₇₁	1	1	yes	characteristic
107	M₃₂	—	2	2	no	—
108	M₃₂	—	2	2	no	—
109	C₂×C₁₆	Q₈×C₇₁	1	4	yes	—
110	#108	—	2	2	no	—
111	#108	—	2	2	no	—
112	#108	—	2	2	no	—
113	#108	—	4	2	no	—
114	#107	—	2	1	no	—
115	#106	D₄×C₇₁	1	2	yes	—
116	C₄⋊C₈	—	4	2	no	—
117	C₄⋊C₈	—	4	2	no	—
118	#97	—	2	2	no	—
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
121	M₁₆	—	4	2	no	—
122	M₁₆	—	4	1	no	—
123	M₁₆	—	4	2	no	—
124	M₁₆	—	4	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	4	2	no	—
128	C₂×C₈	—	4	1	no	—
129	C₂×C₈	SD₁₆×C₇₁	1	2	yes	—
130	C₂×C₈	C₂×D₄×C₇₁	1	1	yes	characteristic
131	C₂×C₈	C₂×Q₈×C₇₁	1	2	yes	—
132	C₂×C₈	#41×C₇₁	1	2	yes	—
133	C₂×C₈	#41×C₇₁	1	1	yes	characteristic
134	C₄⋊C₄	—	2	2	no	—
135	C₂²⋊C₄	—	4	1	no	—
136	C₄²	—	2	1	no	—
137	C₄²	—	4	1	no	—
138	C₄²	SD₁₆×C₇₁	1	2	yes	—
139	C₁₆	—	2	4	no	—
140	C₂³	#137×C₇₁	1	1	yes	characteristic
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	8	1	no	—
144	C₂×C₄	#121×C₇₁	1	1	yes	characteristic
145	C₂×C₄	#123×C₇₁	1	1	yes	characteristic
146	C₂×C₄	C₄⋊Q₈×C₇₁	1	1	yes	characteristic
147	C₂×C₄	C₂×SD₁₆×C₇₁	1	1	yes	characteristic
148	C₂×C₄	C₂×SD₁₆×C₇₁	1	1	yes	characteristic
149	C₂×C₄	AGL(1,ℤ/8)×C₇₁	1	1	yes	characteristic
150	C₈	—	2	2	no	—
151	C₈	—	8	2	no	—
152	C₈	—	8	1	no	—
153	C₈	C₂²⋊Q₈×C₇₁	1	2	yes	—
154	C₈	#125×C₇₁	1	2	yes	—
155	C₂²	—	4	1	no	—
156	C₂²	#461×C₇₁	1	1	yes	characteristic
157	C₂²	#483×C₇₁	1	1	yes	characteristic
158	C₂²	#501×C₇₁	1	1	yes	characteristic
159	C₄	—	4	2	no	—
160	C₄	—	8	2	no	—
161	C₄	#398×C₇₁	1	1	yes	characteristic
162	C₄	#460×C₇₁	1	1	yes	characteristic
163	C₄	#481×C₇₁	1	1	yes	characteristic
164	C₄	#498×C₇₁	1	1	yes	characteristic
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	—	2	1	no	—
168	C₂	C₇₁×#2084	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₁×#2155

C₂

yes

maximal

C₇₁×#1946

C₂

yes

maximal, characteristic

C₇₁×#1849

C₂

yes

maximal, characteristic

C₇₁×#1385

C₂

yes

maximal

C₇₁×#1375

C₂

yes

maximal, characteristic

C₂×M₃₂×C₇₁

C₂²

yes

—

C₂×#108×C₇₁

—

C₂×#108×C₇₁

C₂²

yes

—

C₂×#108×C₇₁

C₂²

yes

characteristic

#427×C₇₁

C₂²

yes

characteristic

#424×C₇₁

—

#423×C₇₁

—

C₂×#97×C₇₁

C₂²

yes

characteristic

#365×C₇₁

—

#339×C₇₁

—

C₂×M₁₆×C₇₁

—

C₂×M₁₆×C₇₁

—

C₂²×C₈×C₇₁

D₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

#117×C₇₁

—

C₂×C₄²×C₇₁

D₄

yes

characteristic

M₃₂×C₇₁

—

M₃₂×C₇₁

—

C₂×C₁₆×C₇₁

Q₈

yes

—

#108×C₇₁

—

#108×C₇₁

—

#108×C₇₁

—

#108×C₇₁

—

#107×C₇₁

—

#106×C₇₁

D₄

yes

—

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

—

#97×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

SD₁₆

yes

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

—

C₂×C₈×C₇₁

#41

yes

—

C₂×C₈×C₇₁

#41

yes

characteristic

C₄⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

SD₁₆

yes

—

C₁₆×C₇₁

—

C₂³×C₇₁

#137

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₁

AGL(1,ℤ/8)

yes

characteristic

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

C₂²⋊Q₈

yes

—

C₈×C₇₁

#125

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₇₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#2084

yes

characteristic

#2155

C₂×C₇₁

yes

—

#1946

C₂×C₇₁

yes

characteristic

#1849

C₂×C₇₁

yes

characteristic

#1385

C₂×C₇₁

yes

—

#1375

C₂×C₇₁

yes

characteristic

C₇₁

#13511

yes

characteristic, direct product factor, Sylow

C₂×M₃₂

C₂²×C₇₁

yes

—

C₂×#108

—

C₂×#108

C₂²×C₇₁

yes

—

C₂×#108

C₂²×C₇₁

yes

characteristic

#427

C₂²×C₇₁

yes

characteristic

#424

—

#423

—

C₂×#97

C₂²×C₇₁

yes

characteristic

#365

—

100

#339

—

101

C₂×M₁₆

—

102

C₂×M₁₆

—

103

C₂²×C₈

D₄×C₇₁

yes

characteristic

104

C₂²×C₈

C₂³×C₇₁

yes

characteristic

105

#117

—

106

C₂×C₄²

D₄×C₇₁

yes

characteristic

107

M₃₂

—

108

M₃₂

—

109

C₂×C₁₆

Q₈×C₇₁

yes

—

110

#108

—

111

#108

—

112

#108

—

113

#108

—

114

#107

—

115

#106

D₄×C₇₁

yes

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

118

#97

—

119

C₂²×C₄

—

120

C₂²×C₄

C₂×D₄×C₇₁

yes

characteristic

121

M₁₆

—

122

M₁₆

—

123

M₁₆

—

124

M₁₆

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

SD₁₆×C₇₁

yes

—

130

C₂×C₈

C₂×D₄×C₇₁

yes

characteristic

131

C₂×C₈

C₂×Q₈×C₇₁

yes

—

132

C₂×C₈

#41×C₇₁

yes

—

133

C₂×C₈

#41×C₇₁

yes

characteristic

134

C₄⋊C₄

—

135

C₂²⋊C₄

—

136

C₄²

—

137

C₄²

—

138

C₄²

SD₁₆×C₇₁

yes

—

139

C₁₆

—

140

C₂³

#137×C₇₁

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

#121×C₇₁

yes

characteristic

145

C₂×C₄

#123×C₇₁

yes

characteristic

146

C₂×C₄

C₄⋊Q₈×C₇₁

yes

characteristic

147

C₂×C₄

C₂×SD₁₆×C₇₁

yes

characteristic

148

C₂×C₄

C₂×SD₁₆×C₇₁

yes

characteristic

149

C₂×C₄

AGL(1,ℤ/8)×C₇₁

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

C₂²⋊Q₈×C₇₁

yes

—

154

C₈

#125×C₇₁

yes

—

155

C₂²

—

156

C₂²

#461×C₇₁

yes

characteristic

157

C₂²

#483×C₇₁

yes

characteristic

158

C₂²

#501×C₇₁

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

#398×C₇₁

yes

characteristic

162

C₄

#460×C₇₁

yes

characteristic

163

C₄

#481×C₇₁

yes

characteristic

164

C₄

#498×C₇₁

yes

characteristic

165

C₂

—

166

C₂

—

167

C₂

—

168

C₂

C₇₁×#2084

yes

characteristic

	subgroup	G/H
center	C₄×C₇₁	#498
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇₁
socle	C₂×C₇₁	#2084
prime radical	C₂³×C₇₁	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇₁ ◁ C₂²×C₄×C₇₁ ◁ C₂×#97×C₇₁ ◁ G

subgroup

G/H

center

C₄×C₇₁

#498

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₇₁

C₂×C₇₁

C₂³×C₇₁

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄×C₇₁ ◁ C₂²×C₄×C₇₁ ◁ C₂×#97×C₇₁ ◁ G

N	H ≅ G/N	type
80. #13511	C₇₁	direct (N × H)
90. C₇₁	#13511	direct (N × H)

H ≅ G/N

type

80. #13511

C₇₁

direct (N × H)

90. C₇₁

#13511

direct (N × H)

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

subgroup	index
C₇₁×#2155	2
C₇₁×#1946	2
C₇₁×#1849	2
C₇₁×#1385	2
C₇₁×#1375	2
#13511	71

subgroup

index

C₇₁×#2155

C₇₁×#1946

C₇₁×#1849

C₇₁×#1385

C₇₁×#1375

#13511

Subgroups — C₇₁×#13511 #31834383

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C71×#13511 #31834383

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#13511 #31834383