Group #31830645 — subgroups

Subgroups — C₇₁×#9773 #31830645

Order 18176.

582 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 174 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	C₂×#493×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₇₁×#1848	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₁×#1545	C₂	1	4	yes	maximal, complemented
4	C₇₁×#1450	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₈×C₇₁	—	2	2	no	—
6	C₂×#127×C₇₁	C₂²	1	2	yes	—
7	#493×C₇₁	—	2	1	no	—
8	#493×C₇₁	—	2	1	no	—
9	#493×C₇₁	C₄	1	4	yes	—
10	#431×C₇₁	C₂²	1	2	yes	—
11	C₂×#108×C₇₁	—	2	2	no	—
12	#423×C₇₁	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
14	#331×C₇₁	—	2	8	no	—
15	#322×C₇₁	—	2	1	no	—
16	C₈²×C₇₁	—	2	1	no	—
17	C₂²×D₄×C₇₁	—	2	2	no	—
18	C₂²×D₄×C₇₁	—	2	4	no	—
19	C₂×D₈×C₇₁	—	2	4	no	—
20	C₂×D₈×C₇₁	—	2	4	no	—
21	C₂×D₈×C₇₁	—	4	4	no	—
22	C₂×D₈×C₇₁	—	4	4	no	—
23	C₂×M₁₆×C₇₁	—	2	2	no	—
24	C₂²×C₈×C₇₁	D₄	1	2	yes	—
25	#127×C₇₁	—	2	2	no	—
26	#127×C₇₁	—	2	4	no	—
27	#127×C₇₁	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
29	#108×C₇₁	—	4	4	no	—
30	C₄⋊C₈×C₇₁	—	2	4	no	—
31	C₄⋊C₈×C₇₁	D₄	1	4	yes	—
32	C₂²⋊C₈×C₇₁	—	2	2	no	—
33	C₄×C₈×C₇₁	—	2	2	no	—
34	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
36	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
37	C₂⁴×C₇₁	—	4	2	no	—
38	C₂×D₄×C₇₁	—	2	4	no	—
39	C₂×D₄×C₇₁	—	2	8	no	—
40	C₂×D₄×C₇₁	—	4	4	no	—
41	C₂×D₄×C₇₁	—	4	2	no	—
42	C₂×D₄×C₇₁	—	4	4	no	—
43	C₂×D₄×C₇₁	—	4	4	no	—
44	C₂×D₄×C₇₁	—	8	8	no	—
45	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₁	#41	1	2	yes	—
47	D₈×C₇₁	—	4	8	no	—
48	D₈×C₇₁	—	4	8	no	—
49	M₁₆×C₇₁	—	4	4	no	—
50	C₂×C₈×C₇₁	—	2	4	no	—
51	C₂×C₈×C₇₁	—	4	2	no	—
52	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
53	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₇₁	D₈	1	4	yes	—
55	C₂×C₈×C₇₁	SD₁₆	1	4	yes	—
56	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₇₁	C₂×D₄	1	2	yes	—
59	C₂³×C₇₁	—	4	4	no	—
60	C₂³×C₇₁	—	8	8	no	—
61	C₂³×C₇₁	—	8	4	no	—
62	C₂³×C₇₁	#124	1	1	yes	characteristic
63	D₄×C₇₁	—	4	8	no	—
64	D₄×C₇₁	—	8	8	no	—
65	D₄×C₇₁	—	8	8	no	—
66	D₄×C₇₁	—	8	2	no	—
67	D₄×C₇₁	—	8	2	no	—
68	C₂×C₄×C₇₁	—	2	4	no	—
69	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
71	C₂×C₄×C₇₁	#121	1	2	yes	—
72	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
73	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
74	C₂×C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
75	C₈×C₇₁	—	4	4	no	—
76	C₈×C₇₁	—	8	2	no	—
77	C₈×C₇₁	#102	1	4	yes	—
78	C₈×C₇₁	#102	1	4	yes	—
79	C₂²×C₇₁	—	2	2	no	—
80	C₂²×C₇₁	—	8	4	no	—
81	C₂²×C₇₁	—	8	8	no	—
82	C₂²×C₇₁	—	16	8	no	—
83	C₂²×C₇₁	#390	1	1	yes	characteristic
84	C₂²×C₇₁	#486	1	2	yes	—
85	C₄×C₇₁	—	2	2	no	—
86	C₄×C₇₁	C₂×#102	1	2	yes	—
87	C₄×C₇₁	#472	1	4	yes	—
88	C₄×C₇₁	#492	1	1	yes	characteristic
89	C₄×C₇₁	#493	1	1	yes	characteristic
90	#9773	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂×C₇₁	—	2	2	no	—
92	C₂×C₇₁	—	16	4	no	—
93	C₂×C₇₁	#1957	1	1	yes	characteristic
94	C₂×C₇₁	#1978	1	2	yes	—
95	C₂×#493	C₂×C₇₁	1	1	yes	characteristic
96	#1848	C₂×C₇₁	1	1	yes	characteristic, complemented
97	#1545	C₂×C₇₁	1	4	yes	complemented
98	#1450	C₂×C₇₁	1	1	yes	characteristic, complemented
99	C₇₁	#9773	1	1	yes	characteristic, direct product factor, Sylow
100	C₂²×D₈	—	2	2	no	—
101	C₂×#127	C₂²×C₇₁	1	2	yes	—
102	#493	—	2	1	no	—
103	#493	—	2	1	no	—
104	#493	C₄×C₇₁	1	4	yes	—
105	#431	C₂²×C₇₁	1	2	yes	—
106	C₂×#108	—	2	2	no	—
107	#423	C₂²×C₇₁	1	2	yes	—
108	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
109	#331	—	2	8	no	—
110	#322	—	2	1	no	—
111	C₈²	—	2	1	no	—
112	C₂²×D₄	—	2	2	no	—
113	C₂²×D₄	—	2	4	no	—
114	C₂×D₈	—	2	4	no	—
115	C₂×D₈	—	2	4	no	—
116	C₂×D₈	—	4	4	no	—
117	C₂×D₈	—	4	4	no	—
118	C₂×M₁₆	—	2	2	no	—
119	C₂²×C₈	D₄×C₇₁	1	2	yes	—
120	#127	—	2	4	no	—
121	#127	—	2	2	no	—
122	#127	C₂×C₄×C₇₁	1	4	yes	—
123	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
124	#108	—	4	4	no	—
125	C₄⋊C₈	—	2	4	no	—
126	C₄⋊C₈	D₄×C₇₁	1	4	yes	—
127	C₂²⋊C₈	—	2	2	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	C₂×C₄×C₇₁	1	2	yes	—
130	C₄×C₈	D₄×C₇₁	1	1	yes	characteristic
131	C₄×C₈	D₄×C₇₁	1	1	yes	characteristic
132	C₂⁴	—	4	2	no	—
133	C₂×D₄	—	2	4	no	—
134	C₂×D₄	—	2	8	no	—
135	C₂×D₄	—	4	4	no	—
136	C₂×D₄	—	4	2	no	—
137	C₂×D₄	—	4	4	no	—
138	C₂×D₄	—	4	4	no	—
139	C₂×D₄	—	8	8	no	—
140	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
141	C₂²×C₄	#41×C₇₁	1	2	yes	—
142	D₈	—	4	8	no	—
143	D₈	—	4	8	no	—
144	M₁₆	—	4	4	no	—
145	C₂×C₈	—	2	4	no	—
146	C₂×C₈	—	4	2	no	—
147	C₂×C₈	C₂²⋊C₄×C₇₁	1	2	yes	—
148	C₂×C₈	C₂²⋊C₄×C₇₁	1	2	yes	—
149	C₂×C₈	D₈×C₇₁	1	4	yes	—
150	C₂×C₈	SD₁₆×C₇₁	1	4	yes	—
151	C₄²	C₂²×C₄×C₇₁	1	1	yes	characteristic
152	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
153	C₄²	C₂×D₄×C₇₁	1	2	yes	—
154	C₂³	—	4	4	no	—
155	C₂³	—	8	8	no	—
156	C₂³	—	8	4	no	—
157	C₂³	#124×C₇₁	1	1	yes	characteristic
158	D₄	—	4	8	no	—
159	D₄	—	8	8	no	—
160	D₄	—	8	8	no	—
161	D₄	—	8	2	no	—
162	D₄	—	8	2	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
165	C₂×C₄	C₄×D₄×C₇₁	1	2	yes	—
166	C₂×C₄	#121×C₇₁	1	2	yes	—
167	C₂×C₄	#127×C₇₁	1	1	yes	characteristic
168	C₂×C₄	C₂×D₈×C₇₁	1	2	yes	—
169	C₂×C₄	C₂×SD₁₆×C₇₁	1	2	yes	—
170	C₈	—	4	4	no	—
171	C₈	—	8	2	no	—
172	C₈	#102×C₇₁	1	4	yes	—
173	C₈	#102×C₇₁	1	4	yes	—
174	C₂²	—	2	2	no	—
175	C₂²	—	8	4	no	—
176	C₂²	—	8	8	no	—
177	C₂²	—	16	8	no	—
178	C₂²	#390×C₇₁	1	1	yes	characteristic
179	C₂²	#486×C₇₁	1	2	yes	—
180	C₄	—	2	2	no	—
181	C₄	C₂×#102×C₇₁	1	2	yes	—
182	C₄	#472×C₇₁	1	4	yes	—
183	C₄	#492×C₇₁	1	1	yes	characteristic
184	C₄	#493×C₇₁	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	—	16	4	no	—
187	C₂	C₇₁×#1957	1	1	yes	characteristic
188	C₂	C₇₁×#1978	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#493×C₇₁

C₂

yes

maximal, characteristic

C₇₁×#1848

C₂

yes

maximal, characteristic, complemented

C₇₁×#1545

C₂

yes

maximal, complemented

C₇₁×#1450

C₂

yes

maximal, characteristic, complemented

C₂²×D₈×C₇₁

—

C₂×#127×C₇₁

C₂²

yes

—

#493×C₇₁

—

#493×C₇₁

—

#493×C₇₁

C₄

yes

—

#431×C₇₁

C₂²

yes

—

C₂×#108×C₇₁

—

#423×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#331×C₇₁

—

#322×C₇₁

—

C₈²×C₇₁

—

C₂²×D₄×C₇₁

—

C₂²×D₄×C₇₁

—

C₂×D₈×C₇₁

—

C₂×D₈×C₇₁

—

C₂×D₈×C₇₁

—

C₂×D₈×C₇₁

—

C₂×M₁₆×C₇₁

—

C₂²×C₈×C₇₁

D₄

yes

—

#127×C₇₁

—

#127×C₇₁

—

#127×C₇₁

C₂×C₄

yes

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

#108×C₇₁

—

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

D₄

yes

—

C₂²⋊C₈×C₇₁

—

C₄×C₈×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

characteristic

C₂⁴×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₁

#41

yes

—

D₈×C₇₁

—

D₈×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₂×C₈×C₇₁

D₈

yes

—

C₂×C₈×C₇₁

SD₁₆

yes

—

C₄²×C₇₁

C₂²×C₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

—

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

#124

yes

characteristic

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄×D₄

yes

—

C₂×C₄×C₇₁

#121

yes

—

C₂×C₄×C₇₁

#127

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

—

C₂×C₄×C₇₁

C₂×SD₁₆

yes

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

#102

yes

—

C₈×C₇₁

#102

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#390

yes

characteristic

C₂²×C₇₁

#486

yes

—

C₄×C₇₁

—

C₄×C₇₁

C₂×#102

yes

—

C₄×C₇₁

#472

yes

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₇₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#1957

yes

characteristic

C₂×C₇₁

#1978

yes

—

C₂×#493

C₂×C₇₁

yes

characteristic

#1848

C₂×C₇₁

yes

characteristic, complemented

#1545

C₂×C₇₁

yes

complemented

#1450

C₂×C₇₁

yes

characteristic, complemented

C₇₁

#9773

yes

characteristic, direct product factor, Sylow

100

C₂²×D₈

—

101

C₂×#127

C₂²×C₇₁

yes

—

102

#493

—

103

#493

—

104

#493

C₄×C₇₁

yes

—

105

#431

C₂²×C₇₁

yes

—

106

C₂×#108

—

107

#423

C₂²×C₇₁

yes

—

108

C₂×C₄×C₈

C₂²×C₇₁

yes

characteristic

109

#331

—

110

#322

—

111

C₈²

—

112

C₂²×D₄

—

113

C₂²×D₄

—

114

C₂×D₈

—

115

C₂×D₈

—

116

C₂×D₈

—

117

C₂×D₈

—

118

C₂×M₁₆

—

119

C₂²×C₈

D₄×C₇₁

yes

—

120

#127

—

121

#127

—

122

#127

C₂×C₄×C₇₁

yes

—

123

C₂×C₄²

C₂³×C₇₁

yes

characteristic

124

#108

—

125

C₄⋊C₈

—

126

C₄⋊C₈

D₄×C₇₁

yes

—

127

C₂²⋊C₈

—

128

C₄×C₈

—

129

C₄×C₈

C₂×C₄×C₇₁

yes

—

130

C₄×C₈

D₄×C₇₁

yes

characteristic

131

C₄×C₈

D₄×C₇₁

yes

characteristic

132

C₂⁴

—

133

C₂×D₄

—

134

C₂×D₄

—

135

C₂×D₄

—

136

C₂×D₄

—

137

C₂×D₄

—

138

C₂×D₄

—

139

C₂×D₄

—

140

C₂²×C₄

C₂×D₄×C₇₁

yes

characteristic

141

C₂²×C₄

#41×C₇₁

yes

—

142

D₈

—

143

D₈

—

144

M₁₆

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

C₂²⋊C₄×C₇₁

yes

—

148

C₂×C₈

C₂²⋊C₄×C₇₁

yes

—

149

C₂×C₈

D₈×C₇₁

yes

—

150

C₂×C₈

SD₁₆×C₇₁

yes

—

151

C₄²

C₂²×C₄×C₇₁

yes

characteristic

152

C₄²

C₂×D₄×C₇₁

yes

characteristic

153

C₄²

C₂×D₄×C₇₁

yes

—

154

C₂³

—

155

C₂³

—

156

C₂³

—

157

C₂³

#124×C₇₁

yes

characteristic

158

D₄

—

159

D₄

—

160

D₄

—

161

D₄

—

162

D₄

—

163

C₂×C₄

—

164

C₂×C₄

C₂×C₂²⋊C₄×C₇₁

yes

characteristic

165

C₂×C₄

C₄×D₄×C₇₁

yes

—

166

C₂×C₄

#121×C₇₁

yes

—

167

C₂×C₄

#127×C₇₁

yes

characteristic

168

C₂×C₄

C₂×D₈×C₇₁

yes

—

169

C₂×C₄

C₂×SD₁₆×C₇₁

yes

—

170

C₈

—

171

C₈

—

172

C₈

#102×C₇₁

yes

—

173

C₈

#102×C₇₁

yes

—

174

C₂²

—

175

C₂²

—

176

C₂²

—

177

C₂²

—

178

C₂²

#390×C₇₁

yes

characteristic

179

C₂²

#486×C₇₁

yes

—

180

C₄

—

181

C₄

C₂×#102×C₇₁

yes

—

182

C₄

#472×C₇₁

yes

—

183

C₄

#492×C₇₁

yes

characteristic

184

C₄

#493×C₇₁

yes

characteristic

185

C₂

—

186

C₂

—

187

C₂

C₇₁×#1957

yes

characteristic

188

C₂

C₇₁×#1978

yes

—

	subgroup	G/H
center, socle	C₂²×C₇₁	#390
commutator	C₄²	C₂²×C₄×C₇₁
Frattini, cosocle	C₂×C₄²	C₂³×C₇₁
prime radical	C₂×#493×C₇₁	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₇₁

#390

commutator

C₄²

C₂²×C₄×C₇₁

Frattini, cosocle

C₂×C₄²

C₂³×C₇₁

prime radical

C₂×#493×C₇₁

C₂

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ G

N	H ≅ G/N	type
2. C₇₁×#1848	C₂	semidirect
3. C₇₁×#1545	C₂	semidirect
4. C₇₁×#1450	C₂	semidirect
90. #9773	C₇₁	direct (N × H)
96. #1848	C₂×C₇₁	semidirect
97. #1545	C₂×C₇₁	semidirect
98. #1450	C₂×C₇₁	semidirect
99. C₇₁	#9773	direct (N × H)

H ≅ G/N

type

2. C₇₁×#1848

C₂

semidirect

3. C₇₁×#1545

C₂

semidirect

4. C₇₁×#1450

C₂

semidirect

90. #9773

C₇₁

direct (N × H)

96. #1848

C₂×C₇₁

semidirect

97. #1545

C₂×C₇₁

semidirect

98. #1450

C₂×C₇₁

semidirect

99. C₇₁

#9773

direct (N × H)

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

subgroup	index
C₂×#493×C₇₁	2
C₇₁×#1848	2
C₇₁×#1545	2
C₇₁×#1450	2
#9773	71

subgroup

index

C₂×#493×C₇₁

C₇₁×#1848

C₇₁×#1545

C₇₁×#1450

#9773

Subgroups — C₇₁×#9773 #31830645

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C71×#9773 #31830645

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#9773 #31830645