Group #31909055 — subgroups

Subgroups — #31909055

Order 18176.

250 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 103 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	#22590142	C₂	1	2	yes	maximal
2	C₈²×D₇₁	C₂	1	1	yes	maximal, characteristic
3	#22586410	C₂	1	1	yes	maximal, characteristic, complemented
4	#22586403	C₂	1	2	yes	maximal, complemented
5	C₇₁×#1370	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×C₈×D₇₁	C₄	1	1	yes	characteristic
7	C₄×C₈×D₇₁	C₄	1	1	yes	characteristic
8	C₄×C₈×D₇₁	C₂²	1	1	yes	characteristic
9	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	2	yes	—
10	(C₂×D₇₁)⋊C₁₆	—	2	2	no	—
11	#17985774	—	2	2	no	—
12	C₈×C₇₁⋊C₈	C₄	1	1	yes	characteristic
13	C₈×C₇₁⋊C₈	C₄	1	1	yes	characteristic
14	C₈×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
15	C₄⋊C₁₆×C₇₁	C₂²	1	2	yes	—
16	C₈²×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₇₁	—	2	1	no	—
18	C₂×C₈×D₇₁	C₈	1	1	yes	characteristic
19	C₂×C₈×D₇₁	C₈	1	1	yes	characteristic
20	C₂×C₈×D₇₁	D₄	1	1	yes	characteristic
21	C₂×C₈×D₇₁	Q₈	1	1	yes	characteristic
22	C₄²×D₇₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₇₁⋊C₁₆	—	2	2	no	—
24	C₄×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₇₁	—	2	1	no	—
28	C₈×Dic₇₁	C₈	1	1	yes	characteristic
29	C₈×Dic₇₁	C₈	1	1	yes	characteristic
30	C₈×Dic₇₁	D₄	1	1	yes	characteristic
31	C₈×Dic₇₁	Q₈	1	1	yes	characteristic
32	C₂×C₁₆×C₇₁	—	2	2	no	—
33	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
36	C₂×C₄×D₇₁	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×D₇₁	C₂×C₈	1	1	yes	characteristic
38	C₂×C₄×D₇₁	M₁₆	1	1	yes	characteristic
39	C₂×C₇₁⋊C₈	—	2	1	no	—
40	C₂×C₇₁⋊C₈	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₇₁⋊C₈	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₇₁⋊C₈	C₂×C₈	1	1	yes	characteristic
43	C₂×C₇₁⋊C₈	C₂×C₈	1	1	yes	characteristic
44	C₈×D₇₁	—	2	4	no	—
45	C₈×D₇₁	—	2	2	no	—
46	C₈×D₇₁	—	2	1	no	—
47	C₈×D₇₁	—	2	1	no	—
48	C₈×D₇₁	—	2	2	no	—
49	C₈×D₇₁	—	2	2	no	—
50	C₄×Dic₇₁	C₄⋊C₄	1	1	yes	characteristic
51	C₄×Dic₇₁	C₂×C₈	1	1	yes	characteristic
52	C₄×Dic₇₁	M₁₆	1	1	yes	characteristic
53	C₇₁⋊C₁₆	—	4	2	no	—
54	C₂×C₈×C₇₁	—	2	1	no	—
55	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
56	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
57	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
60	C₁₆×C₇₁	—	4	2	no	—
61	C₂²×D₇₁	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
65	C₂×Dic₇₁	C₄⋊C₈	1	1	yes	characteristic
66	C₂×Dic₇₁	C₄⋊C₈	1	1	yes	characteristic
67	C₂×Dic₇₁	C₄⋊C₈	1	1	yes	characteristic
68	C₄×D₇₁	—	2	2	no	—
69	C₄×D₇₁	—	2	1	no	—
70	C₄×D₇₁	—	2	1	no	—
71	C₄×D₇₁	—	2	1	no	—
72	C₄×D₇₁	—	2	1	no	—
73	C₈×C₇₁	—	2	2	no	—
74	C₈×C₇₁	—	2	1	no	—
75	C₈×C₇₁	—	2	1	no	—
76	C₈×C₇₁	M₃₂	1	2	yes	—
77	C₈×C₇₁	#135	1	1	yes	characteristic
78	C₈×C₇₁	#135	1	1	yes	characteristic
79	C₇₁⋊C₈	—	2	2	no	—
80	C₇₁⋊C₈	—	2	1	no	—
81	C₇₁⋊C₈	—	2	1	no	—
82	C₇₁⋊C₈	#108	1	2	yes	—
83	C₇₁⋊C₈	M₃₂	1	2	yes	—
84	C₂²×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
85	C₂×D₇₁	—	2	1	no	—
86	C₂×D₇₁	—	2	1	no	—
87	C₂×D₇₁	#364	1	2	yes	—
88	C₄×C₇₁	—	2	1	no	—
89	C₄×C₇₁	#427	1	1	yes	characteristic
90	C₄×C₇₁	C₂×#108	1	1	yes	characteristic
91	C₄×C₇₁	C₂×M₃₂	1	1	yes	characteristic
92	C₄×C₇₁	C₂×M₃₂	1	1	yes	characteristic
93	Dic₇₁	—	2	1	no	—
94	Dic₇₁	—	2	1	no	—
95	Dic₇₁	—	2	1	no	—
96	#12013	—	71	1	no	maximal, Sylow
97	C₂×C₇₁	#1564	1	1	yes	characteristic
98	C₂×C₇₁	#2150	1	1	yes	characteristic
99	C₂×C₇₁	C₂×#364	1	1	yes	characteristic
100	D₇₁	—	2	2	no	—
101	#2163	—	71	2	no	—
102	C₂×C₈²	—	71	1	no	—
103	#1370	—	71	1	no	—
104	#1370	D₇₁	1	1	yes	characteristic, complemented
105	#1367	—	71	2	no	—
106	C₇₁	#12013	1	1	yes	characteristic, complemented, Sylow
107	C₂×C₄×C₈	—	71	1	no	—
108	C₂×C₄×C₈	—	71	1	no	—
109	C₂×C₄×C₈	—	71	1	no	—
110	C₄⋊C₁₆	—	71	2	no	—
111	C₄⋊C₁₆	C₂×D₇₁	1	2	yes	—
112	C₂²⋊C₁₆	—	142	2	no	—
113	#346	—	142	2	no	—
114	C₈²	—	71	1	no	—
115	C₈²	—	71	1	no	—
116	C₈²	—	71	1	no	—
117	C₈²	C₂×D₇₁	1	1	yes	characteristic
118	C₂²×C₈	—	71	1	no	—
119	C₂²×C₈	—	71	1	no	—
120	C₂²×C₈	—	71	1	no	—
121	C₂²×C₈	—	71	1	no	—
122	C₂²×C₈	—	142	1	no	—
123	C₂×C₄²	—	71	1	no	—
124	C₂×C₁₆	—	2	2	no	—
125	C₂×C₁₆	—	142	2	no	—
126	C₄×C₈	—	71	1	no	—
127	C₄×C₈	—	71	1	no	—
128	C₄×C₈	—	71	1	no	—
129	C₄×C₈	—	71	1	no	—
130	C₄×C₈	—	71	1	no	—
131	C₄×C₈	—	71	1	no	—
132	C₄×C₈	—	71	1	no	—
133	C₄×C₈	—	142	1	no	—
134	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
135	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
136	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
137	C₂²×C₄	—	71	1	no	—
138	C₂²×C₄	—	71	1	no	—
139	C₂²×C₄	—	71	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	71	1	no	—
142	C₂×C₈	—	71	1	no	—
143	C₂×C₈	—	71	1	no	—
144	C₂×C₈	—	71	1	no	—
145	C₂×C₈	—	142	2	no	—
146	C₂×C₈	—	142	2	no	—
147	C₂×C₈	—	142	4	no	—
148	C₂×C₈	—	142	1	no	—
149	C₂×C₈	—	142	2	no	—
150	C₂×C₈	—	142	1	no	—
151	C₂×C₈	—	142	1	no	—
152	C₂×C₈	C₈×D₇₁	1	1	yes	characteristic
153	C₂×C₈	C₈×D₇₁	1	1	yes	characteristic
154	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
155	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
156	C₄²	—	71	1	no	—
157	C₄²	—	71	1	no	—
158	C₄²	—	71	1	no	—
159	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
160	C₁₆	—	4	2	no	—
161	C₁₆	—	284	2	no	—
162	C₂³	—	71	1	no	—
163	C₂×C₄	—	71	1	no	—
164	C₂×C₄	—	71	1	no	—
165	C₂×C₄	—	71	1	no	—
166	C₂×C₄	—	142	2	no	—
167	C₂×C₄	—	142	1	no	—
168	C₂×C₄	—	142	1	no	—
169	C₂×C₄	—	142	1	no	—
170	C₂×C₄	—	142	1	no	—
171	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
172	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
173	C₂×C₄	M₁₆×D₇₁	1	1	yes	characteristic
174	C₈	—	2	2	no	—
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	71	2	no	—
178	C₈	—	71	2	no	—
179	C₈	—	142	2	no	—
180	C₈	—	142	1	no	—
181	C₈	—	142	1	no	—
182	C₈	#14556067	1	2	yes	—
183	C₈	#14556147	1	1	yes	characteristic
184	C₈	#14556151	1	1	yes	characteristic
185	C₂²	—	71	2	no	—
186	C₂²	—	142	1	no	—
187	C₂²	—	142	1	no	—
188	C₂²	C₄⋊C₈×D₇₁	1	1	yes	characteristic
189	C₄	—	2	1	no	—
190	C₄	—	142	1	no	—
191	C₄	—	142	1	no	—
192	C₄	—	142	1	no	—
193	C₄	#17986036	1	1	yes	characteristic
194	C₄	#108×D₇₁	1	1	yes	characteristic
195	C₄	M₃₂×D₇₁	1	1	yes	characteristic
196	C₄	C₂×#14556067	1	1	yes	characteristic
197	C₂	—	142	2	no	—
198	C₂	#22587087	1	1	yes	characteristic
199	C₂	#22590094	1	1	yes	characteristic
200	C₂	#364×D₇₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590142

C₂

yes

maximal

C₈²×D₇₁

C₂

yes

maximal, characteristic

#22586410

C₂

yes

maximal, characteristic, complemented

#22586403

C₂

yes

maximal, complemented

C₇₁×#1370

C₂

yes

maximal, characteristic, complemented

C₄×C₈×D₇₁

C₄

yes

characteristic

C₄×C₈×D₇₁

C₄

yes

characteristic

C₄×C₈×D₇₁

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₁₆)

C₂²

yes

—

(C₂×D₇₁)⋊C₁₆

—

#17985774

—

C₈×C₇₁⋊C₈

C₄

yes

characteristic

C₈×C₇₁⋊C₈

C₄

yes

characteristic

C₈×C₇₁⋊C₈

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₁

C₂²

yes

—

C₈²×C₇₁

C₂²

yes

characteristic

C₂×C₈×D₇₁

—

C₂×C₈×D₇₁

C₈

yes

characteristic

C₂×C₈×D₇₁

C₈

yes

characteristic

C₂×C₈×D₇₁

D₄

yes

characteristic

C₂×C₈×D₇₁

Q₈

yes

characteristic

C₄²×D₇₁

C₂×C₄

yes

characteristic

C₂×C₇₁⋊C₁₆

—

C₄×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₄×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₄×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₈×Dic₇₁

—

C₈×Dic₇₁

C₈

yes

characteristic

C₈×Dic₇₁

C₈

yes

characteristic

C₈×Dic₇₁

D₄

yes

characteristic

C₈×Dic₇₁

Q₈

yes

characteristic

C₂×C₁₆×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₄×D₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×D₇₁

C₂×C₈

yes

characteristic

C₂×C₄×D₇₁

M₁₆

yes

characteristic

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₇₁⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₇₁⋊C₈

C₂×C₈

yes

characteristic

C₂×C₇₁⋊C₈

C₂×C₈

yes

characteristic

C₈×D₇₁

—

C₈×D₇₁

—

C₈×D₇₁

—

C₈×D₇₁

—

C₈×D₇₁

—

C₈×D₇₁

—

C₄×Dic₇₁

C₄⋊C₄

yes

characteristic

C₄×Dic₇₁

C₂×C₈

yes

characteristic

C₄×Dic₇₁

M₁₆

yes

characteristic

C₇₁⋊C₁₆

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×C₈

yes

characteristic

C₂×C₈×C₇₁

C₂×C₈

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

characteristic

C₄²×C₇₁

C₂²×C₄

yes

characteristic

C₁₆×C₇₁

—

C₂²×D₇₁

C₄⋊C₈

yes

characteristic

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₁

C₂×M₁₆

yes

characteristic

C₂×Dic₇₁

C₄⋊C₈

yes

characteristic

C₂×Dic₇₁

C₄⋊C₈

yes

characteristic

C₂×Dic₇₁

C₄⋊C₈

yes

characteristic

C₄×D₇₁

—

C₄×D₇₁

—

C₄×D₇₁

—

C₄×D₇₁

—

C₄×D₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

M₃₂

yes

—

C₈×C₇₁

yes

C₈×C₇₁

yes

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

#108

yes

—

C₇₁⋊C₈

M₃₂

yes

—

C₂²×C₇₁

C₂×C₄⋊C₈

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

#364

yes

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

C₂×M₃₂

yes

characteristic

C₄×C₇₁

C₂×M₃₂

yes

characteristic

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

100

D₇₁

—

101

#2163

—

102

C₂×C₈²

—

103

#1370

—

104

#1370

D₇₁

yes

characteristic, complemented

105

#1367

—

106

C₇₁

#12013

yes

characteristic, complemented, Sylow

107

C₂×C₄×C₈

—

108

C₂×C₄×C₈

—

109

C₂×C₄×C₈

—

110

C₄⋊C₁₆

—

111

C₄⋊C₁₆

C₂×D₇₁

yes

—

112

C₂²⋊C₁₆

—

142

—

113

#346

—

142

—

114

C₈²

—

115

C₈²

—

116

C₈²

—

117

C₈²

C₂×D₇₁

yes

characteristic

118

C₂²×C₈

—

119

C₂²×C₈

—

120

C₂²×C₈

—

121

C₂²×C₈

—

122

C₂²×C₈

—

142

—

123

C₂×C₄²

—

124

C₂×C₁₆

—

125

C₂×C₁₆

—

142

—

126

C₄×C₈

—

127

C₄×C₈

—

128

C₄×C₈

—

129

C₄×C₈

—

130

C₄×C₈

—

131

C₄×C₈

—

132

C₄×C₈

—

133

C₄×C₈

—

142

—

134

C₄×C₈

C₄×D₇₁

yes

characteristic

135

C₄×C₈

C₄×D₇₁

yes

characteristic

136

C₄×C₈

C₂²×D₇₁

yes

characteristic

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

142

—

146

C₂×C₈

—

142

—

147

C₂×C₈

—

142

—

148

C₂×C₈

—

142

—

149

C₂×C₈

—

142

—

150

C₂×C₈

—

142

—

151

C₂×C₈

—

142

—

152

C₂×C₈

C₈×D₇₁

yes

characteristic

153

C₂×C₈

C₈×D₇₁

yes

characteristic

154

C₂×C₈

D₄×D₇₁

yes

characteristic

155

C₂×C₈

Q₈×D₇₁

yes

characteristic

156

C₄²

—

157

C₄²

—

158

C₄²

—

159

C₄²

C₂×C₄×D₇₁

yes

characteristic

160

C₁₆

—

161

C₁₆

—

284

—

162

C₂³

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

142

—

167

C₂×C₄

—

142

—

168

C₂×C₄

—

142

—

169

C₂×C₄

—

142

—

170

C₂×C₄

—

142

—

171

C₂×C₄

C₄⋊C₄×D₇₁

yes

characteristic

172

C₂×C₄

C₂×C₈×D₇₁

yes

characteristic

173

C₂×C₄

M₁₆×D₇₁

yes

characteristic

174

C₈

—

175

C₈

—

176

C₈

—

177

C₈

—

178

C₈

—

179

C₈

—

142

—

180

C₈

—

142

—

181

C₈

—

142

—

182

C₈

#14556067

yes

—

183

C₈

#14556147

yes

characteristic

184

C₈

#14556151

yes

characteristic

185

C₂²

—

186

C₂²

—

142

—

187

C₂²

—

142

—

188

C₂²

C₄⋊C₈×D₇₁

yes

characteristic

189

C₄

—

190

C₄

—

142

—

191

C₄

—

142

—

192

C₄

—

142

—

193

C₄

#17986036

yes

characteristic

194

C₄

#108×D₇₁

yes

characteristic

195

C₄

M₃₂×D₇₁

yes

characteristic

196

C₄

C₂×#14556067

yes

characteristic

197

C₂

—

142

—

198

C₂

#22587087

yes

characteristic

199

C₂

#22590094

yes

characteristic

200

C₂

#364×D₇₁

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#1370	C₂
socle	C₂²×C₇₁	C₂×C₄⋊C₈
prime radical	C₂²×D₇₁	C₄⋊C₈
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	#1370	D₇₁
hypocenter	C₇₁	#12013
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1370

subgroup

G/H

center

C₂×C₈

D₄×D₇₁

commutator

C₂×C₄×C₇₁

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₇₁×#1370

C₂

socle

C₂²×C₇₁

C₂×C₄⋊C₈

prime radical

C₂²×D₇₁

C₄⋊C₈

cosocle

C₄×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1370

N	H ≅ G/N	type
3. #22586410	C₂	semidirect
4. #22586403	C₂	semidirect
5. C₇₁×#1370	C₂	semidirect
104. #1370	D₇₁	semidirect
106. C₇₁	#12013	semidirect

H ≅ G/N

type

3. #22586410

C₂

semidirect

4. #22586403

C₂

semidirect

5. C₇₁×#1370

C₂

semidirect

104. #1370

D₇₁

semidirect

106. C₇₁

#12013

semidirect

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

subgroup	index
#22590142	2
C₈²×D₇₁	2
#22586410	2
#22586403	2
C₇₁×#1370	2
#12013	71

subgroup

index

C₈²×D₇₁

C₇₁×#1370

Subgroups — #31909055

Conjugacy classes of subgroups

Complements

Maximal subgroups