Group #31896149 — subgroups

Subgroups — #31896149

Order 18176.

298 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 123 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	#22589113	C₂	1	2	yes	maximal
2	C₂×#17985750	C₂	1	1	yes	maximal, characteristic
3	C₈×#11765849	C₂	1	1	yes	maximal, characteristic
4	#22586235	C₂	1	2	yes	maximal
5	C₇₁×#1566	C₂	1	1	yes	maximal, characteristic
6	#17986052	C₂²	1	1	yes	characteristic
7	C₂×#14556063	—	2	2	no	—
8	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
9	C₄×#11765849	C₂²	1	1	yes	characteristic
10	#17985763	—	2	2	no	—
11	#17985750	C₄	1	2	yes	—
12	#17985750	C₄	1	2	yes	—
13	#17985721	—	2	1	no	—
14	C₈×C₇₁⋊C₈	—	2	1	no	—
15	#423×C₇₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	#334×C₇₁	—	2	1	no	—
18	#334×C₇₁	—	2	1	no	—
19	C₂×#11765849	D₄	1	2	yes	—
20	C₂²×C₇₁⋊C₈	—	2	2	no	—
21	#14556063	—	4	4	no	—
22	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
23	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
24	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	4	yes	—
25	C₂²⋊(C₇₁⋊C₈)	Q₈	1	2	yes	—
26	#14556039	—	2	1	no	—
27	C₄×C₇₁⋊C₈	—	2	1	no	—
28	C₄×C₇₁⋊C₈	—	2	1	no	—
29	C₂×M₁₆×C₇₁	—	2	2	no	—
30	C₂²×C₈×C₇₁	Q₈	1	1	yes	characteristic
31	C₂²×C₈×C₇₁	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₇₁	—	2	2	no	—
34	C₄⋊C₈×C₇₁	—	2	2	no	—
35	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
38	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
39	#11765849	—	2	2	no	—
40	#11765849	SD₁₆	1	4	yes	—
41	C₂×C₇₁⋊C₈	—	2	2	no	—
42	C₂×C₇₁⋊C₈	—	2	4	no	—
43	C₂×C₇₁⋊C₈	—	2	2	no	—
44	C₂×C₇₁⋊C₈	—	4	2	no	—
45	C₂×C₇₁⋊C₈	—	4	2	no	—
46	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
49	M₁₆×C₇₁	—	4	2	no	—
50	M₁₆×C₇₁	—	4	2	no	—
51	C₂×C₈×C₇₁	—	2	2	no	—
52	C₂×C₈×C₇₁	—	4	1	no	—
53	C₂×C₈×C₇₁	—	4	2	no	—
54	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
58	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₇₁	#41	1	1	yes	characteristic
61	C₄²×C₇₁	#41	1	1	yes	characteristic
62	C₂³×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×C₇₁	—	2	1	no	—
64	C₂×C₄×C₇₁	—	2	1	no	—
65	C₂×C₄×C₇₁	—	4	1	no	—
66	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
70	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₇₁	#125	1	1	yes	characteristic
72	C₂×C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
73	C₂×C₄×C₇₁	#135	1	2	yes	—
74	C₈×C₇₁	—	2	1	no	—
75	C₈×C₇₁	—	2	1	no	—
76	C₈×C₇₁	—	2	1	no	—
77	C₈×C₇₁	—	8	2	no	—
78	C₈×C₇₁	#108	1	2	yes	complemented
79	C₇₁⋊C₈	—	2	4	no	—
80	C₇₁⋊C₈	—	4	2	no	—
81	C₇₁⋊C₈	—	4	4	no	—
82	C₂²×C₇₁	—	2	1	no	—
83	C₂²×C₇₁	—	2	1	no	—
84	C₂²×C₇₁	#389	1	1	yes	characteristic
85	C₂²×C₇₁	#498	1	1	yes	characteristic
86	C₂²×C₇₁	#499	1	1	yes	characteristic
87	C₄×C₇₁	—	2	1	no	—
88	C₄×C₇₁	—	2	1	no	—
89	C₄×C₇₁	—	2	1	no	—
90	C₄×C₇₁	C₂×#108	1	1	yes	characteristic
91	C₄×C₇₁	#430	1	1	yes	characteristic
92	C₄×C₇₁	C₄×SD₁₆	1	2	yes	—
93	C₄×C₇₁	#465	1	2	yes	—
94	#9623	—	71	1	no	maximal, Sylow
95	C₂×C₇₁	—	2	1	no	—
96	C₂×C₇₁	—	2	1	no	—
97	C₂×C₇₁	#1941	1	1	yes	characteristic
98	C₂×C₇₁	#1950	1	1	yes	characteristic
99	C₂×C₇₁	#1953	1	1	yes	characteristic
100	#1950	—	71	2	no	—
101	#1566	D₇₁	1	1	yes	characteristic
102	C₂×#334	—	71	1	no	—
103	C₈×M₁₆	—	71	1	no	—
104	#1301	—	71	2	no	—
105	C₇₁	#9623	1	1	yes	characteristic, complemented, Sylow
106	#431	—	71	1	no	—
107	C₂×#108	—	142	2	no	—
108	#423	C₂×D₇₁	1	2	yes	—
109	C₂×C₄⋊C₈	—	71	2	no	—
110	C₄×M₁₆	—	71	1	no	—
111	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
112	#341	—	142	2	no	—
113	#334	—	2	1	no	—
114	#334	—	2	1	no	—
115	#334	—	71	2	no	—
116	#334	—	71	2	no	—
117	#322	—	142	1	no	—
118	C₈²	—	142	1	no	—
119	C₂×M₁₆	—	2	2	no	—
120	C₂×M₁₆	—	71	2	no	—
121	C₂²×C₈	—	142	2	no	—
122	C₂²×C₈	Dic₁₄₂	1	1	yes	characteristic
123	C₂²×C₈	Dic₁₄₂	1	1	yes	characteristic
124	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
125	#108	—	284	4	no	—
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	2	2	no	—
128	C₄⋊C₈	—	71	4	no	—
129	C₄⋊C₈	—	142	2	no	—
130	C₄⋊C₈	—	142	1	no	—
131	C₂²⋊C₈	—	71	2	no	—
132	#97	—	142	1	no	—
133	C₄×C₈	—	142	1	no	—
134	C₄×C₈	—	142	1	no	—
135	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
136	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
137	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
138	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
139	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
140	C₂²×C₄	Q₈×D₇₁	1	1	yes	characteristic
141	C₂²×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
142	M₁₆	—	4	2	no	—
143	M₁₆	—	4	2	no	—
144	M₁₆	—	71	4	no	—
145	M₁₆	—	142	2	no	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	4	1	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	284	2	no	—
150	C₂×C₈	—	284	2	no	—
151	C₂×C₈	—	142	4	no	—
152	C₂×C₈	—	142	2	no	—
153	C₂×C₈	—	142	2	no	—
154	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
155	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
156	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
157	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
158	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
159	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
160	C₄²	#11765864	1	1	yes	characteristic
161	C₄²	#11765866	1	1	yes	characteristic
162	C₂³	#14556127	1	1	yes	characteristic
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	4	1	no	—
166	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
167	C₂×C₄	#14556090	1	1	yes	characteristic
168	C₂×C₄	#14556093	1	1	yes	characteristic
169	C₂×C₄	#14556099	1	1	yes	characteristic
170	C₂×C₄	#14556104	1	1	yes	characteristic
171	C₂×C₄	#14556119	1	1	yes	characteristic
172	C₂×C₄	SD₁₆×D₇₁	1	2	yes	—
173	C₂×C₄	#14556149	1	2	yes	—
174	C₈	—	2	1	no	—
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	8	2	no	—
178	C₈	—	284	2	no	—
179	C₈	—	284	4	no	—
180	C₈	—	142	4	no	—
181	C₈	#14556063	1	2	yes	complemented
182	C₂²	—	2	1	no	—
183	C₂²	—	2	1	no	—
184	C₂²	#17985895	1	1	yes	characteristic
185	C₂²	#17986396	1	1	yes	characteristic
186	C₂²	#17986399	1	1	yes	characteristic
187	C₄	—	2	1	no	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	C₂×#14556063	1	1	yes	characteristic
191	C₄	#17986047	1	1	yes	characteristic
192	C₄	#17986086	1	2	yes	—
193	C₄	#17986225	1	2	yes	—
194	C₂	—	2	1	no	—
195	C₂	—	2	1	no	—
196	C₂	#22589066	1	1	yes	characteristic
197	C₂	#22589113	1	1	yes	characteristic
198	C₂	#22589128	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22589113

C₂

yes

maximal

C₂×#17985750

C₂

yes

maximal, characteristic

C₈×#11765849

C₂

yes

maximal, characteristic

#22586235

C₂

yes

maximal

C₇₁×#1566

C₂

yes

maximal, characteristic

#17986052

C₂²

yes

characteristic

C₂×#14556063

—

C₂×C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

C₄×#11765849

C₂²

yes

characteristic

#17985763

—

#17985750

C₄

yes

—

#17985750

C₄

yes

—

#17985721

—

C₈×C₇₁⋊C₈

—

#423×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#334×C₇₁

—

#334×C₇₁

—

C₂×#11765849

D₄

yes

—

C₂²×C₇₁⋊C₈

—

#14556063

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

C₂×C₄

yes

—

C₂²⋊(C₇₁⋊C₈)

Q₈

yes

—

#14556039

—

C₄×C₇₁⋊C₈

—

C₄×C₇₁⋊C₈

—

C₂×M₁₆×C₇₁

—

C₂²×C₈×C₇₁

Q₈

yes

characteristic

C₂²×C₈×C₇₁

Q₈

yes

characteristic

C₂×C₄²×C₇₁

C₂³

yes

characteristic

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

characteristic

#11765849

—

#11765849

SD₁₆

yes

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₂²×C₄×C₇₁

C₂×Q₈

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₁

C₄⋊C₄

yes

characteristic

C₄²×C₇₁

C₂²×C₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

C₄²×C₇₁

yes

C₄²×C₇₁

yes

C₂³×C₇₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄×D₄

yes

characteristic

C₂×C₄×C₇₁

C₄×Q₈

yes

characteristic

C₂×C₄×C₇₁

#121

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇₁

#125

yes

characteristic

C₂×C₄×C₇₁

C₂×SD₁₆

yes

—

C₂×C₄×C₇₁

#135

yes

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

#108

yes

complemented

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

C₄×SD₁₆

yes

—

C₄×C₇₁

#465

yes

—

#9623

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

100

—

101

#1566

D₇₁

yes

characteristic

102

C₂×#334

—

103

C₈×M₁₆

—

104

#1301

—

105

C₇₁

#9623

yes

characteristic, complemented, Sylow

106

#431

—

107

C₂×#108

—

142

—

108

#423

C₂×D₇₁

yes

—

109

C₂×C₄⋊C₈

—

110

C₄×M₁₆

—

111

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

112

#341

—

142

—

113

#334

—

114

#334

—

115

#334

—

116

#334

—

117

#322

—

142

—

118

C₈²

—

142

—

119

C₂×M₁₆

—

120

C₂×M₁₆

—

121

C₂²×C₈

—

142

—

122

C₂²×C₈

Dic₁₄₂

yes

characteristic

123

C₂²×C₈

Dic₁₄₂

yes

characteristic

124

C₂×C₄²

C₂²×D₇₁

yes

characteristic

125

#108

—

284

—

126

C₄⋊C₈

—

127

C₄⋊C₈

—

128

C₄⋊C₈

—

129

C₄⋊C₈

—

142

—

130

C₄⋊C₈

—

142

—

131

C₂²⋊C₈

—

132

#97

—

142

—

133

C₄×C₈

—

142

—

134

C₄×C₈

—

142

—

135

C₄×C₈

C₄×D₇₁

yes

characteristic

136

C₄×C₈

C₄×D₇₁

yes

characteristic

137

C₄×C₈

C₂²⋊D₇₁

yes

characteristic

138

C₄×C₈

C₂²⋊D₇₁

yes

characteristic

139

C₂²×C₄

D₄×D₇₁

yes

characteristic

140

C₂²×C₄

Q₈×D₇₁

yes

characteristic

141

C₂²×C₄

C₂×Dic₁₄₂

yes

characteristic

142

M₁₆

—

143

M₁₆

—

144

M₁₆

—

145

M₁₆

—

142

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

284

—

150

C₂×C₈

—

284

—

151

C₂×C₈

—

142

—

152

C₂×C₈

—

142

—

153

C₂×C₈

—

142

—

154

C₂×C₈

Dic₇₁⋊C₄

yes

characteristic

155

C₂×C₈

Dic₇₁⋊C₄

yes

characteristic

156

C₂×C₈

Dic₇₁⋊C₄

yes

characteristic

157

C₂×C₈

Dic₇₁⋊C₄

yes

characteristic

158

C₄²

C₂×C₄×D₇₁

yes

characteristic

159

C₄²

C₂×C₂²⋊D₇₁

yes

characteristic

160

C₄²

#11765864

yes

characteristic

161

C₄²

#11765866

yes

characteristic

162

C₂³

#14556127

yes

characteristic

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

C₂×Dic₇₁⋊C₄

yes

characteristic

167

C₂×C₄

#14556090

yes

characteristic

168

C₂×C₄

#14556093

yes

characteristic

169

C₂×C₄

#14556099

yes

characteristic

170

C₂×C₄

#14556104

yes

characteristic

171

C₂×C₄

#14556119

yes

characteristic

172

C₂×C₄

SD₁₆×D₇₁

yes

—

173

C₂×C₄

#14556149

yes

—

174

C₈

—

175

C₈

—

176

C₈

—

177

C₈

—

178

C₈

—

284

—

179

C₈

—

284

—

180

C₈

—

142

—

181

C₈

#14556063

yes

complemented

182

C₂²

—

183

C₂²

—

184

C₂²

#17985895

yes

characteristic

185

C₂²

#17986396

yes

characteristic

186

C₂²

#17986399

yes

characteristic

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

C₂×#14556063

yes

characteristic

191

C₄

#17986047

yes

characteristic

192

C₄

#17986086

yes

—

193

C₄

#17986225

yes

—

194

C₂

—

195

C₂

—

196

C₂

#22589066

yes

characteristic

197

C₂

#22589113

yes

characteristic

198

C₂

#22589128

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14556099
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1566	C₂
socle	C₂²×C₇₁	#389
prime radical	C₂³×C₇₁	C₄⋊Q₈
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1566	D₇₁
hypocenter	C₇₁	#9623
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

subgroup

G/H

center

C₂×C₄

#14556099

commutator

C₄²×C₇₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₇₁×#1566

C₂

socle

C₂²×C₇₁

#389

prime radical

C₂³×C₇₁

C₄⋊Q₈

cosocle

C₂×C₄²×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

N	H ≅ G/N	type
78. C₈×C₇₁	#108	semidirect
105. C₇₁	#9623	semidirect
181. C₈	#14556063	semidirect

H ≅ G/N

type

78. C₈×C₇₁

#108

semidirect

105. C₇₁

#9623

semidirect

181. C₈

#14556063

semidirect

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

subgroup	index
#22589113	2
C₂×#17985750	2
C₈×#11765849	2
#22586235	2
C₇₁×#1566	2
#9623	71

subgroup

index

C₇₁×#1566

Subgroups — #31896149

Conjugacy classes of subgroups

Complements

Maximal subgroups