Group #31897124 — subgroups

Subgroups — #31897124

Order 18176.

434 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 131 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×#17986404	C₂	1	1	yes	maximal, characteristic
2	#22589184	C₂	1	2	yes	maximal
3	#22587096	C₂	1	1	yes	maximal, characteristic
4	#22587011	C₂	1	2	yes	maximal
5	C₇₁×#1451	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556128	C₂²	1	2	yes	—
7	#17986404	—	2	2	no	—
8	#17986404	—	2	1	no	—
9	#17986404	—	2	1	no	—
10	C₂×#14556062	C₄	1	4	yes	—
11	#17986023	C₂²	1	2	yes	—
12	C₂×#14556046	—	2	4	no	—
13	#17985750	—	2	2	no	—
14	#17985744	—	2	4	no	—
15	#431×C₇₁	C₂²	1	1	yes	characteristic
16	C₄×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
18	#322×C₇₁	—	2	1	no	—
19	#322×C₇₁	—	2	1	no	—
20	C₂²×Dic₁₄₂	—	2	2	no	—
21	C₂×#11765849	—	2	2	no	—
22	#14556128	—	2	2	no	—
23	#14556128	—	2	2	no	—
24	#14556128	—	2	2	no	—
25	#14556128	—	2	2	no	—
26	C₂×C₄⋊Dic₇₁	—	2	2	no	—
27	C₂×C₄⋊Dic₇₁	C₂×C₄	1	4	yes	—
28	#14556062	—	2	4	no	—
29	#14556062	—	2	4	no	—
30	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
31	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
32	#14556046	—	4	8	no	—
33	C₂×M₁₆×C₇₁	D₄	1	2	yes	—
34	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₇₁	D₄	1	2	yes	—
37	C₂²⋊C₈×C₇₁	—	2	1	no	—
38	#97×C₇₁	—	2	1	no	—
39	C₄×C₈×C₇₁	—	2	1	no	—
40	C₄×C₈×C₇₁	—	2	1	no	—
41	C₄×C₈×C₇₁	D₄	1	2	yes	—
42	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
43	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
44	C₂×Dic₁₄₂	—	4	2	no	—
45	C₂×Dic₁₄₂	—	4	2	no	—
46	C₂×Dic₁₄₂	—	4	4	no	—
47	C₂×Dic₁₄₂	—	4	4	no	—
48	C₂²×Dic₇₁	—	2	4	no	—
49	#11765849	—	4	4	no	—
50	C₂×C₇₁⋊C₈	—	4	2	no	—
51	C₄⋊Dic₇₁	—	2	4	no	—
52	C₄⋊Dic₇₁	—	2	4	no	—
53	C₄⋊Dic₇₁	—	4	2	no	—
54	C₄⋊Dic₇₁	—	4	2	no	—
55	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
58	M₁₆×C₇₁	—	2	4	no	—
59	C₂×C₈×C₇₁	—	2	4	no	—
60	C₂×C₈×C₇₁	—	2	2	no	—
61	C₂×C₈×C₇₁	—	2	2	no	—
62	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
64	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₇₁	#41	1	1	yes	characteristic
68	C₂³×C₇₁	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₇₁	—	2	2	no	—
70	C₂×C₄×C₇₁	—	2	2	no	—
71	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
74	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
75	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
76	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
77	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	4	yes	—
78	C₂×Dic₇₁	—	4	4	no	—
79	C₂×Dic₇₁	—	4	8	no	—
80	Dic₁₄₂	—	8	4	no	—
81	Dic₁₄₂	—	8	2	no	—
82	Dic₁₄₂	—	8	2	no	—
83	C₈×C₇₁	—	2	2	no	—
84	C₈×C₇₁	—	4	2	no	—
85	C₈×C₇₁	—	4	2	no	—
86	C₈×C₇₁	#101	1	4	yes	complemented
87	C₇₁⋊C₈	—	8	2	no	—
88	C₂²×C₇₁	—	2	1	no	—
89	C₂²×C₇₁	—	2	1	no	—
90	C₂²×C₇₁	#390	1	1	yes	characteristic
91	C₂²×C₇₁	#442	1	2	yes	—
92	C₄×C₇₁	—	2	1	no	—
93	C₄×C₇₁	—	2	1	no	—
94	C₄×C₇₁	C₂×#101	1	2	yes	—
95	C₄×C₇₁	#469	1	2	yes	—
96	C₄×C₇₁	#473	1	2	yes	—
97	C₄×C₇₁	#496	1	2	yes	—
98	Dic₇₁	—	8	4	no	—
99	#9808	—	71	1	no	maximal, Sylow
100	C₂×C₇₁	—	2	2	no	—
101	C₂×C₇₁	#1966	1	1	yes	characteristic
102	C₂×C₇₁	#1972	1	1	yes	characteristic
103	C₂×C₇₁	#1979	1	1	yes	characteristic
104	C₂×#500	—	71	1	no	—
105	#1966	—	71	2	no	—
106	#1566	—	71	1	no	—
107	#1546	—	71	2	no	—
108	#1451	D₇₁	1	1	yes	characteristic
109	C₇₁	#9808	1	1	yes	characteristic, complemented, Sylow
110	C₂×C₄⋊Q₈	—	71	2	no	—
111	#500	—	142	2	no	—
112	#500	—	142	1	no	—
113	#500	—	142	1	no	—
114	#431	C₂×D₇₁	1	1	yes	characteristic
115	C₂×#107	—	71	4	no	—
116	#423	—	71	2	no	—
117	C₂×#101	—	142	4	no	—
118	C₄×M₁₆	C₂×D₇₁	1	1	yes	characteristic
119	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
120	#334	—	142	2	no	—
121	#332	—	142	4	no	—
122	#322	—	2	1	no	—
123	#322	—	2	1	no	—
124	C₂²×Q₈	—	142	2	no	—
125	C₂×M₁₆	—	142	2	no	—
126	C₂×M₁₆	D₂₈₄	1	2	yes	—
127	C₂²×C₈	C₄×D₇₁	1	2	yes	—
128	C₄⋊Q₈	—	142	2	no	—
129	C₄⋊Q₈	—	142	2	no	—
130	C₄⋊Q₈	—	142	2	no	—
131	C₄⋊Q₈	—	142	2	no	—
132	C₂×C₄⋊C₄	—	71	4	no	—
133	C₂×C₄⋊C₄	—	142	2	no	—
134	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
135	#107	—	142	4	no	—
136	#107	—	142	4	no	—
137	C₄⋊C₈	—	142	2	no	—
138	C₄⋊C₈	—	142	2	no	—
139	C₄⋊C₈	D₂₈₄	1	2	yes	—
140	#101	—	284	8	no	—
141	C₂²⋊C₈	—	2	1	no	—
142	#97	—	2	1	no	—
143	C₄×C₈	—	2	1	no	—
144	C₄×C₈	—	2	1	no	—
145	C₄×C₈	D₂₈₄	1	2	yes	—
146	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
147	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
148	C₂×Q₈	—	284	2	no	—
149	C₂×Q₈	—	284	2	no	—
150	C₂×Q₈	—	284	4	no	—
151	C₂×Q₈	—	284	4	no	—
152	C₂²×C₄	—	142	4	no	—
153	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
154	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
155	C₂²×C₄	#11765865	1	1	yes	characteristic
156	M₁₆	—	2	4	no	—
157	M₁₆	—	284	4	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	284	2	no	—
162	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
163	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
164	C₄⋊C₄	—	284	2	no	—
165	C₄⋊C₄	—	284	2	no	—
166	C₄⋊C₄	—	142	4	no	—
167	C₄⋊C₄	—	142	4	no	—
168	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
169	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
170	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
171	C₄²	#11765865	1	1	yes	characteristic
172	C₂³	C₄×D₂₈₄	1	1	yes	characteristic
173	Q₈	—	568	2	no	—
174	Q₈	—	568	2	no	—
175	Q₈	—	568	4	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	284	8	no	—
179	C₂×C₄	—	284	4	no	—
180	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
181	C₂×C₄	C₄×D₂₈₄	1	1	yes	characteristic
182	C₂×C₄	#14556102	1	1	yes	characteristic
183	C₂×C₄	#14556102	1	1	yes	characteristic
184	C₂×C₄	#14556116	1	1	yes	characteristic
185	C₂×C₄	#14556125	1	1	yes	characteristic
186	C₂×C₄	#14556156	1	4	yes	—
187	C₈	—	2	2	no	—
188	C₈	—	4	2	no	—
189	C₈	—	4	2	no	—
190	C₈	—	568	2	no	—
191	C₈	#14556046	1	4	yes	complemented
192	C₂²	—	2	1	no	—
193	C₂²	—	2	1	no	—
194	C₂²	#17985901	1	1	yes	characteristic
195	C₂²	#17986112	1	2	yes	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	568	4	no	—
199	C₄	C₂×#14556046	1	2	yes	—
200	C₄	#17986252	1	2	yes	—
201	C₄	#17986274	1	2	yes	—
202	C₄	#17986388	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	#22589184	1	1	yes	characteristic
205	C₂	#22589216	1	1	yes	characteristic
206	C₂	#22589255	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17986404

C₂

yes

maximal, characteristic

#22589184

C₂

yes

maximal

#22587096

C₂

yes

maximal, characteristic

#22587011

C₂

yes

maximal

C₇₁×#1451

C₂

yes

maximal, characteristic

C₂×#14556128

C₂²

yes

—

—

—

—

C₄

yes

—

#17986023

C₂²

yes

—

C₂×#14556046

—

#17985750

—

#17985744

—

#431×C₇₁

C₂²

yes

characteristic

C₄×M₁₆×C₇₁

C₂²

yes

characteristic

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#322×C₇₁

—

#322×C₇₁

—

C₂²×Dic₁₄₂

—

—

—

—

—

—

C₂×C₄⋊Dic₇₁

—

C₂×C₄⋊Dic₇₁

C₂×C₄

yes

—

#14556062

—

#14556062

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

#14556046

—

C₂×M₁₆×C₇₁

D₄

yes

—

C₂²×C₈×C₇₁

C₂×C₄

yes

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

C₄⋊C₈×C₇₁

D₄

yes

—

C₂²⋊C₈×C₇₁

—

#97×C₇₁

—

C₄×C₈×C₇₁

—

C₄×C₈×C₇₁

—

C₄×C₈×C₇₁

D₄

yes

—

C₄×C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

characteristic

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂²×Dic₇₁

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₁

#41

yes

characteristic

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

#41

yes

characteristic

C₂³×C₇₁

C₄×D₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄×D₄

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

#101

yes

complemented

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#390

yes

characteristic

C₂²×C₇₁

#442

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×#101

yes

—

C₄×C₇₁

#469

yes

—

C₄×C₇₁

#473

yes

—

C₄×C₇₁

#496

yes

—

Dic₇₁

—

#9808

—

maximal, Sylow

100

C₂×C₇₁

—

101

C₂×C₇₁

#1966

yes

characteristic

102

C₂×C₇₁

#1972

yes

characteristic

103

C₂×C₇₁

#1979

yes

characteristic

104

C₂×#500

—

105

#1966

—

106

#1566

—

107

#1546

—

108

#1451

D₇₁

yes

characteristic

109

C₇₁

#9808

yes

characteristic, complemented, Sylow

110

C₂×C₄⋊Q₈

—

111

#500

—

142

—

112

#500

—

142

—

113

#500

—

142

—

114

#431

C₂×D₇₁

yes

characteristic

115

C₂×#107

—

116

#423

—

117

C₂×#101

—

142

—

118

C₄×M₁₆

C₂×D₇₁

yes

characteristic

119

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

120

#334

—

142

—

121

#332

—

142

—

122

#322

—

123

#322

—

124

C₂²×Q₈

—

142

—

125

C₂×M₁₆

—

142

—

126

C₂×M₁₆

D₂₈₄

yes

—

127

C₂²×C₈

C₄×D₇₁

yes

—

128

C₄⋊Q₈

—

142

—

129

C₄⋊Q₈

—

142

—

130

C₄⋊Q₈

—

142

—

131

C₄⋊Q₈

—

142

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

142

—

134

C₂×C₄²

C₂²×D₇₁

yes

characteristic

135

#107

—

142

—

136

#107

—

142

—

137

C₄⋊C₈

—

142

—

138

C₄⋊C₈

—

142

—

139

C₄⋊C₈

D₂₈₄

yes

—

140

#101

—

284

—

141

C₂²⋊C₈

—

142

#97

—

143

C₄×C₈

—

144

C₄×C₈

—

145

C₄×C₈

D₂₈₄

yes

—

146

C₄×C₈

C₂²⋊D₇₁

yes

characteristic

147

C₄×C₈

C₂²⋊D₇₁

yes

characteristic

148

C₂×Q₈

—

284

—

149

C₂×Q₈

—

284

—

150

C₂×Q₈

—

284

—

151

C₂×Q₈

—

284

—

152

C₂²×C₄

—

142

—

153

C₂²×C₄

C₂×C₄×D₇₁

yes

characteristic

154

C₂²×C₄

C₂×D₂₈₄

yes

characteristic

155

C₂²×C₄

#11765865

yes

characteristic

156

M₁₆

—

157

M₁₆

—

284

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

284

—

162

C₂×C₈

(C₂×D₇₁)⋊C₄

yes

—

163

C₂×C₈

(C₂×D₇₁)⋊C₄

yes

—

164

C₄⋊C₄

—

284

—

165

C₄⋊C₄

—

284

—

166

C₄⋊C₄

—

142

—

167

C₄⋊C₄

—

142

—

168

C₄²

C₂×C₂²⋊D₇₁

yes

characteristic

169

C₄²

C₂×D₂₈₄

yes

characteristic

170

C₄²

C₂×D₂₈₄

yes

characteristic

171

C₄²

#11765865

yes

characteristic

172

C₂³

C₄×D₂₈₄

yes

characteristic

173

Q₈

—

568

—

174

Q₈

—

568

—

175

Q₈

—

568

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

284

—

179

C₂×C₄

—

284

—

180

C₂×C₄

C₂×(C₂×D₇₁)⋊C₄

yes

characteristic

181

C₂×C₄

C₄×D₂₈₄

yes

characteristic

182

C₂×C₄

#14556102

yes

characteristic

183

C₂×C₄

#14556102

yes

characteristic

184

C₂×C₄

#14556116

yes

characteristic

185

C₂×C₄

#14556125

yes

characteristic

186

C₂×C₄

#14556156

yes

—

187

C₈

—

188

C₈

—

189

C₈

—

190

C₈

—

568

—

191

C₈

#14556046

yes

complemented

192

C₂²

—

193

C₂²

—

194

C₂²

#17985901

yes

characteristic

195

C₂²

#17986112

yes

—

196

C₄

—

197

C₄

—

198

C₄

—

568

—

199

C₄

C₂×#14556046

yes

—

200

C₄

#17986252

yes

—

201

C₄

#17986274

yes

—

202

C₄

#17986388

yes

—

203

C₂

—

204

C₂

#22589184

yes

characteristic

205

C₂

#22589216

yes

characteristic

206

C₂

#22589255

yes

characteristic

	subgroup	G/H
center	C₂²	#17985901
commutator	C₂²×C₄×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1451	C₂
socle	C₂²×C₇₁	#390
prime radical	C₂³×C₇₁	C₄×D₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1451	D₇₁
hypocenter	C₇₁	#9808
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1451

subgroup

G/H

center

C₂²

#17985901

commutator

C₂²×C₄×C₇₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₇₁×#1451

C₂

socle

C₂²×C₇₁

#390

prime radical

C₂³×C₇₁

C₄×D₄

cosocle

C₂×C₄²×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂²×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1451

N	H ≅ G/N	type
86. C₈×C₇₁	#101	semidirect
109. C₇₁	#9808	semidirect
191. C₈	#14556046	semidirect

H ≅ G/N

type

86. C₈×C₇₁

#101

semidirect

109. C₇₁

#9808

semidirect

191. C₈

#14556046

semidirect

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

subgroup	index
C₂×#17986404	2
#22589184	2
#22587096	2
#22587011	2
C₇₁×#1451	2
#9808	71

subgroup

index

C₇₁×#1451

Subgroups — #31897124

Conjugacy classes of subgroups

Complements

Maximal subgroups