Group #31916009 — subgroups

Subgroups — #31916009

Order 18176.

366 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 127 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×#17986397	C₂	1	1	yes	maximal, characteristic
2	#22588219	C₂	1	1	yes	maximal, characteristic
3	#22586446	C₂	1	2	yes	maximal
4	#22586439	C₂	1	2	yes	maximal
5	C₂×#364×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556128	—	2	1	no	—
7	#17986397	—	2	2	no	—
8	#17986397	C₄	1	4	yes	complemented
9	C₂×#14556064	C₂²	1	1	yes	characteristic
10	#17986037	C₂²	1	1	yes	characteristic
11	C₂×#14556060	—	2	1	no	—
12	C₂×#14556060	C₂²	1	2	yes	—
13	#17985954	—	2	1	no	—
14	#17985759	—	2	1	no	—
15	C₂×M₃₂×C₇₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	#364×C₇₁	—	2	2	no	—
18	C₂²×Dic₁₄₂	—	4	1	no	—
19	C₂×#11765849	—	2	1	no	—
20	#14556128	—	2	4	no	—
21	#14556128	—	4	2	no	—
22	#14556087	—	2	1	no	—
23	C₂×C₄⋊Dic₇₁	—	2	2	no	—
24	C₂×C₄⋊Dic₇₁	—	4	1	no	—
25	#14556064	—	2	1	no	—
26	#14556064	Q₈	1	2	yes	complemented
27	#14556062	Q₈	1	2	yes	complemented
28	#14556060	—	2	4	no	—
29	#14556060	—	2	2	no	—
30	#14556060	—	2	1	no	—
31	#14556060	C₂×C₄	1	4	yes	—
32	#14556038	—	4	1	no	—
33	C₈×Dic₇₁	—	4	1	no	—
34	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
37	M₃₂×C₇₁	—	2	2	no	—
38	M₃₂×C₇₁	—	2	2	no	—
39	C₂×C₁₆×C₇₁	—	2	2	no	—
40	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₇₁	D₄	1	2	yes	—
42	C₂×Dic₁₄₂	—	4	4	no	—
43	C₂×Dic₁₄₂	—	8	4	no	—
44	C₂²×Dic₇₁	—	4	2	no	—
45	#11765849	—	2	2	no	—
46	#11765849	—	4	1	no	—
47	C₂×C₇₁⋊C₈	—	4	1	no	—
48	C₄⋊Dic₇₁	—	2	4	no	—
49	C₄⋊Dic₇₁	—	2	2	no	—
50	C₄⋊Dic₇₁	—	4	4	no	—
51	C₄⋊Dic₇₁	—	4	2	no	—
52	C₂²⋊Dic₇₁	—	4	1	no	—
53	C₄×Dic₇₁	—	4	1	no	—
54	C₂²×C₄×C₇₁	—	2	1	no	—
55	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₇₁	SD₁₆	1	1	yes	characteristic
58	C₂×C₈×C₇₁	SD₁₆	1	1	yes	characteristic
59	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
60	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
65	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
66	C₄²×C₇₁	—	2	1	no	—
67	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
68	C₁₆×C₇₁	—	4	2	no	—
69	C₂³×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
70	C₂×C₄×C₇₁	—	2	4	no	—
71	C₂×C₄×C₇₁	—	2	2	no	—
72	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
77	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂×Dic₇₁	—	4	4	no	—
79	C₂×Dic₇₁	—	4	2	no	—
80	C₂×Dic₇₁	—	8	4	no	—
81	Dic₁₄₂	—	8	8	no	—
82	C₈×C₇₁	#103	1	2	yes	—
83	C₈×C₇₁	#103	1	2	yes	—
84	C₈×C₇₁	C₄×Q₈	1	1	yes	characteristic
85	C₈×C₇₁	C₄×Q₈	1	1	yes	characteristic
86	C₈×C₇₁	#124	1	1	yes	characteristic
87	C₈×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
88	C₇₁⋊C₈	—	4	2	no	—
89	C₂²×C₇₁	—	2	2	no	—
90	C₂²×C₇₁	#418	1	1	yes	characteristic
91	C₂²×C₇₁	#476	1	1	yes	characteristic
92	C₂²×C₇₁	#477	1	1	yes	characteristic
93	C₄×C₇₁	—	2	4	no	—
94	C₄×C₇₁	#391	1	1	yes	characteristic
95	C₄×C₇₁	C₂×#103	1	1	yes	characteristic
96	C₄×C₇₁	#476	1	1	yes	characteristic
97	C₄×C₇₁	#477	1	1	yes	characteristic
98	Dic₇₁	—	8	4	no	—
99	Dic₇₁	—	8	2	no	—
100	#13272	—	71	1	no	maximal, Sylow
101	C₂×C₇₁	—	2	1	no	—
102	C₂×C₇₁	—	2	1	no	—
103	C₂×C₇₁	#1987	1	1	yes	characteristic
104	C₂×C₇₁	#2230	1	2	yes	—
105	C₂×#498	—	71	1	no	—
106	C₂×#364	D₇₁	1	1	yes	characteristic
107	#1772	—	71	1	no	—
108	#1386	—	71	2	no	—
109	#1383	—	71	2	no	—
110	C₇₁	#13272	1	1	yes	characteristic, complemented, Sylow
111	C₂×C₄⋊Q₈	—	142	1	no	—
112	C₂×M₃₂	C₂×D₇₁	1	2	yes	—
113	#498	—	71	4	no	—
114	#498	—	142	2	no	—
115	C₂×#108	—	71	1	no	—
116	#427	—	71	1	no	—
117	C₂×#106	—	71	2	no	—
118	C₂×#106	—	142	1	no	—
119	#405	—	142	1	no	—
120	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
121	#364	—	2	2	no	—
122	#339	—	142	1	no	—
123	C₂²×Q₈	—	284	1	no	—
124	C₂×M₁₆	—	142	1	no	—
125	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
127	C₄⋊Q₈	—	284	2	no	—
128	C₄⋊Q₈	—	142	4	no	—
129	#117	—	142	1	no	—
130	C₂×C₄⋊C₄	—	284	1	no	—
131	C₂×C₄⋊C₄	—	142	2	no	—
132	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
133	M₃₂	—	2	2	no	—
134	M₃₂	—	2	2	no	—
135	C₂×C₁₆	—	2	2	no	—
136	#108	—	71	2	no	—
137	#108	—	142	1	no	—
138	#107	—	71	2	no	—
139	#106	—	71	4	no	—
140	#106	—	142	4	no	—
141	#106	—	142	2	no	—
142	#106	—	142	1	no	—
143	#97	—	284	1	no	—
144	C₄×C₈	—	284	1	no	—
145	C₄×C₈	C₄×D₇₁	1	2	yes	—
146	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
147	C₂×Q₈	—	284	4	no	—
148	C₂×Q₈	—	568	4	no	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	284	2	no	—
151	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
152	M₁₆	—	284	1	no	—
153	M₁₆	—	142	2	no	—
154	C₂×C₈	—	284	1	no	—
155	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
156	C₂×C₈	#11765854	1	1	yes	characteristic
157	C₂×C₈	#11765854	1	1	yes	characteristic
158	C₂×C₈	Dic₂₈₄	1	2	yes	—
159	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
160	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
161	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
162	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
163	C₂×C₈	#11765864	1	1	yes	characteristic
164	C₂×C₈	#11765864	1	1	yes	characteristic
165	C₄⋊C₄	—	284	4	no	—
166	C₄⋊C₄	—	284	2	no	—
167	C₄⋊C₄	—	142	4	no	—
168	C₄⋊C₄	—	142	2	no	—
169	C₂²⋊C₄	—	284	1	no	—
170	C₄²	—	2	1	no	—
171	C₄²	—	284	1	no	—
172	C₄²	(C₂×D₇₁)⋊C₄	1	2	yes	—
173	C₁₆	—	4	2	no	—
174	C₂³	#14556156	1	1	yes	characteristic
175	Q₈	—	568	8	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	284	4	no	—
179	C₂×C₄	—	284	2	no	—
180	C₂×C₄	—	568	4	no	—
181	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
182	C₂×C₄	#14556108	1	1	yes	characteristic
183	C₂×C₄	#14556108	1	1	yes	characteristic
184	C₂×C₄	C₂×#11765854	1	1	yes	characteristic
185	C₂×C₄	C₂×Dic₂₈₄	1	1	yes	characteristic
186	C₂×C₄	#14556156	1	1	yes	characteristic
187	C₈	—	284	2	no	—
188	C₈	#14556053	1	2	yes	—
189	C₈	#14556053	1	2	yes	—
190	C₈	#14556093	1	1	yes	characteristic
191	C₈	#14556093	1	1	yes	characteristic
192	C₈	#14556115	1	1	yes	characteristic
193	C₈	#14556126	1	1	yes	characteristic
194	C₂²	—	2	2	no	—
195	C₂²	#17986005	1	1	yes	characteristic
196	C₂²	#17986290	1	1	yes	characteristic
197	C₂²	#17986295	1	1	yes	characteristic
198	C₄	—	2	4	no	—
199	C₄	—	568	4	no	—
200	C₄	—	568	2	no	—
201	C₄	#17985904	1	1	yes	characteristic
202	C₄	C₂×#14556053	1	1	yes	characteristic
203	C₄	#17986290	1	1	yes	characteristic
204	C₄	#17986295	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	#22589287	1	1	yes	characteristic
208	C₂	#22590512	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17986397

C₂

yes

maximal, characteristic

#22588219

C₂

yes

maximal, characteristic

#22586446

C₂

yes

maximal

#22586439

C₂

yes

maximal

C₂×#364×C₇₁

C₂

yes

maximal, characteristic

C₂×#14556128

—

#17986397

—

#17986397

C₄

yes

complemented

C₂×#14556064

C₂²

yes

characteristic

#17986037

C₂²

yes

characteristic

C₂×#14556060

—

C₂×#14556060

C₂²

yes

—

#17985954

—

#17985759

—

C₂×M₃₂×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#364×C₇₁

—

C₂²×Dic₁₄₂

—

—

—

—

—

C₂×C₄⋊Dic₇₁

—

C₂×C₄⋊Dic₇₁

—

#14556064

—

#14556064

Q₈

yes

complemented

#14556062

Q₈

yes

—

—

—

C₂×C₄

yes

—

#14556038

—

C₈×Dic₇₁

—

C₂²×C₈×C₇₁

D₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₄²×C₇₁

D₄

yes

characteristic

M₃₂×C₇₁

—

M₃₂×C₇₁

—

C₂×C₁₆×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

D₄

yes

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂²×Dic₇₁

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₂×C₈×C₇₁

SD₁₆

yes

characteristic

C₂×C₈×C₇₁

SD₁₆

yes

characteristic

C₂×C₈×C₇₁

Q₁₆

yes

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

C₂×C₈×C₇₁

yes

C₂×C₈×C₇₁

yes

C₄²×C₇₁

—

C₄²×C₇₁

C₂²⋊C₄

yes

—

C₁₆×C₇₁

—

C₂³×C₇₁

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₁

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

C₈×C₇₁

#103

yes

—

C₈×C₇₁

#103

yes

—

C₈×C₇₁

C₄×Q₈

yes

characteristic

C₈×C₇₁

C₄×Q₈

yes

characteristic

C₈×C₇₁

#124

yes

characteristic

C₈×C₇₁

C₄⋊Q₈

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

100

#13272

—

maximal, Sylow

101

C₂×C₇₁

—

102

C₂×C₇₁

—

103

C₂×C₇₁

#1987

yes

characteristic

104

C₂×C₇₁

#2230

yes

—

105

C₂×#498

—

106

C₂×#364

D₇₁

yes

characteristic

107

#1772

—

108

#1386

—

109

#1383

—

110

C₇₁

#13272

yes

characteristic, complemented, Sylow

111

C₂×C₄⋊Q₈

—

142

—

112

C₂×M₃₂

C₂×D₇₁

yes

—

113

#498

—

114

#498

—

142

—

115

C₂×#108

—

116

#427

—

117

C₂×#106

—

118

C₂×#106

—

142

—

119

#405

—

142

—

120

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

121

#364

—

122

#339

—

142

—

123

C₂²×Q₈

—

284

—

124

C₂×M₁₆

—

142

—

125

C₂²×C₈

D₂₈₄

yes

characteristic

126

C₂²×C₈

C₂²×D₇₁

yes

characteristic

127

C₄⋊Q₈

—

284

—

128

C₄⋊Q₈

—

142

—

129

#117

—

142

—

130

C₂×C₄⋊C₄

—

284

—

131

C₂×C₄⋊C₄

—

142

—

132

C₂×C₄²

D₂₈₄

yes

characteristic

133

M₃₂

—

134

M₃₂

—

135

C₂×C₁₆

—

136

#108

—

137

#108

—

142

—

138

#107

—

139

#106

—

140

#106

—

142

—

141

#106

—

142

—

142

#106

—

142

—

143

#97

—

284

—

144

C₄×C₈

—

284

—

145

C₄×C₈

C₄×D₇₁

yes

—

146

C₄×C₈

C₂²⋊D₇₁

yes

—

147

C₂×Q₈

—

284

—

148

C₂×Q₈

—

568

—

149

C₂²×C₄

—

150

C₂²×C₄

—

284

—

151

C₂²×C₄

C₂×D₂₈₄

yes

characteristic

152

M₁₆

—

284

—

153

M₁₆

—

142

—

154

C₂×C₈

—

284

—

155

C₂×C₈

(C₂×D₇₁)⋊C₄

yes

—

156

C₂×C₈

#11765854

yes

characteristic

157

C₂×C₈

#11765854

yes

characteristic

158

C₂×C₈

Dic₂₈₄

yes

—

159

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

160

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

161

C₂×C₈

Q₈×D₇₁

yes

characteristic

162

C₂×C₈

Q₈×D₇₁

yes

characteristic

163

C₂×C₈

#11765864

yes

characteristic

164

C₂×C₈

#11765864

yes

characteristic

165

C₄⋊C₄

—

284

—

166

C₄⋊C₄

—

284

—

167

C₄⋊C₄

—

142

—

168

C₄⋊C₄

—

142

—

169

C₂²⋊C₄

—

284

—

170

C₄²

—

171

C₄²

—

284

—

172

C₄²

(C₂×D₇₁)⋊C₄

yes

—

173

C₁₆

—

174

C₂³

#14556156

yes

characteristic

175

Q₈

—

568

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

284

—

179

C₂×C₄

—

284

—

180

C₂×C₄

—

568

—

181

C₂×C₄

C₂×(C₂×D₇₁)⋊C₄

yes

characteristic

182

C₂×C₄

#14556108

yes

characteristic

183

C₂×C₄

#14556108

yes

characteristic

184

C₂×C₄

C₂×#11765854

yes

characteristic

185

C₂×C₄

C₂×Dic₂₈₄

yes

characteristic

186

C₂×C₄

#14556156

yes

characteristic

187

C₈

—

284

—

188

C₈

#14556053

yes

—

189

C₈

#14556053

yes

—

190

C₈

#14556093

yes

characteristic

191

C₈

#14556093

yes

characteristic

192

C₈

#14556115

yes

characteristic

193

C₈

#14556126

yes

characteristic

194

C₂²

—

195

C₂²

#17986005

yes

characteristic

196

C₂²

#17986290

yes

characteristic

197

C₂²

#17986295

yes

characteristic

198

C₄

—

199

C₄

—

568

—

200

C₄

—

568

—

201

C₄

#17985904

yes

characteristic

202

C₄

C₂×#14556053

yes

characteristic

203

C₄

#17986290

yes

characteristic

204

C₄

#17986295

yes

characteristic

205

C₂

—

206

C₂

—

207

C₂

#22589287

yes

characteristic

208

C₂

#22590512

yes

—

	subgroup	G/H
center	C₂²	#17986290
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#364×C₇₁	C₂
socle	C₂²×C₇₁	#476
prime radical	C₂³×C₇₁	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#364	D₇₁
hypocenter	C₇₁	#13272
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

subgroup

G/H

center

C₂²

#17986290

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₂×#364×C₇₁

C₂

C₂²×C₇₁

C₂³×C₇₁

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

N	H ≅ G/N	type
8. #17986397	C₄	semidirect
26. #14556064	Q₈	semidirect
27. #14556062	Q₈	semidirect
110. C₇₁	#13272	semidirect

H ≅ G/N

type

8. #17986397

C₄

semidirect

26. #14556064

Q₈

semidirect

27. #14556062

Q₈

semidirect

110. C₇₁

#13272

semidirect

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

subgroup	index
C₂×#17986397	2
#22588219	2
#22586446	2
#22586439	2
C₂×#364×C₇₁	2
#13272	71

subgroup

index

C₂×#364×C₇₁

Subgroups — #31916009

Conjugacy classes of subgroups

Complements

Maximal subgroups