Group #31916014 — subgroups

Subgroups — #31916014

Order 18176.

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

normal subgroups only

This group has 334 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×#17986401	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×#14556120	—	2	1	no	—
7	#17986401	—	2	2	no	—
8	#17986401	C₄	1	4	yes	complemented
9	C₂×#14556064	C₂²	1	1	yes	characteristic
10	#17986037	C₂²	1	1	yes	characteristic
11	C₂×#14556062	—	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	C₂×#11765849	—	2	1	no	—
19	#14556120	—	2	4	no	—
20	#14556120	—	4	2	no	—
21	#14556087	—	2	1	no	—
22	C₂×C₄⋊Dic₇₁	—	2	2	no	—
23	C₂×Dic₇₁⋊C₄	—	4	2	no	—
24	#14556064	—	2	1	no	—
25	#14556064	Q₈	1	2	yes	—
26	#14556062	—	2	4	no	—
27	#14556062	Q₈	1	2	yes	—
28	#14556060	—	2	2	no	—
29	#14556060	—	2	1	no	—
30	#14556060	C₂×C₄	1	4	yes	—
31	#14556038	—	4	1	no	—
32	C₈×Dic₇₁	—	4	1	no	—
33	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
36	M₃₂×C₇₁	—	2	2	no	—
37	M₃₂×C₇₁	—	2	2	no	—
38	C₂×C₁₆×C₇₁	—	2	2	no	—
39	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₇₁	D₄	1	2	yes	—
41	C₂²×Dic₇₁	—	4	2	no	—
42	#11765849	—	2	2	no	—
43	#11765849	—	4	1	no	—
44	C₂×C₇₁⋊C₈	—	4	1	no	—
45	C₄⋊Dic₇₁	—	2	4	no	—
46	C₄⋊Dic₇₁	—	2	2	no	—
47	C₄⋊Dic₇₁	—	4	2	no	—
48	Dic₇₁⋊C₄	—	4	8	no	—
49	C₂²⋊Dic₇₁	—	4	1	no	—
50	C₄×Dic₇₁	—	4	1	no	—
51	C₂²×C₄×C₇₁	—	2	1	no	—
52	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₇₁	SD₁₆	1	1	yes	characteristic
55	C₂×C₈×C₇₁	SD₁₆	1	1	yes	characteristic
56	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
57	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
62	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
63	C₄²×C₇₁	—	2	1	no	—
64	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
65	C₁₆×C₇₁	—	4	2	no	—
66	C₂³×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
67	C₂×C₄×C₇₁	—	2	2	no	—
68	C₂×C₄×C₇₁	—	4	2	no	—
69	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
73	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
74	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
75	C₂×Dic₇₁	—	4	4	no	—
76	C₂×Dic₇₁	—	4	2	no	—
77	C₂×Dic₇₁	—	8	4	no	—
78	C₈×C₇₁	#103	1	2	yes	—
79	C₈×C₇₁	#103	1	2	yes	—
80	C₈×C₇₁	C₄×Q₈	1	1	yes	characteristic
81	C₈×C₇₁	C₄×Q₈	1	1	yes	characteristic
82	C₈×C₇₁	#124	1	1	yes	characteristic
83	C₈×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
84	C₇₁⋊C₈	—	4	2	no	—
85	C₂²×C₇₁	—	2	2	no	—
86	C₂²×C₇₁	#418	1	1	yes	characteristic
87	C₂²×C₇₁	#476	1	1	yes	characteristic
88	C₂²×C₇₁	#477	1	1	yes	characteristic
89	C₄×C₇₁	—	4	2	no	—
90	C₄×C₇₁	#391	1	1	yes	characteristic
91	C₄×C₇₁	C₂×#103	1	1	yes	characteristic
92	C₄×C₇₁	#476	1	1	yes	characteristic
93	C₄×C₇₁	#477	1	1	yes	characteristic
94	Dic₇₁	—	8	4	no	—
95	Dic₇₁	—	8	2	no	—
96	#13273	—	71	1	no	maximal, Sylow
97	C₂×C₇₁	—	2	1	no	—
98	C₂×C₇₁	—	2	1	no	—
99	C₂×C₇₁	#1987	1	1	yes	characteristic
100	C₂×C₇₁	#2231	1	2	yes	—
101	C₂×#499	—	71	1	no	—
102	C₂×#364	D₇₁	1	1	yes	characteristic
103	#1772	—	71	1	no	—
104	#1386	—	71	2	no	—
105	#1383	—	71	2	no	—
106	C₇₁	#13273	1	1	yes	characteristic, complemented, Sylow
107	C₂×#125	—	142	1	no	—
108	C₂×M₃₂	C₂×D₇₁	1	2	yes	—
109	#499	—	71	4	no	—
110	#499	—	142	2	no	—
111	C₂×#108	—	71	1	no	—
112	#427	—	71	1	no	—
113	C₂×#107	—	142	1	no	—
114	C₂×#106	—	71	2	no	—
115	#405	—	142	1	no	—
116	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
117	#364	—	2	2	no	—
118	#339	—	142	1	no	—
119	C₂×M₁₆	—	142	1	no	—
120	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
121	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
122	#125	—	284	2	no	—
123	#125	—	142	4	no	—
124	#117	—	142	1	no	—
125	C₂×C₄⋊C₄	—	284	2	no	—
126	C₂×C₄⋊C₄	—	142	2	no	—
127	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
128	M₃₂	—	2	2	no	—
129	M₃₂	—	2	2	no	—
130	C₂×C₁₆	—	2	2	no	—
131	#108	—	71	2	no	—
132	#108	—	142	1	no	—
133	#107	—	71	2	no	—
134	#107	—	142	4	no	—
135	#106	—	71	4	no	—
136	#106	—	142	2	no	—
137	#106	—	142	1	no	—
138	#97	—	284	1	no	—
139	C₄×C₈	—	284	1	no	—
140	C₄×C₈	C₄×D₇₁	1	2	yes	—
141	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
142	C₂²×C₄	—	2	1	no	—
143	C₂²×C₄	—	284	2	no	—
144	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
145	M₁₆	—	284	1	no	—
146	M₁₆	—	142	2	no	—
147	C₂×C₈	—	284	1	no	—
148	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
149	C₂×C₈	#11765854	1	1	yes	characteristic
150	C₂×C₈	#11765854	1	1	yes	characteristic
151	C₂×C₈	Dic₂₈₄	1	2	yes	—
152	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
153	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
154	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
155	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
156	C₂×C₈	#11765864	1	1	yes	characteristic
157	C₂×C₈	#11765864	1	1	yes	characteristic
158	C₄⋊C₄	—	284	8	no	—
159	C₄⋊C₄	—	284	2	no	—
160	C₄⋊C₄	—	142	4	no	—
161	C₄⋊C₄	—	142	2	no	—
162	C₂²⋊C₄	—	284	1	no	—
163	C₄²	—	2	1	no	—
164	C₄²	—	284	1	no	—
165	C₄²	(C₂×D₇₁)⋊C₄	1	2	yes	—
166	C₁₆	—	4	2	no	—
167	C₂³	#14556156	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	284	4	no	—
171	C₂×C₄	—	284	2	no	—
172	C₂×C₄	—	568	4	no	—
173	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
174	C₂×C₄	#14556108	1	1	yes	characteristic
175	C₂×C₄	#14556108	1	1	yes	characteristic
176	C₂×C₄	C₂×#11765854	1	1	yes	characteristic
177	C₂×C₄	C₂×Dic₂₈₄	1	1	yes	characteristic
178	C₂×C₄	#14556156	1	1	yes	characteristic
179	C₈	—	284	2	no	—
180	C₈	#14556053	1	2	yes	—
181	C₈	#14556053	1	2	yes	—
182	C₈	#14556093	1	1	yes	characteristic
183	C₈	#14556093	1	1	yes	characteristic
184	C₈	#14556115	1	1	yes	characteristic
185	C₈	#14556126	1	1	yes	characteristic
186	C₂²	—	2	2	no	—
187	C₂²	#17986005	1	1	yes	characteristic
188	C₂²	#17986290	1	1	yes	characteristic
189	C₂²	#17986295	1	1	yes	characteristic
190	C₄	—	4	2	no	—
191	C₄	—	568	4	no	—
192	C₄	—	568	2	no	—
193	C₄	#17985904	1	1	yes	characteristic
194	C₄	C₂×#14556053	1	1	yes	characteristic
195	C₄	#17986290	1	1	yes	characteristic
196	C₄	#17986295	1	1	yes	characteristic
197	C₂	—	2	1	no	—
198	C₂	—	2	1	no	—
199	C₂	#22589287	1	1	yes	characteristic
200	C₂	#22590517	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17986401

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×#14556120

—

#17986401

—

#17986401

C₄

yes

complemented

C₂×#14556064

C₂²

yes

characteristic

#17986037

C₂²

yes

characteristic

C₂×#14556062

—

C₂×#14556060

C₂²

yes

—

#17985954

—

#17985759

—

C₂×M₃₂×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#364×C₇₁

—

—

—

—

—

C₂×C₄⋊Dic₇₁

—

C₂×Dic₇₁⋊C₄

—

#14556064

—

#14556064

Q₈

yes

—

#14556062

—

#14556062

Q₈

yes

—

#14556060

—

#14556060

—

#14556060

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

—

C₂²×Dic₇₁

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

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

—

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

—

#13273

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#1987

yes

characteristic

100

C₂×C₇₁

#2231

yes

—

101

C₂×#499

—

102

C₂×#364

D₇₁

yes

characteristic

103

#1772

—

104

#1386

—

105

#1383

—

106

C₇₁

#13273

yes

characteristic, complemented, Sylow

107

C₂×#125

—

142

—

108

C₂×M₃₂

C₂×D₇₁

yes

—

109

#499

—

110

#499

—

142

—

111

C₂×#108

—

112

#427

—

113

C₂×#107

—

142

—

114

C₂×#106

—

115

#405

—

142

—

116

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

117

#364

—

118

#339

—

142

—

119

C₂×M₁₆

—

142

—

120

C₂²×C₈

D₂₈₄

yes

characteristic

121

C₂²×C₈

C₂²×D₇₁

yes

characteristic

122

#125

—

284

—

123

#125

—

142

—

124

#117

—

142

—

125

C₂×C₄⋊C₄

—

284

—

126

C₂×C₄⋊C₄

—

142

—

127

C₂×C₄²

D₂₈₄

yes

characteristic

128

M₃₂

—

129

M₃₂

—

130

C₂×C₁₆

—

131

#108

—

132

#108

—

142

—

133

#107

—

134

#107

—

142

—

135

#106

—

136

#106

—

142

—

137

#106

—

142

—

138

#97

—

284

—

139

C₄×C₈

—

284

—

140

C₄×C₈

C₄×D₇₁

yes

—

141

C₄×C₈

C₂²⋊D₇₁

yes

—

142

C₂²×C₄

—

143

C₂²×C₄

—

284

—

144

C₂²×C₄

C₂×D₂₈₄

yes

characteristic

145

M₁₆

—

284

—

146

M₁₆

—

142

—

147

C₂×C₈

—

284

—

148

C₂×C₈

(C₂×D₇₁)⋊C₄

yes

—

149

C₂×C₈

#11765854

yes

characteristic

150

C₂×C₈

#11765854

yes

characteristic

151

C₂×C₈

Dic₂₈₄

yes

—

152

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

153

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

154

C₂×C₈

Q₈×D₇₁

yes

characteristic

155

C₂×C₈

Q₈×D₇₁

yes

characteristic

156

C₂×C₈

#11765864

yes

characteristic

157

C₂×C₈

#11765864

yes

characteristic

158

C₄⋊C₄

—

284

—

159

C₄⋊C₄

—

284

—

160

C₄⋊C₄

—

142

—

161

C₄⋊C₄

—

142

—

162

C₂²⋊C₄

—

284

—

163

C₄²

—

164

C₄²

—

284

—

165

C₄²

(C₂×D₇₁)⋊C₄

yes

—

166

C₁₆

—

167

C₂³

#14556156

yes

characteristic

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

284

—

171

C₂×C₄

—

284

—

172

C₂×C₄

—

568

—

173

C₂×C₄

C₂×(C₂×D₇₁)⋊C₄

yes

characteristic

174

C₂×C₄

#14556108

yes

characteristic

175

C₂×C₄

#14556108

yes

characteristic

176

C₂×C₄

C₂×#11765854

yes

characteristic

177

C₂×C₄

C₂×Dic₂₈₄

yes

characteristic

178

C₂×C₄

#14556156

yes

characteristic

179

C₈

—

284

—

180

C₈

#14556053

yes

—

181

C₈

#14556053

yes

—

182

C₈

#14556093

yes

characteristic

183

C₈

#14556093

yes

characteristic

184

C₈

#14556115

yes

characteristic

185

C₈

#14556126

yes

characteristic

186

C₂²

—

187

C₂²

#17986005

yes

characteristic

188

C₂²

#17986290

yes

characteristic

189

C₂²

#17986295

yes

characteristic

190

C₄

—

191

C₄

—

568

—

192

C₄

—

568

—

193

C₄

#17985904

yes

characteristic

194

C₄

C₂×#14556053

yes

characteristic

195

C₄

#17986290

yes

characteristic

196

C₄

#17986295

yes

characteristic

197

C₂

—

198

C₂

—

199

C₂

#22589287

yes

characteristic

200

C₂

#22590517

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	#13273
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. #17986401	C₄	semidirect
106. C₇₁	#13273	semidirect

H ≅ G/N

type

8. #17986401

C₄

semidirect

106. C₇₁

#13273

semidirect

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

subgroup	index
C₂×#17986401	2
#22588219	2
#22586446	2
#22586439	2
C₂×#364×C₇₁	2
#13273	71

subgroup

index

C₂×#364×C₇₁

Subgroups — #31916014

Conjugacy classes of subgroups

Complements

Maximal subgroups