Group #31917283 — subgroups

Subgroups — #31917283

Order 18176.

302 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 93 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×#17985814	C₂	1	2	yes	maximal
2	#22589211	C₂	1	1	yes	maximal, characteristic
3	#22586440	C₂	1	2	yes	maximal
4	#22586424	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1849	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₈₄	C₂²	1	1	yes	characteristic
7	C₂×#14556068	C₂²	1	2	yes	—
8	C₂×#14556060	C₂²	1	1	yes	characteristic
9	C₂×#14556053	—	2	1	no	—
10	#17985905	—	2	1	no	—
11	#17985814	—	2	4	no	—
12	C₂×#108×C₇₁	—	2	1	no	—
13	C₂×#108×C₇₁	C₂²	1	2	yes	—
14	#423×C₇₁	—	2	1	no	—
15	C₂×#97×C₇₁	C₂²	1	1	yes	characteristic
16	C₂²×Dic₁₄₂	—	2	1	no	—
17	C₂×Dic₂₈₄	—	2	2	no	—
18	C₂×Dic₂₈₄	—	2	2	no	—
19	C₂×Dic₂₈₄	—	2	1	no	—
20	C₂×Dic₂₈₄	D₄	1	2	yes	—
21	C₂×C₄⋊Dic₇₁	—	2	1	no	—
22	#14556068	—	2	2	no	—
23	#14556068	—	2	2	no	—
24	C₂×C₇₁⋊C₁₆	—	2	2	no	—
25	#14556060	—	2	2	no	—
26	#14556053	—	4	2	no	—
27	C₇₁⋊#95	—	4	2	no	—
28	C₂×M₁₆×C₇₁	—	2	2	no	—
29	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
32	#108×C₇₁	—	2	2	no	—
33	#108×C₇₁	—	2	2	no	—
34	#108×C₇₁	—	4	2	no	—
35	C₄⋊C₈×C₇₁	—	4	2	no	—
36	#97×C₇₁	—	2	2	no	—
37	C₂×Dic₁₄₂	—	2	2	no	—
38	C₂×Dic₁₄₂	—	4	1	no	—
39	C₂×Dic₁₄₂	—	4	1	no	—
40	C₂×Dic₁₄₂	—	4	1	no	—
41	C₂×Dic₁₄₂	—	4	1	no	—
42	C₂×Dic₁₄₂	—	4	1	no	—
43	C₂²×Dic₇₁	—	4	1	no	—
44	C₂²×Dic₇₁	—	4	1	no	—
45	Dic₂₈₄	—	2	4	no	—
46	Dic₂₈₄	—	4	2	no	—
47	C₄⋊Dic₇₁	—	4	1	no	—
48	C₄⋊Dic₇₁	—	4	1	no	—
49	C₇₁⋊C₁₆	—	4	2	no	—
50	C₂²×C₄×C₇₁	SD₁₆	1	2	yes	—
51	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
52	M₁₆×C₇₁	—	4	2	no	—
53	M₁₆×C₇₁	—	4	2	no	—
54	C₂×C₈×C₇₁	—	2	1	no	—
55	C₂×C₈×C₇₁	—	2	1	no	—
56	C₂×C₈×C₇₁	—	4	2	no	—
57	C₂×C₈×C₇₁	D₈	1	2	yes	—
58	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
60	C₂×C₈×C₇₁	#41	1	2	yes	—
61	C₂×C₈×C₇₁	#41	1	2	yes	—
62	C₄²×C₇₁	—	2	1	no	—
63	C₄²×C₇₁	—	2	1	no	—
64	C₂³×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×C₄×C₇₁	—	2	2	no	—
66	C₂×C₄×C₇₁	—	4	2	no	—
67	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
68	C₂×C₄×C₇₁	#123	1	1	yes	characteristic
69	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
70	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
72	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
73	C₂×Dic₇₁	—	4	2	no	—
74	C₂×Dic₇₁	—	8	1	no	—
75	C₂×Dic₇₁	—	8	1	no	—
76	C₂×Dic₇₁	—	8	2	no	—
77	C₂×Dic₇₁	—	8	1	no	—
78	Dic₁₄₂	—	4	2	no	—
79	Dic₁₄₂	—	4	2	no	—
80	Dic₁₄₂	—	8	1	no	—
81	Dic₁₄₂	—	8	1	no	—
82	C₈×C₇₁	—	2	2	no	—
83	C₈×C₇₁	—	8	2	no	—
84	C₈×C₇₁	#123	1	2	yes	—
85	C₈×C₇₁	#126	1	2	yes	—
86	C₂²×C₇₁	—	2	2	no	—
87	C₂²×C₇₁	#461	1	1	yes	characteristic
88	C₂²×C₇₁	#481	1	1	yes	characteristic
89	C₂²×C₇₁	#486	1	1	yes	characteristic
90	C₄×C₇₁	—	4	2	no	—
91	C₄×C₇₁	#397	1	1	yes	characteristic
92	C₄×C₇₁	#459	1	1	yes	characteristic
93	C₄×C₇₁	#480	1	1	yes	characteristic
94	C₄×C₇₁	#489	1	1	yes	characteristic
95	Dic₇₁	—	8	2	no	—
96	Dic₇₁	—	16	1	no	—
97	#13488	—	71	1	no	maximal, Sylow
98	C₂×C₇₁	—	2	1	no	—
99	C₂×C₇₁	—	2	1	no	—
100	C₂×C₇₁	#2071	1	1	yes	characteristic
101	C₂×C₇₁	#2239	1	2	yes	—
102	C₂×#362	—	71	2	no	—
103	#1971	—	71	1	no	—
104	#1849	D₇₁	1	1	yes	characteristic
105	#1383	—	71	2	no	—
106	#1377	—	71	1	no	—
107	C₇₁	#13488	1	1	yes	characteristic, complemented, Sylow
108	C₂²×Q₁₆	—	71	1	no	—
109	C₂×M₃₂	—	71	2	no	—
110	C₂×#108	—	2	1	no	—
111	C₂×#108	C₂×D₇₁	1	2	yes	—
112	C₂×#106	—	71	1	no	—
113	#423	—	2	1	no	—
114	C₂×#103	—	142	1	no	—
115	C₂×#97	C₂×D₇₁	1	1	yes	characteristic
116	#391	—	142	1	no	—
117	#362	—	142	4	no	—
118	C₂²×Q₈	—	142	1	no	—
119	C₂×Q₁₆	—	71	2	no	—
120	C₂×Q₁₆	—	142	1	no	—
121	C₂×Q₁₆	—	142	2	no	—
122	C₂×Q₁₆	—	142	2	no	—
123	C₂×M₁₆	—	2	2	no	—
124	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
125	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
126	C₂×C₄⋊C₄	—	142	1	no	—
127	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
128	M₃₂	—	142	2	no	—
129	M₃₂	—	142	2	no	—
130	C₂×C₁₆	—	142	2	no	—
131	#108	—	2	2	no	—
132	#108	—	2	2	no	—
133	#108	—	4	2	no	—
134	#106	—	142	2	no	—
135	C₄⋊C₈	—	4	2	no	—
136	#103	—	284	2	no	—
137	#97	—	2	2	no	—
138	#95	—	284	2	no	—
139	C₂×Q₈	—	284	1	no	—
140	C₂×Q₈	—	284	1	no	—
141	C₂×Q₈	—	284	1	no	—
142	C₂×Q₈	—	284	1	no	—
143	C₂×Q₈	—	284	1	no	—
144	C₂×Q₈	—	142	2	no	—
145	C₂²×C₄	—	284	1	no	—
146	C₂²×C₄	—	284	1	no	—
147	C₂²×C₄	#11765852	1	2	yes	—
148	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
149	Q₁₆	—	284	2	no	—
150	Q₁₆	—	142	4	no	—
151	M₁₆	—	4	2	no	—
152	M₁₆	—	4	2	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	#11765850	1	2	yes	—
157	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
158	C₂×C₈	#11765864	1	1	yes	characteristic
159	C₂×C₈	#11765865	1	2	yes	—
160	C₂×C₈	#11765866	1	2	yes	—
161	C₄⋊C₄	—	284	1	no	—
162	C₄⋊C₄	—	284	1	no	—
163	C₄²	—	2	1	no	—
164	C₄²	—	2	1	no	—
165	C₁₆	—	284	2	no	—
166	C₂³	C₂×#11765852	1	1	yes	characteristic
167	Q₈	—	284	2	no	—
168	Q₈	—	284	2	no	—
169	Q₈	—	568	1	no	—
170	Q₈	—	568	1	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	284	2	no	—
174	C₂×C₄	—	568	1	no	—
175	C₂×C₄	—	568	1	no	—
176	C₂×C₄	—	568	2	no	—
177	C₂×C₄	—	568	1	no	—
178	C₂×C₄	#14556099	1	1	yes	characteristic
179	C₂×C₄	#14556111	1	1	yes	characteristic
180	C₂×C₄	#14556117	1	1	yes	characteristic
181	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
182	C₂×C₄	#14556154	1	1	yes	characteristic
183	C₂×C₄	#14556159	1	1	yes	characteristic
184	C₈	—	2	2	no	—
185	C₈	—	8	2	no	—
186	C₈	#14556113	1	2	yes	—
187	C₈	#14556123	1	2	yes	—
188	C₂²	—	2	2	no	—
189	C₂²	#17986201	1	1	yes	characteristic
190	C₂²	#17986317	1	1	yes	characteristic
191	C₂²	#17986347	1	1	yes	characteristic
192	C₄	—	4	2	no	—
193	C₄	—	568	2	no	—
194	C₄	—	1136	1	no	—
195	C₄	#17985925	1	1	yes	characteristic
196	C₄	#17986191	1	1	yes	characteristic
197	C₄	#17986312	1	1	yes	characteristic
198	C₄	#17986361	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#22589744	1	1	yes	characteristic
202	C₂	#22590559	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17985814

C₂

yes

maximal

#22589211

C₂

yes

maximal, characteristic

#22586440

C₂

yes

maximal

#22586424

C₂

yes

maximal, characteristic

C₇₁×#1849

C₂

yes

maximal, characteristic

C₂²×Dic₂₈₄

C₂²

yes

characteristic

C₂×#14556068

C₂²

yes

—

C₂×#14556060

C₂²

yes

—

—

—

C₂×#108×C₇₁

—

C₂×#108×C₇₁

C₂²

yes

—

#423×C₇₁

—

C₂×#97×C₇₁

C₂²

yes

characteristic

C₂²×Dic₁₄₂

—

C₂×Dic₂₈₄

—

C₂×Dic₂₈₄

—

C₂×Dic₂₈₄

—

C₂×Dic₂₈₄

D₄

yes

—

C₂×C₄⋊Dic₇₁

—

#14556068

—

#14556068

—

C₂×C₇₁⋊C₁₆

—

#14556060

—

#14556053

—

C₇₁⋊#95

—

C₂×M₁₆×C₇₁

—

C₂²×C₈×C₇₁

D₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₄²×C₇₁

D₄

yes

characteristic

#108×C₇₁

—

#108×C₇₁

—

#108×C₇₁

—

C₄⋊C₈×C₇₁

—

#97×C₇₁

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂²×Dic₇₁

—

C₂²×Dic₇₁

—

Dic₂₈₄

—

Dic₂₈₄

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

SD₁₆

yes

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

D₈

yes

—

C₂×C₈×C₇₁

C₂×D₄

yes

C₂×C₈×C₇₁

yes

C₂×C₈×C₇₁

yes

—

C₂×C₈×C₇₁

#41

yes

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₂³×C₇₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

C₂×D₈

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

#123

yes

—

C₈×C₇₁

#126

yes

—

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

—

#13488

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

100

C₂×C₇₁

#2071

yes

characteristic

101

C₂×C₇₁

#2239

yes

—

102

C₂×#362

—

103

#1971

—

104

#1849

D₇₁

yes

characteristic

105

#1383

—

106

#1377

—

107

C₇₁

#13488

yes

characteristic, complemented, Sylow

108

C₂²×Q₁₆

—

109

C₂×M₃₂

—

110

C₂×#108

—

111

C₂×#108

C₂×D₇₁

yes

—

112

C₂×#106

—

113

#423

—

114

C₂×#103

—

142

—

115

C₂×#97

C₂×D₇₁

yes

characteristic

116

#391

—

142

—

117

#362

—

142

—

118

C₂²×Q₈

—

142

—

119

C₂×Q₁₆

—

120

C₂×Q₁₆

—

142

—

121

C₂×Q₁₆

—

142

—

122

C₂×Q₁₆

—

142

—

123

C₂×M₁₆

—

124

C₂²×C₈

C₂²⋊D₇₁

yes

characteristic

125

C₂²×C₈

C₂²×D₇₁

yes

characteristic

126

C₂×C₄⋊C₄

—

142

—

127

C₂×C₄²

C₂²⋊D₇₁

yes

characteristic

128

M₃₂

—

142

—

129

M₃₂

—

142

—

130

C₂×C₁₆

—

142

—

131

#108

—

132

#108

—

133

#108

—

134

#106

—

142

—

135

C₄⋊C₈

—

136

#103

—

284

—

137

#97

—

138

#95

—

284

—

139

C₂×Q₈

—

284

—

140

C₂×Q₈

—

284

—

141

C₂×Q₈

—

284

—

142

C₂×Q₈

—

284

—

143

C₂×Q₈

—

284

—

144

C₂×Q₈

—

142

—

145

C₂²×C₄

—

284

—

146

C₂²×C₄

—

284

—

147

C₂²×C₄

#11765852

yes

—

148

C₂²×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

149

Q₁₆

—

284

—

150

Q₁₆

—

142

—

151

M₁₆

—

152

M₁₆

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

#11765850

yes

—

157

C₂×C₈

D₄×D₇₁

yes

characteristic

158

C₂×C₈

#11765864

yes

characteristic

159

C₂×C₈

#11765865

yes

—

160

C₂×C₈

#11765866

yes

—

161

C₄⋊C₄

—

284

—

162

C₄⋊C₄

—

284

—

163

C₄²

—

164

C₄²

—

165

C₁₆

—

284

—

166

C₂³

C₂×#11765852

yes

characteristic

167

Q₈

—

284

—

168

Q₈

—

284

—

169

Q₈

—

568

—

170

Q₈

—

568

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

284

—

174

C₂×C₄

—

568

—

175

C₂×C₄

—

568

—

176

C₂×C₄

—

568

—

177

C₂×C₄

—

568

—

178

C₂×C₄

#14556099

yes

characteristic

179

C₂×C₄

#14556111

yes

characteristic

180

C₂×C₄

#14556117

yes

characteristic

181

C₂×C₄

C₂×#11765850

yes

characteristic

182

C₂×C₄

#14556154

yes

characteristic

183

C₂×C₄

#14556159

yes

characteristic

184

C₈

—

185

C₈

—

186

C₈

#14556113

yes

—

187

C₈

#14556123

yes

—

188

C₂²

—

189

C₂²

#17986201

yes

characteristic

190

C₂²

#17986317

yes

characteristic

191

C₂²

#17986347

yes

characteristic

192

C₄

—

193

C₄

—

568

—

194

C₄

—

1136

—

195

C₄

#17985925

yes

characteristic

196

C₄

#17986191

yes

characteristic

197

C₄

#17986312

yes

characteristic

198

C₄

#17986361

yes

characteristic

199

C₂

—

200

C₂

—

201

C₂

#22589744

yes

characteristic

202

C₂

#22590559

yes

—

	subgroup	G/H
center	C₂²	#17986317
commutator, cosocle	C₂²×C₈×C₇₁	C₂³
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1849	C₂
socle	C₂²×C₇₁	#481
prime radical	C₂³×C₇₁	C₂×SD₁₆
hypercenter	#1849	D₇₁
hypocenter	C₇₁	#13488
LCS	G ▷ C₂²×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1849

subgroup

G/H

center

C₂²

#17986317

commutator, cosocle

C₂²×C₈×C₇₁

C₂³

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₇₁×#1849

C₂

socle

C₂²×C₇₁

#481

prime radical

C₂³×C₇₁

C₂×SD₁₆

D₇₁

C₇₁

G ▷ C₂²×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1849

N	H ≅ G/N	type
107. C₇₁	#13488	semidirect

H ≅ G/N

type

107. C₇₁

#13488

semidirect

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

subgroup	index
C₂×#17985814	2
#22589211	2
#22586440	2
#22586424	2
C₇₁×#1849	2
#13488	71

subgroup

index

C₇₁×#1849

Subgroups — #31917283

Conjugacy classes of subgroups

Complements

Maximal subgroups