Group #31913643 — subgroups

Subgroups — #31913643

Order 18176.

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

normal subgroups only

This group has 354 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×#17986416	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985828	C₂	1	2	yes	maximal, complemented
3	#22588390	C₂	1	1	yes	maximal, characteristic
4	#22586443	C₂	1	2	yes	maximal, complemented
5	C₇₁×#1813	C₂	1	1	yes	maximal, characteristic
6	#17986416	—	2	4	no	—
7	C₂×#14556068	C₄	1	2	yes	—
8	C₂×#14556068	C₂²	1	1	yes	characteristic
9	C₂²×C₇₁⋊C₁₆	C₄	1	2	yes	—
10	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂×#14556064	C₂²	1	2	yes	—
12	#17985983	—	2	1	no	—
13	#17985828	—	2	4	no	—
14	C₇₁⋊#338	—	2	1	no	—
15	C₂×#131×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×#108×C₇₁	C₂²	1	2	yes	—
17	#413×C₇₁	—	2	1	no	—
18	#338×C₇₁	—	2	1	no	—
19	C₂×#11765849	—	2	2	no	—
20	#14556068	—	2	4	no	—
21	#14556068	—	2	2	no	—
22	C₂×C₇₁⋊C₁₆	—	2	4	no	—
23	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂×C₇₁⋊C₁₆	D₄	1	4	yes	—
27	#14556064	—	2	2	no	—
28	#14556064	—	2	2	no	—
29	#14556047	—	4	1	no	—
30	#14556045	—	4	1	no	—
31	C₂×#41×C₇₁	D₄	1	1	yes	characteristic
32	#131×C₇₁	—	2	2	no	—
33	#131×C₇₁	—	2	2	no	—
34	C₂×M₁₆×C₇₁	—	2	2	no	—
35	C₂×M₁₆×C₇₁	D₄	1	1	yes	characteristic
36	C₂×M₁₆×C₇₁	Q₈	1	2	yes	—
37	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
39	#108×C₇₁	—	2	2	no	—
40	#108×C₇₁	—	2	2	no	—
41	#101×C₇₁	—	4	1	no	—
42	#100×C₇₁	—	4	1	no	—
43	#11765849	—	4	2	no	—
44	#11765849	—	4	2	no	—
45	C₂×C₇₁⋊C₈	—	4	2	no	—
46	C₇₁⋊C₁₆	—	2	2	no	—
47	C₇₁⋊C₁₆	C₂²⋊C₄	1	2	yes	—
48	C₇₁⋊C₁₆	C₂²⋊C₄	1	2	yes	—
49	#41×C₇₁	—	2	2	no	—
50	#41×C₇₁	—	2	2	no	—
51	C₂×Q₈×C₇₁	D₈	1	1	yes	characteristic
52	C₂×D₄×C₇₁	D₈	1	1	yes	characteristic
53	C₂×D₄×C₇₁	Q₁₆	1	2	yes	—
54	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
56	M₁₆×C₇₁	—	2	2	no	—
57	M₁₆×C₇₁	—	2	4	no	—
58	M₁₆×C₇₁	—	4	2	no	—
59	M₁₆×C₇₁	—	4	2	no	—
60	C₂×C₈×C₇₁	—	2	2	no	—
61	C₂×C₈×C₇₁	—	2	2	no	—
62	C₂×C₈×C₇₁	—	4	2	no	—
63	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
64	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₈×C₇₁	#41	1	2	yes	—
68	C₂³×C₇₁	#107	1	2	yes	—
69	C₂³×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
70	Q₈×C₇₁	—	2	2	no	—
71	D₄×C₇₁	—	2	4	no	—
72	D₄×C₇₁	—	2	2	no	—
73	C₂×C₄×C₇₁	—	2	2	no	—
74	C₂×C₄×C₇₁	—	2	2	no	—
75	C₂×C₄×C₇₁	#107	1	2	yes	—
76	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₇₁	#121	1	2	yes	—
78	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₇₁	#135	1	2	yes	—
80	C₈×C₇₁	—	2	2	no	—
81	C₈×C₇₁	—	8	2	no	—
82	C₈×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₈×C₇₁	#117	1	1	yes	characteristic
84	C₈×C₇₁	C₂²⋊Q₈	1	2	yes	—
85	C₇₁⋊C₈	—	8	2	no	—
86	C₂²×C₇₁	—	2	2	no	—
87	C₂²×C₇₁	—	4	1	no	—
88	C₂²×C₇₁	—	4	2	no	—
89	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
90	C₂²×C₇₁	#467	1	2	yes	—
91	C₄×C₇₁	—	2	2	no	—
92	C₄×C₇₁	#380	1	1	yes	characteristic
93	C₄×C₇₁	#427	1	1	yes	characteristic
94	C₄×C₇₁	#466	1	2	yes	—
95	#12840	—	71	1	no	maximal, Sylow
96	C₂×C₇₁	—	2	1	no	—
97	C₂×C₇₁	—	2	2	no	—
98	C₂×C₇₁	—	4	2	no	—
99	C₂×C₇₁	#1808	1	1	yes	characteristic
100	C₂×#504	—	71	1	no	—
101	C₂×#368	—	71	2	no	—
102	#1813	—	71	1	no	—
103	#1813	D₇₁	1	1	yes	characteristic
104	#1385	—	71	2	no	—
105	C₇₁	#12840	1	1	yes	characteristic, complemented, Sylow
106	C₂×#131	C₂×D₇₁	1	1	yes	characteristic
107	#504	—	142	4	no	—
108	C₂×M₃₂	—	71	2	no	—
109	C₂×M₃₂	—	71	1	no	—
110	C₂²×C₁₆	—	71	1	no	—
111	C₂²×C₁₆	—	71	2	no	—
112	C₂×#108	—	71	2	no	—
113	C₂×#108	C₂×D₇₁	1	2	yes	—
114	#413	—	2	1	no	—
115	#413	—	142	1	no	—
116	#368	—	142	4	no	—
117	#338	—	2	1	no	—
118	#338	—	142	1	no	—
119	C₂×#41	C₂²⋊D₇₁	1	1	yes	characteristic
120	#131	—	2	2	no	—
121	#131	—	2	2	no	—
122	C₂×M₁₆	—	2	2	no	—
123	C₂×M₁₆	—	142	2	no	—
124	C₂×M₁₆	Dic₁₄₂	1	2	yes	—
125	C₂×M₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
126	C₂²×C₈	C₄×D₇₁	1	2	yes	—
127	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
128	M₃₂	—	142	4	no	—
129	M₃₂	—	142	2	no	—
130	C₂×C₁₆	—	71	1	no	—
131	C₂×C₁₆	—	71	1	no	—
132	C₂×C₁₆	—	71	4	no	—
133	C₂×C₁₆	—	71	2	no	—
134	C₂×C₁₆	—	142	4	no	—
135	#108	—	2	2	no	—
136	#108	—	2	2	no	—
137	#108	—	142	2	no	—
138	#108	—	142	2	no	—
139	#101	—	4	1	no	—
140	#101	—	284	1	no	—
141	#100	—	4	1	no	—
142	#100	—	284	1	no	—
143	#41	—	2	2	no	—
144	#41	—	2	2	no	—
145	C₂×Q₈	#11765850	1	1	yes	characteristic
146	C₂×D₄	#11765850	1	1	yes	characteristic
147	C₂×D₄	#11765855	1	2	yes	—
148	C₂²×C₄	Dic₇₁⋊C₄	1	2	yes	—
149	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
150	M₁₆	—	2	2	no	—
151	M₁₆	—	2	4	no	—
152	M₁₆	—	4	2	no	—
153	M₁₆	—	4	2	no	—
154	M₁₆	—	284	2	no	—
155	M₁₆	—	284	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	4	2	no	—
159	C₂×C₈	—	284	2	no	—
160	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
161	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
162	C₂×C₈	D₄×D₇₁	1	2	yes	—
163	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
164	C₂×C₈	#11765864	1	2	yes	—
165	C₁₆	—	71	2	no	—
166	C₁₆	—	71	2	no	—
167	C₁₆	—	142	2	no	—
168	C₂³	#14556061	1	2	yes	—
169	C₂³	C₂×#11765855	1	1	yes	characteristic
170	Q₈	—	2	2	no	—
171	D₄	—	2	4	no	—
172	D₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	#14556061	1	2	yes	—
176	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
177	C₂×C₄	#14556100	1	2	yes	—
178	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
179	C₂×C₄	#14556148	1	2	yes	—
180	C₈	—	2	2	no	—
181	C₈	—	8	2	no	—
182	C₈	—	568	2	no	—
183	C₈	C₂²⋊C₄×D₇₁	1	1	yes	characteristic
184	C₈	#14556084	1	1	yes	characteristic
185	C₈	#14556105	1	2	yes	—
186	C₂²	—	2	2	no	—
187	C₂²	—	4	1	no	—
188	C₂²	—	4	2	no	—
189	C₂²	C₂×#14556061	1	1	yes	characteristic
190	C₂²	#17986238	1	2	yes	—
191	C₄	—	2	2	no	—
192	C₄	#17985857	1	1	yes	characteristic
193	C₄	#17986035	1	1	yes	characteristic
194	C₄	#17986233	1	2	yes	—
195	C₂	—	2	1	no	—
196	C₂	—	2	2	no	—
197	C₂	—	4	2	no	—
198	C₂	#22588367	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17986416

C₂

yes

maximal, characteristic

C₂×#17985828

C₂

yes

maximal, complemented

#22588390

C₂

yes

maximal, characteristic

#22586443

C₂

yes

maximal, complemented

C₇₁×#1813

C₂

yes

maximal, characteristic

#17986416

—

C₂×#14556068

C₄

yes

—

C₂×#14556068

C₂²

yes

characteristic

C₂²×C₇₁⋊C₁₆

C₄

yes

—

C₂²×C₇₁⋊C₁₆

C₂²

yes

characteristic

C₂×#14556064

C₂²

yes

—

#17985983

—

#17985828

—

C₇₁⋊#338

—

C₂×#131×C₇₁

C₂²

yes

characteristic

C₂×#108×C₇₁

C₂²

yes

—

#413×C₇₁

—

#338×C₇₁

—

C₂×#11765849

—

#14556068

—

#14556068

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

—

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₇₁⋊C₁₆

D₄

yes

—

—

—

—

—

C₂×#41×C₇₁

D₄

yes

characteristic

#131×C₇₁

—

#131×C₇₁

—

C₂×M₁₆×C₇₁

—

C₂×M₁₆×C₇₁

D₄

yes

characteristic

C₂×M₁₆×C₇₁

Q₈

yes

—

C₂²×C₈×C₇₁

C₂×C₄

yes

—

C₂²×C₈×C₇₁

C₂³

yes

characteristic

#108×C₇₁

—

#108×C₇₁

—

#101×C₇₁

—

#100×C₇₁

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₇₁⋊C₁₆

—

C₇₁⋊C₁₆

C₂²⋊C₄

yes

—

C₇₁⋊C₁₆

C₂²⋊C₄

yes

—

#41×C₇₁

—

#41×C₇₁

—

C₂×Q₈×C₇₁

D₈

yes

characteristic

C₂×D₄×C₇₁

D₈

yes

characteristic

C₂×D₄×C₇₁

Q₁₆

yes

—

C₂²×C₄×C₇₁

C₄⋊C₄

yes

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

—

C₂×C₈×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₁

#41

yes

—

C₂³×C₇₁

#107

yes

—

C₂³×C₇₁

C₂×Q₁₆

yes

characteristic

Q₈×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#107

yes

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

#121

yes

—

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

#135

yes

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₇₁

#117

yes

characteristic

C₈×C₇₁

C₂²⋊Q₈

yes

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×#107

yes

characteristic

C₂²×C₇₁

#467

yes

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

#12840

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#1808

yes

characteristic

100

C₂×#504

—

101

C₂×#368

—

102

#1813

—

103

#1813

D₇₁

yes

characteristic

104

#1385

—

105

C₇₁

#12840

yes

characteristic, complemented, Sylow

106

C₂×#131

C₂×D₇₁

yes

characteristic

107

#504

—

142

—

108

C₂×M₃₂

—

109

C₂×M₃₂

—

110

C₂²×C₁₆

—

111

C₂²×C₁₆

—

112

C₂×#108

—

113

C₂×#108

C₂×D₇₁

yes

—

114

#413

—

115

#413

—

142

—

116

#368

—

142

—

117

#338

—

118

#338

—

142

—

119

C₂×#41

C₂²⋊D₇₁

yes

characteristic

120

#131

—

121

#131

—

122

C₂×M₁₆

—

123

C₂×M₁₆

—

142

—

124

C₂×M₁₆

Dic₁₄₂

yes

—

125

C₂×M₁₆

C₂²⋊D₇₁

yes

characteristic

126

C₂²×C₈

C₄×D₇₁

yes

—

127

C₂²×C₈

C₂²×D₇₁

yes

characteristic

128

M₃₂

—

142

—

129

M₃₂

—

142

—

130

C₂×C₁₆

—

131

C₂×C₁₆

—

132

C₂×C₁₆

—

133

C₂×C₁₆

—

134

C₂×C₁₆

—

142

—

135

#108

—

136

#108

—

137

#108

—

142

—

138

#108

—

142

—

139

#101

—

140

#101

—

284

—

141

#100

—

142

#100

—

284

—

143

#41

—

144

#41

—

145

C₂×Q₈

#11765850

yes

characteristic

146

C₂×D₄

#11765850

yes

characteristic

147

C₂×D₄

#11765855

yes

—

148

C₂²×C₄

Dic₇₁⋊C₄

yes

—

149

C₂²×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

150

M₁₆

—

151

M₁₆

—

152

M₁₆

—

153

M₁₆

—

154

M₁₆

—

284

—

155

M₁₆

—

284

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

284

—

160

C₂×C₈

Dic₇₁⋊C₄

yes

—

161

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

162

C₂×C₈

D₄×D₇₁

yes

—

163

C₂×C₈

C₂×Dic₁₄₂

yes

characteristic

164

C₂×C₈

#11765864

yes

—

165

C₁₆

—

166

C₁₆

—

167

C₁₆

—

142

—

168

C₂³

#14556061

yes

—

169

C₂³

C₂×#11765855

yes

characteristic

170

Q₈

—

171

D₄

—

172

D₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

#14556061

yes

—

176

C₂×C₄

C₂×Dic₇₁⋊C₄

yes

characteristic

177

C₂×C₄

#14556100

yes

—

178

C₂×C₄

C₂×#11765850

yes

characteristic

179

C₂×C₄

#14556148

yes

—

180

C₈

—

181

C₈

—

182

C₈

—

568

—

183

C₈

C₂²⋊C₄×D₇₁

yes

characteristic

184

C₈

#14556084

yes

characteristic

185

C₈

#14556105

yes

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

C₂×#14556061

yes

characteristic

190

C₂²

#17986238

yes

—

191

C₄

—

192

C₄

#17985857

yes

characteristic

193

C₄

#17986035

yes

characteristic

194

C₄

#17986233

yes

—

195

C₂

—

196

C₂

—

197

C₂

—

198

C₂

#22588367

yes

characteristic

	subgroup	G/H
center	C₄	#17986035
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1813	C₂
socle	C₂×C₇₁	#1808
prime radical	C₂×#41×C₇₁	D₄
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1813	D₇₁
hypocenter	C₇₁	#12840
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ #1813

subgroup

G/H

center

C₄

#17986035

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₇₁×#1813

C₂

socle

C₂×C₇₁

#1808

prime radical

C₂×#41×C₇₁

D₄

cosocle

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ #1813

N	H ≅ G/N	type
2. C₂×#17985828	C₂	semidirect
4. #22586443	C₂	semidirect
105. C₇₁	#12840	semidirect

H ≅ G/N

type

2. C₂×#17985828

C₂

semidirect

4. #22586443

C₂

semidirect

105. C₇₁

#12840

semidirect

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

subgroup	index
C₂×#17986416	2
C₂×#17985828	2
#22588390	2
#22586443	2
C₇₁×#1813	2
#12840	71

subgroup

index

C₇₁×#1813

Subgroups — #31913643

Conjugacy classes of subgroups

Complements

Maximal subgroups