Group #31913640 — subgroups

Subgroups — #31913640

Order 18176.

362 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 111 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×#17986417	C₂	1	1	yes	maximal, characteristic
2	#22590122	C₂	1	2	yes	maximal, complemented
3	#22588383	C₂	1	2	yes	maximal
4	#22586442	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₁×#1384	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14556134	C₂²	1	1	yes	characteristic
7	#17986417	—	2	2	no	—
8	#17986417	—	2	2	no	—
9	C₂×#14556067	C₄	1	2	yes	complemented
10	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₁₆×D₇₁	C₄	1	2	yes	complemented
12	#17986037	C₂²	1	2	yes	—
13	#17985972	—	2	2	no	—
14	#17985826	—	2	2	no	—
15	#17985824	—	2	2	no	—
16	#17985755	—	2	2	no	—
17	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
18	#427×C₇₁	C₂²	1	2	yes	—
19	C₂×#11765865	D₄	1	1	yes	characteristic
20	#14556134	—	2	2	no	—
21	#14556134	—	2	2	no	—
22	C₂×#11765849	D₄	1	1	yes	characteristic
23	C₂×#11765848	Q₈	1	2	yes	—
24	C₂×C₈×D₇₁	C₂×C₄	1	2	yes	—
25	#14556087	—	2	2	no	—
26	#14556067	—	2	4	no	—
27	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂×C₇₁⋊C₁₆	D₄	1	2	yes	—
30	C₂×C₇₁⋊C₁₆	D₄	1	2	yes	—
31	C₁₆×D₇₁	—	2	2	no	—
32	C₁₆×D₇₁	—	2	2	no	—
33	#14556062	—	2	2	no	—
34	#14556060	—	2	2	no	—
35	#14556042	—	4	4	no	—
36	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
37	#117×C₇₁	—	2	2	no	—
38	M₃₂×C₇₁	—	2	2	no	—
39	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
40	#107×C₇₁	—	2	2	no	—
41	#106×C₇₁	—	2	2	no	—
42	#11765865	—	2	2	no	—
43	#11765865	—	2	2	no	—
44	C₂×Dic₁₄₂	Q₁₆	1	1	yes	characteristic
45	C₂×D₂₈₄	Q₁₆	1	1	yes	characteristic
46	C₂×C₂²⋊D₇₁	D₈	1	2	yes	—
47	C₂×C₄×D₇₁	C₄⋊C₄	1	2	yes	—
48	#11765849	—	2	2	no	—
49	#11765848	—	2	4	no	—
50	C₂×C₇₁⋊C₈	C₄⋊C₄	1	2	yes	—
51	C₈×D₇₁	—	2	2	no	—
52	C₈×D₇₁	—	2	2	no	—
53	C₄⋊Dic₇₁	—	4	2	no	—
54	C₂²⋊Dic₇₁	—	4	2	no	—
55	C₄×Dic₇₁	—	4	2	no	—
56	C₇₁⋊C₁₆	C₂²⋊C₄	1	1	yes	characteristic
57	C₇₁⋊C₁₆	C₂²⋊C₄	1	1	yes	characteristic
58	C₇₁⋊C₁₆	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₈×C₇₁	#41	1	2	yes	—
64	C₄⋊C₄×C₇₁	—	4	2	no	—
65	C₂²⋊C₄×C₇₁	—	4	2	no	—
66	C₄²×C₇₁	—	4	2	no	—
67	C₁₆×C₇₁	—	2	2	no	—
68	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
69	C₂²×D₇₁	#107	1	2	yes	—
70	C₂×C₄×C₇₁	—	8	2	no	—
71	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₇₁	#121	1	2	yes	—
73	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
74	C₂×C₄×C₇₁	#135	1	2	yes	—
75	C₂²⋊D₇₁	—	2	4	no	—
76	C₂×Dic₇₁	—	8	2	no	—
77	C₂×Dic₇₁	#107	1	2	yes	—
78	D₂₈₄	—	2	2	no	—
79	C₄×D₇₁	—	2	2	no	—
80	C₄×D₇₁	—	2	2	no	—
81	Dic₁₄₂	—	2	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	—	2	2	no	—
86	C₂²×C₇₁	—	4	1	no	—
87	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
88	C₂²×C₇₁	#466	1	2	yes	—
89	C₂×D₇₁	—	2	2	no	—
90	C₂×D₇₁	—	4	2	no	—
91	C₄×C₇₁	—	16	2	no	—
92	C₄×C₇₁	#380	1	1	yes	characteristic
93	C₄×C₇₁	#427	1	1	yes	characteristic
94	C₄×C₇₁	#467	1	2	yes	—
95	Dic₇₁	—	2	2	no	—
96	Dic₇₁	—	16	2	no	—
97	#12839	—	71	1	no	maximal, Sylow
98	C₂×C₇₁	—	2	2	no	—
99	C₂×C₇₁	—	2	1	no	—
100	C₂×C₇₁	#1808	1	1	yes	characteristic
101	D₇₁	—	4	2	no	—
102	C₂×#504	—	71	1	no	—
103	#2158	—	71	2	no	—
104	#1812	—	71	2	no	—
105	#1384	—	71	1	no	—
106	#1384	D₇₁	1	1	yes	characteristic, complemented
107	C₇₁	#12839	1	1	yes	characteristic, complemented, Sylow
108	C₂×#131	—	71	1	no	—
109	#504	—	142	2	no	—
110	#504	—	142	2	no	—
111	C₂×M₃₂	—	71	2	no	—
112	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
113	C₂²×C₁₆	—	71	1	no	—
114	C₂²×C₁₆	—	71	2	no	—
115	#427	—	71	2	no	—
116	#427	C₂×D₇₁	1	2	yes	—
117	#410	—	142	2	no	—
118	#367	—	142	2	no	—
119	#366	—	142	2	no	—
120	H(3,ℤ/4)	—	142	2	no	—
121	C₂×#41	—	71	1	no	—
122	#131	—	142	2	no	—
123	#131	—	142	2	no	—
124	C₂×M₁₆	—	71	1	no	—
125	C₂×M₁₆	—	71	2	no	—
126	C₂²×C₈	—	71	2	no	—
127	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
128	#117	—	2	2	no	—
129	#117	—	142	2	no	—
130	M₃₂	—	2	2	no	—
131	M₃₂	—	142	4	no	—
132	C₂×C₁₆	—	71	1	no	—
133	C₂×C₁₆	—	71	1	no	—
134	C₂×C₁₆	—	71	2	no	—
135	C₂×C₁₆	—	71	2	no	—
136	C₂×C₁₆	—	142	2	no	—
137	C₂×C₁₆	—	142	2	no	—
138	C₂×C₁₆	C₄×D₇₁	1	2	yes	—
139	#107	—	2	2	no	—
140	#107	—	142	2	no	—
141	#106	—	2	2	no	—
142	#106	—	142	2	no	—
143	#99	—	284	4	no	—
144	#41	—	142	2	no	—
145	#41	—	142	2	no	—
146	C₂×Q₈	—	71	1	no	—
147	C₂×D₄	—	71	2	no	—
148	C₂×D₄	—	71	1	no	—
149	C₂²×C₄	—	71	2	no	—
150	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
151	M₁₆	—	142	2	no	—
152	M₁₆	—	142	4	no	—
153	C₂×C₈	—	71	2	no	—
154	C₂×C₈	—	142	2	no	—
155	C₂×C₈	—	142	2	no	—
156	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
157	C₂×C₈	D₄×D₇₁	1	2	yes	—
158	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
159	C₂×C₈	#11765865	1	2	yes	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	284	2	no	—
162	C₂²⋊C₄	—	4	2	no	—
163	C₂²⋊C₄	—	284	2	no	—
164	C₄²	—	4	2	no	—
165	C₄²	—	284	2	no	—
166	C₁₆	—	2	2	no	—
167	C₁₆	—	71	1	no	—
168	C₁₆	—	71	1	no	—
169	C₁₆	—	71	2	no	—
170	C₂³	—	71	2	no	—
171	C₂³	D₈×D₇₁	1	1	yes	characteristic
172	Q₈	—	142	2	no	—
173	D₄	—	142	4	no	—
174	D₄	—	142	2	no	—
175	C₂×C₄	—	8	2	no	—
176	C₂×C₄	—	568	2	no	—
177	C₂×C₄	—	71	2	no	—
178	C₂×C₄	—	142	2	no	—
179	C₂×C₄	—	142	2	no	—
180	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
181	C₂×C₄	#14556101	1	2	yes	—
182	C₂×C₄	Q₁₆×D₇₁	1	1	yes	characteristic
183	C₂×C₄	#14556149	1	2	yes	—
184	C₈	—	142	2	no	—
185	C₈	C₂²⋊C₄×D₇₁	1	1	yes	characteristic
186	C₈	#14556085	1	1	yes	characteristic
187	C₈	#14556106	1	2	yes	—
188	C₂²	—	4	1	no	—
189	C₂²	—	284	2	no	—
190	C₂²	—	142	2	no	—
191	C₂²	#107×D₇₁	1	1	yes	characteristic
192	C₂²	#17986234	1	2	yes	—
193	C₄	—	16	2	no	—
194	C₄	—	1136	2	no	—
195	C₄	—	142	2	no	—
196	C₄	#17985858	1	1	yes	characteristic
197	C₄	#17986034	1	1	yes	characteristic
198	C₄	#17986239	1	2	yes	—
199	C₂	—	2	2	no	—
200	C₂	—	2	1	no	—
201	C₂	—	284	2	no	—
202	C₂	#22588368	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17986417

C₂

yes

maximal, characteristic

#22590122

C₂

yes

maximal, complemented

#22588383

C₂

yes

maximal

#22586442

C₂

yes

maximal, characteristic, complemented

C₇₁×#1384

C₂

yes

maximal, characteristic, complemented

C₂×#14556134

C₂²

yes

—

—

C₄

yes

complemented

C₂²×C₇₁⋊C₁₆

C₂²

yes

characteristic

C₂×C₁₆×D₇₁

C₄

yes

complemented

#17986037

C₂²

yes

—

—

—

—

—

C₂×M₃₂×C₇₁

C₂²

yes

characteristic

#427×C₇₁

C₂²

yes

—

C₂×#11765865

D₄

yes

—

—

D₄

yes

characteristic

C₂×#11765848

Q₈

yes

—

C₂×C₈×D₇₁

C₂×C₄

yes

—

#14556087

—

#14556067

—

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₇₁⋊C₁₆

D₄

yes

—

C₂×C₇₁⋊C₁₆

D₄

yes

—

C₁₆×D₇₁

—

C₁₆×D₇₁

—

#14556062

—

#14556060

—

#14556042

—

C₂²×C₈×C₇₁

C₂³

yes

characteristic

#117×C₇₁

—

M₃₂×C₇₁

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

—

#107×C₇₁

—

#106×C₇₁

—

#11765865

—

#11765865

—

C₂×Dic₁₄₂

Q₁₆

yes

characteristic

C₂×D₂₈₄

Q₁₆

yes

characteristic

C₂×C₂²⋊D₇₁

D₈

yes

—

C₂×C₄×D₇₁

C₄⋊C₄

yes

—

#11765849

—

#11765848

—

C₂×C₇₁⋊C₈

C₄⋊C₄

yes

—

C₈×D₇₁

—

C₈×D₇₁

—

C₄⋊Dic₇₁

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₇₁⋊C₁₆

C₂²⋊C₄

yes

characteristic

C₇₁⋊C₁₆

C₂²⋊C₄

yes

characteristic

C₇₁⋊C₁₆

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

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

—

C₄⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₁₆×C₇₁

—

C₂³×C₇₁

C₂×D₈

yes

characteristic

C₂²×D₇₁

#107

yes

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

#121

yes

—

C₂×C₄×C₇₁

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₁

#135

yes

—

C₂²⋊D₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

#107

yes

—

D₂₈₄

—

C₄×D₇₁

—

C₄×D₇₁

—

Dic₁₄₂

—

C₈×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₇₁

#117

yes

characteristic

C₈×C₇₁

C₂²⋊Q₈

yes

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×#107

yes

characteristic

C₂²×C₇₁

#466

yes

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

Dic₇₁

—

Dic₇₁

—

#12839

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

100

C₂×C₇₁

#1808

yes

characteristic

101

D₇₁

—

102

C₂×#504

—

103

#2158

—

104

#1812

—

105

#1384

—

106

#1384

D₇₁

yes

characteristic, complemented

107

C₇₁

#12839

yes

characteristic, complemented, Sylow

108

C₂×#131

—

109

#504

—

142

—

110

#504

—

142

—

111

C₂×M₃₂

—

112

C₂×M₃₂

C₂×D₇₁

yes

characteristic

113

C₂²×C₁₆

—

114

C₂²×C₁₆

—

115

#427

—

116

#427

C₂×D₇₁

yes

—

117

#410

—

142

—

118

#367

—

142

—

119

#366

—

142

—

120

H(3,ℤ/4)

—

142

—

121

C₂×#41

—

122

#131

—

142

—

123

#131

—

142

—

124

C₂×M₁₆

—

125

C₂×M₁₆

—

126

C₂²×C₈

—

127

C₂²×C₈

C₂²×D₇₁

yes

characteristic

128

#117

—

129

#117

—

142

—

130

M₃₂

—

131

M₃₂

—

142

—

132

C₂×C₁₆

—

133

C₂×C₁₆

—

134

C₂×C₁₆

—

135

C₂×C₁₆

—

136

C₂×C₁₆

—

142

—

137

C₂×C₁₆

—

142

—

138

C₂×C₁₆

C₄×D₇₁

yes

—

139

#107

—

140

#107

—

142

—

141

#106

—

142

#106

—

142

—

143

#99

—

284

—

144

#41

—

142

—

145

#41

—

142

—

146

C₂×Q₈

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂²×C₄

—

150

C₂²×C₄

D₄×D₇₁

yes

characteristic

151

M₁₆

—

142

—

152

M₁₆

—

142

—

153

C₂×C₈

—

154

C₂×C₈

—

142

—

155

C₂×C₈

—

142

—

156

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

157

C₂×C₈

D₄×D₇₁

yes

—

158

C₂×C₈

Q₈×D₇₁

yes

characteristic

159

C₂×C₈

#11765865

yes

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

284

—

162

C₂²⋊C₄

—

163

C₂²⋊C₄

—

284

—

164

C₄²

—

165

C₄²

—

284

—

166

C₁₆

—

167

C₁₆

—

168

C₁₆

—

169

C₁₆

—

170

C₂³

—

171

C₂³

D₈×D₇₁

yes

characteristic

172

Q₈

—

142

—

173

D₄

—

142

—

174

D₄

—

142

—

175

C₂×C₄

—

176

C₂×C₄

—

568

—

177

C₂×C₄

—

178

C₂×C₄

—

142

—

179

C₂×C₄

—

142

—

180

C₂×C₄

C₄⋊C₄×D₇₁

yes

characteristic

181

C₂×C₄

#14556101

yes

—

182

C₂×C₄

Q₁₆×D₇₁

yes

characteristic

183

C₂×C₄

#14556149

yes

—

184

C₈

—

142

—

185

C₈

C₂²⋊C₄×D₇₁

yes

characteristic

186

C₈

#14556085

yes

characteristic

187

C₈

#14556106

yes

—

188

C₂²

—

189

C₂²

—

284

—

190

C₂²

—

142

—

191

C₂²

#107×D₇₁

yes

characteristic

192

C₂²

#17986234

yes

—

193

C₄

—

194

C₄

—

1136

—

195

C₄

—

142

—

196

C₄

#17985858

yes

characteristic

197

C₄

#17986034

yes

characteristic

198

C₄

#17986239

yes

—

199

C₂

—

200

C₂

—

201

C₂

—

284

—

202

C₂

#22588368

yes

characteristic

	subgroup	G/H
center	C₄	#17986034
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1384	C₂
socle	C₂×C₇₁	#1808
prime radical	C₂×#11765865	D₄
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1384	D₇₁
hypocenter	C₇₁	#12839
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1384

subgroup

G/H

center

C₄

#17986034

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₇₁×#1384

C₂

C₂×C₇₁

D₄

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1384

N	H ≅ G/N	type
2. #22590122	C₂	semidirect
4. #22586442	C₂	semidirect
5. C₇₁×#1384	C₂	semidirect
9. C₂×#14556067	C₄	semidirect
11. C₂×C₁₆×D₇₁	C₄	semidirect
106. #1384	D₇₁	semidirect
107. C₇₁	#12839	semidirect

H ≅ G/N

type

2. #22590122

C₂

semidirect

4. #22586442

C₂

semidirect

5. C₇₁×#1384

C₂

semidirect

9. C₂×#14556067

C₄

semidirect

11. C₂×C₁₆×D₇₁

C₄

semidirect

106. #1384

D₇₁

semidirect

107. C₇₁

#12839

semidirect

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

subgroup	index
C₂×#17986417	2
#22590122	2
#22588383	2
#22586442	2
C₇₁×#1384	2
#12839	71

subgroup

index

C₇₁×#1384

Subgroups — #31913640

Conjugacy classes of subgroups

Complements

Maximal subgroups