Group #31897425 — subgroups

Subgroups — #31897425

Order 18176.

514 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 159 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	#22588512	C₂	1	1	yes	maximal, characteristic, complemented
2	#22586791	C₂	1	4	yes	maximal, complemented
3	C₂×#17985720	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₁×#3147	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556064	—	2	2	no	—
6	#17986023	C₂²	1	2	yes	—
7	C₂×C₄×C₇₁⋊C₈	C₂²	1	2	yes	—
8	D₄⋊(C₇₁⋊C₈)	—	2	8	no	—
9	#17985720	—	2	2	no	—
10	C₂×#135×C₇₁	—	2	2	no	—
11	#532×C₇₁	C₂²	1	2	yes	—
12	#495×C₇₁	—	2	2	no	—
13	#494×C₇₁	C₄	1	1	yes	characteristic
14	#493×C₇₁	C₄	1	1	yes	characteristic
15	#492×C₇₁	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×#11765849	—	2	2	no	—
18	C₂²×C₇₁⋊C₈	—	2	2	no	—
19	#14556064	—	4	4	no	—
20	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
21	C₄×C₇₁⋊C₈	—	2	2	no	—
22	C₄×C₇₁⋊C₈	D₄	1	4	yes	—
23	C₂×#41×C₇₁	—	2	2	no	—
24	#135×C₇₁	—	4	4	no	—
25	#135×C₇₁	—	4	4	no	—
26	C₂×Q₁₆×C₇₁	—	2	2	no	—
27	C₂×SD₁₆×C₇₁	—	2	4	no	—
28	C₂×D₈×C₇₁	—	2	2	no	—
29	C₂²×C₈×C₇₁	D₄	1	2	yes	—
30	C₄⋊Q₈×C₇₁	C₂×C₄	1	2	yes	—
31	#127×C₇₁	C₂×C₄	1	2	yes	—
32	#124×C₇₁	—	2	2	no	—
33	#121×C₇₁	—	2	4	no	—
34	C₄×D₄×C₇₁	—	2	4	no	—
35	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
36	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₇₁	D₄	1	2	yes	—
39	#11765849	—	4	4	no	—
40	C₂×C₇₁⋊C₈	—	2	4	no	—
41	C₂×C₇₁⋊C₈	—	4	4	no	—
42	C₂×C₇₁⋊C₈	—	4	2	no	—
43	#41×C₇₁	—	4	4	no	—
44	#41×C₇₁	—	4	4	no	—
45	C₂×Q₈×C₇₁	—	2	2	no	—
46	C₂×D₄×C₇₁	—	2	2	no	—
47	C₂×D₄×C₇₁	—	2	4	no	—
48	C₂×D₄×C₇₁	—	4	2	no	—
49	C₂²×C₄×C₇₁	—	4	2	no	—
50	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₇₁	#41	1	2	yes	—
52	Q₁₆×C₇₁	—	4	4	no	—
53	SD₁₆×C₇₁	—	4	8	no	—
54	D₈×C₇₁	—	4	4	no	—
55	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₇₁	D₈	1	4	yes	—
58	C₂×C₈×C₇₁	SD₁₆	1	4	yes	—
59	C₄⋊C₄×C₇₁	—	2	4	no	—
60	C₂²⋊C₄×C₇₁	—	4	4	no	—
61	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₇₁	C₂×D₄	1	2	yes	—
63	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
64	C₂³×C₇₁	—	4	2	no	—
65	C₂³×C₇₁	#124	1	1	yes	characteristic
66	Q₈×C₇₁	—	4	4	no	—
67	D₄×C₇₁	—	4	4	no	—
68	D₄×C₇₁	—	4	8	no	—
69	D₄×C₇₁	—	8	4	no	—
70	C₂×C₄×C₇₁	—	2	4	no	—
71	C₂×C₄×C₇₁	—	4	2	no	—
72	C₂×C₄×C₇₁	—	8	2	no	—
73	C₂×C₄×C₇₁	—	8	2	no	—
74	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
76	C₂×C₄×C₇₁	#121	1	2	yes	—
77	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
78	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
79	C₂×C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
80	C₈×C₇₁	#102	1	4	yes	—
81	C₈×C₇₁	#102	1	4	yes	—
82	C₇₁⋊C₈	—	4	4	no	—
83	C₇₁⋊C₈	—	8	2	no	—
84	C₂²×C₇₁	—	2	2	no	—
85	C₂²×C₇₁	—	8	2	no	—
86	C₂²×C₇₁	—	8	4	no	—
87	C₂²×C₇₁	#390	1	1	yes	characteristic
88	C₂²×C₇₁	#486	1	2	yes	—
89	C₄×C₇₁	—	2	2	no	—
90	C₄×C₇₁	—	8	2	no	—
91	C₄×C₇₁	C₂×#102	1	2	yes	—
92	C₄×C₇₁	#444	1	4	yes	—
93	C₄×C₇₁	#492	1	1	yes	characteristic
94	C₄×C₇₁	#493	1	1	yes	characteristic
95	#9862	—	71	1	no	maximal, Sylow
96	C₂×C₇₁	—	2	2	no	—
97	C₂×C₇₁	—	16	2	no	—
98	C₂×C₇₁	#1957	1	1	yes	characteristic
99	C₂×C₇₁	#1975	1	2	yes	—
100	#3147	D₇₁	1	1	yes	characteristic
101	#1848	—	71	1	no	—
102	#1492	—	71	4	no	—
103	C₂×#322	—	71	1	no	—
104	C₇₁	#9862	1	1	yes	characteristic, complemented, Sylow
105	C₂×#135	—	2	2	no	—
106	#532	C₂×D₇₁	1	2	yes	—
107	#495	—	2	2	no	—
108	#494	Dic₇₁	1	1	yes	characteristic
109	#493	Dic₇₁	1	1	yes	characteristic
110	#492	Dic₇₁	1	2	yes	—
111	C₂×#108	—	142	2	no	—
112	#423	—	71	2	no	—
113	C₂×C₄×C₈	—	71	2	no	—
114	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
115	D₄⋊C₈	—	142	8	no	—
116	#322	—	142	2	no	—
117	C₂×#41	—	2	2	no	—
118	#135	—	4	4	no	—
119	#135	—	4	4	no	—
120	C₂×Q₁₆	—	2	2	no	—
121	C₂×SD₁₆	—	2	4	no	—
122	C₂×D₈	—	2	2	no	—
123	C₂×M₁₆	—	142	2	no	—
124	C₂²×C₈	—	142	2	no	—
125	C₂²×C₈	C₂²⋊D₇₁	1	2	yes	—
126	C₄⋊Q₈	C₂×Dic₇₁	1	2	yes	—
127	#127	C₂×Dic₇₁	1	2	yes	—
128	#124	—	2	2	no	—
129	#121	—	2	4	no	—
130	C₄×D₄	—	2	4	no	—
131	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
132	#108	—	284	4	no	—
133	C₄⋊C₈	—	142	4	no	—
134	C₄×C₈	—	71	4	no	—
135	C₄×C₈	—	142	2	no	—
136	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
137	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
138	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
139	#41	—	4	4	no	—
140	#41	—	4	4	no	—
141	C₂×Q₈	—	2	2	no	—
142	C₂×D₄	—	2	2	no	—
143	C₂×D₄	—	2	4	no	—
144	C₂×D₄	—	4	2	no	—
145	C₂²×C₄	—	4	2	no	—
146	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
147	C₂²×C₄	#11765864	1	2	yes	—
148	Q₁₆	—	4	4	no	—
149	SD₁₆	—	4	8	no	—
150	D₈	—	4	4	no	—
151	M₁₆	—	284	4	no	—
152	C₂×C₈	—	284	4	no	—
153	C₂×C₈	—	284	2	no	—
154	C₂×C₈	—	142	4	no	—
155	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
156	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
157	C₂×C₈	#11765850	1	4	yes	—
158	C₂×C₈	#11765853	1	4	yes	—
159	C₄⋊C₄	—	2	4	no	—
160	C₂²⋊C₄	—	4	4	no	—
161	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
162	C₄²	D₄×D₇₁	1	2	yes	—
163	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
164	C₂³	—	4	2	no	—
165	C₂³	#14556115	1	1	yes	characteristic
166	Q₈	—	4	4	no	—
167	D₄	—	4	8	no	—
168	D₄	—	4	4	no	—
169	D₄	—	8	4	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
175	C₂×C₄	D₄×Dic₇₁	1	2	yes	—
176	C₂×C₄	#14556099	1	2	yes	—
177	C₂×C₄	#14556124	1	1	yes	characteristic
178	C₂×C₄	C₂×#11765850	1	2	yes	—
179	C₂×C₄	C₂×#11765853	1	2	yes	—
180	C₈	—	284	4	no	—
181	C₈	—	568	2	no	—
182	C₈	#14556049	1	4	yes	—
183	C₈	#14556049	1	4	yes	—
184	C₂²	—	2	2	no	—
185	C₂²	—	8	4	no	—
186	C₂²	—	8	2	no	—
187	C₂²	#17985902	1	1	yes	characteristic
188	C₂²	#17986345	1	2	yes	—
189	C₄	—	2	2	no	—
190	C₄	—	8	2	no	—
191	C₄	C₂×#14556049	1	2	yes	—
192	C₄	#17986121	1	4	yes	—
193	C₄	#17986371	1	1	yes	characteristic
194	C₄	#17986375	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	—	16	2	no	—
197	C₂	#22589147	1	1	yes	characteristic
198	C₂	#22589234	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22588512

C₂

yes

maximal, characteristic, complemented

#22586791

C₂

yes

maximal, complemented

C₂×#17985720

C₂

yes

maximal, characteristic, complemented

C₇₁×#3147

C₂

yes

maximal, characteristic

C₂×#14556064

—

#17986023

C₂²

yes

—

C₂×C₄×C₇₁⋊C₈

C₂²

yes

—

D₄⋊(C₇₁⋊C₈)

—

#17985720

—

C₂×#135×C₇₁

—

#532×C₇₁

C₂²

yes

—

#495×C₇₁

—

#494×C₇₁

C₄

yes

characteristic

#493×C₇₁

C₄

yes

characteristic

#492×C₇₁

C₄

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×#11765849

—

C₂²×C₇₁⋊C₈

—

#14556064

—

C₄⋊(C₇₁⋊C₈)

—

C₄×C₇₁⋊C₈

—

C₄×C₇₁⋊C₈

D₄

yes

—

C₂×#41×C₇₁

—

#135×C₇₁

—

#135×C₇₁

—

C₂×Q₁₆×C₇₁

—

C₂×SD₁₆×C₇₁

—

C₂×D₈×C₇₁

—

C₂²×C₈×C₇₁

D₄

yes

—

C₄⋊Q₈×C₇₁

C₂×C₄

yes

—

#127×C₇₁

C₂×C₄

yes

—

#124×C₇₁

—

#121×C₇₁

—

C₄×D₄×C₇₁

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

#41×C₇₁

—

#41×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₁

#41

yes

—

Q₁₆×C₇₁

—

SD₁₆×C₇₁

—

D₈×C₇₁

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₂×C₈×C₇₁

D₈

yes

—

C₂×C₈×C₇₁

SD₁₆

yes

—

C₄⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

C₂²×C₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

—

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₂³×C₇₁

—

C₂³×C₇₁

#124

yes

characteristic

Q₈×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄×D₄

yes

—

C₂×C₄×C₇₁

#121

yes

—

C₂×C₄×C₇₁

#127

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

—

C₂×C₄×C₇₁

C₂×SD₁₆

yes

—

C₈×C₇₁

#102

yes

—

C₈×C₇₁

#102

yes

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#390

yes

characteristic

C₂²×C₇₁

#486

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×#102

yes

—

C₄×C₇₁

#444

yes

—

C₄×C₇₁

yes

C₄×C₇₁

yes

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#1957

yes

characteristic

C₂×C₇₁

#1975

yes

—

100

#3147

D₇₁

yes

characteristic

101

#1848

—

102

#1492

—

103

C₂×#322

—

104

C₇₁

#9862

yes

characteristic, complemented, Sylow

105

C₂×#135

—

106

#532

C₂×D₇₁

yes

—

107

#495

—

108

#494

Dic₇₁

yes

characteristic

109

#493

Dic₇₁

yes

characteristic

110

#492

Dic₇₁

yes

—

111

C₂×#108

—

142

—

112

#423

—

113

C₂×C₄×C₈

—

114

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

115

D₄⋊C₈

—

142

—

116

#322

—

142

—

117

C₂×#41

—

118

#135

—

119

#135

—

120

C₂×Q₁₆

—

121

C₂×SD₁₆

—

122

C₂×D₈

—

123

C₂×M₁₆

—

142

—

124

C₂²×C₈

—

142

—

125

C₂²×C₈

C₂²⋊D₇₁

yes

—

126

C₄⋊Q₈

C₂×Dic₇₁

yes

—

127

#127

C₂×Dic₇₁

yes

—

128

#124

—

129

#121

—

130

C₄×D₄

—

131

C₂×C₄²

C₂²×D₇₁

yes

characteristic

132

#108

—

284

—

133

C₄⋊C₈

—

142

—

134

C₄×C₈

—

135

C₄×C₈

—

142

—

136

C₄×C₈

C₂×Dic₇₁

yes

characteristic

137

C₄×C₈

C₂×Dic₇₁

yes

characteristic

138

C₄×C₈

C₂²⋊D₇₁

yes

—

139

#41

—

140

#41

—

141

C₂×Q₈

—

142

C₂×D₄

—

143

C₂×D₄

—

144

C₂×D₄

—

145

C₂²×C₄

—

146

C₂²×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

147

C₂²×C₄

#11765864

yes

—

148

Q₁₆

—

149

SD₁₆

—

150

D₈

—

151

M₁₆

—

284

—

152

C₂×C₈

—

284

—

153

C₂×C₈

—

284

—

154

C₂×C₈

—

142

—

155

C₂×C₈

C₂²⋊Dic₇₁

yes

—

156

C₂×C₈

C₂²⋊Dic₇₁

yes

—

157

C₂×C₈

#11765850

yes

—

158

C₂×C₈

#11765853

yes

—

159

C₄⋊C₄

—

160

C₂²⋊C₄

—

161

C₄²

C₂²×Dic₇₁

yes

characteristic

162

C₄²

D₄×D₇₁

yes

—

163

C₄²

C₂×C₂²⋊D₇₁

yes

characteristic

164

C₂³

—

165

C₂³

#14556115

yes

characteristic

166

Q₈

—

167

D₄

—

168

D₄

—

169

D₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

C₂×C₂²⋊Dic₇₁

yes

characteristic

175

C₂×C₄

D₄×Dic₇₁

yes

—

176

C₂×C₄

#14556099

yes

—

177

C₂×C₄

#14556124

yes

characteristic

178

C₂×C₄

C₂×#11765850

yes

—

179

C₂×C₄

C₂×#11765853

yes

—

180

C₈

—

284

—

181

C₈

—

568

—

182

C₈

#14556049

yes

—

183

C₈

#14556049

yes

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

#17985902

yes

characteristic

188

C₂²

#17986345

yes

—

189

C₄

—

190

C₄

—

191

C₄

C₂×#14556049

yes

—

192

C₄

#17986121

yes

—

193

C₄

#17986371

yes

characteristic

194

C₄

#17986375

yes

characteristic

195

C₂

—

196

C₂

—

197

C₂

#22589147

yes

characteristic

198

C₂

#22589234

yes

—

	subgroup	G/H
center	C₂×C₄	#14556124
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting, prime radical	C₇₁×#3147	C₂
socle	C₂²×C₇₁	#390
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#3147	D₇₁
hypocenter	C₇₁	#9862
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3147

subgroup

G/H

center

C₂×C₄

#14556124

commutator

C₄²×C₇₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting, prime radical

C₇₁×#3147

C₂

socle

C₂²×C₇₁

#390

cosocle

C₂×C₄²×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3147

N	H ≅ G/N	type
1. #22588512	C₂	semidirect
2. #22586791	C₂	semidirect
3. C₂×#17985720	C₂	semidirect
104. C₇₁	#9862	semidirect

H ≅ G/N

type

1. #22588512

C₂

semidirect

2. #22586791

C₂

semidirect

3. C₂×#17985720

C₂

semidirect

104. C₇₁

#9862

semidirect

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

subgroup	index
#22588512	2
#22586791	2
C₂×#17985720	2
C₇₁×#3147	2
#9862	71

subgroup

index

C₇₁×#3147

Subgroups — #31897425

Conjugacy classes of subgroups

Complements

Maximal subgroups