Group #21470295 — subgroups

Subgroups — C₃₁×#13536 #21470295

Order 7936.

318 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 114 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×C₃₁×#365	C₂	1	2	yes	maximal
2	C₂×C₃₁×#362	C₂	1	2	yes	maximal
3	C₃₁×#1971	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1849	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1377	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
8	C₂×C₃₁×#108	—	2	1	no	—
9	C₂×C₃₁×#108	C₂²	1	2	yes	—
10	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
11	C₃₁×#423	—	2	1	no	—
12	C₂×C₃₁×#103	—	2	1	no	—
13	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
14	C₃₁×#391	—	2	1	no	—
15	C₃₁×#365	—	2	4	no	—
16	C₃₁×#362	—	2	4	no	—
17	C₂²×Q₈×C₃₁	—	2	1	no	—
18	C₂×Q₁₆×C₃₁	—	2	2	no	—
19	C₂×Q₁₆×C₃₁	—	2	2	no	—
20	C₂×Q₁₆×C₃₁	—	2	1	no	—
21	C₂×Q₁₆×C₃₁	Q₈	1	2	yes	—
22	C₂×M₁₆×C₃₁	—	2	2	no	—
23	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
26	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
27	C₃₁×M₃₂	—	2	2	no	—
28	C₃₁×M₃₂	D₄	1	4	yes	—
29	C₂×C₁₆×C₃₁	—	2	2	no	—
30	C₃₁×#108	—	2	2	no	—
31	C₃₁×#108	—	2	2	no	—
32	C₃₁×#108	—	4	2	no	—
33	C₃₁×#106	—	2	2	no	—
34	C₃₁×C₄⋊C₈	—	4	2	no	—
35	C₃₁×#103	—	4	2	no	—
36	C₃₁×#97	—	2	2	no	—
37	C₃₁×#95	—	4	2	no	—
38	C₂×Q₈×C₃₁	—	2	2	no	—
39	C₂×Q₈×C₃₁	—	4	1	no	—
40	C₂×Q₈×C₃₁	—	4	1	no	—
41	C₂×Q₈×C₃₁	—	4	1	no	—
42	C₂×Q₈×C₃₁	—	4	1	no	—
43	C₂×Q₈×C₃₁	—	4	1	no	—
44	C₂²×C₄×C₃₁	—	4	1	no	—
45	C₂²×C₄×C₃₁	—	4	1	no	—
46	C₂²×C₄×C₃₁	SD₁₆	1	2	yes	—
47	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
48	Q₁₆×C₃₁	—	2	4	no	—
49	Q₁₆×C₃₁	—	4	2	no	—
50	M₁₆×C₃₁	—	4	2	no	—
51	M₁₆×C₃₁	—	4	2	no	—
52	C₂×C₈×C₃₁	—	2	1	no	—
53	C₂×C₈×C₃₁	—	2	1	no	—
54	C₂×C₈×C₃₁	—	4	2	no	—
55	C₂×C₈×C₃₁	SD₁₆	1	2	yes	—
56	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
57	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈×C₃₁	#41	1	2	yes	—
59	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
60	C₄⋊C₄×C₃₁	—	4	1	no	—
61	C₄⋊C₄×C₃₁	—	4	1	no	—
62	C₄²×C₃₁	—	2	1	no	—
63	C₄²×C₃₁	—	2	1	no	—
64	C₁₆×C₃₁	—	2	4	no	—
65	#13536	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂³×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
67	Q₈×C₃₁	—	4	2	no	—
68	Q₈×C₃₁	—	4	2	no	—
69	Q₈×C₃₁	—	8	1	no	—
70	Q₈×C₃₁	—	8	1	no	—
71	C₂×C₄×C₃₁	—	2	2	no	—
72	C₂×C₄×C₃₁	—	4	2	no	—
73	C₂×C₄×C₃₁	—	4	2	no	—
74	C₂×C₄×C₃₁	—	8	1	no	—
75	C₂×C₄×C₃₁	—	8	1	no	—
76	C₂×C₄×C₃₁	—	8	2	no	—
77	C₂×C₄×C₃₁	—	8	1	no	—
78	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
80	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
81	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
82	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
83	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
84	C₈×C₃₁	—	2	2	no	—
85	C₈×C₃₁	—	8	2	no	—
86	C₈×C₃₁	C₂²⋊Q₈	1	2	yes	—
87	C₈×C₃₁	#123	1	2	yes	—
88	C₂×#365	C₂×C₃₁	1	2	yes	—
89	C₂×#362	C₂×C₃₁	1	2	yes	—
90	#1971	C₂×C₃₁	1	1	yes	characteristic
91	#1849	C₂×C₃₁	1	1	yes	characteristic
92	#1377	C₂×C₃₁	1	1	yes	characteristic
93	C₂²×C₃₁	—	2	2	no	—
94	C₂²×C₃₁	#476	1	1	yes	characteristic
95	C₂²×C₃₁	#481	1	1	yes	characteristic
96	C₂²×C₃₁	#492	1	1	yes	characteristic
97	C₄×C₃₁	—	4	2	no	—
98	C₄×C₃₁	—	8	2	no	—
99	C₄×C₃₁	—	16	1	no	—
100	C₄×C₃₁	#399	1	1	yes	characteristic
101	C₄×C₃₁	#476	1	1	yes	characteristic
102	C₄×C₃₁	#481	1	1	yes	characteristic
103	C₄×C₃₁	#496	1	1	yes	characteristic
104	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
105	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
106	C₂×#108	—	2	1	no	—
107	C₂×#108	C₂²×C₃₁	1	2	yes	—
108	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
109	#423	—	2	1	no	—
110	C₂×#103	—	2	1	no	—
111	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
112	#391	—	2	1	no	—
113	#365	—	2	4	no	—
114	#362	—	2	4	no	—
115	C₂×C₃₁	—	2	1	no	—
116	C₂×C₃₁	—	2	1	no	—
117	C₂×C₃₁	#2094	1	1	yes	characteristic
118	C₂×C₃₁	#2239	1	2	yes	—
119	C₂²×Q₈	—	2	1	no	—
120	C₂×Q₁₆	—	2	2	no	—
121	C₂×Q₁₆	—	2	2	no	—
122	C₂×Q₁₆	—	2	1	no	—
123	C₂×Q₁₆	Q₈×C₃₁	1	2	yes	—
124	C₂×M₁₆	—	2	2	no	—
125	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
126	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
127	C₂×C₄⋊C₄	—	2	1	no	—
128	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
129	M₃₂	—	2	2	no	—
130	M₃₂	D₄×C₃₁	1	4	yes	—
131	C₂×C₁₆	—	2	2	no	—
132	#108	—	2	2	no	—
133	#108	—	2	2	no	—
134	#108	—	4	2	no	—
135	#106	—	2	2	no	—
136	C₄⋊C₈	—	4	2	no	—
137	#103	—	4	2	no	—
138	#97	—	2	2	no	—
139	#95	—	4	2	no	—
140	C₃₁	#13536	1	1	yes	characteristic, direct product factor, Sylow
141	C₂×Q₈	—	2	2	no	—
142	C₂×Q₈	—	4	1	no	—
143	C₂×Q₈	—	4	1	no	—
144	C₂×Q₈	—	4	1	no	—
145	C₂×Q₈	—	4	1	no	—
146	C₂×Q₈	—	4	1	no	—
147	C₂²×C₄	—	4	1	no	—
148	C₂²×C₄	—	4	1	no	—
149	C₂²×C₄	SD₁₆×C₃₁	1	2	yes	—
150	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
151	Q₁₆	—	2	4	no	—
152	Q₁₆	—	4	2	no	—
153	M₁₆	—	4	2	no	—
154	M₁₆	—	4	2	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	SD₁₆×C₃₁	1	2	yes	—
159	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
160	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
161	C₂×C₈	#41×C₃₁	1	2	yes	—
162	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
163	C₄⋊C₄	—	4	1	no	—
164	C₄⋊C₄	—	4	1	no	—
165	C₄²	—	2	1	no	—
166	C₄²	—	2	1	no	—
167	C₁₆	—	2	4	no	—
168	C₂³	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
169	Q₈	—	4	2	no	—
170	Q₈	—	4	2	no	—
171	Q₈	—	8	1	no	—
172	Q₈	—	8	1	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	8	1	no	—
177	C₂×C₄	—	8	1	no	—
178	C₂×C₄	—	8	2	no	—
179	C₂×C₄	—	8	1	no	—
180	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
181	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
182	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
183	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
184	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
185	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
186	C₈	—	2	2	no	—
187	C₈	—	8	2	no	—
188	C₈	C₃₁×C₂²⋊Q₈	1	2	yes	—
189	C₈	C₃₁×#123	1	2	yes	—
190	C₂²	—	2	2	no	—
191	C₂²	C₃₁×#476	1	1	yes	characteristic
192	C₂²	C₃₁×#481	1	1	yes	characteristic
193	C₂²	C₃₁×#492	1	1	yes	characteristic
194	C₄	—	4	2	no	—
195	C₄	—	8	2	no	—
196	C₄	—	16	1	no	—
197	C₄	C₃₁×#399	1	1	yes	characteristic
198	C₄	C₃₁×#476	1	1	yes	characteristic
199	C₄	C₃₁×#481	1	1	yes	characteristic
200	C₄	C₃₁×#496	1	1	yes	characteristic
201	C₂	—	2	1	no	—
202	C₂	—	2	1	no	—
203	C₂	C₃₁×#2094	1	1	yes	characteristic
204	C₂	C₃₁×#2239	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃₁×#365

C₂

yes

maximal

C₂×C₃₁×#362

C₂

yes

maximal

C₃₁×#1971

C₂

yes

maximal, characteristic

C₃₁×#1849

C₂

yes

maximal, characteristic

C₃₁×#1377

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₃₁

C₂²

yes

characteristic

C₂×C₃₁×M₃₂

C₂²

yes

—

C₂×C₃₁×#108

—

C₂×C₃₁×#108

C₂²

yes

—

C₂×C₃₁×#106

C₂²

yes

characteristic

C₃₁×#423

—

C₂×C₃₁×#103

—

C₂×C₃₁×#97

C₂²

yes

characteristic

C₃₁×#391

—

C₃₁×#365

—

C₃₁×#362

—

C₂²×Q₈×C₃₁

—

C₂×Q₁₆×C₃₁

—

C₂×Q₁₆×C₃₁

—

C₂×Q₁₆×C₃₁

—

C₂×Q₁₆×C₃₁

Q₈

yes

—

C₂×M₁₆×C₃₁

—

C₂²×C₈×C₃₁

D₄

yes

characteristic

C₂²×C₈×C₃₁

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄²×C₃₁

D₄

yes

characteristic

C₃₁×M₃₂

—

C₃₁×M₃₂

D₄

yes

—

C₂×C₁₆×C₃₁

—

C₃₁×#108

—

C₃₁×#108

—

C₃₁×#108

—

C₃₁×#106

—

C₃₁×C₄⋊C₈

—

C₃₁×#103

—

C₃₁×#97

—

C₃₁×#95

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

SD₁₆

yes

—

C₂²×C₄×C₃₁

C₂×D₄

yes

characteristic

Q₁₆×C₃₁

—

Q₁₆×C₃₁

—

M₁₆×C₃₁

—

M₁₆×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

SD₁₆

yes

—

C₂×C₈×C₃₁

C₂×D₄

yes

—

C₂×C₈×C₃₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₃₁

#41

yes

—

C₂×C₈×C₃₁

#41

yes

characteristic

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

—

C₁₆×C₃₁

—

#13536

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

C₂×SD₁₆

yes

characteristic

Q₈×C₃₁

—

Q₈×C₃₁

—

Q₈×C₃₁

—

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²⋊Q₈

yes

C₂×C₄×C₃₁

yes

C₂×C₄×C₃₁

yes

C₂×C₄×C₃₁

C₂×SD₁₆

yes

C₂×C₄×C₃₁

yes

C₂×C₄×C₃₁

yes

C₈×C₃₁

—

C₈×C₃₁

—

C₈×C₃₁

C₂²⋊Q₈

yes

—

C₈×C₃₁

#123

yes

—

C₂×#365

C₂×C₃₁

yes

—

C₂×#362

C₂×C₃₁

yes

—

#1971

C₂×C₃₁

yes

characteristic

#1849

C₂×C₃₁

yes

characteristic

#1377

C₂×C₃₁

yes

characteristic

C₂²×C₃₁

—

C₂²×C₃₁

yes

C₂²×C₃₁

yes

C₂²×C₃₁

yes

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

100

C₄×C₃₁

#399

yes

characteristic

101

C₄×C₃₁

#476

yes

characteristic

102

C₄×C₃₁

#481

yes

characteristic

103

C₄×C₃₁

#496

yes

characteristic

104

C₂²×Q₁₆

C₂²×C₃₁

yes

characteristic

105

C₂×M₃₂

C₂²×C₃₁

yes

—

106

C₂×#108

—

107

C₂×#108

C₂²×C₃₁

yes

—

108

C₂×#106

C₂²×C₃₁

yes

characteristic

109

#423

—

110

C₂×#103

—

111

C₂×#97

C₂²×C₃₁

yes

characteristic

112

#391

—

113

#365

—

114

#362

—

115

C₂×C₃₁

—

116

C₂×C₃₁

—

117

C₂×C₃₁

#2094

yes

characteristic

118

C₂×C₃₁

#2239

yes

—

119

C₂²×Q₈

—

120

C₂×Q₁₆

—

121

C₂×Q₁₆

—

122

C₂×Q₁₆

—

123

C₂×Q₁₆

Q₈×C₃₁

yes

—

124

C₂×M₁₆

—

125

C₂²×C₈

D₄×C₃₁

yes

characteristic

126

C₂²×C₈

C₂³×C₃₁

yes

characteristic

127

C₂×C₄⋊C₄

—

128

C₂×C₄²

D₄×C₃₁

yes

characteristic

129

M₃₂

—

130

M₃₂

D₄×C₃₁

yes

—

131

C₂×C₁₆

—

132

#108

—

133

#108

—

134

#108

—

135

#106

—

136

C₄⋊C₈

—

137

#103

—

138

#97

—

139

#95

—

140

C₃₁

#13536

yes

characteristic, direct product factor, Sylow

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂×Q₈

—

146

C₂×Q₈

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

SD₁₆×C₃₁

yes

—

150

C₂²×C₄

C₂×D₄×C₃₁

yes

characteristic

151

Q₁₆

—

152

Q₁₆

—

153

M₁₆

—

154

M₁₆

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

SD₁₆×C₃₁

yes

—

159

C₂×C₈

C₂×D₄×C₃₁

yes

—

160

C₂×C₈

C₂×Q₈×C₃₁

yes

characteristic

161

C₂×C₈

#41×C₃₁

yes

—

162

C₂×C₈

#41×C₃₁

yes

characteristic

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄²

—

166

C₄²

—

167

C₁₆

—

168

C₂³

C₂×SD₁₆×C₃₁

yes

characteristic

169

Q₈

—

170

Q₈

—

171

Q₈

—

172

Q₈

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

C₃₁×C₂²⋊Q₈

yes

characteristic

181

C₂×C₄

C₃₁×#123

yes

characteristic

182

C₂×C₄

C₃₁×#127

yes

characteristic

183

C₂×C₄

C₂×SD₁₆×C₃₁

yes

characteristic

184

C₂×C₄

C₃₁×AGL(1,ℤ/8)

yes

characteristic

185

C₂×C₄

C₃₁×AGL(1,ℤ/8)

yes

characteristic

186

C₈

—

187

C₈

—

188

C₈

C₃₁×C₂²⋊Q₈

yes

—

189

C₈

C₃₁×#123

yes

—

190

C₂²

—

191

C₂²

C₃₁×#476

yes

characteristic

192

C₂²

C₃₁×#481

yes

characteristic

193

C₂²

C₃₁×#492

yes

characteristic

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

C₃₁×#399

yes

characteristic

198

C₄

C₃₁×#476

yes

characteristic

199

C₄

C₃₁×#481

yes

characteristic

200

C₄

C₃₁×#496

yes

characteristic

201

C₂

—

202

C₂

—

203

C₂

C₃₁×#2094

yes

characteristic

204

C₂

C₃₁×#2239

yes

—

	subgroup	G/H
center, socle	C₂²×C₃₁	#481
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
prime radical	C₂³×C₃₁	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₂×C₃₁×#97 ◁ G

subgroup

G/H

center, socle

C₂²×C₃₁

#481

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₃₁

prime radical

C₂³×C₃₁

C₂×SD₁₆

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₂×C₃₁×#97 ◁ G

N	H ≅ G/N	type
65. #13536	C₃₁	direct (N × H)
140. C₃₁	#13536	direct (N × H)

H ≅ G/N

type

65. #13536

C₃₁

direct (N × H)

140. C₃₁

#13536

direct (N × H)

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

subgroup	index
C₂×C₃₁×#365	2
C₂×C₃₁×#362	2
C₃₁×#1971	2
C₃₁×#1849	2
C₃₁×#1377	2
#13536	31

subgroup

index

C₂×C₃₁×#365

C₂×C₃₁×#362

C₃₁×#1971

C₃₁×#1849

C₃₁×#1377

#13536

Subgroups — C₃₁×#13536 #21470295

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#13536 #21470295

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13536 #21470295