Group #21465382 — subgroups

Subgroups — C₃₁×#8623 #21465382

Order 7936.

362 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 142 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×C₈×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1568	C₂	1	2	yes	maximal
3	C₃₁×#1467	C₂	1	2	yes	maximal
4	C₃₁×#1290	C₂	1	2	yes	maximal
5	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
6	C₈×Q₈×C₃₁	—	2	2	no	—
7	C₈×Q₈×C₃₁	—	2	2	no	—
8	C₃₁×#432	C₂²	1	4	yes	—
9	C₂×C₃₁×C₄⋊C₈	C₄	1	2	yes	—
10	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₃₁	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#335	—	2	2	no	—
14	C₃₁×#334	—	2	2	no	—
15	C₃₁×#324	—	2	4	no	—
16	C₂²×Q₈×C₃₁	Q₈	1	1	yes	characteristic
17	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
20	C₄×Q₈×C₃₁	—	2	2	no	—
21	C₄×Q₈×C₃₁	—	2	2	no	—
22	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
23	C₂×C₄⋊C₄×C₃₁	Q₈	1	1	yes	characteristic
24	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
26	C₃₁×C₄⋊C₈	—	2	4	no	—
27	C₃₁×C₄⋊C₈	—	2	2	no	—
28	C₃₁×C₄⋊C₈	—	2	4	no	—
29	C₃₁×C₂²⋊C₈	—	2	4	no	—
30	C₃₁×#97	—	2	4	no	—
31	C₄×C₈×C₃₁	—	2	2	no	—
32	C₄×C₈×C₃₁	—	2	2	no	—
33	C₄×C₈×C₃₁	D₄	1	2	yes	—
34	C₄×C₈×C₃₁	D₄	1	2	yes	—
35	C₂×Q₈×C₃₁	—	2	4	no	—
36	C₂×Q₈×C₃₁	—	2	2	no	—
37	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
38	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
39	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₈×C₃₁	—	2	2	no	—
43	C₂×C₈×C₃₁	—	2	2	no	—
44	C₂×C₈×C₃₁	—	4	1	no	—
45	C₂×C₈×C₃₁	—	4	2	no	—
46	C₂×C₈×C₃₁	—	4	4	no	—
47	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
49	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₄⋊C₄×C₃₁	—	2	4	no	—
51	C₄⋊C₄×C₃₁	—	2	2	no	—
52	C₄²×C₃₁	—	2	2	no	—
53	C₄²×C₃₁	—	2	2	no	—
54	C₄²×C₃₁	C₂×D₄	1	2	yes	—
55	C₄²×C₃₁	#41	1	2	yes	—
56	#8623	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
58	Q₈×C₃₁	—	2	8	no	—
59	C₂×C₄×C₃₁	—	2	2	no	—
60	C₂×C₄×C₃₁	—	2	2	no	—
61	C₂×C₄×C₃₁	—	2	2	no	—
62	C₂×C₄×C₃₁	—	2	1	no	—
63	C₂×C₄×C₃₁	—	2	1	no	—
64	C₂×C₄×C₃₁	—	2	2	no	—
65	C₂×C₄×C₃₁	—	4	1	no	—
66	C₂×C₄×C₃₁	—	4	2	no	—
67	C₂×C₄×C₃₁	#108	1	4	yes	—
68	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
70	C₂×C₄×C₃₁	#121	1	2	yes	—
71	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
72	C₈×C₃₁	—	2	1	no	—
73	C₈×C₃₁	—	2	2	no	—
74	C₈×C₃₁	—	4	2	no	—
75	C₈×C₃₁	—	8	4	no	—
76	C₈×C₃₁	#101	1	2	yes	—
77	C₂×C₈×Q₈	C₂×C₃₁	1	1	yes	characteristic
78	#1568	C₂×C₃₁	1	2	yes	—
79	#1467	C₂×C₃₁	1	2	yes	—
80	#1290	C₂×C₃₁	1	2	yes	—
81	C₂²×C₃₁	—	2	1	no	—
82	C₂²×C₃₁	—	2	1	no	—
83	C₂²×C₃₁	#380	1	1	yes	characteristic
84	C₂²×C₃₁	C₂×#108	1	2	yes	—
85	C₄×C₃₁	—	2	1	no	—
86	C₄×C₃₁	—	2	4	no	—
87	C₄×C₃₁	—	2	2	no	—
88	C₄×C₃₁	—	2	2	no	—
89	C₄×C₃₁	—	4	2	no	—
90	C₄×C₃₁	C₂×#101	1	1	yes	characteristic
91	C₄×C₃₁	#413	1	1	yes	characteristic
92	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
93	C₈×Q₈	—	2	2	no	—
94	C₈×Q₈	—	2	2	no	—
95	#432	C₂²×C₃₁	1	4	yes	—
96	C₂×C₄⋊C₈	C₄×C₃₁	1	2	yes	—
97	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
98	C₂×C₄×C₈	C₄×C₃₁	1	2	yes	—
99	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
100	#335	—	2	2	no	—
101	#334	—	2	2	no	—
102	#324	—	2	4	no	—
103	C₂×C₃₁	—	2	2	no	—
104	C₂×C₃₁	#1801	1	1	yes	characteristic
105	C₂×C₃₁	#1810	1	1	yes	characteristic
106	C₂×C₃₁	#1812	1	1	yes	characteristic
107	C₂²×Q₈	Q₈×C₃₁	1	1	yes	characteristic
108	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
109	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
110	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
111	C₄×Q₈	—	2	2	no	—
112	C₄×Q₈	—	2	2	no	—
113	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
114	C₂×C₄⋊C₄	Q₈×C₃₁	1	1	yes	characteristic
115	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
116	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
117	C₄⋊C₈	—	2	4	no	—
118	C₄⋊C₈	—	2	2	no	—
119	C₄⋊C₈	—	2	4	no	—
120	C₂²⋊C₈	—	2	4	no	—
121	#97	—	2	4	no	—
122	C₄×C₈	—	2	2	no	—
123	C₄×C₈	—	2	2	no	—
124	C₄×C₈	D₄×C₃₁	1	2	yes	—
125	C₄×C₈	D₄×C₃₁	1	2	yes	—
126	C₃₁	#8623	1	1	yes	characteristic, direct product factor, Sylow
127	C₂×Q₈	—	2	4	no	—
128	C₂×Q₈	—	2	2	no	—
129	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
130	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
131	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
132	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
133	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	4	1	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	—	4	4	no	—
139	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
140	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
141	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	—	2	2	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	2	2	no	—
146	C₄²	C₂×D₄×C₃₁	1	2	yes	—
147	C₄²	#41×C₃₁	1	2	yes	—
148	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
149	Q₈	—	2	8	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	C₃₁×#108	1	4	yes	—
159	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
160	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
161	C₂×C₄	C₃₁×#121	1	2	yes	—
162	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
163	C₈	—	2	1	no	—
164	C₈	—	2	2	no	—
165	C₈	—	4	2	no	—
166	C₈	—	8	4	no	—
167	C₈	C₃₁×#101	1	2	yes	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	C₃₁×#380	1	1	yes	characteristic
171	C₂²	C₂×C₃₁×#108	1	2	yes	—
172	C₄	—	2	1	no	—
173	C₄	—	2	4	no	—
174	C₄	—	2	2	no	—
175	C₄	—	2	2	no	—
176	C₄	—	4	2	no	—
177	C₄	C₂×C₃₁×#101	1	1	yes	characteristic
178	C₄	C₃₁×#413	1	1	yes	characteristic
179	C₂	—	2	2	no	—
180	C₂	C₃₁×#1801	1	1	yes	characteristic
181	C₂	C₃₁×#1810	1	1	yes	characteristic
182	C₂	C₃₁×#1812	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Q₈×C₃₁

C₂

yes

maximal, characteristic

C₃₁×#1568

C₂

yes

maximal

C₃₁×#1467

C₂

yes

maximal

C₃₁×#1290

C₂

yes

maximal

C₂×C₄×Q₈×C₃₁

C₂²

yes

characteristic

C₈×Q₈×C₃₁

—

C₈×Q₈×C₃₁

—

C₃₁×#432

C₂²

yes

—

C₂×C₃₁×C₄⋊C₈

C₄

yes

—

C₂×C₃₁×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₈×C₃₁

C₄

yes

—

C₂×C₄×C₈×C₃₁

C₂²

yes

characteristic

C₃₁×#335

—

C₃₁×#334

—

C₃₁×#324

—

C₂²×Q₈×C₃₁

Q₈

yes

characteristic

C₂²×C₈×C₃₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₃₁

C₂×C₄

yes

—

C₂²×C₈×C₃₁

C₂×C₄

yes

characteristic

C₄×Q₈×C₃₁

—

C₄×Q₈×C₃₁

—

C₂×C₄⋊C₄×C₃₁

D₄

yes

—

C₂×C₄⋊C₄×C₃₁

Q₈

yes

characteristic

C₂×C₄²×C₃₁

C₂×C₄

yes

—

C₂×C₄²×C₃₁

C₂³

yes

characteristic

C₃₁×C₄⋊C₈

—

C₃₁×C₄⋊C₈

—

C₃₁×C₄⋊C₈

—

C₃₁×C₂²⋊C₈

—

C₃₁×#97

—

C₄×C₈×C₃₁

—

C₄×C₈×C₃₁

—

C₄×C₈×C₃₁

D₄

yes

—

C₄×C₈×C₃₁

D₄

yes

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂²×C₄×C₃₁

C₄⋊C₄

yes

—

C₂²×C₄×C₃₁

C₄⋊C₄

yes

—

C₂²×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₃₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₃₁

C₂²⋊C₄

yes

—

C₂×C₈×C₃₁

C₂²⋊C₄

yes

characteristic

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

C₂×D₄

yes

—

C₄²×C₃₁

#41

yes

—

#8623

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

C₂×C₄⋊C₄

yes

characteristic

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

—

C₂×C₄×C₃₁

#108

yes

—

C₂×C₄×C₃₁

C₂×C₂²⋊C₄

yes

C₂×C₄×C₃₁

yes

C₂×C₄×C₃₁

yes

—

C₂×C₄×C₃₁

C₂²⋊Q₈

yes

—

C₈×C₃₁

—

C₈×C₃₁

—

C₈×C₃₁

—

C₈×C₃₁

—

C₈×C₃₁

#101

yes

—

C₂×C₈×Q₈

C₂×C₃₁

yes

characteristic

#1568

C₂×C₃₁

yes

—

#1467

C₂×C₃₁

yes

—

#1290

C₂×C₃₁

yes

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

#380

yes

characteristic

C₂²×C₃₁

C₂×#108

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

yes

C₄×C₃₁

yes

C₂×C₄×Q₈

C₂²×C₃₁

yes

characteristic

C₈×Q₈

—

C₈×Q₈

—

#432

C₂²×C₃₁

yes

—

C₂×C₄⋊C₈

C₄×C₃₁

yes

—

C₂×C₄⋊C₈

C₂²×C₃₁

yes

characteristic

C₂×C₄×C₈

C₄×C₃₁

yes

—

C₂×C₄×C₈

C₂²×C₃₁

yes

characteristic

100

#335

—

101

#334

—

102

#324

—

103

C₂×C₃₁

—

104

C₂×C₃₁

#1801

yes

characteristic

105

C₂×C₃₁

#1810

yes

characteristic

106

C₂×C₃₁

#1812

yes

characteristic

107

C₂²×Q₈

Q₈×C₃₁

yes

characteristic

108

C₂²×C₈

C₂×C₄×C₃₁

yes

characteristic

109

C₂²×C₈

C₂×C₄×C₃₁

yes

—

110

C₂²×C₈

C₂×C₄×C₃₁

yes

characteristic

111

C₄×Q₈

—

112

C₄×Q₈

—

113

C₂×C₄⋊C₄

D₄×C₃₁

yes

—

114

C₂×C₄⋊C₄

Q₈×C₃₁

yes

characteristic

115

C₂×C₄²

C₂×C₄×C₃₁

yes

—

116

C₂×C₄²

C₂³×C₃₁

yes

characteristic

117

C₄⋊C₈

—

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₂²⋊C₈

—

121

#97

—

122

C₄×C₈

—

123

C₄×C₈

—

124

C₄×C₈

D₄×C₃₁

yes

—

125

C₄×C₈

D₄×C₃₁

yes

—

126

C₃₁

#8623

yes

characteristic, direct product factor, Sylow

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂²×C₄

C₄⋊C₄×C₃₁

yes

—

130

C₂²×C₄

C₄⋊C₄×C₃₁

yes

—

131

C₂²×C₄

C₂²×C₄×C₃₁

yes

characteristic

132

C₂²×C₄

C₂×D₄×C₃₁

yes

characteristic

133

C₂²×C₄

C₂×Q₈×C₃₁

yes

characteristic

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

C₂²⋊C₄×C₃₁

yes

characteristic

140

C₂×C₈

C₂²⋊C₄×C₃₁

yes

—

141

C₂×C₈

C₂²⋊C₄×C₃₁

yes

characteristic

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄²

—

145

C₄²

—

146

C₄²

C₂×D₄×C₃₁

yes

—

147

C₄²

#41×C₃₁

yes

—

148

C₂³

C₂×C₄⋊C₄×C₃₁

yes

characteristic

149

Q₈

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

C₃₁×#108

yes

—

159

C₂×C₄

C₂×C₂²⋊C₄×C₃₁

yes

characteristic

160

C₂×C₄

C₃₁×#117

yes

characteristic

161

C₂×C₄

C₃₁×#121

yes

—

162

C₂×C₄

C₃₁×C₂²⋊Q₈

yes

—

163

C₈

—

164

C₈

—

165

C₈

—

166

C₈

—

167

C₈

C₃₁×#101

yes

—

168

C₂²

—

169

C₂²

—

170

C₂²

C₃₁×#380

yes

characteristic

171

C₂²

C₂×C₃₁×#108

yes

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

C₂×C₃₁×#101

yes

characteristic

178

C₄

C₃₁×#413

yes

characteristic

179

C₂

—

180

C₂

C₃₁×#1801

yes

characteristic

181

C₂

C₃₁×#1810

yes

characteristic

182

C₂

C₃₁×#1812

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	#380
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₂³×C₃₁	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₃₁

#380

commutator

C₂²×C₄

C₂²×C₄×C₃₁

Frattini, cosocle

C₂×C₄²

C₂³×C₃₁

prime radical

C₂³×C₃₁

C₂×C₄⋊C₄

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

N	H ≅ G/N	type
56. #8623	C₃₁	direct (N × H)
126. C₃₁	#8623	direct (N × H)

H ≅ G/N

type

56. #8623

C₃₁

direct (N × H)

126. C₃₁

#8623

direct (N × H)

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

subgroup	index
C₂×C₈×Q₈×C₃₁	2
C₃₁×#1568	2
C₃₁×#1467	2
C₃₁×#1290	2
#8623	31

subgroup

index

C₂×C₈×Q₈×C₃₁

C₃₁×#1568

C₃₁×#1467

C₃₁×#1290

#8623

Subgroups — C₃₁×#8623 #21465382

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#8623 #21465382

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8623 #21465382