Group #21466621 — subgroups

Subgroups — C₃₁×#9862 #21466621

Order 7936.

514 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 174 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	C₃₁×#3147	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1848	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1492	C₂	1	4	yes	maximal, complemented
4	C₂×C₃₁×#322	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₁×#135	—	2	2	no	—
6	C₃₁×#532	C₂²	1	2	yes	—
7	C₃₁×#495	—	2	2	no	—
8	C₃₁×#494	C₄	1	1	yes	characteristic
9	C₃₁×#493	C₄	1	1	yes	characteristic
10	C₃₁×#492	C₄	1	2	yes	—
11	C₂×C₃₁×#108	—	2	2	no	—
12	C₃₁×#423	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×D₄⋊C₈	—	2	8	no	—
16	C₃₁×#322	—	2	2	no	—
17	C₂×#41×C₃₁	—	2	2	no	—
18	C₃₁×#135	—	4	4	no	—
19	C₃₁×#135	—	4	4	no	—
20	C₂×Q₁₆×C₃₁	—	2	2	no	—
21	C₂×SD₁₆×C₃₁	—	2	4	no	—
22	C₂×D₈×C₃₁	—	2	2	no	—
23	C₂×M₁₆×C₃₁	—	2	2	no	—
24	C₂²×C₈×C₃₁	—	2	2	no	—
25	C₂²×C₈×C₃₁	D₄	1	2	yes	—
26	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
27	C₃₁×#127	C₂×C₄	1	2	yes	—
28	C₃₁×#124	—	2	2	no	—
29	C₃₁×#121	—	2	4	no	—
30	C₄×D₄×C₃₁	—	2	4	no	—
31	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
32	C₃₁×#108	—	4	4	no	—
33	C₃₁×C₄⋊C₈	—	2	4	no	—
34	C₄×C₈×C₃₁	—	2	2	no	—
35	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₃₁	D₄	1	4	yes	—
38	C₄×C₈×C₃₁	D₄	1	2	yes	—
39	#41×C₃₁	—	4	4	no	—
40	#41×C₃₁	—	4	4	no	—
41	C₂×Q₈×C₃₁	—	2	2	no	—
42	C₂×D₄×C₃₁	—	2	2	no	—
43	C₂×D₄×C₃₁	—	2	4	no	—
44	C₂×D₄×C₃₁	—	4	2	no	—
45	C₂²×C₄×C₃₁	—	4	2	no	—
46	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₃₁	#41	1	2	yes	—
48	Q₁₆×C₃₁	—	4	4	no	—
49	SD₁₆×C₃₁	—	4	8	no	—
50	D₈×C₃₁	—	4	4	no	—
51	M₁₆×C₃₁	—	4	4	no	—
52	C₂×C₈×C₃₁	—	2	4	no	—
53	C₂×C₈×C₃₁	—	4	4	no	—
54	C₂×C₈×C₃₁	—	4	2	no	—
55	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₃₁	D₈	1	4	yes	—
58	C₂×C₈×C₃₁	SD₁₆	1	4	yes	—
59	C₄⋊C₄×C₃₁	—	2	4	no	—
60	C₂²⋊C₄×C₃₁	—	4	4	no	—
61	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₃₁	C₂×D₄	1	2	yes	—
64	#9862	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂³×C₃₁	—	4	2	no	—
66	C₂³×C₃₁	#124	1	1	yes	characteristic
67	Q₈×C₃₁	—	4	4	no	—
68	D₄×C₃₁	—	4	4	no	—
69	D₄×C₃₁	—	4	8	no	—
70	D₄×C₃₁	—	8	4	no	—
71	C₂×C₄×C₃₁	—	2	4	no	—
72	C₂×C₄×C₃₁	—	4	2	no	—
73	C₂×C₄×C₃₁	—	8	2	no	—
74	C₂×C₄×C₃₁	—	8	2	no	—
75	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
77	C₂×C₄×C₃₁	#121	1	2	yes	—
78	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
79	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
80	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
81	C₈×C₃₁	—	4	4	no	—
82	C₈×C₃₁	—	8	2	no	—
83	C₈×C₃₁	#102	1	4	yes	—
84	C₈×C₃₁	#102	1	4	yes	—
85	#3147	C₂×C₃₁	1	1	yes	characteristic
86	#1848	C₂×C₃₁	1	1	yes	characteristic, complemented
87	#1492	C₂×C₃₁	1	4	yes	complemented
88	C₂×#322	C₂×C₃₁	1	1	yes	characteristic, complemented
89	C₂²×C₃₁	—	2	2	no	—
90	C₂²×C₃₁	—	8	2	no	—
91	C₂²×C₃₁	—	8	4	no	—
92	C₂²×C₃₁	#390	1	1	yes	characteristic
93	C₂²×C₃₁	#486	1	2	yes	—
94	C₄×C₃₁	—	2	2	no	—
95	C₄×C₃₁	—	8	2	no	—
96	C₄×C₃₁	C₂×#102	1	2	yes	—
97	C₄×C₃₁	#444	1	4	yes	—
98	C₄×C₃₁	#492	1	1	yes	characteristic
99	C₄×C₃₁	#493	1	1	yes	characteristic
100	C₂×#135	—	2	2	no	—
101	#532	C₂²×C₃₁	1	2	yes	—
102	#495	—	2	2	no	—
103	#494	C₄×C₃₁	1	1	yes	characteristic
104	#493	C₄×C₃₁	1	1	yes	characteristic
105	#492	C₄×C₃₁	1	2	yes	—
106	C₂×#108	—	2	2	no	—
107	#423	C₂²×C₃₁	1	2	yes	—
108	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
109	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
110	D₄⋊C₈	—	2	8	no	—
111	#322	—	2	2	no	—
112	C₂×C₃₁	—	2	2	no	—
113	C₂×C₃₁	—	16	2	no	—
114	C₂×C₃₁	#1957	1	1	yes	characteristic
115	C₂×C₃₁	#1975	1	2	yes	—
116	C₂×#41	—	2	2	no	—
117	#135	—	4	4	no	—
118	#135	—	4	4	no	—
119	C₂×Q₁₆	—	2	2	no	—
120	C₂×SD₁₆	—	2	4	no	—
121	C₂×D₈	—	2	2	no	—
122	C₂×M₁₆	—	2	2	no	—
123	C₂²×C₈	—	2	2	no	—
124	C₂²×C₈	D₄×C₃₁	1	2	yes	—
125	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
126	#127	C₂×C₄×C₃₁	1	2	yes	—
127	#124	—	2	2	no	—
128	#121	—	2	4	no	—
129	C₄×D₄	—	2	4	no	—
130	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
131	#108	—	4	4	no	—
132	C₄⋊C₈	—	2	4	no	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
135	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
136	C₄×C₈	D₄×C₃₁	1	4	yes	—
137	C₄×C₈	D₄×C₃₁	1	2	yes	—
138	C₃₁	#9862	1	1	yes	characteristic, direct product factor, Sylow
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×D₄×C₃₁	1	1	yes	characteristic
147	C₂²×C₄	#41×C₃₁	1	2	yes	—
148	Q₁₆	—	4	4	no	—
149	SD₁₆	—	4	8	no	—
150	D₈	—	4	4	no	—
151	M₁₆	—	4	4	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	4	4	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
156	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
157	C₂×C₈	D₈×C₃₁	1	4	yes	—
158	C₂×C₈	SD₁₆×C₃₁	1	4	yes	—
159	C₄⋊C₄	—	2	4	no	—
160	C₂²⋊C₄	—	4	4	no	—
161	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
162	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
163	C₄²	C₂×D₄×C₃₁	1	2	yes	—
164	C₂³	—	4	2	no	—
165	C₂³	C₃₁×#124	1	1	yes	characteristic
166	Q₈	—	4	4	no	—
167	D₄	—	4	4	no	—
168	D₄	—	4	8	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⋊C₄×C₃₁	1	1	yes	characteristic
175	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
176	C₂×C₄	C₃₁×#121	1	2	yes	—
177	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
178	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
179	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
180	C₈	—	4	4	no	—
181	C₈	—	8	2	no	—
182	C₈	C₃₁×#102	1	4	yes	—
183	C₈	C₃₁×#102	1	4	yes	—
184	C₂²	—	2	2	no	—
185	C₂²	—	8	2	no	—
186	C₂²	—	8	4	no	—
187	C₂²	C₃₁×#390	1	1	yes	characteristic
188	C₂²	C₃₁×#486	1	2	yes	—
189	C₄	—	2	2	no	—
190	C₄	—	8	2	no	—
191	C₄	C₂×C₃₁×#102	1	2	yes	—
192	C₄	C₃₁×#444	1	4	yes	—
193	C₄	C₃₁×#492	1	1	yes	characteristic
194	C₄	C₃₁×#493	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	—	16	2	no	—
197	C₂	C₃₁×#1957	1	1	yes	characteristic
198	C₂	C₃₁×#1975	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃₁×#3147

C₂

yes

maximal, characteristic

C₃₁×#1848

C₂

yes

maximal, characteristic, complemented

C₃₁×#1492

C₂

yes

maximal, complemented

C₂×C₃₁×#322

C₂

yes

maximal, characteristic, complemented

C₂×C₃₁×#135

—

C₃₁×#532

C₂²

yes

—

C₃₁×#495

—

C₃₁×#494

C₄

yes

characteristic

C₃₁×#493

C₄

yes

characteristic

C₃₁×#492

C₄

yes

—

C₂×C₃₁×#108

—

C₃₁×#423

C₂²

yes

—

C₂×C₄×C₈×C₃₁

C₂²

yes

—

C₂×C₄×C₈×C₃₁

C₂²

yes

characteristic

C₃₁×D₄⋊C₈

—

C₃₁×#322

—

C₂×#41×C₃₁

—

C₃₁×#135

—

C₃₁×#135

—

C₂×Q₁₆×C₃₁

—

C₂×SD₁₆×C₃₁

—

C₂×D₈×C₃₁

—

C₂×M₁₆×C₃₁

—

C₂²×C₈×C₃₁

—

C₂²×C₈×C₃₁

D₄

yes

—

C₃₁×C₄⋊Q₈

C₂×C₄

yes

—

C₃₁×#127

C₂×C₄

yes

—

C₃₁×#124

—

C₃₁×#121

—

C₄×D₄×C₃₁

—

C₂×C₄²×C₃₁

C₂³

yes

characteristic

C₃₁×#108

—

C₃₁×C₄⋊C₈

—

C₄×C₈×C₃₁

—

C₄×C₈×C₃₁

C₂×C₄

yes

characteristic

C₄×C₈×C₃₁

C₂×C₄

yes

characteristic

C₄×C₈×C₃₁

D₄

yes

—

C₄×C₈×C₃₁

D₄

yes

—

#41×C₃₁

—

#41×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₃₁

#41

yes

—

Q₁₆×C₃₁

—

SD₁₆×C₃₁

—

D₈×C₃₁

—

M₁₆×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂²⋊C₄

yes

—

C₂×C₈×C₃₁

C₂²⋊C₄

yes

—

C₂×C₈×C₃₁

D₈

yes

—

C₂×C₈×C₃₁

SD₁₆

yes

—

C₄⋊C₄×C₃₁

—

C₂²⋊C₄×C₃₁

—

C₄²×C₃₁

C₂²×C₄

yes

characteristic

C₄²×C₃₁

C₂×D₄

yes

characteristic

C₄²×C₃₁

C₂×D₄

yes

—

#9862

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

—

C₂³×C₃₁

#124

yes

characteristic

Q₈×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₃₁

C₄×D₄

yes

—

C₂×C₄×C₃₁

#121

yes

—

C₂×C₄×C₃₁

#127

yes

characteristic

C₂×C₄×C₃₁

C₂×D₈

yes

—

C₂×C₄×C₃₁

C₂×SD₁₆

yes

—

C₈×C₃₁

—

C₈×C₃₁

—

C₈×C₃₁

#102

yes

—

C₈×C₃₁

#102

yes

—

#3147

C₂×C₃₁

yes

characteristic

#1848

C₂×C₃₁

yes

characteristic, complemented

#1492

C₂×C₃₁

yes

complemented

C₂×#322

C₂×C₃₁

yes

characteristic, complemented

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

#390

yes

characteristic

C₂²×C₃₁

#486

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂×#102

yes

—

C₄×C₃₁

#444

yes

—

C₄×C₃₁

yes

C₄×C₃₁

yes

100

—

101

#532

C₂²×C₃₁

yes

—

102

#495

—

103

#494

C₄×C₃₁

yes

characteristic

104

#493

C₄×C₃₁

yes

characteristic

105

#492

C₄×C₃₁

yes

—

106

C₂×#108

—

107

#423

C₂²×C₃₁

yes

—

108

C₂×C₄×C₈

C₂²×C₃₁

yes

—

109

C₂×C₄×C₈

C₂²×C₃₁

yes

characteristic

110

D₄⋊C₈

—

111

#322

—

112

C₂×C₃₁

—

113

C₂×C₃₁

—

114

C₂×C₃₁

#1957

yes

characteristic

115

C₂×C₃₁

#1975

yes

—

116

C₂×#41

—

117

#135

—

118

#135

—

119

C₂×Q₁₆

—

120

C₂×SD₁₆

—

121

C₂×D₈

—

122

C₂×M₁₆

—

123

C₂²×C₈

—

124

C₂²×C₈

D₄×C₃₁

yes

—

125

C₄⋊Q₈

C₂×C₄×C₃₁

yes

—

126

#127

C₂×C₄×C₃₁

yes

—

127

#124

—

128

#121

—

129

C₄×D₄

—

130

C₂×C₄²

C₂³×C₃₁

yes

characteristic

131

#108

—

132

C₄⋊C₈

—

133

C₄×C₈

—

134

C₄×C₈

C₂×C₄×C₃₁

yes

characteristic

135

C₄×C₈

C₂×C₄×C₃₁

yes

characteristic

136

C₄×C₈

D₄×C₃₁

yes

—

137

C₄×C₈

D₄×C₃₁

yes

—

138

C₃₁

#9862

yes

characteristic, direct product factor, Sylow

139

#41

—

140

#41

—

141

C₂×Q₈

—

142

C₂×D₄

—

143

C₂×D₄

—

144

C₂×D₄

—

145

C₂²×C₄

—

146

C₂²×C₄

C₂×D₄×C₃₁

yes

characteristic

147

C₂²×C₄

#41×C₃₁

yes

—

148

Q₁₆

—

149

SD₁₆

—

150

D₈

—

151

M₁₆

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

C₂²⋊C₄×C₃₁

yes

—

156

C₂×C₈

C₂²⋊C₄×C₃₁

yes

—

157

C₂×C₈

D₈×C₃₁

yes

—

158

C₂×C₈

SD₁₆×C₃₁

yes

—

159

C₄⋊C₄

—

160

C₂²⋊C₄

—

161

C₄²

C₂²×C₄×C₃₁

yes

characteristic

162

C₄²

C₂×D₄×C₃₁

yes

characteristic

163

C₄²

C₂×D₄×C₃₁

yes

—

164

C₂³

—

165

C₂³

C₃₁×#124

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⋊C₄×C₃₁

yes

characteristic

175

C₂×C₄

C₄×D₄×C₃₁

yes

—

176

C₂×C₄

C₃₁×#121

yes

—

177

C₂×C₄

C₃₁×#127

yes

characteristic

178

C₂×C₄

C₂×D₈×C₃₁

yes

—

179

C₂×C₄

C₂×SD₁₆×C₃₁

yes

—

180

C₈

—

181

C₈

—

182

C₈

C₃₁×#102

yes

—

183

C₈

C₃₁×#102

yes

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

C₃₁×#390

yes

characteristic

188

C₂²

C₃₁×#486

yes

—

189

C₄

—

190

C₄

—

191

C₄

C₂×C₃₁×#102

yes

—

192

C₄

C₃₁×#444

yes

—

193

C₄

C₃₁×#492

yes

characteristic

194

C₄

C₃₁×#493

yes

characteristic

195

C₂

—

196

C₂

—

197

C₂

C₃₁×#1957

yes

characteristic

198

C₂

C₃₁×#1975

yes

—

	subgroup	G/H
center	C₂×C₄×C₃₁	#127
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	#390
prime radical	C₃₁×#3147	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

subgroup

G/H

center

C₂×C₄×C₃₁

#127

commutator

C₄²

C₂²×C₄×C₃₁

Frattini, cosocle

C₂×C₄²

C₂³×C₃₁

socle

C₂²×C₃₁

#390

prime radical

C₃₁×#3147

C₂

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

N	H ≅ G/N	type
2. C₃₁×#1848	C₂	semidirect
3. C₃₁×#1492	C₂	semidirect
4. C₂×C₃₁×#322	C₂	semidirect
64. #9862	C₃₁	direct (N × H)
86. #1848	C₂×C₃₁	semidirect
87. #1492	C₂×C₃₁	semidirect
88. C₂×#322	C₂×C₃₁	semidirect
138. C₃₁	#9862	direct (N × H)

H ≅ G/N

type

2. C₃₁×#1848

C₂

semidirect

3. C₃₁×#1492

C₂

semidirect

4. C₂×C₃₁×#322

C₂

semidirect

64. #9862

C₃₁

direct (N × H)

86. #1848

C₂×C₃₁

semidirect

87. #1492

C₂×C₃₁

semidirect

88. C₂×#322

C₂×C₃₁

semidirect

138. C₃₁

#9862

direct (N × H)

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

subgroup	index
C₃₁×#3147	2
C₃₁×#1848	2
C₃₁×#1492	2
C₂×C₃₁×#322	2
#9862	31

subgroup

index

C₃₁×#3147

C₃₁×#1848

C₃₁×#1492

C₂×C₃₁×#322

#9862

Subgroups — C₃₁×#9862 #21466621

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#9862 #21466621

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9862 #21466621