Group #21466526 — subgroups

Subgroups — C₃₁×#9767 #21466526

Order 7936.

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

normal subgroups only

This group has 414 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	C₂×C₃₁×#498	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1848	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1549	C₂	1	2	yes	maximal
4	C₃₁×#1548	C₂	1	2	yes	maximal
5	C₃₁×#1450	C₂	1	1	yes	maximal, characteristic
6	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₃₁×#498	—	2	1	no	—
8	C₃₁×#498	—	2	1	no	—
9	C₃₁×#498	C₄	1	4	yes	—
10	C₃₁×#431	C₂²	1	2	yes	—
11	C₂×C₃₁×#108	—	2	2	no	—
12	C₂×C₃₁×#106	—	2	2	no	—
13	C₃₁×#423	C₂²	1	1	yes	characteristic
14	C₃₁×#423	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#333	—	2	4	no	—
17	C₃₁×#332	—	2	4	no	—
18	C₃₁×#322	—	2	1	no	—
19	C₈²×C₃₁	—	2	1	no	—
20	C₂²×Q₈×C₃₁	—	2	2	no	—
21	C₂×M₁₆×C₃₁	—	2	1	no	—
22	C₂×M₁₆×C₃₁	—	2	1	no	—
23	C₂²×C₈×C₃₁	D₄	1	2	yes	—
24	C₃₁×C₄⋊Q₈	—	2	2	no	—
25	C₃₁×C₄⋊Q₈	—	2	2	no	—
26	C₃₁×C₄⋊Q₈	—	2	2	no	—
27	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	—
28	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
29	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
30	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
31	C₃₁×#108	—	4	4	no	—
32	C₃₁×#106	—	2	4	no	—
33	C₃₁×#106	—	2	4	no	—
34	C₃₁×C₄⋊C₈	—	2	2	no	—
35	C₃₁×C₄⋊C₈	—	2	2	no	—
36	C₃₁×C₄⋊C₈	D₄	1	4	yes	—
37	C₃₁×C₂²⋊C₈	—	2	2	no	—
38	C₄×C₈×C₃₁	—	2	2	no	—
39	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
41	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
42	C₂×Q₈×C₃₁	—	2	4	no	—
43	C₂×Q₈×C₃₁	—	4	2	no	—
44	C₂×Q₈×C₃₁	—	8	4	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	M₁₆×C₃₁	—	4	2	no	—
49	M₁₆×C₃₁	—	4	2	no	—
50	C₂×C₈×C₃₁	—	2	4	no	—
51	C₂×C₈×C₃₁	—	4	1	no	—
52	C₂×C₈×C₃₁	—	4	1	no	—
53	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
55	C₂×C₈×C₃₁	SD₁₆	1	4	yes	—
56	C₂×C₈×C₃₁	Q₁₆	1	4	yes	—
57	C₄⋊C₄×C₃₁	—	2	4	no	—
58	C₄⋊C₄×C₃₁	—	2	4	no	—
59	C₄⋊C₄×C₃₁	—	4	2	no	—
60	C₄⋊C₄×C₃₁	—	4	2	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	#9767	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂³×C₃₁	#124	1	1	yes	characteristic
66	Q₈×C₃₁	—	8	4	no	—
67	Q₈×C₃₁	—	8	4	no	—
68	C₂×C₄×C₃₁	—	2	4	no	—
69	C₂×C₄×C₃₁	—	4	4	no	—
70	C₂×C₄×C₃₁	—	8	4	no	—
71	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
73	C₂×C₄×C₃₁	#121	1	2	yes	—
74	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
75	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
76	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
77	C₈×C₃₁	—	4	4	no	—
78	C₈×C₃₁	—	8	1	no	—
79	C₈×C₃₁	—	8	1	no	—
80	C₈×C₃₁	#103	1	4	yes	—
81	C₈×C₃₁	#103	1	4	yes	—
82	C₂×#498	C₂×C₃₁	1	1	yes	characteristic
83	#1848	C₂×C₃₁	1	1	yes	characteristic
84	#1549	C₂×C₃₁	1	2	yes	—
85	#1548	C₂×C₃₁	1	2	yes	—
86	#1450	C₂×C₃₁	1	1	yes	characteristic
87	C₂²×C₃₁	—	2	2	no	—
88	C₂²×C₃₁	#390	1	1	yes	characteristic
89	C₂²×C₃₁	#487	1	2	yes	—
90	C₄×C₃₁	—	2	2	no	—
91	C₄×C₃₁	—	8	4	no	—
92	C₄×C₃₁	C₂×#103	1	2	yes	—
93	C₄×C₃₁	#471	1	4	yes	—
94	C₄×C₃₁	#492	1	1	yes	characteristic
95	C₄×C₃₁	#494	1	1	yes	characteristic
96	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
97	#498	—	2	1	no	—
98	#498	—	2	1	no	—
99	#498	C₄×C₃₁	1	4	yes	—
100	#431	C₂²×C₃₁	1	2	yes	—
101	C₂×#108	—	2	2	no	—
102	C₂×#106	—	2	2	no	—
103	#423	C₂²×C₃₁	1	1	yes	characteristic
104	#423	C₂²×C₃₁	1	1	yes	characteristic
105	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
106	#333	—	2	4	no	—
107	#332	—	2	4	no	—
108	#322	—	2	1	no	—
109	C₈²	—	2	1	no	—
110	C₂×C₃₁	—	2	2	no	—
111	C₂×C₃₁	#1956	1	1	yes	characteristic
112	C₂×C₃₁	#1977	1	2	yes	—
113	C₂²×Q₈	—	2	2	no	—
114	C₂×M₁₆	—	2	1	no	—
115	C₂×M₁₆	—	2	1	no	—
116	C₂²×C₈	D₄×C₃₁	1	2	yes	—
117	C₄⋊Q₈	—	2	2	no	—
118	C₄⋊Q₈	—	2	2	no	—
119	C₄⋊Q₈	—	2	2	no	—
120	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
124	#108	—	4	4	no	—
125	#106	—	2	4	no	—
126	#106	—	2	4	no	—
127	C₄⋊C₈	—	2	2	no	—
128	C₄⋊C₈	—	2	2	no	—
129	C₄⋊C₈	D₄×C₃₁	1	4	yes	—
130	C₂²⋊C₈	—	2	2	no	—
131	C₄×C₈	—	2	2	no	—
132	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
133	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
134	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
135	C₃₁	#9767	1	1	yes	characteristic, direct product factor, Sylow
136	C₂×Q₈	—	2	4	no	—
137	C₂×Q₈	—	4	2	no	—
138	C₂×Q₈	—	8	4	no	—
139	C₂²×C₄	—	4	2	no	—
140	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
141	C₂²×C₄	#41×C₃₁	1	2	yes	—
142	M₁₆	—	4	2	no	—
143	M₁₆	—	4	2	no	—
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	—	4	1	no	—
146	C₂×C₈	—	4	1	no	—
147	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
148	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
149	C₂×C₈	SD₁₆×C₃₁	1	4	yes	—
150	C₂×C₈	Q₁₆×C₃₁	1	4	yes	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄⋊C₄	—	4	2	no	—
155	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
156	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
157	C₄²	C₂×D₄×C₃₁	1	2	yes	—
158	C₂³	C₃₁×#124	1	1	yes	characteristic
159	Q₈	—	8	4	no	—
160	Q₈	—	8	4	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	—	8	4	no	—
164	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
165	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
166	C₂×C₄	C₃₁×#121	1	2	yes	—
167	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
168	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
169	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
170	C₈	—	4	4	no	—
171	C₈	—	8	1	no	—
172	C₈	—	8	1	no	—
173	C₈	C₃₁×#103	1	4	yes	—
174	C₈	C₃₁×#103	1	4	yes	—
175	C₂²	—	2	2	no	—
176	C₂²	C₃₁×#390	1	1	yes	characteristic
177	C₂²	C₃₁×#487	1	2	yes	—
178	C₄	—	2	2	no	—
179	C₄	—	8	4	no	—
180	C₄	C₂×C₃₁×#103	1	2	yes	—
181	C₄	C₃₁×#471	1	4	yes	—
182	C₄	C₃₁×#492	1	1	yes	characteristic
183	C₄	C₃₁×#494	1	1	yes	characteristic
184	C₂	—	2	2	no	—
185	C₂	C₃₁×#1956	1	1	yes	characteristic
186	C₂	C₃₁×#1977	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃₁×#498

C₂

yes

maximal, characteristic

C₃₁×#1848

C₂

yes

maximal, characteristic

C₃₁×#1549

C₂

yes

maximal

C₃₁×#1548

C₂

yes

maximal

C₃₁×#1450

C₂

yes

maximal, characteristic

C₂×C₃₁×C₄⋊Q₈

C₂²

yes

—

C₃₁×#498

—

C₃₁×#498

—

C₃₁×#498

C₄

yes

—

C₃₁×#431

C₂²

yes

—

C₂×C₃₁×#108

—

C₂×C₃₁×#106

—

C₃₁×#423

C₂²

yes

characteristic

C₃₁×#423

C₂²

yes

characteristic

C₂×C₄×C₈×C₃₁

C₂²

yes

characteristic

C₃₁×#333

—

C₃₁×#332

—

C₃₁×#322

—

C₈²×C₃₁

—

C₂²×Q₈×C₃₁

—

C₂×M₁₆×C₃₁

—

C₂×M₁₆×C₃₁

—

C₂²×C₈×C₃₁

D₄

yes

—

C₃₁×C₄⋊Q₈

—

C₃₁×C₄⋊Q₈

—

C₃₁×C₄⋊Q₈

—

C₃₁×C₄⋊Q₈

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄²×C₃₁

C₂³

yes

characteristic

C₃₁×#108

—

C₃₁×#106

—

C₃₁×#106

—

C₃₁×C₄⋊C₈

—

C₃₁×C₄⋊C₈

—

C₃₁×C₄⋊C₈

D₄

yes

—

C₃₁×C₂²⋊C₈

—

C₄×C₈×C₃₁

—

C₄×C₈×C₃₁

C₂×C₄

yes

—

C₄×C₈×C₃₁

D₄

yes

characteristic

C₄×C₈×C₃₁

D₄

yes

characteristic

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₃₁

#41

yes

—

M₁₆×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

SD₁₆

yes

—

C₂×C₈×C₃₁

Q₁₆

yes

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

C₂²×C₄

yes

characteristic

C₄²×C₃₁

C₂×D₄

yes

characteristic

C₄²×C₃₁

C₂×D₄

yes

—

#9767

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

#124

yes

characteristic

Q₈×C₃₁

—

Q₈×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×SD₁₆

yes

—

C₂×C₄×C₃₁

C₂×Q₁₆

yes

—

C₈×C₃₁

—

C₈×C₃₁

—

C₈×C₃₁

—

C₈×C₃₁

#103

yes

—

C₈×C₃₁

#103

yes

—

C₂×#498

C₂×C₃₁

yes

characteristic

#1848

C₂×C₃₁

yes

characteristic

#1549

C₂×C₃₁

yes

—

#1548

C₂×C₃₁

yes

—

#1450

C₂×C₃₁

yes

characteristic

C₂²×C₃₁

—

C₂²×C₃₁

#390

yes

characteristic

C₂²×C₃₁

#487

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂×#103

yes

—

C₄×C₃₁

#471

yes

—

C₄×C₃₁

yes

C₄×C₃₁

yes

C₂×C₄⋊Q₈

C₂²×C₃₁

yes

—

#498

—

#498

—

#498

C₄×C₃₁

yes

—

100

#431

C₂²×C₃₁

yes

—

101

C₂×#108

—

102

C₂×#106

—

103

#423

C₂²×C₃₁

yes

characteristic

104

#423

C₂²×C₃₁

yes

characteristic

105

C₂×C₄×C₈

C₂²×C₃₁

yes

characteristic

106

#333

—

107

#332

—

108

#322

—

109

C₈²

—

110

C₂×C₃₁

—

111

C₂×C₃₁

#1956

yes

characteristic

112

C₂×C₃₁

#1977

yes

—

113

C₂²×Q₈

—

114

C₂×M₁₆

—

115

C₂×M₁₆

—

116

C₂²×C₈

D₄×C₃₁

yes

—

117

C₄⋊Q₈

—

118

C₄⋊Q₈

—

119

C₄⋊Q₈

—

120

C₄⋊Q₈

C₂×C₄×C₃₁

yes

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄²

C₂³×C₃₁

yes

characteristic

124

#108

—

125

#106

—

126

#106

—

127

C₄⋊C₈

—

128

C₄⋊C₈

—

129

C₄⋊C₈

D₄×C₃₁

yes

—

130

C₂²⋊C₈

—

131

C₄×C₈

—

132

C₄×C₈

C₂×C₄×C₃₁

yes

—

133

C₄×C₈

D₄×C₃₁

yes

characteristic

134

C₄×C₈

D₄×C₃₁

yes

characteristic

135

C₃₁

#9767

yes

characteristic, direct product factor, Sylow

136

C₂×Q₈

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂²×C₄

—

140

C₂²×C₄

C₂×D₄×C₃₁

yes

characteristic

141

C₂²×C₄

#41×C₃₁

yes

—

142

M₁₆

—

143

M₁₆

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

C₂²⋊C₄×C₃₁

yes

—

148

C₂×C₈

C₂²⋊C₄×C₃₁

yes

—

149

C₂×C₈

SD₁₆×C₃₁

yes

—

150

C₂×C₈

Q₁₆×C₃₁

yes

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄²

C₂²×C₄×C₃₁

yes

characteristic

156

C₄²

C₂×D₄×C₃₁

yes

characteristic

157

C₄²

C₂×D₄×C₃₁

yes

—

158

C₂³

C₃₁×#124

yes

characteristic

159

Q₈

—

160

Q₈

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

C₂×C₂²⋊C₄×C₃₁

yes

characteristic

165

C₂×C₄

C₄×D₄×C₃₁

yes

—

166

C₂×C₄

C₃₁×#121

yes

—

167

C₂×C₄

C₃₁×#127

yes

characteristic

168

C₂×C₄

C₂×SD₁₆×C₃₁

yes

—

169

C₂×C₄

C₂×Q₁₆×C₃₁

yes

—

170

C₈

—

171

C₈

—

172

C₈

—

173

C₈

C₃₁×#103

yes

—

174

C₈

C₃₁×#103

yes

—

175

C₂²

—

176

C₂²

C₃₁×#390

yes

characteristic

177

C₂²

C₃₁×#487

yes

—

178

C₄

—

179

C₄

—

180

C₄

C₂×C₃₁×#103

yes

—

181

C₄

C₃₁×#471

yes

—

182

C₄

C₃₁×#492

yes

characteristic

183

C₄

C₃₁×#494

yes

characteristic

184

C₂

—

185

C₂

C₃₁×#1956

yes

characteristic

186

C₂

C₃₁×#1977

yes

—

	subgroup	G/H
center, socle	C₂²×C₃₁	#390
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₂³×C₃₁	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₃₁

#390

commutator

C₄²

C₂²×C₄×C₃₁

Frattini, cosocle

C₂×C₄²

C₂³×C₃₁

C₂³×C₃₁

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

N	H ≅ G/N	type
64. #9767	C₃₁	direct (N × H)
135. C₃₁	#9767	direct (N × H)

H ≅ G/N

type

64. #9767

C₃₁

direct (N × H)

135. C₃₁

#9767

direct (N × H)

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

subgroup	index
C₂×C₃₁×#498	2
C₃₁×#1848	2
C₃₁×#1549	2
C₃₁×#1548	2
C₃₁×#1450	2
#9767	31

subgroup

index

C₂×C₃₁×#498

C₃₁×#1848

C₃₁×#1549

C₃₁×#1548

C₃₁×#1450

#9767

Subgroups — C₃₁×#9767 #21466526

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#9767 #21466526

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9767 #21466526