Group #21467419 — subgroups

Subgroups — C₃₁×#10660 #21467419

Order 7936.

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

normal subgroups only

This group has 372 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	C₃₁×#1955	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1849	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1571	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1486	C₂	1	2	yes	maximal
5	C₃₁×#1275	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
7	C₃₁×#431	C₂²	1	2	yes	—
8	C₂×C₃₁×#108	—	2	2	no	—
9	C₃₁×#423	C₂²	1	2	yes	—
10	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×C₂²⋊C₈	—	2	2	no	—
12	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
13	C₃₁×#335	—	2	4	no	—
14	C₃₁×D₄⋊C₈	—	2	4	no	—
15	C₂²×D₄×C₃₁	Q₈	1	1	yes	characteristic
16	C₂³×C₄×C₃₁	—	2	1	no	—
17	C₂×M₁₆×C₃₁	—	2	2	no	—
18	C₂²×C₈×C₃₁	—	2	1	no	—
19	C₂²×C₈×C₃₁	D₄	1	2	yes	—
20	C₄×D₄×C₃₁	—	2	2	no	—
21	C₄×D₄×C₃₁	—	2	2	no	—
22	C₂×C₄⋊C₄×C₃₁	Q₈	1	1	yes	characteristic
23	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
24	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
25	C₃₁×#108	—	4	4	no	—
26	C₃₁×C₄⋊C₈	—	2	2	no	—
27	C₃₁×C₄⋊C₈	—	2	2	no	—
28	C₃₁×C₄⋊C₈	—	2	2	no	—
29	C₃₁×C₄⋊C₈	—	2	2	no	—
30	C₃₁×C₂²⋊C₈	—	2	2	no	—
31	C₃₁×C₂²⋊C₈	—	4	4	no	—
32	C₃₁×#97	—	2	2	no	—
33	C₄×C₈×C₃₁	D₄	1	4	yes	—
34	C₂⁴×C₃₁	—	2	1	no	—
35	C₂×D₄×C₃₁	—	2	2	no	—
36	C₂×D₄×C₃₁	—	4	2	no	—
37	C₂²×C₄×C₃₁	—	2	1	no	—
38	C₂²×C₄×C₃₁	—	4	1	no	—
39	C₂²×C₄×C₃₁	—	4	1	no	—
40	C₂²×C₄×C₃₁	—	4	2	no	—
41	C₂²×C₄×C₃₁	—	4	2	no	—
42	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
44	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
45	M₁₆×C₃₁	—	4	4	no	—
46	C₂×C₈×C₃₁	—	2	2	no	—
47	C₂×C₈×C₃₁	—	2	2	no	—
48	C₂×C₈×C₃₁	—	2	4	no	—
49	C₂×C₈×C₃₁	—	4	1	no	—
50	C₂×C₈×C₃₁	—	4	2	no	—
51	C₂×C₈×C₃₁	—	4	2	no	—
52	C₂×C₈×C₃₁	D₈	1	4	yes	—
53	C₄⋊C₄×C₃₁	—	2	2	no	—
54	C₂²⋊C₄×C₃₁	—	4	2	no	—
55	C₄²×C₃₁	C₂×D₄	1	2	yes	—
56	C₄²×C₃₁	#41	1	2	yes	—
57	#10660	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂³×C₃₁	—	2	2	no	—
59	C₂³×C₃₁	—	4	1	no	—
60	C₂³×C₃₁	—	4	2	no	—
61	C₂³×C₃₁	—	4	1	no	—
62	C₂³×C₃₁	—	4	2	no	—
63	C₂³×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
64	D₄×C₃₁	—	2	4	no	—
65	D₄×C₃₁	—	4	2	no	—
66	C₂×C₄×C₃₁	—	4	1	no	—
67	C₂×C₄×C₃₁	—	4	1	no	—
68	C₂×C₄×C₃₁	—	4	1	no	—
69	C₂×C₄×C₃₁	—	4	2	no	—
70	C₂×C₄×C₃₁	—	8	2	no	—
71	C₂×C₄×C₃₁	—	8	1	no	—
72	C₂×C₄×C₃₁	—	8	1	no	—
73	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
74	C₂×C₄×C₃₁	#123	1	2	yes	—
75	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
76	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
77	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
78	C₂×C₄×C₃₁	#137	1	2	yes	—
79	C₈×C₃₁	—	2	4	no	—
80	C₈×C₃₁	—	4	4	no	—
81	C₈×C₃₁	—	8	1	no	—
82	C₈×C₃₁	—	8	2	no	—
83	#1955	C₂×C₃₁	1	1	yes	characteristic
84	#1849	C₂×C₃₁	1	1	yes	characteristic, complemented
85	#1571	C₂×C₃₁	1	2	yes	complemented
86	#1486	C₂×C₃₁	1	2	yes	—
87	#1275	C₂×C₃₁	1	1	yes	characteristic, complemented
88	C₂²×C₃₁	—	2	1	no	—
89	C₂²×C₃₁	—	2	1	no	—
90	C₂²×C₃₁	—	4	4	no	—
91	C₂²×C₃₁	—	4	2	no	—
92	C₂²×C₃₁	—	8	2	no	—
93	C₂²×C₃₁	—	8	1	no	—
94	C₂²×C₃₁	—	8	1	no	—
95	C₂²×C₃₁	#399	1	1	yes	characteristic
96	C₂²×C₃₁	#474	1	2	yes	—
97	C₄×C₃₁	—	2	2	no	—
98	C₄×C₃₁	—	2	2	no	—
99	C₄×C₃₁	—	8	1	no	—
100	C₄×C₃₁	#480	1	2	yes	—
101	C₄×C₃₁	#493	1	1	yes	characteristic
102	C₄×C₃₁	#497	1	1	yes	characteristic
103	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
104	#431	C₂²×C₃₁	1	2	yes	—
105	C₂×#108	—	2	2	no	—
106	#423	C₂²×C₃₁	1	2	yes	—
107	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
108	C₂×C₂²⋊C₈	—	2	2	no	—
109	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
110	#335	—	2	4	no	—
111	D₄⋊C₈	—	2	4	no	—
112	C₂×C₃₁	—	2	2	no	—
113	C₂×C₃₁	—	8	2	no	—
114	C₂×C₃₁	#2091	1	1	yes	characteristic
115	C₂×C₃₁	#2093	1	1	yes	characteristic
116	C₂×C₃₁	#2098	1	1	yes	characteristic
117	C₂²×D₄	Q₈×C₃₁	1	1	yes	characteristic
118	C₂³×C₄	—	2	1	no	—
119	C₂×M₁₆	—	2	2	no	—
120	C₂²×C₈	—	2	1	no	—
121	C₂²×C₈	D₄×C₃₁	1	2	yes	—
122	C₄×D₄	—	2	2	no	—
123	C₄×D₄	—	2	2	no	—
124	C₂×C₄⋊C₄	Q₈×C₃₁	1	1	yes	characteristic
125	C₂×C₂²⋊C₄	—	2	1	no	—
126	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
127	#108	—	4	4	no	—
128	C₄⋊C₈	—	2	2	no	—
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	2	2	no	—
131	C₄⋊C₈	—	2	2	no	—
132	C₂²⋊C₈	—	2	2	no	—
133	C₂²⋊C₈	—	4	4	no	—
134	#97	—	2	2	no	—
135	C₄×C₈	D₄×C₃₁	1	4	yes	—
136	C₃₁	#10660	1	1	yes	characteristic, direct product factor, Sylow
137	C₂⁴	—	2	1	no	—
138	C₂×D₄	—	2	2	no	—
139	C₂×D₄	—	4	2	no	—
140	C₂²×C₄	—	2	1	no	—
141	C₂²×C₄	—	4	1	no	—
142	C₂²×C₄	—	4	1	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	—	4	2	no	—
145	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
146	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
147	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
148	M₁₆	—	4	4	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	4	1	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	D₈×C₃₁	1	4	yes	—
156	C₄⋊C₄	—	2	2	no	—
157	C₂²⋊C₄	—	4	2	no	—
158	C₄²	C₂×D₄×C₃₁	1	2	yes	—
159	C₄²	#41×C₃₁	1	2	yes	—
160	C₂³	—	2	2	no	—
161	C₂³	—	4	1	no	—
162	C₂³	—	4	2	no	—
163	C₂³	—	4	1	no	—
164	C₂³	—	4	2	no	—
165	C₂³	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
166	D₄	—	2	4	no	—
167	D₄	—	4	2	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	1	no	—
174	C₂×C₄	—	8	1	no	—
175	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
176	C₂×C₄	C₃₁×#123	1	2	yes	—
177	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
178	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
179	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
180	C₂×C₄	C₃₁×#137	1	2	yes	—
181	C₈	—	2	4	no	—
182	C₈	—	4	4	no	—
183	C₈	—	8	1	no	—
184	C₈	—	8	2	no	—
185	C₂²	—	2	1	no	—
186	C₂²	—	2	1	no	—
187	C₂²	—	4	4	no	—
188	C₂²	—	4	2	no	—
189	C₂²	—	8	2	no	—
190	C₂²	—	8	1	no	—
191	C₂²	—	8	1	no	—
192	C₂²	C₃₁×#399	1	1	yes	characteristic
193	C₂²	C₃₁×#474	1	2	yes	—
194	C₄	—	2	2	no	—
195	C₄	—	2	2	no	—
196	C₄	—	8	1	no	—
197	C₄	C₃₁×#480	1	2	yes	—
198	C₄	C₃₁×#493	1	1	yes	characteristic
199	C₄	C₃₁×#497	1	1	yes	characteristic
200	C₂	—	2	2	no	—
201	C₂	—	8	2	no	—
202	C₂	C₃₁×#2091	1	1	yes	characteristic
203	C₂	C₃₁×#2093	1	1	yes	characteristic
204	C₂	C₃₁×#2098	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃₁×#1955

C₂

yes

maximal, characteristic

C₃₁×#1849

C₂

yes

maximal, characteristic, complemented

C₃₁×#1571

C₂

yes

maximal, complemented

C₃₁×#1486

C₂

yes

maximal

C₃₁×#1275

C₂

yes

maximal, characteristic, complemented

C₂×C₄×D₄×C₃₁

C₂²

yes

characteristic

C₃₁×#431

C₂²

yes

—

C₂×C₃₁×#108

—

C₃₁×#423

C₂²

yes

—

C₂×C₃₁×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₃₁×C₂²⋊C₈

—

C₂×C₃₁×#97

C₂²

yes

characteristic

C₃₁×#335

—

C₃₁×D₄⋊C₈

—

C₂²×D₄×C₃₁

Q₈

yes

characteristic

C₂³×C₄×C₃₁

—

C₂×M₁₆×C₃₁

—

C₂²×C₈×C₃₁

—

C₂²×C₈×C₃₁

D₄

yes

—

C₄×D₄×C₃₁

—

C₄×D₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

Q₈

yes

characteristic

C₂×C₂²⋊C₄×C₃₁

—

C₂×C₄²×C₃₁

C₂³

yes

characteristic

C₃₁×#108

—

C₃₁×C₄⋊C₈

—

C₃₁×C₄⋊C₈

—

C₃₁×C₄⋊C₈

—

C₃₁×C₄⋊C₈

—

C₃₁×C₂²⋊C₈

—

C₃₁×C₂²⋊C₈

—

C₃₁×#97

—

C₄×C₈×C₃₁

D₄

yes

—

C₂⁴×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₃₁

C₂×Q₈

yes

characteristic

C₂²×C₄×C₃₁

#41

yes

characteristic

M₁₆×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

D₈

yes

—

C₄⋊C₄×C₃₁

—

C₂²⋊C₄×C₃₁

—

C₄²×C₃₁

C₂×D₄

yes

—

C₄²×C₃₁

#41

yes

—

#10660

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

C₂²⋊Q₈

yes

characteristic

D₄×C₃₁

—

D₄×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

#123

yes

—

C₂×C₄×C₃₁

yes

C₂×C₄×C₃₁

yes

C₂×C₄×C₃₁

C₂×D₈

yes

—

C₂×C₄×C₃₁

#137

yes

—

C₈×C₃₁

—

C₈×C₃₁

—

C₈×C₃₁

—

C₈×C₃₁

—

#1955

C₂×C₃₁

yes

characteristic

#1849

C₂×C₃₁

yes

characteristic, complemented

#1571

C₂×C₃₁

yes

complemented

#1486

C₂×C₃₁

yes

—

#1275

C₂×C₃₁

yes

characteristic, complemented

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

#399

yes

characteristic

C₂²×C₃₁

#474

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

100

C₄×C₃₁

#480

yes

—

101

C₄×C₃₁

#493

yes

characteristic

102

C₄×C₃₁

#497

yes

characteristic

103

C₂×C₄×D₄

C₂²×C₃₁

yes

characteristic

104

#431

C₂²×C₃₁

yes

—

105

C₂×#108

—

106

#423

C₂²×C₃₁

yes

—

107

C₂×C₄⋊C₈

C₂²×C₃₁

yes

characteristic

108

C₂×C₂²⋊C₈

—

109

C₂×#97

C₂²×C₃₁

yes

characteristic

110

#335

—

111

D₄⋊C₈

—

112

C₂×C₃₁

—

113

C₂×C₃₁

—

114

C₂×C₃₁

#2091

yes

characteristic

115

C₂×C₃₁

#2093

yes

characteristic

116

C₂×C₃₁

#2098

yes

characteristic

117

C₂²×D₄

Q₈×C₃₁

yes

characteristic

118

C₂³×C₄

—

119

C₂×M₁₆

—

120

C₂²×C₈

—

121

C₂²×C₈

D₄×C₃₁

yes

—

122

C₄×D₄

—

123

C₄×D₄

—

124

C₂×C₄⋊C₄

Q₈×C₃₁

yes

characteristic

125

C₂×C₂²⋊C₄

—

126

C₂×C₄²

C₂³×C₃₁

yes

characteristic

127

#108

—

128

C₄⋊C₈

—

129

C₄⋊C₈

—

130

C₄⋊C₈

—

131

C₄⋊C₈

—

132

C₂²⋊C₈

—

133

C₂²⋊C₈

—

134

#97

—

135

C₄×C₈

D₄×C₃₁

yes

—

136

C₃₁

#10660

yes

characteristic, direct product factor, Sylow

137

C₂⁴

—

138

C₂×D₄

—

139

C₂×D₄

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

C₂×D₄×C₃₁

yes

characteristic

146

C₂²×C₄

C₂×Q₈×C₃₁

yes

characteristic

147

C₂²×C₄

#41×C₃₁

yes

characteristic

148

M₁₆

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

D₈×C₃₁

yes

—

156

C₄⋊C₄

—

157

C₂²⋊C₄

—

158

C₄²

C₂×D₄×C₃₁

yes

—

159

C₄²

#41×C₃₁

yes

—

160

C₂³

—

161

C₂³

—

162

C₂³

—

163

C₂³

—

164

C₂³

—

165

C₂³

C₃₁×C₂²⋊Q₈

yes

characteristic

166

D₄

—

167

D₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

C₃₁×C₂²⋊Q₈

yes

—

176

C₂×C₄

C₃₁×#123

yes

—

177

C₂×C₄

C₃₁×#123

yes

characteristic

178

C₂×C₄

C₃₁×#127

yes

characteristic

179

C₂×C₄

C₂×D₈×C₃₁

yes

—

180

C₂×C₄

C₃₁×#137

yes

—

181

C₈

—

182

C₈

—

183

C₈

—

184

C₈

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

192

C₂²

C₃₁×#399

yes

characteristic

193

C₂²

C₃₁×#474

yes

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

C₃₁×#480

yes

—

198

C₄

C₃₁×#493

yes

characteristic

199

C₄

C₃₁×#497

yes

characteristic

200

C₂

—

201

C₂

—

202

C₂

C₃₁×#2091

yes

characteristic

203

C₂

C₃₁×#2093

yes

characteristic

204

C₂

C₃₁×#2098

yes

characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	#399
prime radical	C₂²×D₄×C₃₁	Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

subgroup

G/H

center

C₂×C₄×C₃₁

#127

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₃₁

socle

C₂²×C₃₁

#399

prime radical

C₂²×D₄×C₃₁

Q₈

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

N	H ≅ G/N	type
2. C₃₁×#1849	C₂	semidirect
3. C₃₁×#1571	C₂	semidirect
5. C₃₁×#1275	C₂	semidirect
57. #10660	C₃₁	direct (N × H)
84. #1849	C₂×C₃₁	semidirect
85. #1571	C₂×C₃₁	semidirect
87. #1275	C₂×C₃₁	semidirect
136. C₃₁	#10660	direct (N × H)

H ≅ G/N

type

2. C₃₁×#1849

C₂

semidirect

3. C₃₁×#1571

C₂

semidirect

5. C₃₁×#1275

C₂

semidirect

57. #10660

C₃₁

direct (N × H)

84. #1849

C₂×C₃₁

semidirect

85. #1571

C₂×C₃₁

semidirect

87. #1275

C₂×C₃₁

semidirect

136. C₃₁

#10660

direct (N × H)

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

subgroup	index
C₃₁×#1955	2
C₃₁×#1849	2
C₃₁×#1571	2
C₃₁×#1486	2
C₃₁×#1275	2
#10660	31

subgroup

index

C₃₁×#1955

C₃₁×#1849

C₃₁×#1571

C₃₁×#1486

C₃₁×#1275

#10660

Subgroups — C₃₁×#10660 #21467419

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#10660 #21467419

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#10660 #21467419