Group #21469605 — subgroups

Subgroups — C₃₁×#12846 #21469605

Order 7936.

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

normal subgroups only

This group has 478 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	C₂²×C₃₁×#97	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2112	C₂	1	2	yes	maximal
3	C₃₁×#1377	C₂	1	4	yes	maximal, complemented
4	C₂³×C₈×C₃₁	C₄	1	1	yes	characteristic
5	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₃₁	C₄	1	1	yes	characteristic
7	C₂×C₃₁×M₃₂	C₂²	1	4	yes	—
8	C₂×C₃₁×#97	—	2	4	no	—
9	C₂×C₃₁×#97	C₄	1	2	yes	—
10	C₂×C₃₁×#97	C₂²	1	2	yes	—
11	C₃₁×C₂²⋊C₁₆	—	2	4	no	—
12	C₂³×C₄×C₃₁	C₈	1	2	yes	—
13	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈×C₃₁	—	2	2	no	—
15	C₂²×C₈×C₃₁	—	2	4	no	—
16	C₂²×C₈×C₃₁	—	2	4	no	—
17	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₃₁	D₄	1	4	yes	—
20	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₃₁	—	2	2	no	—
22	C₂×C₄²×C₃₁	—	2	2	no	—
23	C₂×C₄²×C₃₁	C₈	1	2	yes	—
24	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
25	C₃₁×M₃₂	—	2	8	no	—
26	C₂×C₁₆×C₃₁	—	2	4	no	—
27	C₃₁×#97	—	2	4	no	—
28	C₃₁×#97	—	2	4	no	—
29	C₂⁴×C₃₁	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄×C₃₁	—	2	2	no	—
31	C₂²×C₄×C₃₁	—	2	2	no	—
32	C₂²×C₄×C₃₁	—	2	2	no	—
33	C₂²×C₄×C₃₁	—	2	2	no	—
34	C₂²×C₄×C₃₁	—	4	4	no	—
35	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
36	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₃₁	C₂×C₈	1	4	yes	—
38	C₂²×C₄×C₃₁	M₁₆	1	2	yes	—
39	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₃₁	—	2	8	no	—
41	C₂×C₈×C₃₁	—	2	2	no	—
42	C₂×C₈×C₃₁	—	2	2	no	—
43	C₂×C₈×C₃₁	—	2	2	no	—
44	C₂×C₈×C₃₁	—	2	8	no	—
45	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
46	C₂×C₈×C₃₁	M₁₆	1	4	yes	—
47	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
48	C₂×C₈×C₃₁	#41	1	4	yes	—
49	C₄²×C₃₁	—	2	4	no	—
50	C₄²×C₃₁	—	2	2	no	—
51	C₄²×C₃₁	—	2	2	no	—
52	C₁₆×C₃₁	—	4	4	no	—
53	#12846	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂³×C₃₁	—	2	4	no	—
55	C₂³×C₃₁	—	2	2	no	—
56	C₂³×C₃₁	C₂²⋊C₈	1	2	yes	—
57	C₂³×C₃₁	C₂²×C₈	1	1	yes	characteristic
58	C₂×C₄×C₃₁	—	2	2	no	—
59	C₂×C₄×C₃₁	—	2	2	no	—
60	C₂×C₄×C₃₁	—	2	2	no	—
61	C₂×C₄×C₃₁	—	2	4	no	—
62	C₂×C₄×C₃₁	—	2	4	no	—
63	C₂×C₄×C₃₁	—	4	4	no	—
64	C₂×C₄×C₃₁	—	4	4	no	—
65	C₂×C₄×C₃₁	—	4	4	no	—
66	C₂×C₄×C₃₁	C₂²⋊C₈	1	2	yes	—
67	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₃₁	#117	1	2	yes	—
69	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
70	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
71	C₈×C₃₁	—	2	2	no	—
72	C₈×C₃₁	—	2	4	no	—
73	C₈×C₃₁	#123	1	4	yes	—
74	C₂²×#97	C₂×C₃₁	1	1	yes	characteristic
75	#2112	C₂×C₃₁	1	2	yes	—
76	#1377	C₂×C₃₁	1	4	yes	complemented
77	C₂²×C₃₁	—	2	8	no	—
78	C₂²×C₃₁	—	2	2	no	—
79	C₂²×C₃₁	—	2	2	no	—
80	C₂²×C₃₁	—	2	2	no	—
81	C₂²×C₃₁	#349	1	4	yes	—
82	C₂²×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
83	C₂²×C₃₁	#431	1	2	yes	—
84	C₄×C₃₁	—	2	2	no	—
85	C₄×C₃₁	—	4	4	no	—
86	C₄×C₃₁	#381	1	1	yes	characteristic
87	C₄×C₃₁	#407	1	1	yes	characteristic
88	C₄×C₃₁	#432	1	2	yes	—
89	C₂³×C₈	C₄×C₃₁	1	1	yes	characteristic
90	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
91	C₂²×C₄²	C₄×C₃₁	1	1	yes	characteristic
92	C₂×M₃₂	C₂²×C₃₁	1	4	yes	—
93	C₂×#97	—	2	4	no	—
94	C₂×#97	C₄×C₃₁	1	2	yes	—
95	C₂×#97	C₂²×C₃₁	1	2	yes	—
96	C₂²⋊C₁₆	—	2	4	no	—
97	C₂×C₃₁	—	2	4	no	—
98	C₂×C₃₁	—	2	2	no	—
99	C₂×C₃₁	#1817	1	1	yes	characteristic
100	C₂×C₃₁	C₂×#349	1	2	yes	—
101	C₂³×C₄	C₈×C₃₁	1	2	yes	—
102	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
103	C₂²×C₈	—	2	2	no	—
104	C₂²×C₈	—	2	4	no	—
105	C₂²×C₈	—	2	4	no	—
106	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
107	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
108	C₂²×C₈	D₄×C₃₁	1	4	yes	—
109	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
110	C₂×C₄²	—	2	2	no	—
111	C₂×C₄²	—	2	2	no	—
112	C₂×C₄²	C₈×C₃₁	1	2	yes	—
113	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
114	M₃₂	—	2	8	no	—
115	C₂×C₁₆	—	2	4	no	—
116	#97	—	2	4	no	—
117	#97	—	2	4	no	—
118	C₃₁	#12846	1	1	yes	characteristic, direct product factor, Sylow
119	C₂⁴	C₂×C₈×C₃₁	1	1	yes	characteristic
120	C₂²×C₄	—	2	2	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	2	2	no	—
123	C₂²×C₄	—	2	2	no	—
124	C₂²×C₄	—	4	4	no	—
125	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
126	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
127	C₂²×C₄	C₂×C₈×C₃₁	1	4	yes	—
128	C₂²×C₄	M₁₆×C₃₁	1	2	yes	—
129	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
130	C₂×C₈	—	2	8	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	8	no	—
135	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
136	C₂×C₈	M₁₆×C₃₁	1	4	yes	—
137	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
138	C₂×C₈	#41×C₃₁	1	4	yes	—
139	C₄²	—	2	4	no	—
140	C₄²	—	2	2	no	—
141	C₄²	—	2	2	no	—
142	C₁₆	—	4	4	no	—
143	C₂³	—	2	2	no	—
144	C₂³	—	2	4	no	—
145	C₂³	C₃₁×C₂²⋊C₈	1	2	yes	—
146	C₂³	C₂²×C₈×C₃₁	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	4	4	no	—
153	C₂×C₄	—	4	4	no	—
154	C₂×C₄	—	4	4	no	—
155	C₂×C₄	C₃₁×C₂²⋊C₈	1	2	yes	—
156	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
157	C₂×C₄	C₃₁×#117	1	2	yes	—
158	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
159	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
160	C₈	—	2	4	no	—
161	C₈	—	2	2	no	—
162	C₈	C₃₁×#123	1	4	yes	—
163	C₂²	—	2	8	no	—
164	C₂²	—	2	2	no	—
165	C₂²	—	2	2	no	—
166	C₂²	—	2	2	no	—
167	C₂²	C₃₁×#349	1	4	yes	—
168	C₂²	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
169	C₂²	C₃₁×#431	1	2	yes	—
170	C₄	—	2	2	no	—
171	C₄	—	4	4	no	—
172	C₄	C₃₁×#381	1	1	yes	characteristic
173	C₄	C₃₁×#407	1	1	yes	characteristic
174	C₄	C₃₁×#432	1	2	yes	—
175	C₂	—	2	2	no	—
176	C₂	—	2	4	no	—
177	C₂	C₃₁×#1817	1	1	yes	characteristic
178	C₂	C₂×C₃₁×#349	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₃₁×#97

C₂

yes

maximal, characteristic

C₃₁×#2112

C₂

yes

maximal

C₃₁×#1377

C₂

yes

maximal, complemented

C₂³×C₈×C₃₁

C₄

yes

characteristic

C₂³×C₈×C₃₁

C₂²

yes

characteristic

C₂²×C₄²×C₃₁

C₄

yes

characteristic

C₂×C₃₁×M₃₂

C₂²

yes

—

C₂×C₃₁×#97

—

C₂×C₃₁×#97

C₄

yes

—

C₂×C₃₁×#97

C₂²

yes

—

C₃₁×C₂²⋊C₁₆

—

C₂³×C₄×C₃₁

C₈

yes

—

C₂³×C₄×C₃₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₃₁

—

C₂²×C₈×C₃₁

—

C₂²×C₈×C₃₁

—

C₂²×C₈×C₃₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₃₁

C₂×C₄

yes

—

C₂²×C₈×C₃₁

D₄

yes

—

C₂²×C₈×C₃₁

C₂³

yes

characteristic

C₂×C₄²×C₃₁

—

C₂×C₄²×C₃₁

—

C₂×C₄²×C₃₁

C₈

yes

—

C₂×C₄²×C₃₁

C₂×C₄

yes

—

C₃₁×M₃₂

—

C₂×C₁₆×C₃₁

—

C₃₁×#97

—

C₃₁×#97

—

C₂⁴×C₃₁

C₂×C₈

yes

characteristic

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₃₁

C₂×C₈

yes

characteristic

C₂²×C₄×C₃₁

C₂×C₈

yes

—

C₂²×C₄×C₃₁

M₁₆

yes

—

C₂²×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂²⋊C₄

yes

—

C₂×C₈×C₃₁

M₁₆

yes

—

C₂×C₈×C₃₁

C₂×D₄

yes

—

C₂×C₈×C₃₁

#41

yes

—

C₄²×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

—

C₁₆×C₃₁

—

#12846

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

C₂²⋊C₈

yes

—

C₂³×C₃₁

C₂²×C₈

yes

characteristic

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

C₂²⋊C₈

yes

—

C₂×C₄×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₃₁

#117

yes

—

C₂×C₄×C₃₁

C₂×M₁₆

yes

—

C₂×C₄×C₃₁

C₂×M₁₆

yes

characteristic

C₈×C₃₁

—

C₈×C₃₁

—

C₈×C₃₁

#123

yes

—

C₂²×#97

C₂×C₃₁

yes

characteristic

#2112

C₂×C₃₁

yes

—

#1377

C₂×C₃₁

yes

complemented

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

#349

yes

—

C₂²×C₃₁

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₃₁

#431

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

—

C₂³×C₈

C₄×C₃₁

yes

characteristic

C₂³×C₈

C₂²×C₃₁

yes

characteristic

C₂²×C₄²

C₄×C₃₁

yes

characteristic

C₂×M₃₂

C₂²×C₃₁

yes

—

C₂×#97

—

C₂×#97

C₄×C₃₁

yes

—

C₂×#97

C₂²×C₃₁

yes

—

C₂²⋊C₁₆

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

#1817

yes

characteristic

100

C₂×C₃₁

C₂×#349

yes

—

101

C₂³×C₄

C₈×C₃₁

yes

—

102

C₂³×C₄

C₂×C₄×C₃₁

yes

characteristic

103

C₂²×C₈

—

104

C₂²×C₈

—

105

C₂²×C₈

—

106

C₂²×C₈

C₂×C₄×C₃₁

yes

characteristic

107

C₂²×C₈

C₂×C₄×C₃₁

yes

—

108

C₂²×C₈

D₄×C₃₁

yes

—

109

C₂²×C₈

C₂³×C₃₁

yes

characteristic

110

C₂×C₄²

—

111

C₂×C₄²

—

112

C₂×C₄²

C₈×C₃₁

yes

—

113

C₂×C₄²

C₂×C₄×C₃₁

yes

—

114

M₃₂

—

115

C₂×C₁₆

—

116

#97

—

117

#97

—

118

C₃₁

#12846

yes

characteristic, direct product factor, Sylow

119

C₂⁴

C₂×C₈×C₃₁

yes

characteristic

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

C₂²⋊C₄×C₃₁

yes

—

126

C₂²×C₄

C₂×C₈×C₃₁

yes

characteristic

127

C₂²×C₄

C₂×C₈×C₃₁

yes

—

128

C₂²×C₄

M₁₆×C₃₁

yes

—

129

C₂²×C₄

C₂²×C₄×C₃₁

yes

characteristic

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

C₂²⋊C₄×C₃₁

yes

—

136

C₂×C₈

M₁₆×C₃₁

yes

—

137

C₂×C₈

C₂×D₄×C₃₁

yes

—

138

C₂×C₈

#41×C₃₁

yes

—

139

C₄²

—

140

C₄²

—

141

C₄²

—

142

C₁₆

—

143

C₂³

—

144

C₂³

—

145

C₂³

C₃₁×C₂²⋊C₈

yes

—

146

C₂³

C₂²×C₈×C₃₁

yes

characteristic

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

C₃₁×C₂²⋊C₈

yes

—

156

C₂×C₄

C₂×C₂²⋊C₄×C₃₁

yes

characteristic

157

C₂×C₄

C₃₁×#117

yes

—

158

C₂×C₄

C₂×M₁₆×C₃₁

yes

—

159

C₂×C₄

C₂×M₁₆×C₃₁

yes

characteristic

160

C₈

—

161

C₈

—

162

C₈

C₃₁×#123

yes

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

167

C₂²

C₃₁×#349

yes

—

168

C₂²

C₂×C₃₁×C₂²⋊C₈

yes

characteristic

169

C₂²

C₃₁×#431

yes

—

170

C₄

—

171

C₄

—

172

C₄

C₃₁×#381

yes

characteristic

173

C₄

C₃₁×#407

yes

characteristic

174

C₄

C₃₁×#432

yes

—

175

C₂

—

176

C₂

—

177

C₂

C₃₁×#1817

yes

characteristic

178

C₂

C₂×C₃₁×#349

yes

—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₈×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₃₁	C₂×C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

subgroup

G/H

center

C₂×C₄×C₃₁

C₂×C₂²⋊C₄

commutator

C₂³

C₂²×C₈×C₃₁

Frattini, cosocle

C₂²×C₈

C₂³×C₃₁

socle

C₂²×C₃₁

C₂×C₂²⋊C₈

prime radical

C₂⁴×C₃₁

C₂×C₈

LCS

G ▷ C₂³ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

N	H ≅ G/N	type
3. C₃₁×#1377	C₂	semidirect
53. #12846	C₃₁	direct (N × H)
76. #1377	C₂×C₃₁	semidirect
118. C₃₁	#12846	direct (N × H)

H ≅ G/N

type

3. C₃₁×#1377

C₂

semidirect

53. #12846

C₃₁

direct (N × H)

76. #1377

C₂×C₃₁

semidirect

118. C₃₁

#12846

direct (N × H)

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

subgroup	index
C₂²×C₃₁×#97	2
C₃₁×#2112	2
C₃₁×#1377	2
#12846	31

subgroup

index

C₂²×C₃₁×#97

C₃₁×#2112

C₃₁×#1377

#12846

Subgroups — C₃₁×#12846 #21469605

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#12846 #21469605

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12846 #21469605