Group #21470015 — subgroups

Subgroups — C₃₁×#13256 #21470015

Order 7936.

462 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 158 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×#494	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2150	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#362	C₂	1	4	yes	maximal
4	C₄×C₃₁×#108	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₁₆×C₃₁	—	2	1	no	—
6	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
7	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
8	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
9	C₃₁×#494	—	2	2	no	—
10	C₃₁×#494	C₄	1	4	yes	—
11	C₂×C₃₁×#108	C₂²	1	2	yes	—
12	C₄×M₁₆×C₃₁	—	2	1	no	—
13	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×C₄⋊C₁₆	—	2	2	no	—
15	C₃₁×#362	—	2	8	no	—
16	C₃₁×#341	—	2	1	no	—
17	C₂²×Q₈×C₃₁	—	2	2	no	—
18	C₂×Q₁₆×C₃₁	—	2	4	no	—
19	C₂×Q₁₆×C₃₁	—	2	4	no	—
20	C₂×Q₁₆×C₃₁	—	2	2	no	—
21	C₂×Q₁₆×C₃₁	—	2	4	no	—
22	C₂×Q₁₆×C₃₁	—	4	4	no	—
23	C₂×Q₁₆×C₃₁	C₂×C₄	1	4	yes	—
24	C₂×M₁₆×C₃₁	—	2	2	no	—
25	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
27	C₃₁×C₄⋊Q₈	—	2	4	no	—
28	C₃₁×C₄⋊Q₈	—	4	1	no	—
29	C₃₁×C₄⋊Q₈	—	4	1	no	—
30	C₂×C₄⋊C₄×C₃₁	—	4	2	no	—
31	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
32	C₃₁×M₃₂	—	2	4	no	—
33	C₂×C₁₆×C₃₁	—	2	2	no	—
34	C₃₁×#108	—	2	2	no	—
35	C₃₁×#108	D₄	1	4	yes	—
36	C₃₁×#97	—	4	1	no	—
37	C₄×C₈×C₃₁	—	4	1	no	—
38	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₃₁	D₄	1	2	yes	—
40	C₂×Q₈×C₃₁	—	2	4	no	—
41	C₂×Q₈×C₃₁	—	4	2	no	—
42	C₂×Q₈×C₃₁	—	4	4	no	—
43	C₂×Q₈×C₃₁	—	4	4	no	—
44	C₂²×C₄×C₃₁	—	4	2	no	—
45	C₂²×C₄×C₃₁	SD₁₆	1	2	yes	—
46	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
47	Q₁₆×C₃₁	—	2	8	no	—
48	Q₁₆×C₃₁	—	4	8	no	—
49	Q₁₆×C₃₁	—	4	2	no	—
50	Q₁₆×C₃₁	—	4	2	no	—
51	M₁₆×C₃₁	—	2	4	no	—
52	M₁₆×C₃₁	—	4	2	no	—
53	C₂×C₈×C₃₁	—	2	2	no	—
54	C₂×C₈×C₃₁	—	4	2	no	—
55	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₃₁	#41	1	2	yes	—
60	C₄⋊C₄×C₃₁	—	4	4	no	—
61	C₄⋊C₄×C₃₁	—	4	4	no	—
62	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
63	C₄²×C₃₁	D₈	1	1	yes	characteristic
64	C₄²×C₃₁	D₈	1	1	yes	characteristic
65	C₁₆×C₃₁	—	4	2	no	—
66	#13256	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂³×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
68	Q₈×C₃₁	—	4	4	no	—
69	Q₈×C₃₁	—	4	4	no	—
70	Q₈×C₃₁	—	8	4	no	—
71	C₂×C₄×C₃₁	—	2	4	no	—
72	C₂×C₄×C₃₁	—	4	4	no	—
73	C₂×C₄×C₃₁	—	8	4	no	—
74	C₂×C₄×C₃₁	#102	1	2	yes	—
75	C₂×C₄×C₃₁	#102	1	2	yes	—
76	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₃₁	#121	1	2	yes	—
78	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
80	C₈×C₃₁	—	2	2	no	—
81	C₈×C₃₁	—	4	4	no	—
82	C₈×C₃₁	C₄×D₄	1	2	yes	—
83	C₈×C₃₁	#124	1	1	yes	characteristic
84	C₈×C₃₁	#127	1	1	yes	characteristic
85	C₂×#494	C₂×C₃₁	1	1	yes	characteristic
86	#2150	C₂×C₃₁	1	1	yes	characteristic
87	C₂×#362	C₂×C₃₁	1	4	yes	—
88	C₄×#108	C₂×C₃₁	1	1	yes	characteristic
89	C₂²×C₃₁	—	2	2	no	—
90	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
91	C₂²×C₃₁	#460	1	2	yes	—
92	C₄×C₃₁	—	2	2	no	—
93	C₄×C₃₁	—	8	4	no	—
94	C₄×C₃₁	#362	1	4	yes	complemented
95	C₄×C₃₁	#390	1	1	yes	characteristic
96	C₄×C₃₁	#418	1	1	yes	characteristic
97	C₄×C₃₁	#459	1	2	yes	—
98	C₂²×Q₁₆	—	2	1	no	—
99	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
100	C₂×C₄⋊Q₈	—	2	1	no	—
101	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
102	#494	—	2	2	no	—
103	#494	C₄×C₃₁	1	4	yes	—
104	C₂×#108	C₂²×C₃₁	1	2	yes	—
105	C₄×M₁₆	—	2	1	no	—
106	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
107	C₄⋊C₁₆	—	2	2	no	—
108	#362	—	2	8	no	—
109	#341	—	2	1	no	—
110	C₂×C₃₁	—	2	2	no	—
111	C₂×C₃₁	#1982	1	1	yes	characteristic
112	C₂×C₃₁	C₂×#362	1	2	yes	—
113	C₂²×Q₈	—	2	2	no	—
114	C₂×Q₁₆	—	2	4	no	—
115	C₂×Q₁₆	—	2	4	no	—
116	C₂×Q₁₆	—	2	2	no	—
117	C₂×Q₁₆	—	2	4	no	—
118	C₂×Q₁₆	—	4	4	no	—
119	C₂×Q₁₆	C₂×C₄×C₃₁	1	4	yes	—
120	C₂×M₁₆	—	2	2	no	—
121	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
122	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
123	C₄⋊Q₈	—	2	4	no	—
124	C₄⋊Q₈	—	4	1	no	—
125	C₄⋊Q₈	—	4	1	no	—
126	C₂×C₄⋊C₄	—	4	2	no	—
127	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
128	M₃₂	—	2	4	no	—
129	C₂×C₁₆	—	2	2	no	—
130	#108	—	2	2	no	—
131	#108	D₄×C₃₁	1	4	yes	—
132	#97	—	4	1	no	—
133	C₄×C₈	—	4	1	no	—
134	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
135	C₄×C₈	D₄×C₃₁	1	2	yes	—
136	C₃₁	#13256	1	1	yes	characteristic, direct product factor, Sylow
137	C₂×Q₈	—	2	4	no	—
138	C₂×Q₈	—	4	2	no	—
139	C₂×Q₈	—	4	4	no	—
140	C₂×Q₈	—	4	4	no	—
141	C₂²×C₄	—	4	2	no	—
142	C₂²×C₄	SD₁₆×C₃₁	1	2	yes	—
143	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
144	Q₁₆	—	2	8	no	—
145	Q₁₆	—	4	8	no	—
146	Q₁₆	—	4	2	no	—
147	Q₁₆	—	4	2	no	—
148	M₁₆	—	2	4	no	—
149	M₁₆	—	4	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	4	2	no	—
152	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
153	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
154	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
155	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
156	C₂×C₈	#41×C₃₁	1	2	yes	—
157	C₄⋊C₄	—	4	4	no	—
158	C₄⋊C₄	—	4	4	no	—
159	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
160	C₄²	D₈×C₃₁	1	1	yes	characteristic
161	C₄²	D₈×C₃₁	1	1	yes	characteristic
162	C₁₆	—	4	2	no	—
163	C₂³	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
164	Q₈	—	4	4	no	—
165	Q₈	—	4	4	no	—
166	Q₈	—	8	4	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	—	8	4	no	—
170	C₂×C₄	C₃₁×#102	1	2	yes	—
171	C₂×C₄	C₃₁×#102	1	2	yes	—
172	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
173	C₂×C₄	C₃₁×#121	1	2	yes	—
174	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
175	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
176	C₈	—	2	2	no	—
177	C₈	—	4	4	no	—
178	C₈	C₄×D₄×C₃₁	1	2	yes	—
179	C₈	C₃₁×#124	1	1	yes	characteristic
180	C₈	C₃₁×#127	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
183	C₂²	C₃₁×#460	1	2	yes	—
184	C₄	—	2	2	no	—
185	C₄	—	8	4	no	—
186	C₄	C₃₁×#362	1	4	yes	complemented
187	C₄	C₃₁×#390	1	1	yes	characteristic
188	C₄	C₃₁×#418	1	1	yes	characteristic
189	C₄	C₃₁×#459	1	2	yes	—
190	C₂	—	2	2	no	—
191	C₂	C₃₁×#1982	1	1	yes	characteristic
192	C₂	C₂×C₃₁×#362	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃₁×#494

C₂

yes

maximal, characteristic

C₃₁×#2150

C₂

yes

maximal, characteristic

C₂×C₃₁×#362

C₂

yes

maximal

C₄×C₃₁×#108

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₃₁

—

C₂²×Q₁₆×C₃₁

C₂²

yes

—

C₂×C₃₁×C₄⋊Q₈

—

C₂×C₃₁×M₃₂

C₂²

yes

—

C₃₁×#494

—

C₃₁×#494

C₄

yes

—

C₂×C₃₁×#108

C₂²

yes

—

C₄×M₁₆×C₃₁

—

C₂×C₄×C₈×C₃₁

C₂²

yes

characteristic

C₃₁×C₄⋊C₁₆

—

C₃₁×#362

—

C₃₁×#341

—

C₂²×Q₈×C₃₁

—

C₂×Q₁₆×C₃₁

—

C₂×Q₁₆×C₃₁

—

C₂×Q₁₆×C₃₁

—

C₂×Q₁₆×C₃₁

—

C₂×Q₁₆×C₃₁

—

C₂×Q₁₆×C₃₁

C₂×C₄

yes

—

C₂×M₁₆×C₃₁

—

C₂²×C₈×C₃₁

D₄

yes

characteristic

C₂²×C₈×C₃₁

C₂³

yes

characteristic

C₃₁×C₄⋊Q₈

—

C₃₁×C₄⋊Q₈

—

C₃₁×C₄⋊Q₈

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄²×C₃₁

D₄

yes

characteristic

C₃₁×M₃₂

—

C₂×C₁₆×C₃₁

—

C₃₁×#108

—

C₃₁×#108

D₄

yes

—

C₃₁×#97

—

C₄×C₈×C₃₁

—

C₄×C₈×C₃₁

C₂×C₄

yes

—

C₄×C₈×C₃₁

D₄

yes

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

SD₁₆

yes

—

C₂²×C₄×C₃₁

C₂×D₄

yes

characteristic

Q₁₆×C₃₁

—

Q₁₆×C₃₁

—

Q₁₆×C₃₁

—

Q₁₆×C₃₁

—

M₁₆×C₃₁

—

M₁₆×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂²⋊C₄

yes

—

C₂×C₈×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₃₁

C₂×D₄

yes

—

C₂×C₈×C₃₁

C₂×D₄

yes

characteristic

C₂×C₈×C₃₁

#41

yes

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

C₂²⋊C₄

yes

—

C₄²×C₃₁

D₈

yes

characteristic

C₄²×C₃₁

D₈

yes

characteristic

C₁₆×C₃₁

—

#13256

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

C₂×SD₁₆

yes

characteristic

Q₈×C₃₁

—

Q₈×C₃₁

—

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

#102

yes

—

C₂×C₄×C₃₁

#102

yes

—

C₂×C₄×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₃₁

#121

yes

—

C₂×C₄×C₃₁

C₂×D₈

yes

characteristic

C₂×C₄×C₃₁

AGL(1,ℤ/8)

yes

—

C₈×C₃₁

—

C₈×C₃₁

—

C₈×C₃₁

C₄×D₄

yes

—

C₈×C₃₁

yes

C₈×C₃₁

yes

C₂×C₃₁

yes

characteristic

#2150

C₂×C₃₁

yes

characteristic

C₂×#362

C₂×C₃₁

yes

—

C₄×#108

C₂×C₃₁

yes

characteristic

C₂²×C₃₁

—

C₂²×C₃₁

C₂×#102

yes

characteristic

C₂²×C₃₁

#460

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

—

C₂²×Q₁₆

—

C₂²×Q₁₆

C₂²×C₃₁

yes

—

100

C₂×C₄⋊Q₈

—

101

C₂×M₃₂

C₂²×C₃₁

yes

—

102

#494

—

103

#494

C₄×C₃₁

yes

—

104

C₂×#108

C₂²×C₃₁

yes

—

105

C₄×M₁₆

—

106

C₂×C₄×C₈

C₂²×C₃₁

yes

characteristic

107

C₄⋊C₁₆

—

108

#362

—

109

#341

—

110

C₂×C₃₁

—

111

C₂×C₃₁

#1982

yes

characteristic

112

C₂×C₃₁

C₂×#362

yes

—

113

C₂²×Q₈

—

114

C₂×Q₁₆

—

115

C₂×Q₁₆

—

116

C₂×Q₁₆

—

117

C₂×Q₁₆

—

118

C₂×Q₁₆

—

119

C₂×Q₁₆

C₂×C₄×C₃₁

yes

—

120

C₂×M₁₆

—

121

C₂²×C₈

D₄×C₃₁

yes

characteristic

122

C₂²×C₈

C₂³×C₃₁

yes

characteristic

123

C₄⋊Q₈

—

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄²

D₄×C₃₁

yes

characteristic

128

M₃₂

—

129

C₂×C₁₆

—

130

#108

—

131

#108

D₄×C₃₁

yes

—

132

#97

—

133

C₄×C₈

—

134

C₄×C₈

C₂×C₄×C₃₁

yes

—

135

C₄×C₈

D₄×C₃₁

yes

—

136

C₃₁

#13256

yes

characteristic, direct product factor, Sylow

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂²×C₄

—

142

C₂²×C₄

SD₁₆×C₃₁

yes

—

143

C₂²×C₄

C₂×D₄×C₃₁

yes

characteristic

144

Q₁₆

—

145

Q₁₆

—

146

Q₁₆

—

147

Q₁₆

—

148

M₁₆

—

149

M₁₆

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

C₂²⋊C₄×C₃₁

yes

—

153

C₂×C₈

C₂²×C₄×C₃₁

yes

characteristic

154

C₂×C₈

C₂×D₄×C₃₁

yes

—

155

C₂×C₈

C₂×D₄×C₃₁

yes

characteristic

156

C₂×C₈

#41×C₃₁

yes

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄²

C₂²⋊C₄×C₃₁

yes

—

160

C₄²

D₈×C₃₁

yes

characteristic

161

C₄²

D₈×C₃₁

yes

characteristic

162

C₁₆

—

163

C₂³

C₂×SD₁₆×C₃₁

yes

characteristic

164

Q₈

—

165

Q₈

—

166

Q₈

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

C₃₁×#102

yes

—

171

C₂×C₄

C₃₁×#102

yes

—

172

C₂×C₄

C₂×C₂²⋊C₄×C₃₁

yes

characteristic

173

C₂×C₄

C₃₁×#121

yes

—

174

C₂×C₄

C₂×D₈×C₃₁

yes

characteristic

175

C₂×C₄

C₃₁×AGL(1,ℤ/8)

yes

—

176

C₈

—

177

C₈

—

178

C₈

C₄×D₄×C₃₁

yes

—

179

C₈

C₃₁×#124

yes

characteristic

180

C₈

C₃₁×#127

yes

characteristic

181

C₂²

—

182

C₂²

C₂×C₃₁×#102

yes

characteristic

183

C₂²

C₃₁×#460

yes

—

184

C₄

—

185

C₄

—

186

C₄

C₃₁×#362

yes

complemented

187

C₄

C₃₁×#390

yes

characteristic

188

C₄

C₃₁×#418

yes

characteristic

189

C₄

C₃₁×#459

yes

—

190

C₂

—

191

C₂

C₃₁×#1982

yes

characteristic

192

C₂

C₂×C₃₁×#362

yes

—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
prime radical	C₂³×C₃₁	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₃₁

C₂×#102

commutator

C₂×C₈

C₂²×C₄×C₃₁

Frattini, cosocle

C₂²×C₈

C₂³×C₃₁

prime radical

C₂³×C₃₁

C₂×SD₁₆

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

N	H ≅ G/N	type
66. #13256	C₃₁	direct (N × H)
94. C₄×C₃₁	#362	semidirect
136. C₃₁	#13256	direct (N × H)
186. C₄	C₃₁×#362	semidirect

H ≅ G/N

type

66. #13256

C₃₁

direct (N × H)

94. C₄×C₃₁

#362

semidirect

136. C₃₁

#13256

direct (N × H)

186. C₄

C₃₁×#362

semidirect

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

subgroup	index
C₂×C₃₁×#494	2
C₃₁×#2150	2
C₂×C₃₁×#362	2
C₄×C₃₁×#108	2
#13256	31

subgroup

index

C₂×C₃₁×#494

C₃₁×#2150

C₂×C₃₁×#362

C₄×C₃₁×#108

#13256

Subgroups — C₃₁×#13256 #21470015

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#13256 #21470015

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13256 #21470015