Group #21465397 — subgroups

Subgroups — C₃₁×#8638 #21465397

Order 7936.

404 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 138 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×#2931	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1567	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1560	C₂	1	2	yes	maximal
4	C₃₁×#1309	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×#131	C₄	1	2	yes	—
6	C₃₁×#532	C₂²	1	1	yes	characteristic
7	C₃₁×#436	—	2	2	no	—
8	C₈×D₄×C₃₁	—	2	2	no	—
9	C₃₁×#423	C₂²	1	4	yes	—
10	C₃₁×#423	C₂²	1	1	yes	characteristic
11	C₃₁×#408	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#335	—	2	2	no	—
14	C₃₁×#335	—	2	2	no	—
15	C₃₁×#333	—	2	2	no	—
16	C₃₁×#331	—	2	2	no	—
17	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
18	C₃₁×#131	—	4	4	no	—
19	C₂×M₁₆×C₃₁	—	2	2	no	—
20	C₂×M₁₆×C₃₁	—	2	4	no	—
21	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₃₁	—	2	2	no	—
23	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
25	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
26	C₃₁×#127	D₄	1	1	yes	characteristic
27	C₃₁×#124	Q₈	1	2	yes	—
28	C₃₁×#121	—	2	2	no	—
29	C₄×D₄×C₃₁	—	2	2	no	—
30	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
31	C₃₁×C₄⋊C₈	—	2	2	no	—
32	C₃₁×C₄⋊C₈	—	2	4	no	—
33	C₃₁×C₄⋊C₈	—	2	4	no	—
34	C₃₁×C₂²⋊C₈	—	2	4	no	—
35	C₄×C₈×C₃₁	D₄	1	2	yes	—
36	C₄×C₈×C₃₁	D₄	1	2	yes	—
37	#41×C₃₁	—	4	2	no	—
38	#41×C₃₁	—	4	2	no	—
39	C₂×Q₈×C₃₁	C₄⋊C₄	1	2	yes	—
40	C₂×D₄×C₃₁	—	2	2	no	—
41	C₂×D₄×C₃₁	—	2	2	no	—
42	C₂×D₄×C₃₁	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₃₁	—	2	2	no	—
44	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₃₁	#41	1	2	yes	—
46	M₁₆×C₃₁	—	4	4	no	—
47	M₁₆×C₃₁	—	4	2	no	—
48	M₁₆×C₃₁	—	8	4	no	—
49	C₂×C₈×C₃₁	—	2	2	no	—
50	C₂×C₈×C₃₁	—	2	2	no	—
51	C₂×C₈×C₃₁	—	4	1	no	—
52	C₂×C₈×C₃₁	—	4	4	no	—
53	C₂×C₈×C₃₁	—	4	4	no	—
54	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
55	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
57	C₄⋊C₄×C₃₁	—	2	2	no	—
58	C₂²⋊C₄×C₃₁	—	2	4	no	—
59	C₄²×C₃₁	C₂×D₄	1	2	yes	—
60	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
62	#8638	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂³×C₃₁	—	2	2	no	—
64	C₂³×C₃₁	#117	1	1	yes	characteristic
65	Q₈×C₃₁	—	4	2	no	—
66	D₄×C₃₁	—	4	2	no	—
67	D₄×C₃₁	—	4	4	no	—
68	D₄×C₃₁	—	4	2	no	—
69	D₄×C₃₁	—	4	2	no	—
70	C₂×C₄×C₃₁	—	2	1	no	—
71	C₂×C₄×C₃₁	—	2	1	no	—
72	C₂×C₄×C₃₁	—	2	2	no	—
73	C₂×C₄×C₃₁	—	2	2	no	—
74	C₂×C₄×C₃₁	—	2	2	no	—
75	C₂×C₄×C₃₁	—	4	2	no	—
76	C₂×C₄×C₃₁	—	4	2	no	—
77	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₃₁	#121	1	2	yes	—
80	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
81	C₈×C₃₁	—	2	1	no	—
82	C₈×C₃₁	—	2	2	no	—
83	C₈×C₃₁	—	4	2	no	—
84	C₈×C₃₁	—	8	4	no	—
85	C₈×C₃₁	#100	1	2	yes	—
86	#2931	C₂×C₃₁	1	1	yes	characteristic
87	#1567	C₂×C₃₁	1	2	yes	complemented
88	#1560	C₂×C₃₁	1	2	yes	—
89	#1309	C₂×C₃₁	1	2	yes	complemented
90	C₂²×C₃₁	—	2	1	no	—
91	C₂²×C₃₁	—	2	2	no	—
92	C₂²×C₃₁	—	4	2	no	—
93	C₂²×C₃₁	—	4	4	no	—
94	C₂²×C₃₁	#380	1	1	yes	characteristic
95	C₄×C₃₁	—	2	1	no	—
96	C₄×C₃₁	—	2	2	no	—
97	C₄×C₃₁	—	4	2	no	—
98	C₄×C₃₁	C₂×#100	1	1	yes	characteristic
99	C₄×C₃₁	#413	1	1	yes	characteristic
100	C₄×C₃₁	#472	1	2	yes	—
101	C₄×C₃₁	#473	1	2	yes	—
102	C₂×#131	C₄×C₃₁	1	2	yes	—
103	#532	C₂²×C₃₁	1	1	yes	characteristic
104	#436	—	2	2	no	—
105	C₈×D₄	—	2	2	no	—
106	#423	C₂²×C₃₁	1	4	yes	—
107	#423	C₂²×C₃₁	1	1	yes	characteristic
108	#408	C₄×C₃₁	1	2	yes	—
109	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
110	#335	—	2	2	no	—
111	#335	—	2	2	no	—
112	#333	—	2	2	no	—
113	#331	—	2	2	no	—
114	C₂×C₃₁	—	4	1	no	—
115	C₂×C₃₁	—	8	2	no	—
116	C₂×C₃₁	#1800	1	1	yes	characteristic
117	C₂×C₃₁	#1816	1	2	yes	—
118	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
119	#131	—	4	4	no	—
120	C₂×M₁₆	—	2	2	no	—
121	C₂×M₁₆	—	2	4	no	—
122	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
123	C₂²×C₈	—	2	2	no	—
124	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
125	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
126	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
127	#127	D₄×C₃₁	1	1	yes	characteristic
128	#124	Q₈×C₃₁	1	2	yes	—
129	#121	—	2	2	no	—
130	C₄×D₄	—	2	2	no	—
131	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
132	C₄⋊C₈	—	2	2	no	—
133	C₄⋊C₈	—	2	4	no	—
134	C₄⋊C₈	—	2	4	no	—
135	C₂²⋊C₈	—	2	4	no	—
136	C₄×C₈	D₄×C₃₁	1	2	yes	—
137	C₄×C₈	D₄×C₃₁	1	2	yes	—
138	C₃₁	#8638	1	1	yes	characteristic, direct product factor, Sylow
139	#41	—	4	2	no	—
140	#41	—	4	2	no	—
141	C₂×Q₈	C₄⋊C₄×C₃₁	1	2	yes	—
142	C₂×D₄	—	2	2	no	—
143	C₂×D₄	—	2	2	no	—
144	C₂×D₄	C₄⋊C₄×C₃₁	1	2	yes	—
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
147	C₂²×C₄	#41×C₃₁	1	2	yes	—
148	M₁₆	—	4	4	no	—
149	M₁₆	—	4	2	no	—
150	M₁₆	—	8	4	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	4	1	no	—
154	C₂×C₈	—	4	4	no	—
155	C₂×C₈	—	4	4	no	—
156	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
157	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
158	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
159	C₄⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	—	2	4	no	—
161	C₄²	C₂×D₄×C₃₁	1	2	yes	—
162	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
163	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
164	C₂³	—	2	2	no	—
165	C₂³	C₃₁×#117	1	1	yes	characteristic
166	Q₈	—	4	2	no	—
167	D₄	—	4	2	no	—
168	D₄	—	4	4	no	—
169	D₄	—	4	2	no	—
170	D₄	—	4	2	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
180	C₂×C₄	C₃₁×#121	1	2	yes	—
181	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
182	C₈	—	2	1	no	—
183	C₈	—	2	2	no	—
184	C₈	—	4	2	no	—
185	C₈	—	8	4	no	—
186	C₈	C₃₁×#100	1	2	yes	—
187	C₂²	—	2	1	no	—
188	C₂²	—	2	2	no	—
189	C₂²	—	4	2	no	—
190	C₂²	—	4	4	no	—
191	C₂²	C₃₁×#380	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	2	2	no	—
194	C₄	—	4	2	no	—
195	C₄	C₂×C₃₁×#100	1	1	yes	characteristic
196	C₄	C₃₁×#413	1	1	yes	characteristic
197	C₄	C₃₁×#472	1	2	yes	—
198	C₄	C₃₁×#473	1	2	yes	—
199	C₂	—	4	1	no	—
200	C₂	—	8	2	no	—
201	C₂	C₃₁×#1800	1	1	yes	characteristic
202	C₂	C₃₁×#1816	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃₁×#2931

C₂

yes

maximal, characteristic

C₃₁×#1567

C₂

yes

maximal, complemented

C₃₁×#1560

C₂

yes

maximal

C₃₁×#1309

C₂

yes

maximal, complemented

C₂×C₃₁×#131

C₄

yes

—

C₃₁×#532

C₂²

yes

characteristic

C₃₁×#436

—

C₈×D₄×C₃₁

—

C₃₁×#423

C₂²

yes

—

C₃₁×#423

C₂²

yes

characteristic

C₃₁×#408

C₄

yes

—

C₂×C₄×C₈×C₃₁

C₂²

yes

characteristic

C₃₁×#335

—

C₃₁×#335

—

C₃₁×#333

—

C₃₁×#331

—

C₂×#41×C₃₁

C₂×C₄

yes

—

C₃₁×#131

—

C₂×M₁₆×C₃₁

—

C₂×M₁₆×C₃₁

—

C₂×M₁₆×C₃₁

C₂×C₄

yes

—

C₂²×C₈×C₃₁

—

C₂²×C₈×C₃₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₃₁

C₂×C₄

yes

characteristic

C₃₁×C₄⋊Q₈

D₄

yes

characteristic

C₃₁×#127

D₄

yes

characteristic

C₃₁×#124

Q₈

yes

—

C₃₁×#121

—

C₄×D₄×C₃₁

—

C₂×C₄²×C₃₁

C₂³

yes

characteristic

C₃₁×C₄⋊C₈

—

C₃₁×C₄⋊C₈

—

C₃₁×C₄⋊C₈

—

C₃₁×C₂²⋊C₈

—

C₄×C₈×C₃₁

D₄

yes

—

C₄×C₈×C₃₁

D₄

yes

—

#41×C₃₁

—

#41×C₃₁

—

C₂×Q₈×C₃₁

C₄⋊C₄

yes

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

C₄⋊C₄

yes

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₃₁

#41

yes

—

M₁₆×C₃₁

—

M₁₆×C₃₁

—

M₁₆×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂²⋊C₄

yes

—

C₂×C₈×C₃₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₃₁

C₂²⋊C₄

yes

characteristic

C₄⋊C₄×C₃₁

—

C₂²⋊C₄×C₃₁

—

C₄²×C₃₁

C₂×D₄

yes

—

C₄²×C₃₁

C₂×D₄

yes

characteristic

C₄²×C₃₁

C₂×Q₈

yes

characteristic

#8638

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

—

C₂³×C₃₁

#117

yes

characteristic

Q₈×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

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

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₃₁

#121

yes

—

C₂×C₄×C₃₁

C₂²⋊Q₈

yes

—

C₈×C₃₁

—

C₈×C₃₁

—

C₈×C₃₁

—

C₈×C₃₁

—

C₈×C₃₁

#100

yes

—

#2931

C₂×C₃₁

yes

characteristic

#1567

C₂×C₃₁

yes

complemented

#1560

C₂×C₃₁

yes

—

#1309

C₂×C₃₁

yes

complemented

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

#380

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

yes

C₄×C₃₁

yes

100

C₄×C₃₁

#472

yes

—

101

C₄×C₃₁

#473

yes

—

102

C₂×#131

C₄×C₃₁

yes

—

103

#532

C₂²×C₃₁

yes

characteristic

104

#436

—

105

C₈×D₄

—

106

#423

C₂²×C₃₁

yes

—

107

#423

C₂²×C₃₁

yes

characteristic

108

#408

C₄×C₃₁

yes

—

109

C₂×C₄×C₈

C₂²×C₃₁

yes

characteristic

110

#335

—

111

#335

—

112

#333

—

113

#331

—

114

C₂×C₃₁

—

115

C₂×C₃₁

—

116

C₂×C₃₁

#1800

yes

characteristic

117

C₂×C₃₁

#1816

yes

—

118

C₂×#41

C₂×C₄×C₃₁

yes

—

119

#131

—

120

C₂×M₁₆

—

121

C₂×M₁₆

—

122

C₂×M₁₆

C₂×C₄×C₃₁

yes

—

123

C₂²×C₈

—

124

C₂²×C₈

C₂×C₄×C₃₁

yes

characteristic

125

C₂²×C₈

C₂×C₄×C₃₁

yes

characteristic

126

C₄⋊Q₈

D₄×C₃₁

yes

characteristic

127

#127

D₄×C₃₁

yes

characteristic

128

#124

Q₈×C₃₁

yes

—

129

#121

—

130

C₄×D₄

—

131

C₂×C₄²

C₂³×C₃₁

yes

characteristic

132

C₄⋊C₈

—

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

C₂²⋊C₈

—

136

C₄×C₈

D₄×C₃₁

yes

—

137

C₄×C₈

D₄×C₃₁

yes

—

138

C₃₁

#8638

yes

characteristic, direct product factor, Sylow

139

#41

—

140

#41

—

141

C₂×Q₈

C₄⋊C₄×C₃₁

yes

—

142

C₂×D₄

—

143

C₂×D₄

—

144

C₂×D₄

C₄⋊C₄×C₃₁

yes

—

145

C₂²×C₄

—

146

C₂²×C₄

C₂²×C₄×C₃₁

yes

characteristic

147

C₂²×C₄

#41×C₃₁

yes

—

148

M₁₆

—

149

M₁₆

—

150

M₁₆

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

C₂²⋊C₄×C₃₁

yes

—

157

C₂×C₈

C₂²⋊C₄×C₃₁

yes

characteristic

158

C₂×C₈

C₂²⋊C₄×C₃₁

yes

characteristic

159

C₄⋊C₄

—

160

C₂²⋊C₄

—

161

C₄²

C₂×D₄×C₃₁

yes

—

162

C₄²

C₂×D₄×C₃₁

yes

characteristic

163

C₄²

C₂×Q₈×C₃₁

yes

characteristic

164

C₂³

—

165

C₂³

C₃₁×#117

yes

characteristic

166

Q₈

—

167

D₄

—

168

D₄

—

169

D₄

—

170

D₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

C₂×C₂²⋊C₄×C₃₁

yes

characteristic

179

C₂×C₄

C₂×C₄⋊C₄×C₃₁

yes

characteristic

180

C₂×C₄

C₃₁×#121

yes

—

181

C₂×C₄

C₃₁×C₂²⋊Q₈

yes

—

182

C₈

—

183

C₈

—

184

C₈

—

185

C₈

—

186

C₈

C₃₁×#100

yes

—

187

C₂²

—

188

C₂²

—

189

C₂²

—

190

C₂²

—

191

C₂²

C₃₁×#380

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

C₂×C₃₁×#100

yes

characteristic

196

C₄

C₃₁×#413

yes

characteristic

197

C₄

C₃₁×#472

yes

—

198

C₄

C₃₁×#473

yes

—

199

C₂

—

200

C₂

—

201

C₂

C₃₁×#1800

yes

characteristic

202

C₂

C₃₁×#1816

yes

—

	subgroup	G/H
center, socle	C₂²×C₃₁	#380
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₃₁×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₃₁

#380

commutator

C₂²×C₄

C₂²×C₄×C₃₁

Frattini, cosocle

C₂×C₄²

C₂³×C₃₁

prime radical

C₃₁×#532

C₂²

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

N	H ≅ G/N	type
2. C₃₁×#1567	C₂	semidirect
4. C₃₁×#1309	C₂	semidirect
62. #8638	C₃₁	direct (N × H)
87. #1567	C₂×C₃₁	semidirect
89. #1309	C₂×C₃₁	semidirect
138. C₃₁	#8638	direct (N × H)

H ≅ G/N

type

2. C₃₁×#1567

C₂

semidirect

4. C₃₁×#1309

C₂

semidirect

62. #8638

C₃₁

direct (N × H)

87. #1567

C₂×C₃₁

semidirect

89. #1309

C₂×C₃₁

semidirect

138. C₃₁

#8638

direct (N × H)

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

subgroup	index
C₃₁×#2931	2
C₃₁×#1567	2
C₃₁×#1560	2
C₃₁×#1309	2
#8638	31

subgroup

index

C₃₁×#2931

C₃₁×#1567

C₃₁×#1560

C₃₁×#1309

#8638

Subgroups — C₃₁×#8638 #21465397

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#8638 #21465397

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8638 #21465397