Group #21468253 — subgroups

Subgroups — C₃₁×#11494 #21468253

Order 7936.

330 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 142 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×C₈×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2163	C₂	1	2	yes	maximal
3	C₃₁×#1322	C₂	1	4	yes	maximal
4	C₂×C₄×Q₈×C₃₁	C₄	1	1	yes	characteristic
5	C₈×Q₈×C₃₁	—	2	2	no	—
6	C₈×Q₈×C₃₁	—	2	2	no	—
7	C₂×C₃₁×C₄⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₃₁	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×C₄⋊C₁₆	—	2	2	no	—
12	C₃₁×C₂²⋊C₁₆	C₂²	1	4	yes	—
13	C₃₁×#346	—	2	2	no	—
14	C₂²×Q₈×C₃₁	C₈	1	1	yes	characteristic
15	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
18	C₄×Q₈×C₃₁	—	2	2	no	—
19	C₄×Q₈×C₃₁	—	2	2	no	—
20	C₂×C₄⋊C₄×C₃₁	C₈	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₃₁	C₈	1	2	yes	—
22	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₆×C₃₁	—	2	4	no	—
25	C₃₁×C₄⋊C₈	—	2	4	no	—
26	C₃₁×C₄⋊C₈	—	2	2	no	—
27	C₄×C₈×C₃₁	—	2	2	no	—
28	C₄×C₈×C₃₁	—	2	2	no	—
29	C₄×C₈×C₃₁	D₄	1	2	yes	—
30	C₄×C₈×C₃₁	D₄	1	2	yes	—
31	C₂×Q₈×C₃₁	—	2	2	no	—
32	C₂×Q₈×C₃₁	—	2	4	no	—
33	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
34	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
35	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
36	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₃₁	—	2	2	no	—
39	C₂×C₈×C₃₁	—	2	2	no	—
40	C₂×C₈×C₃₁	—	4	1	no	—
41	C₂×C₈×C₃₁	—	4	2	no	—
42	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
45	C₄⋊C₄×C₃₁	—	2	4	no	—
46	C₄⋊C₄×C₃₁	—	2	2	no	—
47	C₄²×C₃₁	—	2	2	no	—
48	C₄²×C₃₁	—	2	2	no	—
49	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
50	C₄²×C₃₁	M₁₆	1	2	yes	—
51	C₁₆×C₃₁	—	4	4	no	—
52	#11494	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂³×C₃₁	C₂²×C₈	1	1	yes	characteristic
54	Q₈×C₃₁	—	2	8	no	—
55	C₂×C₄×C₃₁	—	2	2	no	—
56	C₂×C₄×C₃₁	—	2	2	no	—
57	C₂×C₄×C₃₁	—	2	2	no	—
58	C₂×C₄×C₃₁	—	2	1	no	—
59	C₂×C₄×C₃₁	—	2	1	no	—
60	C₂×C₄×C₃₁	—	2	2	no	—
61	C₂×C₄×C₃₁	—	4	1	no	—
62	C₂×C₄×C₃₁	—	4	2	no	—
63	C₂×C₄×C₃₁	C₂²⋊C₈	1	2	yes	—
64	C₂×C₄×C₃₁	C₂²⋊C₈	1	2	yes	—
65	C₂×C₄×C₃₁	M₃₂	1	4	yes	—
66	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
68	C₈×C₃₁	—	2	2	no	—
69	C₈×C₃₁	—	2	1	no	—
70	C₈×C₃₁	—	4	2	no	—
71	C₈×C₃₁	#101	1	2	yes	—
72	C₂×C₈×Q₈	C₂×C₃₁	1	1	yes	characteristic
73	#2163	C₂×C₃₁	1	2	yes	—
74	#1322	C₂×C₃₁	1	4	yes	—
75	C₂²×C₃₁	—	2	1	no	—
76	C₂²×C₃₁	—	2	1	no	—
77	C₂²×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
78	C₂²×C₃₁	C₂×M₃₂	1	2	yes	—
79	C₄×C₃₁	—	2	2	no	—
80	C₄×C₃₁	—	2	4	no	—
81	C₄×C₃₁	—	2	2	no	—
82	C₄×C₃₁	—	2	1	no	—
83	C₄×C₃₁	—	4	2	no	—
84	C₄×C₃₁	#410	1	1	yes	characteristic
85	C₄×C₃₁	C₂×#101	1	1	yes	characteristic
86	C₂×C₄×Q₈	C₄×C₃₁	1	1	yes	characteristic
87	C₈×Q₈	—	2	2	no	—
88	C₈×Q₈	—	2	2	no	—
89	C₂×C₄⋊C₈	C₄×C₃₁	1	1	yes	characteristic
90	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
91	C₂×C₄×C₈	C₄×C₃₁	1	2	yes	—
92	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
93	C₄⋊C₁₆	—	2	2	no	—
94	C₂²⋊C₁₆	C₂²×C₃₁	1	4	yes	—
95	#346	—	2	2	no	—
96	C₂×C₃₁	—	2	2	no	—
97	C₂×C₃₁	#1461	1	1	yes	characteristic
98	C₂×C₃₁	#2112	1	1	yes	characteristic
99	C₂×C₃₁	#2115	1	1	yes	characteristic
100	C₂²×Q₈	C₈×C₃₁	1	1	yes	characteristic
101	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
102	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
103	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
104	C₄×Q₈	—	2	2	no	—
105	C₄×Q₈	—	2	2	no	—
106	C₂×C₄⋊C₄	C₈×C₃₁	1	1	yes	characteristic
107	C₂×C₄⋊C₄	C₈×C₃₁	1	2	yes	—
108	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
109	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
110	C₂×C₁₆	—	2	4	no	—
111	C₄⋊C₈	—	2	4	no	—
112	C₄⋊C₈	—	2	2	no	—
113	C₄×C₈	—	2	2	no	—
114	C₄×C₈	—	2	2	no	—
115	C₄×C₈	D₄×C₃₁	1	2	yes	—
116	C₄×C₈	D₄×C₃₁	1	2	yes	—
117	C₃₁	#11494	1	1	yes	characteristic, direct product factor, Sylow
118	C₂×Q₈	—	2	2	no	—
119	C₂×Q₈	—	2	4	no	—
120	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
121	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
122	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
123	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
124	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	4	1	no	—
128	C₂×C₈	—	4	2	no	—
129	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
130	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
131	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
132	C₄⋊C₄	—	2	4	no	—
133	C₄⋊C₄	—	2	2	no	—
134	C₄²	—	2	2	no	—
135	C₄²	—	2	2	no	—
136	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
137	C₄²	M₁₆×C₃₁	1	2	yes	—
138	C₁₆	—	4	4	no	—
139	C₂³	C₂²×C₈×C₃₁	1	1	yes	characteristic
140	Q₈	—	2	8	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	C₃₁×C₂²⋊C₈	1	2	yes	—
150	C₂×C₄	C₃₁×C₂²⋊C₈	1	2	yes	—
151	C₂×C₄	C₃₁×M₃₂	1	4	yes	—
152	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
153	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
154	C₈	—	2	2	no	—
155	C₈	—	2	1	no	—
156	C₈	—	4	2	no	—
157	C₈	C₃₁×#101	1	2	yes	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
161	C₂²	C₂×C₃₁×M₃₂	1	2	yes	—
162	C₄	—	2	2	no	—
163	C₄	—	2	4	no	—
164	C₄	—	2	2	no	—
165	C₄	—	2	1	no	—
166	C₄	—	4	2	no	—
167	C₄	C₃₁×#410	1	1	yes	characteristic
168	C₄	C₂×C₃₁×#101	1	1	yes	characteristic
169	C₂	—	2	2	no	—
170	C₂	C₃₁×#1461	1	1	yes	characteristic
171	C₂	C₃₁×#2112	1	1	yes	characteristic
172	C₂	C₃₁×#2115	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Q₈×C₃₁

C₂

yes

maximal, characteristic

C₃₁×#2163

C₂

yes

maximal

C₃₁×#1322

C₂

yes

maximal

C₂×C₄×Q₈×C₃₁

C₄

yes

characteristic

C₈×Q₈×C₃₁

—

C₈×Q₈×C₃₁

—

C₂×C₃₁×C₄⋊C₈

C₄

yes

characteristic

C₂×C₃₁×C₄⋊C₈

C₂²

yes

—

C₂×C₄×C₈×C₃₁

C₄

yes

—

C₂×C₄×C₈×C₃₁

C₂²

yes

characteristic

C₃₁×C₄⋊C₁₆

—

C₃₁×C₂²⋊C₁₆

C₂²

yes

—

C₃₁×#346

—

C₂²×Q₈×C₃₁

C₈

yes

characteristic

C₂²×C₈×C₃₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₃₁

C₂×C₄

yes

—

C₂²×C₈×C₃₁

C₂³

yes

characteristic

C₄×Q₈×C₃₁

—

C₄×Q₈×C₃₁

—

C₂×C₄⋊C₄×C₃₁

C₈

yes

characteristic

C₂×C₄⋊C₄×C₃₁

C₈

yes

—

C₂×C₄²×C₃₁

C₂×C₄

yes

—

C₂×C₄²×C₃₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₃₁

—

C₃₁×C₄⋊C₈

—

C₃₁×C₄⋊C₈

—

C₄×C₈×C₃₁

—

C₄×C₈×C₃₁

—

C₄×C₈×C₃₁

D₄

yes

—

C₄×C₈×C₃₁

D₄

yes

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂²×C₄×C₃₁

C₂×C₈

yes

characteristic

C₂²×C₄×C₃₁

C₂×C₈

yes

—

C₂²×C₄×C₃₁

C₂×C₈

yes

—

C₂²×C₄×C₃₁

C₂×C₈

yes

characteristic

C₂²×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₃₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₃₁

C₂×D₄

yes

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

C₂²⋊C₄

yes

—

C₄²×C₃₁

M₁₆

yes

—

C₁₆×C₃₁

—

#11494

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

C₂²×C₈

yes

characteristic

Q₈×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×C₄×C₃₁

C₂²⋊C₈

yes

—

C₂×C₄×C₃₁

C₂²⋊C₈

yes

—

C₂×C₄×C₃₁

M₃₂

yes

—

C₂×C₄×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×M₁₆

yes

characteristic

C₈×C₃₁

—

C₈×C₃₁

—

C₈×C₃₁

—

C₈×C₃₁

#101

yes

—

C₂×C₈×Q₈

C₂×C₃₁

yes

characteristic

#2163

C₂×C₃₁

yes

—

#1322

C₂×C₃₁

yes

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₃₁

C₂×M₃₂

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

yes

C₄×C₃₁

yes

C₂×C₄×Q₈

C₄×C₃₁

yes

characteristic

C₈×Q₈

—

C₈×Q₈

—

C₂×C₄⋊C₈

C₄×C₃₁

yes

characteristic

C₂×C₄⋊C₈

C₂²×C₃₁

yes

—

C₂×C₄×C₈

C₄×C₃₁

yes

—

C₂×C₄×C₈

C₂²×C₃₁

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

C₂²×C₃₁

yes

—

#346

—

C₂×C₃₁

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

100

C₂²×Q₈

C₈×C₃₁

yes

characteristic

101

C₂²×C₈

C₂×C₄×C₃₁

yes

characteristic

102

C₂²×C₈

C₂×C₄×C₃₁

yes

—

103

C₂²×C₈

C₂³×C₃₁

yes

characteristic

104

C₄×Q₈

—

105

C₄×Q₈

—

106

C₂×C₄⋊C₄

C₈×C₃₁

yes

characteristic

107

C₂×C₄⋊C₄

C₈×C₃₁

yes

—

108

C₂×C₄²

C₂×C₄×C₃₁

yes

—

109

C₂×C₄²

C₂×C₄×C₃₁

yes

characteristic

110

C₂×C₁₆

—

111

C₄⋊C₈

—

112

C₄⋊C₈

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₄×C₈

D₄×C₃₁

yes

—

116

C₄×C₈

D₄×C₃₁

yes

—

117

C₃₁

#11494

yes

characteristic, direct product factor, Sylow

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂²×C₄

C₂×C₈×C₃₁

yes

characteristic

121

C₂²×C₄

C₂×C₈×C₃₁

yes

—

122

C₂²×C₄

C₂×C₈×C₃₁

yes

—

123

C₂²×C₄

C₂×C₈×C₃₁

yes

characteristic

124

C₂²×C₄

C₂²×C₄×C₃₁

yes

characteristic

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

C₂²⋊C₄×C₃₁

yes

characteristic

130

C₂×C₈

C₂²⋊C₄×C₃₁

yes

characteristic

131

C₂×C₈

C₂×D₄×C₃₁

yes

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄²

—

135

C₄²

—

136

C₄²

C₂²⋊C₄×C₃₁

yes

—

137

C₄²

M₁₆×C₃₁

yes

—

138

C₁₆

—

139

C₂³

C₂²×C₈×C₃₁

yes

characteristic

140

Q₈

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

C₃₁×C₂²⋊C₈

yes

—

150

C₂×C₄

C₃₁×C₂²⋊C₈

yes

—

151

C₂×C₄

C₃₁×M₃₂

yes

—

152

C₂×C₄

C₂×C₂²⋊C₄×C₃₁

yes

characteristic

153

C₂×C₄

C₂×M₁₆×C₃₁

yes

characteristic

154

C₈

—

155

C₈

—

156

C₈

—

157

C₈

C₃₁×#101

yes

—

158

C₂²

—

159

C₂²

—

160

C₂²

C₂×C₃₁×C₂²⋊C₈

yes

characteristic

161

C₂²

C₂×C₃₁×M₃₂

yes

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

C₃₁×#410

yes

characteristic

168

C₄

C₂×C₃₁×#101

yes

characteristic

169

C₂

—

170

C₂

C₃₁×#1461

yes

characteristic

171

C₂

C₃₁×#2112

yes

characteristic

172

C₂

C₃₁×#2115

yes

characteristic

	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	C₂³×C₃₁	C₂²×C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄×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

C₂³×C₃₁

C₂²×C₈

LCS

G ▷ C₂³ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

N	H ≅ G/N	type
52. #11494	C₃₁	direct (N × H)
117. C₃₁	#11494	direct (N × H)

H ≅ G/N

type

52. #11494

C₃₁

direct (N × H)

117. C₃₁

#11494

direct (N × H)

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

subgroup	index
C₂×C₈×Q₈×C₃₁	2
C₃₁×#2163	2
C₃₁×#1322	2
#11494	31

subgroup

index

C₂×C₈×Q₈×C₃₁

C₃₁×#2163

C₃₁×#1322

#11494

Subgroups — C₃₁×#11494 #21468253

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#11494 #21468253

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11494 #21468253