Group #21469604 — subgroups

Subgroups — C₃₁×#12845 #21469604

Order 7936.

454 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 182 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×#405	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×C₂²⋊C₁₆	C₂	1	2	yes	maximal
3	C₃₁×#1374	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₃₁	C₄	1	1	yes	characteristic
5	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
6	C₂×C₃₁×#117	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₃₁	C₂²	1	4	yes	—
8	C₃₁×#405	—	2	4	no	—
9	C₂×C₃₁×#97	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
11	C₃₁×C₂²⋊C₁₆	—	2	4	no	—
12	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
13	C₂×M₁₆×C₃₁	—	2	2	no	—
14	C₂×M₁₆×C₃₁	—	2	4	no	—
15	C₂²×C₈×C₃₁	—	2	4	no	—
16	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₃₁	D₄	1	4	yes	—
19	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
20	C₃₁×#117	—	2	2	no	—
21	C₃₁×#117	—	2	2	no	—
22	C₂×C₄⋊C₄×C₃₁	C₈	1	2	yes	—
23	C₂×C₂²⋊C₄×C₃₁	C₈	1	2	yes	—
24	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
25	C₂×C₁₆×C₃₁	—	2	4	no	—
26	C₂×C₁₆×C₃₁	—	2	8	no	—
27	C₃₁×#97	—	2	4	no	—
28	C₄×C₈×C₃₁	—	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	C₂²⋊C₄	1	2	yes	—
33	C₂²×C₄×C₃₁	C₂×C₈	1	4	yes	—
34	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄×C₃₁	M₁₆	1	2	yes	—
36	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
37	M₁₆×C₃₁	—	2	8	no	—
38	C₂×C₈×C₃₁	—	2	2	no	—
39	C₂×C₈×C₃₁	—	2	2	no	—
40	C₂×C₈×C₃₁	—	2	2	no	—
41	C₂×C₈×C₃₁	—	2	8	no	—
42	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
43	C₂×C₈×C₃₁	M₁₆	1	4	yes	—
44	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
45	C₂×C₈×C₃₁	#41	1	4	yes	—
46	C₄⋊C₄×C₃₁	—	2	4	no	—
47	C₂²⋊C₄×C₃₁	—	2	4	no	—
48	C₄²×C₃₁	—	2	2	no	—
49	C₄²×C₃₁	—	2	2	no	—
50	C₁₆×C₃₁	—	4	4	no	—
51	#12845	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂³×C₃₁	—	2	2	no	—
53	C₂³×C₃₁	—	2	4	no	—
54	C₂³×C₃₁	C₂²⋊C₈	1	2	yes	—
55	C₂³×C₃₁	C₂²×C₈	1	1	yes	characteristic
56	C₂×C₄×C₃₁	—	2	2	no	—
57	C₂×C₄×C₃₁	—	2	2	no	—
58	C₂×C₄×C₃₁	—	2	2	no	—
59	C₂×C₄×C₃₁	—	2	4	no	—
60	C₂×C₄×C₃₁	—	2	4	no	—
61	C₂×C₄×C₃₁	—	2	8	no	—
62	C₂×C₄×C₃₁	C₂²⋊C₈	1	2	yes	—
63	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₃₁	#117	1	2	yes	—
65	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
66	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
67	C₈×C₃₁	—	2	2	no	—
68	C₈×C₃₁	—	2	4	no	—
69	C₈×C₃₁	#123	1	4	yes	—
70	C₂×#405	C₂×C₃₁	1	1	yes	characteristic
71	C₂×C₂²⋊C₁₆	C₂×C₃₁	1	2	yes	—
72	#1374	C₂×C₃₁	1	4	yes	complemented
73	C₂²×C₃₁	—	2	2	no	—
74	C₂²×C₃₁	—	2	4	no	—
75	C₂²×C₃₁	—	4	4	no	—
76	C₂²×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
77	C₂²×C₃₁	#431	1	2	yes	—
78	C₂²×C₃₁	#504	1	4	yes	—
79	C₄×C₃₁	—	2	2	no	—
80	C₄×C₃₁	—	4	4	no	—
81	C₄×C₃₁	#381	1	1	yes	characteristic
82	C₄×C₃₁	#407	1	1	yes	characteristic
83	C₄×C₃₁	#432	1	2	yes	—
84	C₂²×M₁₆	C₄×C₃₁	1	1	yes	characteristic
85	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
86	C₂×#117	C₄×C₃₁	1	1	yes	characteristic
87	C₂²×C₁₆	C₂²×C₃₁	1	4	yes	—
88	#405	—	2	4	no	—
89	C₂×#97	C₄×C₃₁	1	2	yes	—
90	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
91	C₂²⋊C₁₆	—	2	4	no	—
92	C₂×C₃₁	—	4	2	no	—
93	C₂×C₃₁	#1817	1	1	yes	characteristic
94	C₂×C₃₁	#2113	1	2	yes	—
95	C₂×C₃₁	#2164	1	4	yes	—
96	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
97	C₂×M₁₆	—	2	2	no	—
98	C₂×M₁₆	—	2	4	no	—
99	C₂²×C₈	—	2	4	no	—
100	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
101	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
102	C₂²×C₈	D₄×C₃₁	1	4	yes	—
103	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
104	#117	—	2	2	no	—
105	#117	—	2	2	no	—
106	C₂×C₄⋊C₄	C₈×C₃₁	1	2	yes	—
107	C₂×C₂²⋊C₄	C₈×C₃₁	1	2	yes	—
108	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
109	C₂×C₁₆	—	2	4	no	—
110	C₂×C₁₆	—	2	8	no	—
111	#97	—	2	4	no	—
112	C₄×C₈	—	2	4	no	—
113	C₃₁	#12845	1	1	yes	characteristic, direct product factor, Sylow
114	C₂⁴	C₂×C₈×C₃₁	1	1	yes	characteristic
115	C₂²×C₄	—	2	2	no	—
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
118	C₂²×C₄	C₂×C₈×C₃₁	1	4	yes	—
119	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
120	C₂²×C₄	M₁₆×C₃₁	1	2	yes	—
121	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
122	M₁₆	—	2	8	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	8	no	—
127	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
128	C₂×C₈	M₁₆×C₃₁	1	4	yes	—
129	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
130	C₂×C₈	#41×C₃₁	1	4	yes	—
131	C₄⋊C₄	—	2	4	no	—
132	C₂²⋊C₄	—	2	4	no	—
133	C₄²	—	2	2	no	—
134	C₄²	—	2	2	no	—
135	C₁₆	—	4	4	no	—
136	C₂³	—	2	2	no	—
137	C₂³	—	2	4	no	—
138	C₂³	C₃₁×C₂²⋊C₈	1	2	yes	—
139	C₂³	C₂²×C₈×C₃₁	1	1	yes	characteristic
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	8	no	—
146	C₂×C₄	C₃₁×C₂²⋊C₈	1	2	yes	—
147	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
148	C₂×C₄	C₃₁×#117	1	2	yes	—
149	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
150	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
151	C₈	—	2	4	no	—
152	C₈	—	2	2	no	—
153	C₈	C₃₁×#123	1	4	yes	—
154	C₂²	—	2	2	no	—
155	C₂²	—	2	4	no	—
156	C₂²	—	4	4	no	—
157	C₂²	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
158	C₂²	C₃₁×#431	1	2	yes	—
159	C₂²	C₃₁×#504	1	4	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	4	4	no	—
162	C₄	C₃₁×#381	1	1	yes	characteristic
163	C₄	C₃₁×#407	1	1	yes	characteristic
164	C₄	C₃₁×#432	1	2	yes	—
165	C₂	—	4	2	no	—
166	C₂	C₃₁×#1817	1	1	yes	characteristic
167	C₂	C₃₁×#2113	1	2	yes	—
168	C₂	C₃₁×#2164	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃₁×#405

C₂

yes

maximal, characteristic

C₂×C₃₁×C₂²⋊C₁₆

C₂

yes

maximal

C₃₁×#1374

C₂

yes

maximal, complemented

C₂²×M₁₆×C₃₁

C₄

yes

characteristic

C₂³×C₈×C₃₁

C₂²

yes

characteristic

C₂×C₃₁×#117

C₄

yes

characteristic

C₂²×C₁₆×C₃₁

C₂²

yes

—

C₃₁×#405

—

C₂×C₃₁×#97

C₄

yes

—

C₂×C₄×C₈×C₃₁

C₂²

yes

—

C₃₁×C₂²⋊C₁₆

—

C₂³×C₄×C₃₁

C₂×C₄

yes

characteristic

C₂×M₁₆×C₃₁

—

C₂×M₁₆×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

C₃₁×#117

—

C₃₁×#117

—

C₂×C₄⋊C₄×C₃₁

C₈

yes

—

C₂×C₂²⋊C₄×C₃₁

C₈

yes

—

C₂×C₄²×C₃₁

C₂×C₄

yes

—

C₂×C₁₆×C₃₁

—

C₂×C₁₆×C₃₁

—

C₃₁×#97

—

C₄×C₈×C₃₁

—

C₂⁴×C₃₁

C₂×C₈

yes

characteristic

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₃₁

C₂×C₈

yes

—

C₂²×C₄×C₃₁

C₂×C₈

yes

characteristic

C₂²×C₄×C₃₁

M₁₆

yes

—

C₂²×C₄×C₃₁

C₂²×C₄

yes

characteristic

M₁₆×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

—

C₄⋊C₄×C₃₁

—

C₂²⋊C₄×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

—

C₁₆×C₃₁

—

#12845

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

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

—

C₂×#405

C₂×C₃₁

yes

characteristic

C₂×C₂²⋊C₁₆

C₂×C₃₁

yes

—

#1374

C₂×C₃₁

yes

complemented

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₃₁

#431

yes

—

C₂²×C₃₁

#504

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

—

C₂²×M₁₆

C₄×C₃₁

yes

characteristic

C₂³×C₈

C₂²×C₃₁

yes

characteristic

C₂×#117

C₄×C₃₁

yes

characteristic

C₂²×C₁₆

C₂²×C₃₁

yes

—

#405

—

C₂×#97

C₄×C₃₁

yes

—

C₂×C₄×C₈

C₂²×C₃₁

yes

—

C₂²⋊C₁₆

—

C₂×C₃₁

—

C₂×C₃₁

#1817

yes

characteristic

C₂×C₃₁

#2113

yes

—

C₂×C₃₁

#2164

yes

—

C₂³×C₄

C₂×C₄×C₃₁

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

100

C₂²×C₈

C₂×C₄×C₃₁

yes

characteristic

101

C₂²×C₈

C₂×C₄×C₃₁

yes

—

102

C₂²×C₈

D₄×C₃₁

yes

—

103

C₂²×C₈

C₂³×C₃₁

yes

characteristic

104

#117

—

105

#117

—

106

C₂×C₄⋊C₄

C₈×C₃₁

yes

—

107

C₂×C₂²⋊C₄

C₈×C₃₁

yes

—

108

C₂×C₄²

C₂×C₄×C₃₁

yes

—

109

C₂×C₁₆

—

110

C₂×C₁₆

—

111

#97

—

112

C₄×C₈

—

113

C₃₁

#12845

yes

characteristic, direct product factor, Sylow

114

C₂⁴

C₂×C₈×C₃₁

yes

characteristic

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

C₂²⋊C₄×C₃₁

yes

—

118

C₂²×C₄

C₂×C₈×C₃₁

yes

—

119

C₂²×C₄

C₂×C₈×C₃₁

yes

characteristic

120

C₂²×C₄

M₁₆×C₃₁

yes

—

121

C₂²×C₄

C₂²×C₄×C₃₁

yes

characteristic

122

M₁₆

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

C₂²⋊C₄×C₃₁

yes

—

128

C₂×C₈

M₁₆×C₃₁

yes

—

129

C₂×C₈

C₂×D₄×C₃₁

yes

—

130

C₂×C₈

#41×C₃₁

yes

—

131

C₄⋊C₄

—

132

C₂²⋊C₄

—

133

C₄²

—

134

C₄²

—

135

C₁₆

—

136

C₂³

—

137

C₂³

—

138

C₂³

C₃₁×C₂²⋊C₈

yes

—

139

C₂³

C₂²×C₈×C₃₁

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

C₃₁×C₂²⋊C₈

yes

—

147

C₂×C₄

C₂×C₂²⋊C₄×C₃₁

yes

characteristic

148

C₂×C₄

C₃₁×#117

yes

—

149

C₂×C₄

C₂×M₁₆×C₃₁

yes

—

150

C₂×C₄

C₂×M₁₆×C₃₁

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

C₃₁×#123

yes

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

C₂×C₃₁×C₂²⋊C₈

yes

characteristic

158

C₂²

C₃₁×#431

yes

—

159

C₂²

C₃₁×#504

yes

—

160

C₄

—

161

C₄

—

162

C₄

C₃₁×#381

yes

characteristic

163

C₄

C₃₁×#407

yes

characteristic

164

C₄

C₃₁×#432

yes

—

165

C₂

—

166

C₂

C₃₁×#1817

yes

characteristic

167

C₂

C₃₁×#2113

yes

—

168

C₂

C₃₁×#2164

yes

—

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

subgroup

G/H

center

C₂²×C₈×C₃₁

C₂³

commutator

C₂³

C₂²×C₈×C₃₁

Frattini, cosocle

C₂²×C₈

C₂³×C₃₁

socle

C₂³×C₃₁

C₂²×C₈

prime radical

C₂⁴×C₃₁

C₂×C₈

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂²×C₈×C₃₁ ◁ G

N	H ≅ G/N	type
3. C₃₁×#1374	C₂	semidirect
51. #12845	C₃₁	direct (N × H)
72. #1374	C₂×C₃₁	semidirect
113. C₃₁	#12845	direct (N × H)

H ≅ G/N

type

3. C₃₁×#1374

C₂

semidirect

51. #12845

C₃₁

direct (N × H)

72. #1374

C₂×C₃₁

semidirect

113. C₃₁

#12845

direct (N × H)

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

subgroup	index
C₂×C₃₁×#405	2
C₂×C₃₁×C₂²⋊C₁₆	2
C₃₁×#1374	2
#12845	31

subgroup

index

C₂×C₃₁×#405

C₂×C₃₁×C₂²⋊C₁₆

C₃₁×#1374

#12845

Subgroups — C₃₁×#12845 #21469604

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#12845 #21469604

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12845 #21469604