Group #21467444 — subgroups

Subgroups — C₃₁×#10685 #21467444

Order 7936.

442 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 178 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×#1990	C₂	1	2	yes	maximal
2	C₃₁×#1956	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1844	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1844	C₂	1	2	yes	maximal
5	C₃₁×#1273	C₂	1	1	yes	maximal, characteristic
6	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₃₁×#106	—	2	2	no	—
8	C₂×C₃₁×#106	—	2	4	no	—
9	C₂×C₃₁×#103	—	2	2	no	—
10	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
12	C₃₁×#384	C₂²	1	1	yes	characteristic
13	C₃₁×#384	C₂²	1	2	yes	—
14	C₃₁×#340	—	2	4	no	—
15	C₂²×Q₈×C₃₁	—	2	1	no	—
16	C₂²×C₈×C₃₁	D₄	1	2	yes	—
17	C₂²×C₈×C₃₁	D₄	1	4	yes	—
18	C₃₁×C₄⋊Q₈	—	2	2	no	—
19	C₃₁×C₄⋊Q₈	—	2	1	no	—
20	C₃₁×C₄⋊Q₈	Q₈	1	2	yes	—
21	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
22	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
23	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
24	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
25	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
26	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
27	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
28	C₃₁×#106	—	4	4	no	—
29	C₃₁×#106	—	4	8	no	—
30	C₃₁×#103	—	4	4	no	—
31	C₄×C₈×C₃₁	—	2	2	no	—
32	C₄×C₈×C₃₁	—	2	2	no	—
33	C₄×C₈×C₃₁	—	2	2	no	—
34	C₂×Q₈×C₃₁	—	2	2	no	—
35	C₂×Q₈×C₃₁	—	4	1	no	—
36	C₂×Q₈×C₃₁	—	4	2	no	—
37	C₂×Q₈×C₃₁	—	4	2	no	—
38	C₂²×C₄×C₃₁	—	4	1	no	—
39	C₂²×C₄×C₃₁	—	4	1	no	—
40	C₂²×C₄×C₃₁	—	4	2	no	—
41	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
43	C₂×C₈×C₃₁	—	2	2	no	—
44	C₂×C₈×C₃₁	—	2	4	no	—
45	C₂×C₈×C₃₁	—	2	2	no	—
46	C₂×C₈×C₃₁	—	2	4	no	—
47	C₂×C₈×C₃₁	SD₁₆	1	4	yes	—
48	C₂×C₈×C₃₁	SD₁₆	1	8	yes	—
49	C₄⋊C₄×C₃₁	—	2	4	no	—
50	C₄⋊C₄×C₃₁	—	4	2	no	—
51	C₄⋊C₄×C₃₁	—	4	2	no	—
52	C₄⋊C₄×C₃₁	—	4	4	no	—
53	C₄⋊C₄×C₃₁	—	4	4	no	—
54	C₄⋊C₄×C₃₁	—	4	4	no	—
55	C₄⋊C₄×C₃₁	—	4	4	no	—
56	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
57	C₄²×C₃₁	#41	1	2	yes	—
58	C₄²×C₃₁	#41	1	1	yes	characteristic
59	#10685	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂³×C₃₁	#127	1	1	yes	characteristic
61	Q₈×C₃₁	—	4	4	no	—
62	Q₈×C₃₁	—	8	2	no	—
63	C₂×C₄×C₃₁	—	4	2	no	—
64	C₂×C₄×C₃₁	—	8	1	no	—
65	C₂×C₄×C₃₁	—	8	1	no	—
66	C₂×C₄×C₃₁	—	8	1	no	—
67	C₂×C₄×C₃₁	—	8	2	no	—
68	C₂×C₄×C₃₁	—	8	2	no	—
69	C₂×C₄×C₃₁	—	8	2	no	—
70	C₂×C₄×C₃₁	—	8	2	no	—
71	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
72	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
74	C₂×C₄×C₃₁	#123	1	2	yes	—
75	C₂×C₄×C₃₁	C₂×SD₁₆	1	4	yes	—
76	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
77	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
78	C₂×C₄×C₃₁	#137	1	4	yes	—
79	C₈×C₃₁	—	2	4	no	—
80	C₈×C₃₁	—	2	8	no	—
81	#1990	C₂×C₃₁	1	2	yes	—
82	#1956	C₂×C₃₁	1	1	yes	characteristic
83	#1844	C₂×C₃₁	1	2	yes	—
84	#1844	C₂×C₃₁	1	1	yes	characteristic
85	#1273	C₂×C₃₁	1	1	yes	characteristic
86	C₂²×C₃₁	#399	1	1	yes	characteristic
87	C₂²×C₃₁	#492	1	1	yes	characteristic
88	C₂²×C₃₁	#492	1	2	yes	—
89	C₂²×C₃₁	#496	1	1	yes	characteristic
90	C₂²×C₃₁	#497	1	2	yes	—
91	C₄×C₃₁	—	8	2	no	—
92	C₄×C₃₁	—	16	1	no	—
93	C₄×C₃₁	—	16	2	no	—
94	C₄×C₃₁	#475	1	4	yes	—
95	C₄×C₃₁	#476	1	2	yes	—
96	C₄×C₃₁	#481	1	2	yes	—
97	C₄×C₃₁	#483	1	4	yes	—
98	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
99	C₂×#106	—	2	2	no	—
100	C₂×#106	—	2	4	no	—
101	C₂×#103	—	2	2	no	—
102	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
103	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
104	#384	C₂²×C₃₁	1	1	yes	characteristic
105	#384	C₂²×C₃₁	1	2	yes	—
106	#340	—	2	4	no	—
107	C₂×C₃₁	#1705	1	1	yes	characteristic
108	C₂×C₃₁	#1714	1	2	yes	—
109	C₂×C₃₁	#1716	1	1	yes	characteristic
110	C₂×C₃₁	#2094	1	1	yes	characteristic
111	C₂×C₃₁	#2096	1	2	yes	—
112	C₂²×Q₈	—	2	1	no	—
113	C₂²×C₈	D₄×C₃₁	1	4	yes	—
114	C₂²×C₈	D₄×C₃₁	1	2	yes	—
115	C₄⋊Q₈	—	2	2	no	—
116	C₄⋊Q₈	—	2	1	no	—
117	C₄⋊Q₈	Q₈×C₃₁	1	2	yes	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	1	no	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
125	#106	—	4	4	no	—
126	#106	—	4	8	no	—
127	#103	—	4	4	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	2	2	no	—
131	C₃₁	#10685	1	1	yes	characteristic, direct product factor, Sylow
132	C₂×Q₈	—	2	2	no	—
133	C₂×Q₈	—	4	1	no	—
134	C₂×Q₈	—	4	2	no	—
135	C₂×Q₈	—	4	2	no	—
136	C₂²×C₄	—	4	1	no	—
137	C₂²×C₄	—	4	1	no	—
138	C₂²×C₄	—	4	2	no	—
139	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
140	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	SD₁₆×C₃₁	1	4	yes	—
146	C₂×C₈	SD₁₆×C₃₁	1	8	yes	—
147	C₄⋊C₄	—	2	4	no	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄⋊C₄	—	4	4	no	—
152	C₄⋊C₄	—	4	4	no	—
153	C₄⋊C₄	—	4	4	no	—
154	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
155	C₄²	#41×C₃₁	1	2	yes	—
156	C₄²	#41×C₃₁	1	1	yes	characteristic
157	C₂³	C₃₁×#127	1	1	yes	characteristic
158	Q₈	—	4	4	no	—
159	Q₈	—	8	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	—	8	1	no	—
163	C₂×C₄	—	8	1	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	—	8	2	no	—
167	C₂×C₄	—	8	2	no	—
168	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
169	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
170	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
171	C₂×C₄	C₃₁×#123	1	2	yes	—
172	C₂×C₄	C₂×SD₁₆×C₃₁	1	4	yes	—
173	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
174	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
175	C₂×C₄	C₃₁×#137	1	4	yes	—
176	C₈	—	2	4	no	—
177	C₈	—	2	8	no	—
178	C₂²	C₃₁×#399	1	1	yes	characteristic
179	C₂²	C₃₁×#492	1	1	yes	characteristic
180	C₂²	C₃₁×#492	1	2	yes	—
181	C₂²	C₃₁×#496	1	1	yes	characteristic
182	C₂²	C₃₁×#497	1	2	yes	—
183	C₄	—	8	2	no	—
184	C₄	—	16	1	no	—
185	C₄	—	16	2	no	—
186	C₄	C₃₁×#475	1	4	yes	—
187	C₄	C₃₁×#476	1	2	yes	—
188	C₄	C₃₁×#481	1	2	yes	—
189	C₄	C₃₁×#483	1	4	yes	—
190	C₂	C₃₁×#1705	1	1	yes	characteristic
191	C₂	C₃₁×#1714	1	2	yes	—
192	C₂	C₃₁×#1716	1	1	yes	characteristic
193	C₂	C₃₁×#2094	1	1	yes	characteristic
194	C₂	C₃₁×#2096	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃₁×#1990

C₂

yes

maximal

C₃₁×#1956

C₂

yes

maximal, characteristic

C₃₁×#1844

C₂

yes

maximal, characteristic

C₃₁×#1844

C₂

yes

maximal

C₃₁×#1273

C₂

yes

maximal, characteristic

C₂×C₃₁×C₄⋊Q₈

C₂²

yes

characteristic

C₂×C₃₁×#106

—

C₂×C₃₁×#106

—

C₂×C₃₁×#103

—

C₂×C₄×C₈×C₃₁

C₂²

yes

characteristic

C₂×C₄×C₈×C₃₁

C₂²

yes

—

C₃₁×#384

C₂²

yes

characteristic

C₃₁×#384

C₂²

yes

—

C₃₁×#340

—

C₂²×Q₈×C₃₁

—

C₂²×C₈×C₃₁

D₄

yes

—

C₂²×C₈×C₃₁

D₄

yes

—

C₃₁×C₄⋊Q₈

—

C₃₁×C₄⋊Q₈

—

C₃₁×C₄⋊Q₈

Q₈

yes

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄²×C₃₁

C₂³

yes

characteristic

C₃₁×#106

—

C₃₁×#106

—

C₃₁×#103

—

C₄×C₈×C₃₁

—

C₄×C₈×C₃₁

—

C₄×C₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

SD₁₆

yes

—

C₂×C₈×C₃₁

SD₁₆

yes

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

C₂×Q₈

yes

characteristic

C₄²×C₃₁

#41

yes

—

C₄²×C₃₁

#41

yes

characteristic

#10685

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

#127

yes

characteristic

Q₈×C₃₁

—

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⋊Q₈

yes

—

C₂×C₄×C₃₁

C₂²⋊Q₈

yes

C₂×C₄×C₃₁

yes

C₂×C₄×C₃₁

yes

—

C₂×C₄×C₃₁

C₂×SD₁₆

yes

—

C₂×C₄×C₃₁

C₂×SD₁₆

yes

—

C₂×C₄×C₃₁

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₃₁

#137

yes

—

C₈×C₃₁

—

C₈×C₃₁

—

#1990

C₂×C₃₁

yes

—

#1956

C₂×C₃₁

yes

characteristic

#1844

C₂×C₃₁

yes

—

#1844

C₂×C₃₁

yes

characteristic

#1273

C₂×C₃₁

yes

C₂²×C₃₁

yes

C₂²×C₃₁

yes

C₂²×C₃₁

yes

—

C₂²×C₃₁

#496

yes

characteristic

C₂²×C₃₁

#497

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#475

yes

—

C₄×C₃₁

#476

yes

—

C₄×C₃₁

#481

yes

—

C₄×C₃₁

#483

yes

—

C₂×C₄⋊Q₈

C₂²×C₃₁

yes

characteristic

C₂×#106

—

100

C₂×#106

—

101

C₂×#103

—

102

C₂×C₄×C₈

C₂²×C₃₁

yes

characteristic

103

C₂×C₄×C₈

C₂²×C₃₁

yes

—

104

#384

C₂²×C₃₁

yes

characteristic

105

#384

C₂²×C₃₁

yes

—

106

#340

—

107

C₂×C₃₁

#1705

yes

characteristic

108

C₂×C₃₁

#1714

yes

—

109

C₂×C₃₁

#1716

yes

characteristic

110

C₂×C₃₁

#2094

yes

characteristic

111

C₂×C₃₁

#2096

yes

—

112

C₂²×Q₈

—

113

C₂²×C₈

D₄×C₃₁

yes

—

114

C₂²×C₈

D₄×C₃₁

yes

—

115

C₄⋊Q₈

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

Q₈×C₃₁

yes

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄²

C₂³×C₃₁

yes

characteristic

125

#106

—

126

#106

—

127

#103

—

128

C₄×C₈

—

129

C₄×C₈

—

130

C₄×C₈

—

131

C₃₁

#10685

yes

characteristic, direct product factor, Sylow

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

C₂×D₄×C₃₁

yes

characteristic

140

C₂²×C₄

C₂×D₄×C₃₁

yes

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

SD₁₆×C₃₁

yes

—

146

C₂×C₈

SD₁₆×C₃₁

yes

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄²

C₂×Q₈×C₃₁

yes

characteristic

155

C₄²

#41×C₃₁

yes

—

156

C₄²

#41×C₃₁

yes

characteristic

157

C₂³

C₃₁×#127

yes

characteristic

158

Q₈

—

159

Q₈

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

C₃₁×C₂²⋊Q₈

yes

—

169

C₂×C₄

C₃₁×C₂²⋊Q₈

yes

characteristic

170

C₂×C₄

C₃₁×#123

yes

characteristic

171

C₂×C₄

C₃₁×#123

yes

—

172

C₂×C₄

C₂×SD₁₆×C₃₁

yes

—

173

C₂×C₄

C₂×SD₁₆×C₃₁

yes

—

174

C₂×C₄

C₃₁×AGL(1,ℤ/8)

yes

—

175

C₂×C₄

C₃₁×#137

yes

—

176

C₈

—

177

C₈

—

178

C₂²

C₃₁×#399

yes

characteristic

179

C₂²

C₃₁×#492

yes

characteristic

180

C₂²

C₃₁×#492

yes

—

181

C₂²

C₃₁×#496

yes

characteristic

182

C₂²

C₃₁×#497

yes

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

C₃₁×#475

yes

—

187

C₄

C₃₁×#476

yes

—

188

C₄

C₃₁×#481

yes

—

189

C₄

C₃₁×#483

yes

—

190

C₂

C₃₁×#1705

yes

characteristic

191

C₂

C₃₁×#1714

yes

—

192

C₂

C₃₁×#1716

yes

characteristic

193

C₂

C₃₁×#2094

yes

characteristic

194

C₂

C₃₁×#2096

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₃₁

#127

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₃₁

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

N	H ≅ G/N	type
59. #10685	C₃₁	direct (N × H)
131. C₃₁	#10685	direct (N × H)

H ≅ G/N

type

59. #10685

C₃₁

direct (N × H)

131. C₃₁

#10685

direct (N × H)

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

subgroup	index
C₃₁×#1990	2
C₃₁×#1956	2
C₃₁×#1844	2
C₃₁×#1273	2
#10685	31

subgroup

index

C₃₁×#1990

C₃₁×#1956

C₃₁×#1844

C₃₁×#1273

#10685

Subgroups — C₃₁×#10685 #21467444

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#10685 #21467444

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#10685 #21467444