Group #31913608 — subgroups

Subgroups — #31913608

Order 18176.

358 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 143 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	#22590113	C₂	1	2	yes	maximal, complemented
2	#22588390	C₂	1	1	yes	maximal, characteristic
3	#22588386	C₂	1	1	yes	maximal, characteristic
4	#22586436	C₂	1	2	yes	maximal, complemented
5	C₂×#504×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556064	C₂²	1	2	yes	—
7	#17986037	C₂²	1	2	yes	—
8	#17985983	—	2	1	no	—
9	#17985973	—	2	1	no	—
10	#17985822	—	2	4	no	—
11	C₇₁⋊#338	—	2	1	no	—
12	#17985756	—	2	1	no	—
13	C₂×#131×C₇₁	C₂²	1	1	yes	characteristic
14	#504×C₇₁	—	2	4	no	—
15	C₂×M₃₂×C₇₁	C₄	1	2	yes	complemented
16	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₇₁	C₄	1	2	yes	complemented
18	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×#11765849	—	2	2	no	—
20	#14556087	—	2	2	no	—
21	#14556064	—	2	2	no	—
22	#14556064	—	2	2	no	—
23	#14556062	—	2	2	no	—
24	#14556060	—	2	2	no	—
25	#14556047	—	4	1	no	—
26	#14556045	—	4	1	no	—
27	#14556043	—	4	2	no	—
28	C₂×#41×C₇₁	D₄	1	1	yes	characteristic
29	#131×C₇₁	—	2	2	no	—
30	#131×C₇₁	—	2	2	no	—
31	C₂×M₁₆×C₇₁	D₄	1	1	yes	characteristic
32	C₂×M₁₆×C₇₁	Q₈	1	2	yes	—
33	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
35	M₃₂×C₇₁	—	2	4	no	—
36	M₃₂×C₇₁	D₄	1	4	yes	—
37	C₂×C₁₆×C₇₁	—	2	2	no	—
38	C₂×C₁₆×C₇₁	—	2	4	no	—
39	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
40	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
41	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
42	#11765849	—	4	2	no	—
43	#11765849	—	4	2	no	—
44	C₂×C₇₁⋊C₈	—	4	2	no	—
45	C₄⋊Dic₇₁	—	4	2	no	—
46	C₂²⋊Dic₇₁	—	4	2	no	—
47	C₄×Dic₇₁	—	4	2	no	—
48	#41×C₇₁	—	2	2	no	—
49	#41×C₇₁	—	2	2	no	—
50	C₂×Q₈×C₇₁	SD₁₆	1	1	yes	characteristic
51	C₂×D₄×C₇₁	SD₁₆	1	2	yes	—
52	C₂×D₄×C₇₁	SD₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
55	M₁₆×C₇₁	—	2	2	no	—
56	M₁₆×C₇₁	—	2	4	no	—
57	C₂×C₈×C₇₁	—	2	2	no	—
58	C₂×C₈×C₇₁	—	2	2	no	—
59	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
60	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₈×C₇₁	#41	1	2	yes	—
64	C₁₆×C₇₁	—	2	1	no	—
65	C₁₆×C₇₁	—	2	1	no	—
66	C₁₆×C₇₁	C₂²⋊C₄	1	4	yes	—
67	C₂³×C₇₁	#106	1	2	yes	—
68	C₂³×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
69	Q₈×C₇₁	—	2	2	no	—
70	D₄×C₇₁	—	2	4	no	—
71	D₄×C₇₁	—	2	2	no	—
72	C₂×C₄×C₇₁	—	2	2	no	—
73	C₂×C₄×C₇₁	—	2	2	no	—
74	C₂×C₄×C₇₁	#106	1	2	yes	—
75	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₇₁	#121	1	2	yes	—
77	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₇₁	#135	1	2	yes	—
79	C₂×Dic₇₁	—	8	2	no	—
80	C₈×C₇₁	—	2	2	no	—
81	C₈×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₈×C₇₁	#117	1	1	yes	characteristic
83	C₈×C₇₁	C₂²⋊Q₈	1	2	yes	—
84	C₇₁⋊C₈	—	8	2	no	—
85	C₂²×C₇₁	—	2	2	no	—
86	C₂²×C₇₁	—	4	1	no	—
87	C₂²×C₇₁	—	4	2	no	—
88	C₂²×C₇₁	C₂×#106	1	1	yes	characteristic
89	C₂²×C₇₁	#465	1	2	yes	—
90	C₄×C₇₁	—	2	2	no	—
91	C₄×C₇₁	#380	1	1	yes	characteristic
92	C₄×C₇₁	#427	1	1	yes	characteristic
93	C₄×C₇₁	#465	1	2	yes	—
94	Dic₇₁	—	16	2	no	—
95	#12832	—	71	1	no	maximal, Sylow
96	C₂×C₇₁	—	2	2	no	—
97	C₂×C₇₁	—	2	1	no	—
98	C₂×C₇₁	—	4	2	no	—
99	C₂×C₇₁	#1807	1	1	yes	characteristic
100	C₂×#504	D₇₁	1	1	yes	characteristic
101	#2155	—	71	2	no	—
102	#1813	—	71	1	no	—
103	#1812	—	71	1	no	—
104	#1382	—	71	2	no	—
105	C₇₁	#12832	1	1	yes	characteristic, complemented, Sylow
106	C₂×#131	C₂×D₇₁	1	1	yes	characteristic
107	#504	—	2	4	no	—
108	C₂×M₃₂	Dic₇₁	1	2	yes	complemented
109	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
110	C₂²×C₁₆	Dic₇₁	1	2	yes	complemented
111	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
112	C₂×#108	—	71	2	no	—
113	#427	—	71	2	no	—
114	#413	—	142	1	no	—
115	#410	—	142	1	no	—
116	#365	—	142	4	no	—
117	#338	—	142	1	no	—
118	H(3,ℤ/4)	—	142	1	no	—
119	C₂×#41	D₂₈₄	1	1	yes	characteristic
120	#131	—	2	2	no	—
121	#131	—	2	2	no	—
122	C₂×M₁₆	—	142	2	no	—
123	C₂×M₁₆	Dic₁₄₂	1	2	yes	—
124	C₂×M₁₆	D₂₈₄	1	1	yes	characteristic
125	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
126	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
127	#117	—	142	2	no	—
128	M₃₂	—	2	4	no	—
129	M₃₂	C₂²⋊D₇₁	1	4	yes	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	2	4	no	—
132	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
133	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
134	C₂×C₁₆	C₂×Dic₇₁	1	2	yes	—
135	#108	—	142	2	no	—
136	#108	—	142	2	no	—
137	#107	—	142	2	no	—
138	#106	—	142	2	no	—
139	#101	—	284	1	no	—
140	#100	—	284	1	no	—
141	#99	—	284	2	no	—
142	#41	—	2	2	no	—
143	#41	—	2	2	no	—
144	C₂×Q₈	#11765854	1	1	yes	characteristic
145	C₂×D₄	#11765854	1	2	yes	—
146	C₂×D₄	#11765854	1	1	yes	characteristic
147	C₂²×C₄	C₄⋊Dic₇₁	1	2	yes	—
148	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
149	M₁₆	—	2	2	no	—
150	M₁₆	—	2	4	no	—
151	M₁₆	—	284	2	no	—
152	M₁₆	—	284	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	284	2	no	—
156	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
157	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
158	C₂×C₈	C₂×C₂²⋊D₇₁	1	2	yes	—
159	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
160	C₂×C₈	#11765865	1	2	yes	—
161	C₄⋊C₄	—	284	2	no	—
162	C₂²⋊C₄	—	284	2	no	—
163	C₄²	—	284	2	no	—
164	C₁₆	—	2	1	no	—
165	C₁₆	—	2	1	no	—
166	C₁₆	C₂²⋊Dic₇₁	1	4	yes	—
167	C₂³	#14556060	1	2	yes	—
168	C₂³	C₂×#11765854	1	1	yes	characteristic
169	Q₈	—	2	2	no	—
170	D₄	—	2	4	no	—
171	D₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	568	2	no	—
175	C₂×C₄	#14556060	1	2	yes	—
176	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
177	C₂×C₄	#14556102	1	2	yes	—
178	C₂×C₄	C₂×#11765854	1	1	yes	characteristic
179	C₂×C₄	#14556150	1	2	yes	—
180	C₈	—	2	2	no	—
181	C₈	—	568	2	no	—
182	C₈	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
183	C₈	#14556087	1	1	yes	characteristic
184	C₈	#14556107	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	—	4	1	no	—
187	C₂²	—	4	2	no	—
188	C₂²	C₂×#14556060	1	1	yes	characteristic
189	C₂²	#17986228	1	2	yes	—
190	C₄	—	2	2	no	—
191	C₄	—	1136	2	no	—
192	C₄	#17985859	1	1	yes	characteristic
193	C₄	#17986037	1	1	yes	characteristic
194	C₄	#17986228	1	2	yes	—
195	C₂	—	2	2	no	—
196	C₂	—	2	1	no	—
197	C₂	—	4	2	no	—
198	C₂	#22588365	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590113

C₂

yes

maximal, complemented

#22588390

C₂

yes

maximal, characteristic

#22588386

C₂

yes

maximal, characteristic

#22586436

C₂

yes

maximal, complemented

C₂×#504×C₇₁

C₂

yes

maximal, characteristic

C₂×#14556064

C₂²

yes

—

#17986037

C₂²

yes

—

#17985983

—

#17985973

—

#17985822

—

C₇₁⋊#338

—

#17985756

—

C₂×#131×C₇₁

C₂²

yes

characteristic

#504×C₇₁

—

C₂×M₃₂×C₇₁

C₄

yes

complemented

C₂×M₃₂×C₇₁

C₂²

yes

characteristic

C₂²×C₁₆×C₇₁

C₄

yes

complemented

C₂²×C₁₆×C₇₁

C₂²

yes

—

—

—

—

—

—

—

—

—

C₂×#41×C₇₁

D₄

yes

characteristic

#131×C₇₁

—

#131×C₇₁

—

C₂×M₁₆×C₇₁

D₄

yes

characteristic

C₂×M₁₆×C₇₁

Q₈

yes

—

C₂²×C₈×C₇₁

C₂×C₄

yes

—

C₂²×C₈×C₇₁

C₂³

yes

characteristic

M₃₂×C₇₁

—

M₃₂×C₇₁

D₄

yes

—

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

C₂×C₄

yes

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

#41×C₇₁

—

#41×C₇₁

—

C₂×Q₈×C₇₁

SD₁₆

yes

characteristic

C₂×D₄×C₇₁

SD₁₆

yes

—

C₂×D₄×C₇₁

SD₁₆

yes

characteristic

C₂²×C₄×C₇₁

C₄⋊C₄

yes

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

—

C₂×C₈×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₁

#41

yes

—

C₁₆×C₇₁

—

C₁₆×C₇₁

—

C₁₆×C₇₁

C₂²⋊C₄

yes

—

C₂³×C₇₁

#106

yes

—

C₂³×C₇₁

C₂×SD₁₆

yes

characteristic

Q₈×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#106

yes

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

#121

yes

—

C₂×C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₁

#135

yes

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₈×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₇₁

#117

yes

characteristic

C₈×C₇₁

C₂²⋊Q₈

yes

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×#106

yes

characteristic

C₂²×C₇₁

#465

yes

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

Dic₇₁

—

#12832

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#1807

yes

characteristic

100

C₂×#504

D₇₁

yes

characteristic

101

#2155

—

102

#1813

—

103

#1812

—

104

#1382

—

105

C₇₁

#12832

yes

characteristic, complemented, Sylow

106

C₂×#131

C₂×D₇₁

yes

characteristic

107

#504

—

108

C₂×M₃₂

Dic₇₁

yes

complemented

109

C₂×M₃₂

C₂×D₇₁

yes

characteristic

110

C₂²×C₁₆

Dic₇₁

yes

complemented

111

C₂²×C₁₆

C₂×D₇₁

yes

characteristic

112

C₂×#108

—

113

#427

—

114

#413

—

142

—

115

#410

—

142

—

116

#365

—

142

—

117

#338

—

142

—

118

H(3,ℤ/4)

—

142

—

119

C₂×#41

D₂₈₄

yes

characteristic

120

#131

—

121

#131

—

122

C₂×M₁₆

—

142

—

123

C₂×M₁₆

Dic₁₄₂

yes

—

124

C₂×M₁₆

D₂₈₄

yes

characteristic

125

C₂²×C₈

C₂×Dic₇₁

yes

—

126

C₂²×C₈

C₂²×D₇₁

yes

characteristic

127

#117

—

142

—

128

M₃₂

—

129

M₃₂

C₂²⋊D₇₁

yes

—

130

C₂×C₁₆

—

131

C₂×C₁₆

—

132

C₂×C₁₆

C₂×Dic₇₁

yes

characteristic

133

C₂×C₁₆

C₂×Dic₇₁

yes

characteristic

134

C₂×C₁₆

C₂×Dic₇₁

yes

—

135

#108

—

142

—

136

#108

—

142

—

137

#107

—

142

—

138

#106

—

142

—

139

#101

—

284

—

140

#100

—

284

—

141

#99

—

284

—

142

#41

—

143

#41

—

144

C₂×Q₈

#11765854

yes

characteristic

145

C₂×D₄

#11765854

yes

—

146

C₂×D₄

#11765854

yes

characteristic

147

C₂²×C₄

C₄⋊Dic₇₁

yes

—

148

C₂²×C₄

C₂×D₂₈₄

yes

characteristic

149

M₁₆

—

150

M₁₆

—

151

M₁₆

—

284

—

152

M₁₆

—

284

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

284

—

156

C₂×C₈

C₄⋊Dic₇₁

yes

—

157

C₂×C₈

C₂²×Dic₇₁

yes

characteristic

158

C₂×C₈

C₂×C₂²⋊D₇₁

yes

—

159

C₂×C₈

C₂×Dic₁₄₂

yes

characteristic

160

C₂×C₈

#11765865

yes

—

161

C₄⋊C₄

—

284

—

162

C₂²⋊C₄

—

284

—

163

C₄²

—

284

—

164

C₁₆

—

165

C₁₆

—

166

C₁₆

C₂²⋊Dic₇₁

yes

—

167

C₂³

#14556060

yes

—

168

C₂³

C₂×#11765854

yes

characteristic

169

Q₈

—

170

D₄

—

171

D₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

568

—

175

C₂×C₄

#14556060

yes

—

176

C₂×C₄

C₂×C₄⋊Dic₇₁

yes

characteristic

177

C₂×C₄

#14556102

yes

—

178

C₂×C₄

C₂×#11765854

yes

characteristic

179

C₂×C₄

#14556150

yes

—

180

C₈

—

181

C₈

—

568

—

182

C₈

C₂×C₂²⋊Dic₇₁

yes

characteristic

183

C₈

#14556087

yes

characteristic

184

C₈

#14556107

yes

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

C₂×#14556060

yes

characteristic

189

C₂²

#17986228

yes

—

190

C₄

—

191

C₄

—

1136

—

192

C₄

#17985859

yes

characteristic

193

C₄

#17986037

yes

characteristic

194

C₄

#17986228

yes

—

195

C₂

—

196

C₂

—

197

C₂

—

198

C₂

#22588365

yes

characteristic

	subgroup	G/H
center	C₄	#17986037
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#504×C₇₁	C₂
socle	C₂×C₇₁	#1807
prime radical	C₂×#41×C₇₁	D₄
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#504	D₇₁
hypocenter	C₇₁	#12832
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ C₂×#504

subgroup

G/H

center

C₄

#17986037

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₂×#504×C₇₁

C₂

socle

C₂×C₇₁

#1807

prime radical

C₂×#41×C₇₁

D₄

cosocle

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ C₂×#504

N	H ≅ G/N	type
1. #22590113	C₂	semidirect
4. #22586436	C₂	semidirect
15. C₂×M₃₂×C₇₁	C₄	semidirect
17. C₂²×C₁₆×C₇₁	C₄	semidirect
105. C₇₁	#12832	semidirect
108. C₂×M₃₂	Dic₇₁	semidirect
110. C₂²×C₁₆	Dic₇₁	semidirect

H ≅ G/N

type

1. #22590113

C₂

semidirect

4. #22586436

C₂

semidirect

15. C₂×M₃₂×C₇₁

C₄

semidirect

17. C₂²×C₁₆×C₇₁

C₄

semidirect

105. C₇₁

#12832

semidirect

108. C₂×M₃₂

Dic₇₁

semidirect

110. C₂²×C₁₆

Dic₇₁

semidirect

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

subgroup	index
#22590113	2
#22588390	2
#22588386	2
#22586436	2
C₂×#504×C₇₁	2
#12832	71

subgroup

index

C₂×#504×C₇₁

Subgroups — #31913608

Conjugacy classes of subgroups

Complements

Maximal subgroups