Group #32152769 — subgroups

Subgroups — C₂×#22590585 #32152769

Order 18176.

582 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 203 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×#17986401	C₂	1	1	yes	maximal, characteristic
2	#22590585	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17985803	C₂	1	4	yes	maximal
4	C₂×#17985775	C₂	1	1	yes	maximal, characteristic
5	C₂×#494×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556120	—	2	1	no	—
7	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
8	#17986401	—	2	2	no	—
9	#17986401	C₂²	1	4	yes	—
10	C₂×#14556062	C₂²	1	2	yes	—
11	C₂×#14556060	—	2	1	no	—
12	#17985803	C₂²	1	16	yes	—
13	#17985775	C₂²	1	4	yes	—
14	C₂²×Q₁₆×C₇₁	—	2	1	no	—
15	C₂²×Q₁₆×C₇₁	C₂²	1	2	yes	—
16	C₂×C₄⋊Q₈×C₇₁	—	2	1	no	—
17	#494×C₇₁	—	2	2	no	—
18	#494×C₇₁	C₂²	1	4	yes	—
19	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
20	#14556120	—	2	4	no	—
21	#14556120	—	4	2	no	—
22	C₂×C₄⋊Dic₇₁	—	2	2	no	—
23	C₂×Dic₇₁⋊C₄	—	4	2	no	—
24	C₂×C₇₁⋊C₁₆	—	2	4	no	—
25	C₂×C₇₁⋊C₁₆	C₂³	1	4	yes	—
26	#14556062	—	2	2	no	—
27	#14556062	C₂³	1	4	yes	—
28	#14556060	—	2	4	no	—
29	C₂²×Q₈×C₇₁	—	2	2	no	—
30	C₂×Q₁₆×C₇₁	—	2	2	no	—
31	C₂×Q₁₆×C₇₁	—	2	4	no	—
32	C₂×Q₁₆×C₇₁	—	2	8	no	—
33	C₂×Q₁₆×C₇₁	—	4	4	no	—
34	C₂×Q₁₆×C₇₁	C₂³	1	4	yes	—
35	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
37	C₄⋊Q₈×C₇₁	—	2	4	no	—
38	C₄⋊Q₈×C₇₁	—	4	2	no	—
39	C₂×C₄⋊C₄×C₇₁	—	4	2	no	—
40	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
41	C₄×C₈×C₇₁	D₄	1	2	yes	—
42	C₄×C₈×C₇₁	C₂³	1	2	yes	—
43	C₂²×Dic₇₁	—	4	2	no	—
44	C₄⋊Dic₇₁	—	2	4	no	—
45	C₄⋊Dic₇₁	—	4	2	no	—
46	Dic₇₁⋊C₄	—	4	8	no	—
47	C₇₁⋊C₁₆	—	2	4	no	—
48	C₂×Q₈×C₇₁	—	2	4	no	—
49	C₂×Q₈×C₇₁	—	4	2	no	—
50	C₂×Q₈×C₇₁	—	4	4	no	—
51	C₂×Q₈×C₇₁	—	4	4	no	—
52	C₂²×C₄×C₇₁	—	4	2	no	—
53	C₂²×C₄×C₇₁	D₈	1	2	yes	—
54	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
55	Q₁₆×C₇₁	—	2	8	no	—
56	Q₁₆×C₇₁	—	4	4	no	—
57	Q₁₆×C₇₁	—	4	8	no	—
58	C₂×C₈×C₇₁	SD₁₆	1	4	yes	—
59	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₇₁	#41	1	4	yes	—
62	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
63	C₄⋊C₄×C₇₁	—	4	8	no	—
64	C₄²×C₇₁	D₈	1	2	yes	—
65	C₄²×C₇₁	C₂×D₄	1	2	yes	—
66	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
67	Q₈×C₇₁	—	4	4	no	—
68	Q₈×C₇₁	—	4	4	no	—
69	Q₈×C₇₁	—	8	4	no	—
70	C₂×C₄×C₇₁	—	2	4	no	—
71	C₂×C₄×C₇₁	—	4	4	no	—
72	C₂×C₄×C₇₁	—	8	4	no	—
73	C₂×C₄×C₇₁	#124	1	2	yes	—
74	C₂×C₄×C₇₁	C₂×D₈	1	4	yes	—
75	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
77	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
78	C₂×Dic₇₁	—	4	4	no	—
79	C₂×Dic₇₁	—	8	4	no	—
80	C₈×C₇₁	#124	1	2	yes	—
81	C₈×C₇₁	C₂×SD₁₆	1	4	yes	—
82	C₈×C₇₁	C₂×#41	1	2	yes	—
83	C₂²×C₇₁	#486	1	2	yes	—
84	C₂²×C₇₁	#510	1	4	yes	—
85	C₂²×C₇₁	C₂²×D₈	1	1	yes	characteristic
86	C₄×C₇₁	—	2	4	no	—
87	C₄×C₇₁	—	8	4	no	—
88	C₄×C₇₁	#486	1	2	yes	—
89	C₄×C₇₁	C₂×#124	1	1	yes	characteristic
90	C₄×C₇₁	C₂²×SD₁₆	1	1	yes	characteristic
91	Dic₇₁	—	8	4	no	—
92	C₂×#2245	—	71	1	no	maximal, Sylow
93	C₂×C₇₁	#2245	1	4	yes	complemented
94	C₂×C₇₁	C₂×#486	1	1	yes	characteristic
95	C₂×C₇₁	C₂×#510	1	2	yes	—
96	C₂×#499	—	71	1	no	—
97	C₂×#494	D₇₁	1	1	yes	characteristic
98	#2245	—	71	8	no	—
99	C₂×#358	—	71	4	no	—
100	C₂×#346	—	71	1	no	—
101	C₇₁	C₂×#2245	1	1	yes	characteristic, complemented, Sylow
102	C₂²×Q₁₆	—	2	1	no	—
103	C₂²×Q₁₆	C₂×D₇₁	1	2	yes	—
104	C₂×C₄⋊Q₈	—	2	1	no	—
105	C₂×#125	—	142	1	no	—
106	C₂²×C₁₆	—	71	2	no	—
107	#499	—	71	4	no	—
108	#499	—	142	2	no	—
109	#494	—	2	2	no	—
110	#494	C₂×D₇₁	1	4	yes	—
111	C₂×#107	—	71	2	no	—
112	C₂×#106	—	142	1	no	—
113	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
114	#358	—	71	16	no	—
115	#346	—	71	4	no	—
116	C₂²×Q₈	—	2	2	no	—
117	C₂×Q₁₆	—	2	4	no	—
118	C₂×Q₁₆	—	2	2	no	—
119	C₂×Q₁₆	—	2	8	no	—
120	C₂×Q₁₆	—	4	4	no	—
121	C₂×Q₁₆	C₂²×D₇₁	1	4	yes	—
122	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
124	C₄⋊Q₈	—	2	4	no	—
125	C₄⋊Q₈	—	4	2	no	—
126	#125	—	284	2	no	—
127	#125	—	142	4	no	—
128	C₂×C₄⋊C₄	—	4	2	no	—
129	C₂×C₄⋊C₄	—	284	2	no	—
130	C₂×C₄⋊C₄	—	142	2	no	—
131	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
132	C₂×C₁₆	—	71	4	no	—
133	C₂×C₁₆	—	142	4	no	—
134	#107	—	71	4	no	—
135	#107	—	142	2	no	—
136	#106	—	142	4	no	—
137	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
138	C₄×C₈	C₂²×D₇₁	1	2	yes	—
139	C₂×Q₈	—	2	4	no	—
140	C₂×Q₈	—	4	2	no	—
141	C₂×Q₈	—	4	4	no	—
142	C₂×Q₈	—	4	4	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	—	284	2	no	—
145	C₂²×C₄	#11765850	1	2	yes	—
146	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
147	Q₁₆	—	2	8	no	—
148	Q₁₆	—	4	4	no	—
149	Q₁₆	—	4	8	no	—
150	C₂×C₈	#11765853	1	4	yes	—
151	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
152	C₂×C₈	C₂×C₂²⋊D₇₁	1	2	yes	—
153	C₂×C₈	#11765864	1	4	yes	—
154	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
155	C₄⋊C₄	—	4	8	no	—
156	C₄⋊C₄	—	284	8	no	—
157	C₄⋊C₄	—	284	2	no	—
158	C₄⋊C₄	—	142	4	no	—
159	C₄²	#11765850	1	2	yes	—
160	C₄²	C₂×C₂²⋊D₇₁	1	2	yes	—
161	C₁₆	—	142	4	no	—
162	C₂³	C₂×#11765850	1	1	yes	characteristic
163	Q₈	—	4	4	no	—
164	Q₈	—	4	4	no	—
165	Q₈	—	8	4	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	—	8	4	no	—
169	C₂×C₄	—	284	4	no	—
170	C₂×C₄	—	568	4	no	—
171	C₂×C₄	#14556115	1	2	yes	—
172	C₂×C₄	C₂×#11765850	1	4	yes	—
173	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
174	C₂×C₄	C₂×#11765853	1	2	yes	—
175	C₂×C₄	C₂²×C₂²⋊D₇₁	1	1	yes	characteristic
176	C₈	#14556115	1	2	yes	—
177	C₈	C₂×#11765853	1	4	yes	—
178	C₈	C₂×#11765864	1	2	yes	—
179	C₂²	#17986345	1	2	yes	—
180	C₂²	#17986440	1	4	yes	—
181	C₂²	C₂²×#11765850	1	1	yes	characteristic
182	C₄	—	2	4	no	—
183	C₄	—	8	4	no	—
184	C₄	—	568	4	no	—
185	C₄	#17986345	1	2	yes	—
186	C₄	C₂×#14556115	1	1	yes	characteristic
187	C₄	C₂²×#11765853	1	1	yes	characteristic
188	C₂	#22590585	1	4	yes	direct product factor
189	C₂	C₂×#17986345	1	1	yes	characteristic
190	C₂	C₂×#17986440	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17986401

C₂

yes

maximal, characteristic

#22590585

C₂

yes

maximal, direct product factor

C₂×#17985803

C₂

yes

maximal

C₂×#17985775

C₂

yes

maximal, characteristic

C₂×#494×C₇₁

C₂

yes

maximal, characteristic

C₂×#14556120

—

C₂²×C₇₁⋊C₁₆

C₂²

yes

—

#17986401

—

#17986401

C₂²

yes

—

C₂×#14556062

C₂²

yes

—

C₂×#14556060

—

#17985803

C₂²

yes

—

#17985775

C₂²

yes

—

C₂²×Q₁₆×C₇₁

—

C₂²×Q₁₆×C₇₁

C₂²

yes

—

C₂×C₄⋊Q₈×C₇₁

—

#494×C₇₁

—

#494×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#14556120

—

#14556120

—

C₂×C₄⋊Dic₇₁

—

C₂×Dic₇₁⋊C₄

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

C₂³

yes

—

#14556062

—

#14556062

C₂³

yes

—

#14556060

—

C₂²×Q₈×C₇₁

—

C₂×Q₁₆×C₇₁

—

C₂×Q₁₆×C₇₁

—

C₂×Q₁₆×C₇₁

—

C₂×Q₁₆×C₇₁

—

C₂×Q₁₆×C₇₁

C₂³

yes

—

C₂²×C₈×C₇₁

D₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄²×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

—

C₄×C₈×C₇₁

C₂³

yes

—

C₂²×Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

C₇₁⋊C₁₆

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

D₈

yes

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

C₂×C₈×C₇₁

SD₁₆

yes

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

—

C₂×C₈×C₇₁

#41

yes

—

C₂×C₈×C₇₁

C₂⁴

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

D₈

yes

—

C₄²×C₇₁

C₂×D₄

yes

—

C₂³×C₇₁

C₂×D₈

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#124

yes

—

C₂×C₄×C₇₁

C₂×D₈

yes

—

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

C₂×SD₁₆

yes

—

C₂×C₄×C₇₁

C₂²×D₄

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

#124

yes

—

C₈×C₇₁

C₂×SD₁₆

yes

—

C₈×C₇₁

C₂×#41

yes

—

C₂²×C₇₁

#486

yes

—

C₂²×C₇₁

#510

yes

—

C₂²×C₇₁

C₂²×D₈

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#486

yes

—

C₄×C₇₁

C₂×#124

yes

characteristic

C₄×C₇₁

C₂²×SD₁₆

yes

characteristic

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₂×#499

—

C₂×#494

D₇₁

yes

characteristic

#2245

—

C₂×#358

—

100

C₂×#346

—

101

C₇₁

C₂×#2245

yes

characteristic, complemented, Sylow

102

C₂²×Q₁₆

—

103

C₂²×Q₁₆

C₂×D₇₁

yes

—

104

C₂×C₄⋊Q₈

—

105

C₂×#125

—

142

—

106

C₂²×C₁₆

—

107

#499

—

108

#499

—

142

—

109

#494

—

110

#494

C₂×D₇₁

yes

—

111

C₂×#107

—

112

C₂×#106

—

142

—

113

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

114

#358

—

115

#346

—

116

C₂²×Q₈

—

117

C₂×Q₁₆

—

118

C₂×Q₁₆

—

119

C₂×Q₁₆

—

120

C₂×Q₁₆

—

121

C₂×Q₁₆

C₂²×D₇₁

yes

—

122

C₂²×C₈

C₂²⋊D₇₁

yes

characteristic

123

C₂²×C₈

C₂²×D₇₁

yes

characteristic

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

126

#125

—

284

—

127

#125

—

142

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

284

—

130

C₂×C₄⋊C₄

—

142

—

131

C₂×C₄²

C₂²⋊D₇₁

yes

characteristic

132

C₂×C₁₆

—

133

C₂×C₁₆

—

142

—

134

#107

—

135

#107

—

142

—

136

#106

—

142

—

137

C₄×C₈

C₂²⋊D₇₁

yes

—

138

C₄×C₈

C₂²×D₇₁

yes

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂²×C₄

—

144

C₂²×C₄

—

284

—

145

C₂²×C₄

#11765850

yes

—

146

C₂²×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

147

Q₁₆

—

148

Q₁₆

—

149

Q₁₆

—

150

C₂×C₈

#11765853

yes

—

151

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

152

C₂×C₈

C₂×C₂²⋊D₇₁

yes

—

153

C₂×C₈

#11765864

yes

—

154

C₂×C₈

C₂³×D₇₁

yes

characteristic

155

C₄⋊C₄

—

156

C₄⋊C₄

—

284

—

157

C₄⋊C₄

—

284

—

158

C₄⋊C₄

—

142

—

159

C₄²

#11765850

yes

—

160

C₄²

C₂×C₂²⋊D₇₁

yes

—

161

C₁₆

—

142

—

162

C₂³

C₂×#11765850

yes

characteristic

163

Q₈

—

164

Q₈

—

165

Q₈

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

284

—

170

C₂×C₄

—

568

—

171

C₂×C₄

#14556115

yes

—

172

C₂×C₄

C₂×#11765850

yes

—

173

C₂×C₄

C₂×#11765850

yes

characteristic

174

C₂×C₄

C₂×#11765853

yes

—

175

C₂×C₄

C₂²×C₂²⋊D₇₁

yes

characteristic

176

C₈

#14556115

yes

—

177

C₈

C₂×#11765853

yes

—

178

C₈

C₂×#11765864

yes

—

179

C₂²

#17986345

yes

—

180

C₂²

#17986440

yes

—

181

C₂²

C₂²×#11765850

yes

characteristic

182

C₄

—

183

C₄

—

184

C₄

—

568

—

185

C₄

#17986345

yes

—

186

C₄

C₂×#14556115

yes

characteristic

187

C₄

C₂²×#11765853

yes

characteristic

188

C₂

#22590585

yes

direct product factor

189

C₂

C₂×#17986345

yes

characteristic

190

C₂

C₂×#17986440

yes

—

	subgroup	G/H
center	C₂³	C₂×#11765850
commutator, cosocle	C₂×C₈×C₇₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₂×#494×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×D₈
hypercenter	C₂×#494	D₇₁
hypocenter	C₇₁	C₂×#2245
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

subgroup

G/H

center

C₂³

C₂×#11765850

commutator, cosocle

C₂×C₈×C₇₁

C₂⁴

Frattini

C₂×C₈

C₂³×D₇₁

Fitting

C₂×#494×C₇₁

C₂

socle, prime radical

C₂³×C₇₁

C₂×D₈

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

N	H ≅ G/N	type
2. #22590585	C₂	direct (N × H)
93. C₂×C₇₁	#2245	semidirect
101. C₇₁	C₂×#2245	semidirect
188. C₂	#22590585	direct (N × H)

H ≅ G/N

type

2. #22590585

C₂

direct (N × H)

93. C₂×C₇₁

#2245

semidirect

101. C₇₁

C₂×#2245

semidirect

188. C₂

#22590585

direct (N × H)

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

subgroup	index
C₂×#17986401	2
#22590585	2
C₂×#17985803	2
C₂×#17985775	2
C₂×#494×C₇₁	2
C₂×#2245	71

subgroup

index

C₂×#494×C₇₁

Subgroups — C₂×#22590585 #32152769

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#22590585 #32152769

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22590585 #32152769