Group #34443582 — subgroups

Subgroups — C₂×#23765642 #34443582

Order 20224.

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

normal subgroups only

This group has 686 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×#19098965	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19098940	C₂	1	1	yes	maximal, characteristic
3	#23765642	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19098365	C₂	1	4	yes	maximal, complemented
5	C₂×#346×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₆₃₂	—	2	1	no	—
7	C₂²×D₆₃₂	C₂²	1	2	yes	—
8	C₂×#14579158	—	2	1	no	—
9	C₂×#14579153	—	2	1	no	—
10	#19098965	—	2	2	no	—
11	#19098965	C₂²	1	4	yes	complemented
12	#19098940	—	2	2	no	—
13	#19098940	C₂²	1	4	yes	—
14	C₂×#14579095	C₂²	1	2	yes	—
15	C₂×#14579093	—	2	1	no	—
16	#19098365	C₂²	1	16	yes	complemented
17	C₂²×C₁₆×C₇₉	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
19	#346×C₇₉	C₂²	1	4	yes	complemented
20	C₂²×D₃₁₆	—	2	2	no	—
21	C₂²×D₃₁₆	—	4	2	no	—
22	C₂×D₆₃₂	—	2	8	no	—
23	C₂×D₆₃₂	—	2	2	no	—
24	C₂×D₆₃₂	—	2	4	no	—
25	C₂×D₆₃₂	—	4	4	no	—
26	C₂×D₆₃₂	C₂³	1	4	yes	—
27	#14579158	—	2	4	no	—
28	#14579158	—	4	2	no	—
29	#14579153	—	2	4	no	—
30	#14579153	—	4	2	no	—
31	C₂×C₄⋊Dic₇₉	—	2	2	no	—
32	C₂×Dic₇₉⋊C₄	—	4	2	no	—
33	#14579095	—	2	2	no	—
34	#14579095	C₂³	1	4	yes	—
35	#14579093	—	2	4	no	—
36	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
39	C₂×C₁₆×C₇₉	—	2	4	no	—
40	C₂×C₁₆×C₇₉	C₂³	1	4	yes	—
41	C₄×C₈×C₇₉	D₄	1	2	yes	—
42	C₄×C₈×C₇₉	C₂³	1	2	yes	—
43	C₂³×D₇₉	—	4	2	no	—
44	C₂×D₃₁₆	—	2	4	no	—
45	C₂×D₃₁₆	—	4	4	no	—
46	C₂×D₃₁₆	—	4	8	no	—
47	C₂×D₃₁₆	—	4	2	no	—
48	C₂×D₃₁₆	—	4	4	no	—
49	C₂×D₃₁₆	—	8	8	no	—
50	C₂²×Dic₇₉	—	4	2	no	—
51	D₆₃₂	—	2	8	no	—
52	D₆₃₂	—	4	4	no	—
53	D₆₃₂	—	4	8	no	—
54	C₄⋊Dic₇₉	—	2	4	no	—
55	C₄⋊Dic₇₉	—	4	2	no	—
56	Dic₇₉⋊C₄	—	4	8	no	—
57	C₂²×C₄×C₇₉	D₈	1	2	yes	—
58	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₇₉	SD₁₆	1	4	yes	—
60	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₇₉	#41	1	4	yes	—
63	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
64	C₄²×C₇₉	D₈	1	2	yes	—
65	C₄²×C₇₉	C₂×D₄	1	2	yes	—
66	C₁₆×C₇₉	—	2	4	no	—
67	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
68	C₂²×D₇₉	—	4	4	no	—
69	C₂²×D₇₉	—	8	4	no	—
70	C₂²×D₇₉	—	8	8	no	—
71	C₂×C₄×C₇₉	—	2	4	no	—
72	C₂×C₄×C₇₉	#124	1	2	yes	—
73	C₂×C₄×C₇₉	C₂×D₈	1	4	yes	—
74	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
76	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
77	C₂×Dic₇₉	—	4	4	no	—
78	C₂×Dic₇₉	—	8	4	no	—
79	D₃₁₆	—	4	4	no	—
80	D₃₁₆	—	4	4	no	—
81	D₃₁₆	—	8	16	no	—
82	D₃₁₆	—	8	4	no	—
83	C₈×C₇₉	#124	1	2	yes	—
84	C₈×C₇₉	C₂×SD₁₆	1	4	yes	—
85	C₈×C₇₉	C₂×#41	1	2	yes	—
86	C₂²×C₇₉	#486	1	2	yes	—
87	C₂²×C₇₉	#509	1	4	yes	—
88	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
89	C₂×D₇₉	—	8	4	no	—
90	C₂×D₇₉	—	8	8	no	—
91	C₂×D₇₉	—	16	8	no	—
92	C₄×C₇₉	—	2	4	no	—
93	C₄×C₇₉	#486	1	2	yes	—
94	C₄×C₇₉	C₂×#124	1	1	yes	characteristic
95	C₄×C₇₉	C₂²×SD₁₆	1	1	yes	characteristic
96	Dic₇₉	—	8	4	no	—
97	C₂×#2244	—	79	1	no	maximal, Sylow
98	C₂×C₇₉	#2244	1	4	yes	complemented
99	C₂×C₇₉	C₂×#486	1	1	yes	characteristic
100	C₂×C₇₉	C₂×#509	1	2	yes	—
101	D₇₉	—	16	4	no	—
102	C₂×#499	—	79	1	no	—
103	C₂×#493	—	79	1	no	—
104	#2244	—	79	8	no	—
105	C₂×#357	—	79	4	no	—
106	C₂×#346	D₇₉	1	1	yes	characteristic, complemented
107	C₇₉	C₂×#2244	1	1	yes	characteristic, complemented, Sylow
108	C₂²×D₈	—	79	2	no	—
109	C₂²×D₈	—	158	1	no	—
110	C₂×#127	—	158	1	no	—
111	C₂×#125	—	158	1	no	—
112	C₂²×C₁₆	C₂×D₇₉	1	2	yes	—
113	#499	—	79	4	no	—
114	#499	—	158	2	no	—
115	#493	—	79	4	no	—
116	#493	—	158	2	no	—
117	C₂×#107	—	79	2	no	—
118	C₂×#106	—	158	1	no	—
119	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
120	#357	—	79	16	no	—
121	#346	C₂×D₇₉	1	4	yes	complemented
122	C₂²×D₄	—	316	2	no	—
123	C₂²×D₄	—	158	2	no	—
124	C₂×D₈	—	316	4	no	—
125	C₂×D₈	—	79	4	no	—
126	C₂×D₈	—	158	2	no	—
127	C₂×D₈	—	158	4	no	—
128	C₂×D₈	—	158	8	no	—
129	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
130	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
131	#127	—	316	2	no	—
132	#127	—	158	4	no	—
133	#125	—	316	2	no	—
134	#125	—	158	4	no	—
135	C₂×C₄⋊C₄	—	316	2	no	—
136	C₂×C₄⋊C₄	—	158	2	no	—
137	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
138	C₂×C₁₆	—	2	4	no	—
139	C₂×C₁₆	C₂²×D₇₉	1	4	yes	—
140	#107	—	79	4	no	—
141	#107	—	158	2	no	—
142	#106	—	158	4	no	—
143	C₄×C₈	D₃₁₆	1	2	yes	—
144	C₄×C₈	C₂²×D₇₉	1	2	yes	—
145	C₂⁴	—	316	2	no	—
146	C₂×D₄	—	316	4	no	—
147	C₂×D₄	—	316	2	no	—
148	C₂×D₄	—	316	8	no	—
149	C₂×D₄	—	316	4	no	—
150	C₂×D₄	—	632	8	no	—
151	C₂×D₄	—	158	4	no	—
152	C₂²×C₄	—	316	2	no	—
153	C₂²×C₄	D₆₃₂	1	2	yes	—
154	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
155	D₈	—	316	4	no	—
156	D₈	—	316	8	no	—
157	D₈	—	158	8	no	—
158	C₂×C₈	#11929613	1	4	yes	—
159	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
160	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
161	C₂×C₈	#11929624	1	4	yes	—
162	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
163	C₄⋊C₄	—	316	8	no	—
164	C₄⋊C₄	—	316	2	no	—
165	C₄⋊C₄	—	158	4	no	—
166	C₄²	D₆₃₂	1	2	yes	—
167	C₄²	C₂×D₃₁₆	1	2	yes	—
168	C₁₆	—	2	4	no	—
169	C₂³	—	316	4	no	—
170	C₂³	—	632	4	no	—
171	C₂³	—	632	8	no	—
172	C₂³	C₂×D₆₃₂	1	1	yes	characteristic
173	D₄	—	316	4	no	—
174	D₄	—	316	4	no	—
175	D₄	—	632	16	no	—
176	D₄	—	632	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	316	4	no	—
179	C₂×C₄	—	632	4	no	—
180	C₂×C₄	#14579149	1	2	yes	—
181	C₂×C₄	C₂×D₆₃₂	1	4	yes	—
182	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
183	C₂×C₄	C₂×#11929613	1	2	yes	—
184	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
185	C₈	#14579149	1	2	yes	—
186	C₈	C₂×#11929613	1	4	yes	—
187	C₈	C₂×#11929624	1	2	yes	—
188	C₂²	—	632	4	no	—
189	C₂²	—	632	8	no	—
190	C₂²	—	1264	8	no	—
191	C₂²	#19098910	1	2	yes	—
192	C₂²	#19099001	1	4	yes	—
193	C₂²	C₂²×D₆₃₂	1	1	yes	characteristic
194	C₄	—	2	4	no	—
195	C₄	—	632	4	no	—
196	C₄	#19098910	1	2	yes	—
197	C₄	C₂×#14579149	1	1	yes	characteristic
198	C₄	C₂²×#11929613	1	1	yes	characteristic
199	C₂	—	1264	4	no	—
200	C₂	#23765642	1	4	yes	direct product factor
201	C₂	C₂×#19098910	1	1	yes	characteristic
202	C₂	C₂×#19099001	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19098965

C₂

yes

maximal, characteristic, complemented

C₂×#19098940

C₂

yes

maximal, characteristic

#23765642

C₂

yes

maximal, direct product factor

C₂×#19098365

C₂

yes

maximal, complemented

C₂×#346×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₆₃₂

—

C₂²×D₆₃₂

C₂²

yes

—

—

—

—

C₂²

yes

complemented

#19098940

—

#19098940

C₂²

yes

—

C₂×#14579095

C₂²

yes

—

C₂×#14579093

—

#19098365

C₂²

yes

complemented

C₂²×C₁₆×C₇₉

C₂²

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#346×C₇₉

C₂²

yes

complemented

C₂²×D₃₁₆

—

C₂²×D₃₁₆

—

C₂×D₆₃₂

—

C₂×D₆₃₂

—

C₂×D₆₃₂

—

C₂×D₆₃₂

—

C₂×D₆₃₂

C₂³

yes

—

—

—

—

—

C₂×C₄⋊Dic₇₉

—

C₂×Dic₇₉⋊C₄

—

#14579095

—

#14579095

C₂³

yes

—

#14579093

—

C₂²×C₈×C₇₉

D₄

yes

characteristic

C₂²×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

C₂³

yes

—

C₄×C₈×C₇₉

D₄

yes

—

C₄×C₈×C₇₉

C₂³

yes

—

C₂³×D₇₉

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂²×Dic₇₉

—

D₆₃₂

—

D₆₃₂

—

D₆₃₂

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₂²×C₄×C₇₉

D₈

yes

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

SD₁₆

yes

—

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

—

C₂×C₈×C₇₉

#41

yes

—

C₂×C₈×C₇₉

C₂⁴

yes

characteristic

C₄²×C₇₉

D₈

yes

—

C₄²×C₇₉

C₂×D₄

yes

—

C₁₆×C₇₉

—

C₂³×C₇₉

C₂×D₈

yes

characteristic

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#124

yes

—

C₂×C₄×C₇₉

C₂×D₈

yes

—

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

C₂×SD₁₆

yes

—

C₂×C₄×C₇₉

C₂²×D₄

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

C₈×C₇₉

#124

yes

—

C₈×C₇₉

C₂×SD₁₆

yes

—

C₈×C₇₉

C₂×#41

yes

—

C₂²×C₇₉

#486

yes

—

C₂²×C₇₉

#509

yes

—

C₂²×C₇₉

C₂²×D₈

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#486

yes

—

C₄×C₇₉

C₂×#124

yes

characteristic

C₄×C₇₉

C₂²×SD₁₆

yes

characteristic

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

100

C₂×C₇₉

C₂×#509

yes

—

101

D₇₉

—

102

C₂×#499

—

103

C₂×#493

—

104

#2244

—

105

C₂×#357

—

106

C₂×#346

D₇₉

yes

characteristic, complemented

107

C₇₉

C₂×#2244

yes

characteristic, complemented, Sylow

108

C₂²×D₈

—

109

C₂²×D₈

—

158

—

110

C₂×#127

—

158

—

111

C₂×#125

—

158

—

112

C₂²×C₁₆

C₂×D₇₉

yes

—

113

#499

—

114

#499

—

158

—

115

#493

—

116

#493

—

158

—

117

C₂×#107

—

118

C₂×#106

—

158

—

119

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

120

#357

—

121

#346

C₂×D₇₉

yes

complemented

122

C₂²×D₄

—

316

—

123

C₂²×D₄

—

158

—

124

C₂×D₈

—

316

—

125

C₂×D₈

—

126

C₂×D₈

—

158

—

127

C₂×D₈

—

158

—

128

C₂×D₈

—

158

—

129

C₂²×C₈

D₃₁₆

yes

characteristic

130

C₂²×C₈

C₂²×D₇₉

yes

characteristic

131

#127

—

316

—

132

#127

—

158

—

133

#125

—

316

—

134

#125

—

158

—

135

C₂×C₄⋊C₄

—

316

—

136

C₂×C₄⋊C₄

—

158

—

137

C₂×C₄²

D₃₁₆

yes

characteristic

138

C₂×C₁₆

—

139

C₂×C₁₆

C₂²×D₇₉

yes

—

140

#107

—

141

#107

—

158

—

142

#106

—

158

—

143

C₄×C₈

D₃₁₆

yes

—

144

C₄×C₈

C₂²×D₇₉

yes

—

145

C₂⁴

—

316

—

146

C₂×D₄

—

316

—

147

C₂×D₄

—

316

—

148

C₂×D₄

—

316

—

149

C₂×D₄

—

316

—

150

C₂×D₄

—

632

—

151

C₂×D₄

—

158

—

152

C₂²×C₄

—

316

—

153

C₂²×C₄

D₆₃₂

yes

—

154

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

155

D₈

—

316

—

156

D₈

—

316

—

157

D₈

—

158

—

158

C₂×C₈

#11929613

yes

—

159

C₂×C₈

C₂×D₃₁₆

yes

characteristic

160

C₂×C₈

C₂×D₃₁₆

yes

—

161

C₂×C₈

#11929624

yes

—

162

C₂×C₈

C₂³×D₇₉

yes

characteristic

163

C₄⋊C₄

—

316

—

164

C₄⋊C₄

—

316

—

165

C₄⋊C₄

—

158

—

166

C₄²

D₆₃₂

yes

—

167

C₄²

C₂×D₃₁₆

yes

—

168

C₁₆

—

169

C₂³

—

316

—

170

C₂³

—

632

—

171

C₂³

—

632

—

172

C₂³

C₂×D₆₃₂

yes

characteristic

173

D₄

—

316

—

174

D₄

—

316

—

175

D₄

—

632

—

176

D₄

—

632

—

177

C₂×C₄

—

178

C₂×C₄

—

316

—

179

C₂×C₄

—

632

—

180

C₂×C₄

#14579149

yes

—

181

C₂×C₄

C₂×D₆₃₂

yes

—

182

C₂×C₄

C₂×D₆₃₂

yes

characteristic

183

C₂×C₄

C₂×#11929613

yes

—

184

C₂×C₄

C₂²×D₃₁₆

yes

characteristic

185

C₈

#14579149

yes

—

186

C₈

C₂×#11929613

yes

—

187

C₈

C₂×#11929624

yes

—

188

C₂²

—

632

—

189

C₂²

—

632

—

190

C₂²

—

1264

—

191

C₂²

#19098910

yes

—

192

C₂²

#19099001

yes

—

193

C₂²

C₂²×D₆₃₂

yes

characteristic

194

C₄

—

195

C₄

—

632

—

196

C₄

#19098910

yes

—

197

C₄

C₂×#14579149

yes

characteristic

198

C₄

C₂²×#11929613

yes

characteristic

199

C₂

—

1264

—

200

C₂

#23765642

yes

direct product factor

201

C₂

C₂×#19098910

yes

characteristic

202

C₂

C₂×#19099001

yes

—

	subgroup	G/H
center	C₂³	C₂×D₆₃₂
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂×#346×C₇₉	C₂
socle	C₂³×C₇₉	C₂×D₈
prime radical	C₂×#19098940	C₂
hypercenter	C₂×#346	D₇₉
hypocenter	C₇₉	C₂×#2244
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂³

C₂×D₆₃₂

commutator, cosocle

C₂×C₈×C₇₉

C₂⁴

Frattini

C₂×C₈

C₂³×D₇₉

Fitting

C₂×#346×C₇₉

C₂

socle

C₂³×C₇₉

C₂×D₈

C₂

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
1. C₂×#19098965	C₂	semidirect
3. #23765642	C₂	direct (N × H)
4. C₂×#19098365	C₂	semidirect
5. C₂×#346×C₇₉	C₂	semidirect
11. #19098965	C₂²	semidirect
16. #19098365	C₂²	semidirect
19. #346×C₇₉	C₂²	semidirect
98. C₂×C₇₉	#2244	semidirect
106. C₂×#346	D₇₉	semidirect
107. C₇₉	C₂×#2244	semidirect
121. #346	C₂×D₇₉	semidirect
200. C₂	#23765642	direct (N × H)

H ≅ G/N

type

1. C₂×#19098965

C₂

semidirect

3. #23765642

C₂

direct (N × H)

4. C₂×#19098365

C₂

semidirect

5. C₂×#346×C₇₉

C₂

semidirect

11. #19098965

C₂²

semidirect

16. #19098365

C₂²

semidirect

19. #346×C₇₉

C₂²

semidirect

98. C₂×C₇₉

#2244

semidirect

106. C₂×#346

D₇₉

semidirect

107. C₇₉

C₂×#2244

semidirect

121. #346

C₂×D₇₉

semidirect

200. C₂

#23765642

direct (N × H)

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

subgroup	index
C₂×#19098965	2
C₂×#19098940	2
#23765642	2
C₂×#19098365	2
C₂×#346×C₇₉	2
C₂×#2244	79

subgroup

index

C₂×#346×C₇₉

Subgroups — C₂×#23765642 #34443582

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#23765642 #34443582

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23765642 #34443582