Group #34441646 — subgroups

Subgroups — #34441646

Order 20224.

646 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 211 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	C₂²×Dic₆₃₂	C₂	1	1	yes	maximal, characteristic
2	#23772267	C₂	1	2	yes	maximal
3	#23765270	C₂	1	4	yes	maximal, complemented
4	#23765258	C₂	1	4	yes	maximal, complemented
5	#23765119	C₂	1	2	yes	maximal
6	#23764996	C₂	1	1	yes	maximal, characteristic
7	C₇₉×#2158	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₃₁₆	C₂²	1	2	yes	—
9	#19099040	—	2	2	no	—
10	C₂×Dic₆₃₂	C₄	1	8	yes	complemented
11	C₂×Dic₆₃₂	C₂²	1	2	yes	—
12	C₂×Dic₆₃₂	C₂²	1	2	yes	—
13	C₂×#14579101	C₂²	1	1	yes	characteristic
14	#19098669	—	2	4	no	—
15	#19098657	C₂²	1	8	yes	—
16	#19098571	—	2	2	no	—
17	#19098518	C₂²	1	1	yes	characteristic
18	#19098366	C₂²	1	8	yes	—
19	#19098338	C₂²	1	2	yes	—
20	C₁₆×Dic₇₉	C₂²	1	2	yes	—
21	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
22	#427×C₇₉	C₂²	1	2	yes	—
23	#367×C₇₉	C₂²	1	2	yes	—
24	#366×C₇₉	C₂²	1	2	yes	—
25	C₂²×Dic₁₅₈	—	2	2	no	—
26	C₂×Dic₃₁₆	—	2	2	no	—
27	C₂×Dic₃₁₆	—	2	4	no	—
28	C₂×Dic₃₁₆	C₂×C₄	1	8	yes	—
29	C₂×Dic₃₁₆	C₂³	1	4	yes	—
30	C₂×#11929608	D₄	1	1	yes	characteristic
31	C₄×Dic₁₅₈	—	4	4	no	—
32	#14579126	—	2	8	no	—
33	#14579120	D₄	1	1	yes	characteristic
34	#14579117	—	4	4	no	—
35	Dic₆₃₂	C₂×C₄	1	8	yes	—
36	Dic₆₃₂	C₂×C₄	1	8	yes	—
37	#14579101	—	2	2	no	—
38	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
39	#14579084	—	2	8	no	—
40	#14579071	D₄	1	2	yes	—
41	C₈×Dic₇₉	C₂³	1	2	yes	—
42	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
43	#117×C₇₉	—	2	2	no	—
44	C₂×C₁₆×C₇₉	C₂×C₄	1	4	yes	—
45	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
46	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
47	#107×C₇₉	C₂³	1	4	yes	—
48	#106×C₇₉	—	2	2	no	—
49	C₂×Dic₁₅₈	—	2	8	no	—
50	C₂×Dic₁₅₈	—	2	4	no	—
51	C₂×Dic₁₅₈	—	4	2	no	—
52	C₂×Dic₁₅₈	—	4	4	no	—
53	C₂²×Dic₇₉	—	4	2	no	—
54	Dic₃₁₆	—	2	8	no	—
55	Dic₃₁₆	—	2	4	no	—
56	Dic₃₁₆	C₂²×C₄	1	8	yes	—
57	#11929608	—	2	2	no	—
58	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
59	C₄⋊Dic₇₉	D₈	1	2	yes	—
60	Dic₇₉⋊C₄	—	4	8	no	—
61	C₂²⋊Dic₇₉	D₈	1	2	yes	—
62	C₄×Dic₇₉	—	4	8	no	—
63	C₄×Dic₇₉	C₂×D₄	1	2	yes	—
64	C₇₉⋊C₁₆	—	2	2	no	—
65	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₇₉	#41	1	2	yes	—
69	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
70	C₄⋊C₄×C₇₉	—	2	4	no	—
71	C₂²⋊C₄×C₇₉	—	4	2	no	—
72	C₄²×C₇₉	—	4	2	no	—
73	C₁₆×C₇₉	C₂²×C₄	1	2	yes	—
74	C₁₆×C₇₉	C₂²×C₄	1	2	yes	—
75	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₇₉	—	4	4	no	—
77	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
78	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₇₉	#135	1	2	yes	—
80	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
81	C₂×Dic₇₉	—	4	4	no	—
82	C₂×Dic₇₉	—	4	8	no	—
83	C₂×Dic₇₉	C₂×D₈	1	4	yes	—
84	Dic₁₅₈	—	2	8	no	—
85	Dic₁₅₈	—	4	4	no	—
86	Dic₁₅₈	—	4	8	no	—
87	C₈×C₇₉	C₄×D₄	1	2	yes	—
88	C₈×C₇₉	C₂³×C₄	1	1	yes	characteristic
89	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
90	C₇₉⋊C₈	—	2	2	no	—
91	C₂²×C₇₉	—	2	2	no	—
92	C₂²×C₇₉	C₄×D₈	1	2	yes	—
93	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
94	C₄×C₇₉	—	8	4	no	—
95	C₄×C₇₉	C₄×D₈	1	2	yes	—
96	C₄×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
97	C₄×C₇₉	C₂×#135	1	1	yes	characteristic
98	Dic₇₉	—	2	4	no	—
99	Dic₇₉	—	4	8	no	—
100	#33458	—	79	1	no	maximal, Sylow
101	C₂×C₇₉	—	2	2	no	—
102	C₂×C₇₉	C₂×C₄×D₈	1	1	yes	characteristic
103	C₂×C₇₉	#3417	1	2	yes	—
104	C₂²×Q₃₂	—	79	1	no	—
105	#2950	—	79	2	no	—
106	#2176	—	79	4	no	—
107	C₄×Q₃₂	—	79	4	no	—
108	#2158	D₇₉	1	1	yes	characteristic
109	#2141	—	79	2	no	—
110	#2108	—	79	1	no	—
111	C₇₉	#33458	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₁₆	—	79	2	no	—
113	#519	—	158	2	no	—
114	C₂×Q₃₂	—	79	2	no	—
115	C₂×Q₃₂	—	79	2	no	—
116	C₂×Q₃₂	—	79	8	no	—
117	C₂×M₃₂	—	79	1	no	—
118	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
119	#441	—	158	4	no	—
120	C₄×Q₁₆	—	79	8	no	—
121	#427	C₂×D₇₉	1	2	yes	—
122	#419	—	158	2	no	—
123	#405	—	79	1	no	—
124	#367	C₂×D₇₉	1	2	yes	—
125	#366	C₂×D₇₉	1	2	yes	—
126	#358	—	79	8	no	—
127	#346	—	79	2	no	—
128	C₄×C₁₆	—	79	2	no	—
129	C₂²×Q₈	—	158	2	no	—
130	C₂×Q₁₆	—	79	8	no	—
131	C₂×Q₁₆	—	79	4	no	—
132	C₂×Q₁₆	—	158	2	no	—
133	C₂×Q₁₆	—	158	4	no	—
134	C₂×M₁₆	—	79	1	no	—
135	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
136	C₄×Q₈	—	316	4	no	—
137	C₄×Q₈	—	158	8	no	—
138	#117	—	2	2	no	—
139	#117	—	316	4	no	—
140	#117	—	79	1	no	—
141	Q₃₂	—	79	8	no	—
142	Q₃₂	—	79	8	no	—
143	M₃₂	—	158	2	no	—
144	C₂×C₁₆	—	79	2	no	—
145	C₂×C₁₆	C₄×D₇₉	1	4	yes	—
146	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
147	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
148	#107	C₂²×D₇₉	1	4	yes	—
149	#106	—	2	2	no	—
150	#103	—	158	8	no	—
151	#97	—	79	2	no	—
152	C₄×C₈	—	79	2	no	—
153	C₂×Q₈	—	316	2	no	—
154	C₂×Q₈	—	316	4	no	—
155	C₂×Q₈	—	158	8	no	—
156	C₂×Q₈	—	158	4	no	—
157	C₂²×C₄	—	316	2	no	—
158	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
159	Q₁₆	—	79	8	no	—
160	Q₁₆	—	158	4	no	—
161	Q₁₆	—	158	8	no	—
162	M₁₆	—	158	2	no	—
163	C₂×C₈	—	79	2	no	—
164	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
165	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
166	C₂×C₈	#11929625	1	2	yes	—
167	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
168	C₄⋊C₄	—	2	4	no	—
169	C₄⋊C₄	—	316	8	no	—
170	C₄⋊C₄	—	79	2	no	—
171	C₂²⋊C₄	—	4	2	no	—
172	C₂²⋊C₄	—	79	2	no	—
173	C₄²	—	4	2	no	—
174	C₄²	—	316	8	no	—
175	C₄²	—	79	2	no	—
176	C₁₆	—	158	2	no	—
177	C₁₆	C₂×C₄×D₇₉	1	2	yes	—
178	C₁₆	C₂×C₄×D₇₉	1	2	yes	—
179	C₂³	D₈×D₇₉	1	1	yes	characteristic
180	Q₈	—	316	8	no	—
181	Q₈	—	316	4	no	—
182	Q₈	—	158	8	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	316	8	no	—
185	C₂×C₄	—	316	4	no	—
186	C₂×C₄	—	79	4	no	—
187	C₂×C₄	#14579123	1	2	yes	—
188	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
189	C₂×C₄	#14579184	1	2	yes	—
190	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
191	C₈	—	158	2	no	—
192	C₈	#14579123	1	2	yes	—
193	C₈	C₂²×C₄×D₇₉	1	1	yes	characteristic
194	C₈	C₂×#11929625	1	1	yes	characteristic
195	C₂²	—	2	2	no	—
196	C₂²	#19098645	1	2	yes	—
197	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
198	C₄	—	8	4	no	—
199	C₄	—	316	8	no	—
200	C₄	—	158	4	no	—
201	C₄	#19098645	1	2	yes	—
202	C₄	C₂×#14579123	1	1	yes	characteristic
203	C₄	C₂×#14579184	1	1	yes	characteristic
204	C₂	—	2	2	no	—
205	C₂	C₂×#19098645	1	1	yes	characteristic
206	C₂	#23776550	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₆₃₂

C₂

yes

maximal, characteristic

#23772267

C₂

yes

maximal

#23765270

C₂

yes

maximal, complemented

#23765258

C₂

yes

maximal, complemented

#23765119

C₂

yes

maximal

#23764996

C₂

yes

maximal, characteristic

C₇₉×#2158

C₂

yes

maximal, characteristic

C₂²×Dic₃₁₆

C₂²

yes

—

#19099040

—

C₂×Dic₆₃₂

C₄

yes

complemented

C₂×Dic₆₃₂

C₂²

yes

—

C₂×Dic₆₃₂

C₂²

yes

—

C₂×#14579101

C₂²

yes

characteristic

#19098669

—

#19098657

C₂²

yes

—

#19098571

—

#19098518

C₂²

yes

characteristic

#19098366

C₂²

yes

—

#19098338

C₂²

yes

—

C₁₆×Dic₇₉

C₂²

yes

—

C₂²×C₁₆×C₇₉

C₂²

yes

characteristic

#427×C₇₉

C₂²

yes

—

#367×C₇₉

C₂²

yes

—

#366×C₇₉

C₂²

yes

—

C₂²×Dic₁₅₈

—

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

C₂×C₄

yes

—

C₂×Dic₃₁₆

C₂³

yes

—

C₂×#11929608

D₄

yes

characteristic

C₄×Dic₁₅₈

—

#14579126

—

#14579120

D₄

yes

characteristic

#14579117

—

Dic₆₃₂

C₂×C₄

yes

—

Dic₆₃₂

C₂×C₄

yes

—

#14579101

—

C₂×C₇₉⋊C₁₆

C₂³

yes

—

#14579084

—

#14579071

D₄

yes

—

C₈×Dic₇₉

C₂³

yes

—

C₂²×C₈×C₇₉

C₂³

yes

characteristic

#117×C₇₉

—

C₂×C₁₆×C₇₉

C₂×C₄

yes

—

C₂×C₁₆×C₇₉

C₂³

yes

characteristic

C₂×C₁₆×C₇₉

C₂³

yes

characteristic

#107×C₇₉

C₂³

yes

—

#106×C₇₉

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂²×Dic₇₉

—

Dic₃₁₆

—

Dic₃₁₆

—

Dic₃₁₆

C₂²×C₄

yes

—

#11929608

—

C₂×C₇₉⋊C₈

C₂×D₄

yes

—

C₄⋊Dic₇₉

D₈

yes

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

D₈

yes

—

C₄×Dic₇₉

—

C₄×Dic₇₉

C₂×D₄

yes

—

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂²×C₄

yes

—

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

#41

yes

—

C₂×C₈×C₇₉

C₂⁴

yes

characteristic

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₁₆×C₇₉

C₂²×C₄

yes

—

C₁₆×C₇₉

C₂²×C₄

yes

—

C₂³×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₄×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

#135

yes

—

C₂×C₄×C₇₉

C₂²×D₄

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

C₂×D₈

yes

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

C₄×D₄

yes

—

C₈×C₇₉

C₂³×C₄

yes

characteristic

C₈×C₇₉

C₂×#41

yes

characteristic

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

C₄×D₈

yes

—

C₂²×C₇₉

C₂²×D₈

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

C₄×D₈

yes

—

C₄×C₇₉

C₂×C₄×D₄

yes

characteristic

C₄×C₇₉

C₂×#135

yes

characteristic

Dic₇₉

—

Dic₇₉

—

100

#33458

—

maximal, Sylow

101

C₂×C₇₉

—

102

C₂×C₇₉

C₂×C₄×D₈

yes

characteristic

103

C₂×C₇₉

#3417

yes

—

104

C₂²×Q₃₂

—

105

#2950

—

106

#2176

—

107

C₄×Q₃₂

—

108

#2158

D₇₉

yes

characteristic

109

#2141

—

110

#2108

—

111

C₇₉

#33458

yes

characteristic, complemented, Sylow

112

C₂²×Q₁₆

—

113

#519

—

158

—

114

C₂×Q₃₂

—

115

C₂×Q₃₂

—

116

C₂×Q₃₂

—

117

C₂×M₃₂

—

118

C₂²×C₁₆

C₂×D₇₉

yes

characteristic

119

#441

—

158

—

120

C₄×Q₁₆

—

121

#427

C₂×D₇₉

yes

—

122

#419

—

158

—

123

#405

—

124

#367

C₂×D₇₉

yes

—

125

#366

C₂×D₇₉

yes

—

126

#358

—

127

#346

—

128

C₄×C₁₆

—

129

C₂²×Q₈

—

158

—

130

C₂×Q₁₆

—

131

C₂×Q₁₆

—

132

C₂×Q₁₆

—

158

—

133

C₂×Q₁₆

—

158

—

134

C₂×M₁₆

—

135

C₂²×C₈

C₂²×D₇₉

yes

characteristic

136

C₄×Q₈

—

316

—

137

C₄×Q₈

—

158

—

138

#117

—

139

#117

—

316

—

140

#117

—

141

Q₃₂

—

142

Q₃₂

—

143

M₃₂

—

158

—

144

C₂×C₁₆

—

145

C₂×C₁₆

C₄×D₇₉

yes

—

146

C₂×C₁₆

C₂²×D₇₉

yes

characteristic

147

C₂×C₁₆

C₂²×D₇₉

yes

characteristic

148

#107

C₂²×D₇₉

yes

—

149

#106

—

150

#103

—

158

—

151

#97

—

152

C₄×C₈

—

153

C₂×Q₈

—

316

—

154

C₂×Q₈

—

316

—

155

C₂×Q₈

—

158

—

156

C₂×Q₈

—

158

—

157

C₂²×C₄

—

316

—

158

C₂²×C₄

D₄×D₇₉

yes

characteristic

159

Q₁₆

—

160

Q₁₆

—

158

—

161

Q₁₆

—

158

—

162

M₁₆

—

158

—

163

C₂×C₈

—

164

C₂×C₈

C₂×C₄×D₇₉

yes

—

165

C₂×C₈

D₄×D₇₉

yes

characteristic

166

C₂×C₈

#11929625

yes

—

167

C₂×C₈

C₂³×D₇₉

yes

characteristic

168

C₄⋊C₄

—

169

C₄⋊C₄

—

316

—

170

C₄⋊C₄

—

171

C₂²⋊C₄

—

172

C₂²⋊C₄

—

173

C₄²

—

174

C₄²

—

316

—

175

C₄²

—

176

C₁₆

—

158

—

177

C₁₆

C₂×C₄×D₇₉

yes

—

178

C₁₆

C₂×C₄×D₇₉

yes

—

179

C₂³

D₈×D₇₉

yes

characteristic

180

Q₈

—

316

—

181

Q₈

—

316

—

182

Q₈

—

158

—

183

C₂×C₄

—

184

C₂×C₄

—

316

—

185

C₂×C₄

—

316

—

186

C₂×C₄

—

187

C₂×C₄

#14579123

yes

—

188

C₂×C₄

D₈×D₇₉

yes

characteristic

189

C₂×C₄

#14579184

yes

—

190

C₂×C₄

C₂×D₄×D₇₉

yes

characteristic

191

C₈

—

158

—

192

C₈

#14579123

yes

—

193

C₈

C₂²×C₄×D₇₉

yes

characteristic

194

C₈

C₂×#11929625

yes

characteristic

195

C₂²

—

196

C₂²

#19098645

yes

—

197

C₂²

C₂×D₈×D₇₉

yes

characteristic

198

C₄

—

199

C₄

—

316

—

200

C₄

—

158

—

201

C₄

#19098645

yes

—

202

C₄

C₂×#14579123

yes

characteristic

203

C₄

C₂×#14579184

yes

characteristic

204

C₂

—

205

C₂

C₂×#19098645

yes

characteristic

206

C₂

#23776550

yes

—

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₉
commutator	C₈×C₇₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₇₉×#2158	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	C₂³×C₇₉	C₂×D₈
cosocle	C₂×C₈×C₇₉	C₂⁴
hypercenter	#2158	D₇₉
hypocenter	C₇₉	#33458
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

subgroup

G/H

center

C₂²

C₂×D₈×D₇₉

commutator

C₈×C₇₉

C₂³×C₄

Frattini

C₂×C₈

C₂³×D₇₉

Fitting

C₇₉×#2158

C₂

socle

C₂²×C₇₉

C₂²×D₈

prime radical

C₂³×C₇₉

C₂×D₈

cosocle

C₂×C₈×C₇₉

C₂⁴

D₇₉

C₇₉

G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

N	H ≅ G/N	type
3. #23765270	C₂	semidirect
4. #23765258	C₂	semidirect
10. C₂×Dic₆₃₂	C₄	semidirect
111. C₇₉	#33458	semidirect

H ≅ G/N

type

3. #23765270

C₂

semidirect

4. #23765258

C₂

semidirect

10. C₂×Dic₆₃₂

C₄

semidirect

111. C₇₉

#33458

semidirect

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

subgroup	index
C₂²×Dic₆₃₂	2
#23772267	2
#23765270	2
#23765258	2
#23765119	2
#23764996	2
C₇₉×#2158	2
#33458	79

subgroup

index

C₂²×Dic₆₃₂

C₇₉×#2158

Subgroups — #34441646

Conjugacy classes of subgroups

Complements

Maximal subgroups