Group #34206307 — subgroups

Subgroups — #34206307

Order 20224.

318 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 131 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×#19098391	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098338	C₂	1	1	yes	maximal, characteristic
3	#23764131	C₂	1	2	yes	maximal
4	#23761491	C₂	1	2	yes	maximal
5	C₂×#366×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#14579095	C₂²	1	2	yes	—
8	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
9	#19098459	—	2	2	no	—
10	#19098391	C₄	1	4	yes	complemented
11	#19098338	—	2	2	no	—
12	#19098325	—	2	2	no	—
13	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
14	C₂×#107×C₇₉	C₂²	1	2	yes	—
15	#366×C₇₉	—	2	2	no	—
16	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
17	C₂×C₄⋊Dic₇₉	—	2	2	no	—
18	C₂×Dic₇₉⋊C₄	—	4	2	no	—
19	C₂×C₄×Dic₇₉	D₄	1	1	yes	characteristic
20	C₂×C₇₉⋊C₁₆	—	2	1	no	—
21	C₂×C₇₉⋊C₁₆	—	2	1	no	—
22	C₂×C₇₉⋊C₁₆	Q₈	1	2	yes	—
23	#14579095	—	2	2	no	—
24	#14579095	C₂×C₄	1	4	yes	—
25	C₈×Dic₇₉	—	2	1	no	—
26	C₈×Dic₇₉	—	2	1	no	—
27	C₇₉⋊#95	—	4	2	no	—
28	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
30	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
32	C₂×C₁₆×C₇₉	Q₈	1	2	yes	—
33	#107×C₇₉	—	2	2	no	—
34	#107×C₇₉	—	2	2	no	—
35	C₂²×Dic₇₉	—	4	2	no	—
36	C₂²×Dic₇₉	D₈	1	1	yes	characteristic
37	C₂²×Dic₇₉	D₈	1	1	yes	characteristic
38	C₂×C₇₉⋊C₈	—	2	1	no	—
39	C₂×C₇₉⋊C₈	—	2	1	no	—
40	C₂×C₇₉⋊C₈	Q₁₆	1	2	yes	—
41	C₄⋊Dic₇₉	—	2	4	no	—
42	C₄⋊Dic₇₉	—	4	2	no	—
43	Dic₇₉⋊C₄	—	8	4	no	—
44	C₄×Dic₇₉	—	2	1	no	—
45	C₄×Dic₇₉	—	2	1	no	—
46	C₇₉⋊C₁₆	—	2	2	no	—
47	C₂²×C₄×C₇₉	—	4	2	no	—
48	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
54	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
55	C₄⋊C₄×C₇₉	—	4	2	no	—
56	C₄⋊C₄×C₇₉	—	4	2	no	—
57	C₁₆×C₇₉	C₄⋊C₄	1	4	yes	complemented
58	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₇₉	—	8	2	no	—
60	C₂×C₄×C₇₉	—	8	2	no	—
61	C₂×C₄×C₇₉	—	8	2	no	—
62	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
64	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
67	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
68	C₂×Dic₇₉	—	2	1	no	—
69	C₂×Dic₇₉	—	2	1	no	—
70	C₂×Dic₇₉	—	2	2	no	—
71	C₂×Dic₇₉	—	2	2	no	—
72	C₂×Dic₇₉	—	4	4	no	—
73	C₂×Dic₇₉	—	8	2	no	—
74	C₂×Dic₇₉	—	8	2	no	—
75	C₈×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₈×C₇₉	C₄×Q₈	1	1	yes	characteristic
77	C₈×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
78	C₈×C₇₉	#125	1	1	yes	characteristic
79	C₇₉⋊C₈	—	2	2	no	—
80	C₂²×C₇₉	C₄×D₈	1	1	yes	characteristic
81	C₂²×C₇₉	#466	1	1	yes	characteristic
82	C₂²×C₇₉	#500	1	1	yes	characteristic
83	C₂²×C₇₉	#509	1	2	yes	—
84	C₂²×C₇₉	#510	1	2	yes	—
85	C₄×C₇₉	—	16	2	no	—
86	C₄×C₇₉	#389	1	1	yes	characteristic
87	C₄×C₇₉	C₄×Q₁₆	1	1	yes	characteristic
88	C₄×C₇₉	#467	1	1	yes	characteristic
89	C₄×C₇₉	#499	1	1	yes	characteristic
90	Dic₇₉	—	4	1	no	—
91	Dic₇₉	—	4	1	no	—
92	Dic₇₉	—	8	4	no	—
93	#13167	—	79	1	no	maximal, Sylow
94	C₂×C₇₉	#1942	1	1	yes	characteristic
95	C₂×C₇₉	#2175	1	2	yes	—
96	C₂×C₇₉	#2218	1	1	yes	characteristic
97	C₂×C₇₉	#2219	1	1	yes	characteristic
98	C₂×C₇₉	#2255	1	2	yes	—
99	C₂×#367	—	79	1	no	—
100	C₂×#366	D₇₉	1	1	yes	characteristic
101	C₂×#346	—	79	1	no	—
102	#1942	—	79	2	no	—
103	#1380	—	79	2	no	—
104	C₇₉	#13167	1	1	yes	characteristic, complemented, Sylow
105	C₂²×C₁₆	—	79	1	no	—
106	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
107	C₂×#107	—	79	2	no	—
108	C₂×#107	C₂×D₇₉	1	2	yes	—
109	C₂×C₄×C₈	—	79	1	no	—
110	#389	—	158	2	no	—
111	#367	—	79	4	no	—
112	#366	—	2	2	no	—
113	#346	—	158	2	no	—
114	#340	—	158	2	no	—
115	C₂²×C₈	—	79	1	no	—
116	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	—	316	2	no	—
119	C₂×C₄⋊C₄	—	158	2	no	—
120	C₂×C₄²	—	79	1	no	—
121	C₂×C₁₆	—	79	2	no	—
122	C₂×C₁₆	—	158	1	no	—
123	C₂×C₁₆	—	158	1	no	—
124	C₂×C₁₆	Dic₁₅₈	1	2	yes	—
125	C₂×C₁₆	C₄×D₇₉	1	2	yes	—
126	C₂×C₁₆	C₂²⋊D₇₉	1	2	yes	—
127	#107	—	2	2	no	—
128	#107	—	2	2	no	—
129	#107	—	79	4	no	—
130	#107	—	158	2	no	—
131	C₄×C₈	—	158	1	no	—
132	C₄×C₈	—	158	1	no	—
133	#95	—	316	2	no	—
134	C₂²×C₄	—	4	2	no	—
135	C₂²×C₄	—	316	2	no	—
136	C₂²×C₄	—	79	1	no	—
137	C₂²×C₄	—	79	1	no	—
138	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
139	C₂×C₈	—	79	2	no	—
140	C₂×C₈	—	158	1	no	—
141	C₂×C₈	—	158	1	no	—
142	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
143	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
144	C₂×C₈	Q₈×D₇₉	1	1	yes	characteristic
145	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
146	C₂×C₈	#11929623	1	1	yes	characteristic
147	C₂×C₈	#11929625	1	1	yes	characteristic
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	316	2	no	—
151	C₄⋊C₄	—	632	4	no	—
152	C₄⋊C₄	—	158	4	no	—
153	C₄²	—	158	1	no	—
154	C₄²	—	158	1	no	—
155	C₁₆	—	158	2	no	—
156	C₁₆	Dic₇₉⋊C₄	1	4	yes	complemented
157	C₂³	D₈×D₇₉	1	1	yes	characteristic
158	C₂×C₄	—	8	2	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	—	316	4	no	—
162	C₂×C₄	—	632	2	no	—
163	C₂×C₄	—	632	2	no	—
164	C₂×C₄	—	158	2	no	—
165	C₂×C₄	—	158	2	no	—
166	C₂×C₄	—	158	1	no	—
167	C₂×C₄	—	158	1	no	—
168	C₂×C₄	#14579123	1	1	yes	characteristic
169	C₂×C₄	#14579132	1	1	yes	characteristic
170	C₂×C₄	#14579160	1	1	yes	characteristic
171	C₂×C₄	Q₁₆×D₇₉	1	1	yes	characteristic
172	C₂×C₄	#14579180	1	1	yes	characteristic
173	C₂×C₄	#14579184	1	1	yes	characteristic
174	C₈	—	158	2	no	—
175	C₈	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
176	C₈	#14579126	1	1	yes	characteristic
177	C₈	#14579137	1	1	yes	characteristic
178	C₈	#14579152	1	1	yes	characteristic
179	C₂²	#19098645	1	1	yes	characteristic
180	C₂²	#19098796	1	1	yes	characteristic
181	C₂²	#19098967	1	1	yes	characteristic
182	C₂²	#19098997	1	2	yes	—
183	C₂²	#19099005	1	2	yes	—
184	C₄	—	16	2	no	—
185	C₄	—	316	1	no	—
186	C₄	—	316	1	no	—
187	C₄	—	632	4	no	—
188	C₄	#19098459	1	1	yes	characteristic
189	C₄	#19098657	1	1	yes	characteristic
190	C₄	#19098801	1	1	yes	characteristic
191	C₄	#19098964	1	1	yes	characteristic
192	C₂	#23764131	1	1	yes	characteristic
193	C₂	#23765263	1	2	yes	—
194	C₂	#23765500	1	1	yes	characteristic
195	C₂	#23765505	1	1	yes	characteristic
196	C₂	#23765684	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19098391

C₂

yes

maximal, characteristic

C₂×#19098338

C₂

yes

maximal, characteristic

#23764131

C₂

yes

maximal

#23761491

C₂

yes

maximal

C₂×#366×C₇₉

C₂

yes

maximal, characteristic

C₂²×C₇₉⋊C₁₆

C₂²

yes

characteristic

C₂×#14579095

C₂²

yes

—

C₂×C₈×Dic₇₉

C₂²

yes

characteristic

#19098459

—

#19098391

C₄

yes

complemented

#19098338

—

#19098325

—

C₂²×C₁₆×C₇₉

C₂²

yes

characteristic

C₂×#107×C₇₉

C₂²

yes

—

#366×C₇₉

—

C₂²×C₇₉⋊C₈

D₄

yes

characteristic

C₂×C₄⋊Dic₇₉

—

C₂×Dic₇₉⋊C₄

—

C₂×C₄×Dic₇₉

D₄

yes

characteristic

C₂×C₇₉⋊C₁₆

—

C₂×C₇₉⋊C₁₆

—

C₂×C₇₉⋊C₁₆

Q₈

yes

—

#14579095

—

#14579095

C₂×C₄

yes

—

C₈×Dic₇₉

—

C₈×Dic₇₉

—

C₇₉⋊#95

—

C₂²×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₇₉

—

C₂×C₁₆×C₇₉

C₂×C₄

yes

—

C₂×C₁₆×C₇₉

D₄

yes

—

C₂×C₁₆×C₇₉

Q₈

yes

—

#107×C₇₉

—

#107×C₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

D₈

yes

characteristic

C₂²×Dic₇₉

D₈

yes

characteristic

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

Q₁₆

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₉

C₂×Q₈

yes

C₂×C₈×C₇₉

yes

C₂×C₈×C₇₉

yes

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₁₆×C₇₉

C₄⋊C₄

yes

complemented

C₂³×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₄×D₄

yes

characteristic

C₂×C₄×C₇₉

#121

yes

characteristic

C₂×C₄×C₇₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₁₆

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₈×C₇₉

C₄×Q₈

yes

characteristic

C₈×C₇₉

C₂²⋊Q₈

yes

characteristic

C₈×C₇₉

#125

yes

characteristic

C₇₉⋊C₈

—

C₂²×C₇₉

C₄×D₈

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

—

C₂²×C₇₉

#510

yes

—

C₄×C₇₉

—

C₄×C₇₉

#389

yes

characteristic

C₄×C₇₉

C₄×Q₁₆

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₂×#367

—

100

C₂×#366

D₇₉

yes

characteristic

101

C₂×#346

—

102

#1942

—

103

#1380

—

104

C₇₉

#13167

yes

characteristic, complemented, Sylow

105

C₂²×C₁₆

—

106

C₂²×C₁₆

C₂×D₇₉

yes

characteristic

107

C₂×#107

—

108

C₂×#107

C₂×D₇₉

yes

—

109

C₂×C₄×C₈

—

110

#389

—

158

—

111

#367

—

112

#366

—

113

#346

—

158

—

114

#340

—

158

—

115

C₂²×C₈

—

116

C₂²×C₈

C₂²×D₇₉

yes

characteristic

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

316

—

119

C₂×C₄⋊C₄

—

158

—

120

C₂×C₄²

—

121

C₂×C₁₆

—

122

C₂×C₁₆

—

158

—

123

C₂×C₁₆

—

158

—

124

C₂×C₁₆

Dic₁₅₈

yes

—

125

C₂×C₁₆

C₄×D₇₉

yes

—

126

C₂×C₁₆

C₂²⋊D₇₉

yes

—

127

#107

—

128

#107

—

129

#107

—

130

#107

—

158

—

131

C₄×C₈

—

158

—

132

C₄×C₈

—

158

—

133

#95

—

316

—

134

C₂²×C₄

—

135

C₂²×C₄

—

316

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

D₄×D₇₉

yes

characteristic

139

C₂×C₈

—

140

C₂×C₈

—

158

—

141

C₂×C₈

—

158

—

142

C₂×C₈

C₂×C₄×D₇₉

yes

characteristic

143

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

144

C₂×C₈

Q₈×D₇₉

yes

characteristic

145

C₂×C₈

C₂×Dic₁₅₈

yes

characteristic

146

C₂×C₈

#11929623

yes

characteristic

147

C₂×C₈

#11929625

yes

characteristic

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

316

—

151

C₄⋊C₄

—

632

—

152

C₄⋊C₄

—

158

—

153

C₄²

—

158

—

154

C₄²

—

158

—

155

C₁₆

—

158

—

156

C₁₆

Dic₇₉⋊C₄

yes

complemented

157

C₂³

D₈×D₇₉

yes

characteristic

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

316

—

162

C₂×C₄

—

632

—

163

C₂×C₄

—

632

—

164

C₂×C₄

—

158

—

165

C₂×C₄

—

158

—

166

C₂×C₄

—

158

—

167

C₂×C₄

—

158

—

168

C₂×C₄

#14579123

yes

characteristic

169

C₂×C₄

#14579132

yes

characteristic

170

C₂×C₄

#14579160

yes

characteristic

171

C₂×C₄

Q₁₆×D₇₉

yes

characteristic

172

C₂×C₄

#14579180

yes

characteristic

173

C₂×C₄

#14579184

yes

characteristic

174

C₈

—

158

—

175

C₈

C₂×Dic₇₉⋊C₄

yes

characteristic

176

C₈

#14579126

yes

characteristic

177

C₈

#14579137

yes

characteristic

178

C₈

#14579152

yes

characteristic

179

C₂²

#19098645

yes

characteristic

180

C₂²

#19098796

yes

characteristic

181

C₂²

#19098967

yes

characteristic

182

C₂²

#19098997

yes

—

183

C₂²

#19099005

yes

—

184

C₄

—

185

C₄

—

316

—

186

C₄

—

316

—

187

C₄

—

632

—

188

C₄

#19098459

yes

characteristic

189

C₄

#19098657

yes

characteristic

190

C₄

#19098801

yes

characteristic

191

C₄

#19098964

yes

characteristic

192

C₂

#23764131

yes

characteristic

193

C₂

#23765263

yes

—

194

C₂

#23765500

yes

characteristic

195

C₂

#23765505

yes

characteristic

196

C₂

#23765684

yes

—

	subgroup	G/H
center	C₂³	D₈×D₇₉
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#366×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#366	D₇₉
hypocenter	C₇₉	#13167
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

subgroup

G/H

center

C₂³

D₈×D₇₉

commutator

C₂×C₈×C₇₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₂×#366×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂×D₈

cosocle

C₂²×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

N	H ≅ G/N	type
10. #19098391	C₄	semidirect
57. C₁₆×C₇₉	C₄⋊C₄	semidirect
104. C₇₉	#13167	semidirect
156. C₁₆	Dic₇₉⋊C₄	semidirect

H ≅ G/N

type

10. #19098391

C₄

semidirect

57. C₁₆×C₇₉

C₄⋊C₄

semidirect

104. C₇₉

#13167

semidirect

156. C₁₆

Dic₇₉⋊C₄

semidirect

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

subgroup	index
C₂×#19098391	2
C₂×#19098338	2
#23764131	2
#23761491	2
C₂×#366×C₇₉	2
#13167	79

subgroup

index

C₂×#366×C₇₉

Subgroups — #34206307

Conjugacy classes of subgroups

Complements

Maximal subgroups