Group #34206813 — subgroups

Subgroups — #34206813

Order 20224.

308 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 115 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	#23772338	C₂	1	1	yes	maximal, characteristic
2	#23765166	C₂	1	1	yes	maximal, characteristic
3	#23761485	C₂	1	2	yes	maximal
4	#23761479	C₂	1	2	yes	maximal
5	C₇₉×#2110	C₂	1	1	yes	maximal, characteristic
6	#19099116	C₄	1	1	yes	characteristic
7	C₂×#14579101	C₂²	1	2	yes	—
8	#19098693	—	2	2	no	—
9	#19098691	—	2	2	no	—
10	#19098591	C₄	1	1	yes	characteristic
11	#19098590	C₄	1	1	yes	characteristic
12	#19098586	C₄	1	1	yes	characteristic
13	#19098518	C₂²	1	1	yes	characteristic
14	C₂×#14579071	C₂²	1	1	yes	characteristic
15	#19098384	—	2	2	no	—
16	C₂×M₃₂×C₇₉	C₂²	1	2	yes	—
17	#405×C₇₉	C₂²	1	1	yes	characteristic
18	#347×C₇₉	—	2	2	no	—
19	C₂×#11929608	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
21	#14579160	—	2	1	no	—
22	#14579152	—	2	1	no	—
23	#14579138	—	2	2	no	—
24	C₄×Dic₁₅₈	—	2	2	no	—
25	#14579120	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
27	#14579101	—	2	2	no	—
28	#14579101	—	2	2	no	—
29	C₂×C₇₉⋊C₁₆	—	2	2	no	—
30	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
31	Dic₇₉⋊C₈	—	2	2	no	—
32	Dic₇₉⋊C₈	—	2	2	no	—
33	#14579071	—	2	1	no	—
34	#14579071	—	2	1	no	—
35	#14579071	—	2	1	no	—
36	C₈×Dic₇₉	—	2	1	no	—
37	C₂×M₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
39	#117×C₇₉	C₂×C₄	1	1	yes	characteristic
40	M₃₂×C₇₉	—	2	2	no	—
41	M₃₂×C₇₉	—	2	2	no	—
42	C₂×C₁₆×C₇₉	Q₈	1	4	yes	complemented
43	#97×C₇₉	D₄	1	2	yes	—
44	C₄×C₈×C₇₉	D₄	1	2	yes	—
45	C₂×Dic₁₅₈	—	2	2	no	—
46	C₂²×Dic₇₉	—	2	1	no	—
47	#11929608	—	2	2	no	—
48	C₂×C₇₉⋊C₈	—	2	1	no	—
49	C₂×C₇₉⋊C₈	—	2	1	no	—
50	C₂×C₇₉⋊C₈	—	2	2	no	—
51	C₄⋊Dic₇₉	—	2	2	no	—
52	C₄⋊Dic₇₉	—	2	1	no	—
53	Dic₇₉⋊C₄	—	2	2	no	—
54	Dic₇₉⋊C₄	—	2	2	no	—
55	C₂²⋊Dic₇₉	M₁₆	1	2	yes	—
56	C₄×Dic₇₉	—	2	1	no	—
57	C₄×Dic₇₉	—	2	1	no	—
58	C₄×Dic₇₉	M₁₆	1	1	yes	characteristic
59	C₄×Dic₇₉	M₁₆	1	1	yes	characteristic
60	C₇₉⋊C₁₆	—	4	2	no	—
61	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
62	M₁₆×C₇₉	—	2	2	no	—
63	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
67	C₂×C₈×C₇₉	#41	1	2	yes	—
68	C₄⋊C₄×C₇₉	—	2	1	no	—
69	C₂²⋊C₄×C₇₉	—	2	1	no	—
70	C₄²×C₇₉	C₂²⋊C₄	1	2	yes	—
71	C₁₆×C₇₉	—	2	4	no	—
72	C₂³×C₇₉	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₄×C₇₉	—	2	2	no	—
74	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₇₉	C₄×Q₈	1	2	yes	—
76	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₇₉	#131	1	2	yes	—
78	C₂×Dic₇₉	—	2	1	no	—
79	C₂×Dic₇₉	—	2	1	no	—
80	C₂×Dic₇₉	—	2	2	no	—
81	C₂×Dic₇₉	—	8	1	no	—
82	Dic₁₅₈	—	8	2	no	—
83	C₈×C₇₉	—	2	2	no	—
84	C₈×C₇₉	C₂²⋊Q₈	1	2	yes	—
85	C₈×C₇₉	#124	1	1	yes	characteristic
86	C₈×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
87	C₇₉⋊C₈	—	4	1	no	—
88	C₇₉⋊C₈	—	4	1	no	—
89	C₂²×C₇₉	—	4	1	no	—
90	C₂²×C₇₉	#407	1	1	yes	characteristic
91	C₂²×C₇₉	#446	1	2	yes	—
92	C₄×C₇₉	—	4	2	no	—
93	C₄×C₇₉	#391	1	1	yes	characteristic
94	C₄×C₇₉	#408	1	1	yes	characteristic
95	C₄×C₇₉	#446	1	2	yes	—
96	Dic₇₉	—	4	1	no	—
97	Dic₇₉	—	4	2	no	—
98	Dic₇₉	—	4	1	no	—
99	#13269	—	79	1	no	maximal, Sylow
100	C₂×C₇₉	—	2	1	no	—
101	C₂×C₇₉	—	2	2	no	—
102	C₂×C₇₉	#1985	1	1	yes	characteristic
103	#2961	—	79	1	no	—
104	#2153	—	79	1	no	—
105	#2110	D₇₉	1	1	yes	characteristic
106	#1378	—	79	2	no	—
107	#1375	—	79	2	no	—
108	C₇₉	#13269	1	1	yes	characteristic, complemented, Sylow
109	#533	—	79	1	no	—
110	C₂×M₃₂	—	79	2	no	—
111	C₂×M₃₂	C₂×D₇₉	1	2	yes	—
112	#446	—	158	2	no	—
113	#446	—	158	2	no	—
114	#424	—	79	1	no	—
115	#424	—	79	1	no	—
116	#423	—	79	1	no	—
117	#405	—	79	1	no	—
118	#405	C₂×D₇₉	1	1	yes	characteristic
119	C₂×#97	—	79	1	no	—
120	#364	—	158	2	no	—
121	#347	—	2	2	no	—
122	C₂×M₁₆	—	79	1	no	—
123	C₂×M₁₆	C₄×D₇₉	1	1	yes	characteristic
124	C₂²×C₈	—	79	1	no	—
125	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
126	C₄⋊Q₈	—	158	1	no	—
127	#125	—	158	1	no	—
128	C₂²⋊Q₈	—	158	2	no	—
129	C₄×Q₈	—	158	2	no	—
130	#117	—	79	1	no	—
131	#117	C₄×D₇₉	1	1	yes	characteristic
132	C₂×C₄²	—	79	1	no	—
133	M₃₂	—	2	2	no	—
134	M₃₂	—	2	2	no	—
135	M₃₂	—	158	2	no	—
136	M₃₂	—	158	2	no	—
137	C₂×C₁₆	—	158	2	no	—
138	C₂×C₁₆	Dic₁₅₈	1	4	yes	complemented
139	C₄⋊C₈	—	158	2	no	—
140	C₄⋊C₈	—	158	2	no	—
141	C₄⋊C₈	—	158	2	no	—
142	#97	—	158	1	no	—
143	#97	—	158	1	no	—
144	#97	—	158	1	no	—
145	#97	C₂²⋊D₇₉	1	2	yes	—
146	C₄×C₈	—	158	1	no	—
147	C₄×C₈	D₃₁₆	1	2	yes	—
148	C₂×Q₈	—	158	2	no	—
149	C₂²×C₄	—	158	1	no	—
150	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
151	M₁₆	—	2	2	no	—
152	M₁₆	—	158	2	no	—
153	C₂×C₈	—	158	1	no	—
154	C₂×C₈	—	158	1	no	—
155	C₂×C₈	—	158	2	no	—
156	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
157	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
158	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
159	C₂×C₈	C₂×Dic₁₅₈	1	2	yes	—
160	C₂×C₈	#11929624	1	2	yes	—
161	C₄⋊C₄	—	2	1	no	—
162	C₄⋊C₄	—	158	2	no	—
163	C₄⋊C₄	—	158	2	no	—
164	C₄⋊C₄	—	158	2	no	—
165	C₄⋊C₄	—	158	1	no	—
166	C₂²⋊C₄	—	2	1	no	—
167	C₂²⋊C₄	—	79	2	no	—
168	C₄²	—	79	1	no	—
169	C₄²	—	79	1	no	—
170	C₄²	—	158	1	no	—
171	C₄²	—	158	1	no	—
172	C₄²	(C₂×D₇₉)⋊C₄	1	2	yes	—
173	C₁₆	—	2	4	no	—
174	C₁₆	—	316	2	no	—
175	C₂³	M₁₆×D₇₉	1	1	yes	characteristic
176	Q₈	—	632	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	632	1	no	—
179	C₂×C₄	—	158	1	no	—
180	C₂×C₄	—	158	1	no	—
181	C₂×C₄	—	158	2	no	—
182	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
183	C₂×C₄	C₄×Dic₁₅₈	1	2	yes	—
184	C₂×C₄	M₁₆×D₇₉	1	1	yes	characteristic
185	C₂×C₄	#14579169	1	2	yes	—
186	C₈	—	2	2	no	—
187	C₈	—	316	1	no	—
188	C₈	—	316	1	no	—
189	C₈	#14579140	1	2	yes	—
190	C₈	#14579149	1	1	yes	characteristic
191	C₈	#14579161	1	1	yes	characteristic
192	C₂²	—	4	1	no	—
193	C₂²	#19098526	1	1	yes	characteristic
194	C₂²	#19098691	1	2	yes	—
195	C₄	—	4	2	no	—
196	C₄	—	316	1	no	—
197	C₄	—	316	1	no	—
198	C₄	—	316	2	no	—
199	C₄	#19098469	1	1	yes	characteristic
200	C₄	#19098531	1	1	yes	characteristic
201	C₄	#19098691	1	2	yes	—
202	C₂	—	2	1	no	—
203	C₂	—	2	2	no	—
204	C₂	#23764340	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23772338

C₂

yes

maximal, characteristic

#23765166

C₂

yes

maximal, characteristic

#23761485

C₂

yes

maximal

#23761479

C₂

yes

maximal

C₇₉×#2110

C₂

yes

maximal, characteristic

#19099116

C₄

yes

characteristic

C₂×#14579101

C₂²

yes

—

#19098693

—

#19098691

—

#19098591

C₄

yes

characteristic

#19098590

C₄

yes

characteristic

#19098586

C₄

yes

characteristic

#19098518

C₂²

yes

characteristic

C₂×#14579071

C₂²

yes

characteristic

#19098384

—

C₂×M₃₂×C₇₉

C₂²

yes

—

#405×C₇₉

C₂²

yes

characteristic

#347×C₇₉

—

C₂×#11929608

C₂×C₄

yes

characteristic

C₂²×C₇₉⋊C₈

C₂×C₄

yes

—

—

—

C₄×Dic₁₅₈

—

#14579120

C₂×C₄

yes

characteristic

C₂×C₄×Dic₇₉

C₂×C₄

yes

characteristic

#14579101

—

#14579101

—

C₂×C₇₉⋊C₁₆

—

C₄⋊(C₇₉⋊C₈)

—

Dic₇₉⋊C₈

—

Dic₇₉⋊C₈

—

#14579071

—

#14579071

—

#14579071

—

C₈×Dic₇₉

—

C₂×M₁₆×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₉

C₂³

yes

characteristic

#117×C₇₉

C₂×C₄

yes

characteristic

M₃₂×C₇₉

—

M₃₂×C₇₉

—

C₂×C₁₆×C₇₉

Q₈

yes

complemented

#97×C₇₉

D₄

yes

—

C₄×C₈×C₇₉

D₄

yes

—

C₂×Dic₁₅₈

—

C₂²×Dic₇₉

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

M₁₆

yes

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

M₁₆

yes

characteristic

C₄×Dic₇₉

M₁₆

yes

characteristic

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

characteristic

M₁₆×C₇₉

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×Q₈

yes

—

C₂×C₈×C₇₉

#41

yes

—

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

C₂²⋊C₄

yes

—

C₁₆×C₇₉

—

C₂³×C₇₉

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄×Q₈

yes

—

C₂×C₄×C₇₉

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₉

#131

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

C₂²⋊Q₈

yes

—

C₈×C₇₉

#124

yes

characteristic

C₈×C₇₉

C₄⋊Q₈

yes

characteristic

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

#407

yes

characteristic

C₂²×C₇₉

#446

yes

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

#13269

—

maximal, Sylow

100

C₂×C₇₉

—

101

C₂×C₇₉

—

102

C₂×C₇₉

#1985

yes

characteristic

103

#2961

—

104

#2153

—

105

#2110

D₇₉

yes

characteristic

106

#1378

—

107

#1375

—

108

C₇₉

#13269

yes

characteristic, complemented, Sylow

109

#533

—

110

C₂×M₃₂

—

111

C₂×M₃₂

C₂×D₇₉

yes

—

112

#446

—

158

—

113

#446

—

158

—

114

#424

—

115

#424

—

116

#423

—

117

#405

—

118

#405

C₂×D₇₉

yes

characteristic

119

C₂×#97

—

120

#364

—

158

—

121

#347

—

122

C₂×M₁₆

—

123

C₂×M₁₆

C₄×D₇₉

yes

characteristic

124

C₂²×C₈

—

125

C₂²×C₈

C₂²×D₇₉

yes

characteristic

126

C₄⋊Q₈

—

158

—

127

#125

—

158

—

128

C₂²⋊Q₈

—

158

—

129

C₄×Q₈

—

158

—

130

#117

—

131

#117

C₄×D₇₉

yes

characteristic

132

C₂×C₄²

—

133

M₃₂

—

134

M₃₂

—

135

M₃₂

—

158

—

136

M₃₂

—

158

—

137

C₂×C₁₆

—

158

—

138

C₂×C₁₆

Dic₁₅₈

yes

complemented

139

C₄⋊C₈

—

158

—

140

C₄⋊C₈

—

158

—

141

C₄⋊C₈

—

158

—

142

#97

—

158

—

143

#97

—

158

—

144

#97

—

158

—

145

#97

C₂²⋊D₇₉

yes

—

146

C₄×C₈

—

158

—

147

C₄×C₈

D₃₁₆

yes

—

148

C₂×Q₈

—

158

—

149

C₂²×C₄

—

158

—

150

C₂²×C₄

C₂×C₄×D₇₉

yes

characteristic

151

M₁₆

—

152

M₁₆

—

158

—

153

C₂×C₈

—

158

—

154

C₂×C₈

—

158

—

155

C₂×C₈

—

158

—

156

C₂×C₈

(C₂×D₇₉)⋊C₄

yes

—

157

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

158

C₂×C₈

C₂×D₃₁₆

yes

characteristic

159

C₂×C₈

C₂×Dic₁₅₈

yes

—

160

C₂×C₈

#11929624

yes

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

158

—

163

C₄⋊C₄

—

158

—

164

C₄⋊C₄

—

158

—

165

C₄⋊C₄

—

158

—

166

C₂²⋊C₄

—

167

C₂²⋊C₄

—

168

C₄²

—

169

C₄²

—

170

C₄²

—

158

—

171

C₄²

—

158

—

172

C₄²

(C₂×D₇₉)⋊C₄

yes

—

173

C₁₆

—

174

C₁₆

—

316

—

175

C₂³

M₁₆×D₇₉

yes

characteristic

176

Q₈

—

632

—

177

C₂×C₄

—

178

C₂×C₄

—

632

—

179

C₂×C₄

—

158

—

180

C₂×C₄

—

158

—

181

C₂×C₄

—

158

—

182

C₂×C₄

C₂×(C₂×D₇₉)⋊C₄

yes

characteristic

183

C₂×C₄

C₄×Dic₁₅₈

yes

—

184

C₂×C₄

M₁₆×D₇₉

yes

characteristic

185

C₂×C₄

#14579169

yes

—

186

C₈

—

187

C₈

—

316

—

188

C₈

—

316

—

189

C₈

#14579140

yes

—

190

C₈

#14579149

yes

characteristic

191

C₈

#14579161

yes

characteristic

192

C₂²

—

193

C₂²

#19098526

yes

characteristic

194

C₂²

#19098691

yes

—

195

C₄

—

196

C₄

—

316

—

197

C₄

—

316

—

198

C₄

—

316

—

199

C₄

#19098469

yes

characteristic

200

C₄

#19098531

yes

characteristic

201

C₄

#19098691

yes

—

202

C₂

—

203

C₂

—

204

C₂

#23764340

yes

characteristic

	subgroup	G/H
center	C₄	#19098469
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#2110	C₂
socle	C₂×C₇₉	#1985
prime radical	C₂³×C₇₉	C₂×M₁₆
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#2110	D₇₉
hypocenter	C₇₉	#13269
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #2110

subgroup

G/H

center

C₄

#19098469

commutator

C₂²×C₄×C₇₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₇₉×#2110

C₂

socle

C₂×C₇₉

#1985

prime radical

C₂³×C₇₉

C₂×M₁₆

cosocle

C₂²×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #2110

N	H ≅ G/N	type
42. C₂×C₁₆×C₇₉	Q₈	semidirect
108. C₇₉	#13269	semidirect
138. C₂×C₁₆	Dic₁₅₈	semidirect

H ≅ G/N

type

42. C₂×C₁₆×C₇₉

Q₈

semidirect

108. C₇₉

#13269

semidirect

138. C₂×C₁₆

Dic₁₅₈

semidirect

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

subgroup	index
#23772338	2
#23765166	2
#23761485	2
#23761479	2
C₇₉×#2110	2
#13269	79

subgroup

index

C₇₉×#2110

Subgroups — #34206813

Conjugacy classes of subgroups

Complements

Maximal subgroups