Group #34188212 — subgroups

Subgroups — #34188212

Order 20224.

530 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 159 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	#23763572	C₂	1	1	yes	maximal, characteristic, complemented
2	#23761851	C₂	1	4	yes	maximal, complemented
3	C₂×C₈×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₉×#3147	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579097	—	2	2	no	—
6	#19098587	C₂²	1	2	yes	—
7	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
8	D₄⋊(C₇₉⋊C₈)	—	2	8	no	—
9	C₈×C₇₉⋊C₈	—	2	2	no	—
10	C₂×#135×C₇₉	—	2	2	no	—
11	#532×C₇₉	C₂²	1	2	yes	—
12	#495×C₇₉	—	2	2	no	—
13	#494×C₇₉	C₄	1	1	yes	characteristic
14	#493×C₇₉	C₄	1	1	yes	characteristic
15	#492×C₇₉	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×#11929608	—	2	2	no	—
18	C₂²×C₇₉⋊C₈	—	2	2	no	—
19	#14579097	—	4	4	no	—
20	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
21	C₄×C₇₉⋊C₈	—	2	2	no	—
22	C₄×C₇₉⋊C₈	D₄	1	4	yes	—
23	C₂×#41×C₇₉	—	2	2	no	—
24	#135×C₇₉	—	4	4	no	—
25	#135×C₇₉	—	4	4	no	—
26	C₂×Q₁₆×C₇₉	—	2	2	no	—
27	C₂×SD₁₆×C₇₉	—	2	4	no	—
28	C₂×D₈×C₇₉	—	2	2	no	—
29	C₂²×C₈×C₇₉	D₄	1	2	yes	—
30	C₄⋊Q₈×C₇₉	C₂×C₄	1	2	yes	—
31	#127×C₇₉	C₂×C₄	1	2	yes	—
32	#124×C₇₉	—	2	2	no	—
33	#121×C₇₉	—	2	4	no	—
34	C₄×D₄×C₇₉	—	2	4	no	—
35	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
36	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₇₉	D₄	1	2	yes	—
39	#11929608	—	4	4	no	—
40	C₂×C₇₉⋊C₈	—	2	8	no	—
41	C₂×C₇₉⋊C₈	—	2	4	no	—
42	C₂×C₇₉⋊C₈	—	4	2	no	—
43	#41×C₇₉	—	4	4	no	—
44	#41×C₇₉	—	4	4	no	—
45	C₂×Q₈×C₇₉	—	2	2	no	—
46	C₂×D₄×C₇₉	—	2	2	no	—
47	C₂×D₄×C₇₉	—	2	4	no	—
48	C₂×D₄×C₇₉	—	4	2	no	—
49	C₂²×C₄×C₇₉	—	4	2	no	—
50	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₇₉	#41	1	2	yes	—
52	Q₁₆×C₇₉	—	4	4	no	—
53	SD₁₆×C₇₉	—	4	8	no	—
54	D₈×C₇₉	—	4	4	no	—
55	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₇₉	D₈	1	4	yes	—
58	C₂×C₈×C₇₉	SD₁₆	1	4	yes	—
59	C₄⋊C₄×C₇₉	—	2	4	no	—
60	C₂²⋊C₄×C₇₉	—	4	4	no	—
61	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₇₉	C₂×D₄	1	2	yes	—
64	C₂³×C₇₉	—	4	2	no	—
65	C₂³×C₇₉	#124	1	1	yes	characteristic
66	Q₈×C₇₉	—	4	4	no	—
67	D₄×C₇₉	—	4	4	no	—
68	D₄×C₇₉	—	4	8	no	—
69	D₄×C₇₉	—	8	4	no	—
70	C₂×C₄×C₇₉	—	2	4	no	—
71	C₂×C₄×C₇₉	—	4	2	no	—
72	C₂×C₄×C₇₉	—	8	2	no	—
73	C₂×C₄×C₇₉	—	8	2	no	—
74	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
76	C₂×C₄×C₇₉	#121	1	2	yes	—
77	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
78	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
79	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
80	C₈×C₇₉	#102	1	4	yes	—
81	C₈×C₇₉	#102	1	4	yes	—
82	C₇₉⋊C₈	—	2	8	no	—
83	C₇₉⋊C₈	—	8	2	no	—
84	C₂²×C₇₉	—	2	2	no	—
85	C₂²×C₇₉	—	8	2	no	—
86	C₂²×C₇₉	—	8	4	no	—
87	C₂²×C₇₉	#390	1	1	yes	characteristic
88	C₂²×C₇₉	#486	1	2	yes	—
89	C₄×C₇₉	—	2	2	no	—
90	C₄×C₇₉	—	8	2	no	—
91	C₄×C₇₉	C₂×#102	1	2	yes	—
92	C₄×C₇₉	#443	1	4	yes	—
93	C₄×C₇₉	#492	1	1	yes	characteristic
94	C₄×C₇₉	#493	1	1	yes	characteristic
95	#9856	—	79	1	no	maximal, Sylow
96	C₂×C₇₉	—	2	2	no	—
97	C₂×C₇₉	—	16	2	no	—
98	C₂×C₇₉	#1957	1	1	yes	characteristic
99	C₂×C₇₉	#1974	1	2	yes	—
100	#3147	D₇₉	1	1	yes	characteristic
101	#1848	—	79	1	no	—
102	#1492	—	79	4	no	—
103	C₂×C₈²	—	79	1	no	—
104	C₇₉	#9856	1	1	yes	characteristic, complemented, Sylow
105	C₂×#135	—	2	2	no	—
106	#532	C₂×D₇₉	1	2	yes	—
107	#495	—	2	2	no	—
108	#494	Dic₇₉	1	1	yes	characteristic
109	#493	Dic₇₉	1	1	yes	characteristic
110	#492	Dic₇₉	1	2	yes	—
111	C₂×#108	—	158	2	no	—
112	#423	—	79	2	no	—
113	C₂×C₄×C₈	—	79	2	no	—
114	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
115	D₄⋊C₈	—	158	8	no	—
116	C₈²	—	158	2	no	—
117	C₂×#41	—	2	2	no	—
118	#135	—	4	4	no	—
119	#135	—	4	4	no	—
120	C₂×Q₁₆	—	2	2	no	—
121	C₂×SD₁₆	—	2	4	no	—
122	C₂×D₈	—	2	2	no	—
123	C₂×M₁₆	—	158	2	no	—
124	C₂²×C₈	—	158	2	no	—
125	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
126	C₄⋊Q₈	C₂×Dic₇₉	1	2	yes	—
127	#127	C₂×Dic₇₉	1	2	yes	—
128	#124	—	2	2	no	—
129	#121	—	2	4	no	—
130	C₄×D₄	—	2	4	no	—
131	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
132	#108	—	316	4	no	—
133	C₄⋊C₈	—	158	4	no	—
134	C₄×C₈	—	79	4	no	—
135	C₄×C₈	—	158	2	no	—
136	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
137	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
138	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
139	#41	—	4	4	no	—
140	#41	—	4	4	no	—
141	C₂×Q₈	—	2	2	no	—
142	C₂×D₄	—	2	2	no	—
143	C₂×D₄	—	2	4	no	—
144	C₂×D₄	—	4	2	no	—
145	C₂²×C₄	—	4	2	no	—
146	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
147	C₂²×C₄	#11929623	1	2	yes	—
148	Q₁₆	—	4	4	no	—
149	SD₁₆	—	4	8	no	—
150	D₈	—	4	4	no	—
151	M₁₆	—	316	4	no	—
152	C₂×C₈	—	316	2	no	—
153	C₂×C₈	—	158	8	no	—
154	C₂×C₈	—	158	4	no	—
155	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
156	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
157	C₂×C₈	#11929609	1	4	yes	—
158	C₂×C₈	#11929612	1	4	yes	—
159	C₄⋊C₄	—	2	4	no	—
160	C₂²⋊C₄	—	4	4	no	—
161	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
162	C₄²	D₄×D₇₉	1	2	yes	—
163	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
164	C₂³	—	4	2	no	—
165	C₂³	#14579148	1	1	yes	characteristic
166	Q₈	—	4	4	no	—
167	D₄	—	4	8	no	—
168	D₄	—	4	4	no	—
169	D₄	—	8	4	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
175	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
176	C₂×C₄	#14579132	1	2	yes	—
177	C₂×C₄	#14579157	1	1	yes	characteristic
178	C₂×C₄	C₂×#11929609	1	2	yes	—
179	C₂×C₄	C₂×#11929612	1	2	yes	—
180	C₈	—	632	2	no	—
181	C₈	—	158	8	no	—
182	C₈	#14579082	1	4	yes	—
183	C₈	#14579082	1	4	yes	—
184	C₂²	—	2	2	no	—
185	C₂²	—	8	4	no	—
186	C₂²	—	8	2	no	—
187	C₂²	#19098466	1	1	yes	characteristic
188	C₂²	#19098909	1	2	yes	—
189	C₄	—	2	2	no	—
190	C₄	—	8	2	no	—
191	C₄	C₂×#14579082	1	2	yes	—
192	C₄	#19098680	1	4	yes	—
193	C₄	#19098935	1	1	yes	characteristic
194	C₄	#19098939	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	—	16	2	no	—
197	C₂	#23764207	1	1	yes	characteristic
198	C₂	#23764289	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23763572

C₂

yes

maximal, characteristic, complemented

#23761851

C₂

yes

maximal, complemented

C₂×C₈×C₇₉⋊C₈

C₂

yes

maximal, characteristic, complemented

C₇₉×#3147

C₂

yes

maximal, characteristic

C₂×#14579097

—

#19098587

C₂²

yes

—

C₂×C₄×C₇₉⋊C₈

C₂²

yes

—

D₄⋊(C₇₉⋊C₈)

—

C₈×C₇₉⋊C₈

—

C₂×#135×C₇₉

—

#532×C₇₉

C₂²

yes

—

#495×C₇₉

—

#494×C₇₉

C₄

yes

characteristic

#493×C₇₉

C₄

yes

characteristic

#492×C₇₉

C₄

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×#11929608

—

C₂²×C₇₉⋊C₈

—

#14579097

—

C₄⋊(C₇₉⋊C₈)

—

C₄×C₇₉⋊C₈

—

C₄×C₇₉⋊C₈

D₄

yes

—

C₂×#41×C₇₉

—

#135×C₇₉

—

#135×C₇₉

—

C₂×Q₁₆×C₇₉

—

C₂×SD₁₆×C₇₉

—

C₂×D₈×C₇₉

—

C₂²×C₈×C₇₉

D₄

yes

—

C₄⋊Q₈×C₇₉

C₂×C₄

yes

—

#127×C₇₉

C₂×C₄

yes

—

#124×C₇₉

—

#121×C₇₉

—

C₄×D₄×C₇₉

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

#41×C₇₉

—

#41×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

#41

yes

—

Q₁₆×C₇₉

—

SD₁₆×C₇₉

—

D₈×C₇₉

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

D₈

yes

—

C₂×C₈×C₇₉

SD₁₆

yes

—

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

—

C₂³×C₇₉

—

C₂³×C₇₉

#124

yes

characteristic

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄×D₄

yes

—

C₂×C₄×C₇₉

#121

yes

—

C₂×C₄×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

—

C₂×C₄×C₇₉

C₂×SD₁₆

yes

—

C₈×C₇₉

#102

yes

—

C₈×C₇₉

#102

yes

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#390

yes

characteristic

C₂²×C₇₉

#486

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×#102

yes

—

C₄×C₇₉

#443

yes

—

C₄×C₇₉

yes

C₄×C₇₉

yes

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#1957

yes

characteristic

C₂×C₇₉

#1974

yes

—

100

#3147

D₇₉

yes

characteristic

101

#1848

—

102

#1492

—

103

C₂×C₈²

—

104

C₇₉

#9856

yes

characteristic, complemented, Sylow

105

C₂×#135

—

106

#532

C₂×D₇₉

yes

—

107

#495

—

108

#494

Dic₇₉

yes

characteristic

109

#493

Dic₇₉

yes

characteristic

110

#492

Dic₇₉

yes

—

111

C₂×#108

—

158

—

112

#423

—

113

C₂×C₄×C₈

—

114

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

115

D₄⋊C₈

—

158

—

116

C₈²

—

158

—

117

C₂×#41

—

118

#135

—

119

#135

—

120

C₂×Q₁₆

—

121

C₂×SD₁₆

—

122

C₂×D₈

—

123

C₂×M₁₆

—

158

—

124

C₂²×C₈

—

158

—

125

C₂²×C₈

C₂²⋊D₇₉

yes

—

126

C₄⋊Q₈

C₂×Dic₇₉

yes

—

127

#127

C₂×Dic₇₉

yes

—

128

#124

—

129

#121

—

130

C₄×D₄

—

131

C₂×C₄²

C₂²×D₇₉

yes

characteristic

132

#108

—

316

—

133

C₄⋊C₈

—

158

—

134

C₄×C₈

—

135

C₄×C₈

—

158

—

136

C₄×C₈

C₂×Dic₇₉

yes

characteristic

137

C₄×C₈

C₂×Dic₇₉

yes

characteristic

138

C₄×C₈

C₂²⋊D₇₉

yes

—

139

#41

—

140

#41

—

141

C₂×Q₈

—

142

C₂×D₄

—

143

C₂×D₄

—

144

C₂×D₄

—

145

C₂²×C₄

—

146

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

147

C₂²×C₄

#11929623

yes

—

148

Q₁₆

—

149

SD₁₆

—

150

D₈

—

151

M₁₆

—

316

—

152

C₂×C₈

—

316

—

153

C₂×C₈

—

158

—

154

C₂×C₈

—

158

—

155

C₂×C₈

C₂²⋊Dic₇₉

yes

—

156

C₂×C₈

C₂²⋊Dic₇₉

yes

—

157

C₂×C₈

#11929609

yes

—

158

C₂×C₈

#11929612

yes

—

159

C₄⋊C₄

—

160

C₂²⋊C₄

—

161

C₄²

C₂²×Dic₇₉

yes

characteristic

162

C₄²

D₄×D₇₉

yes

—

163

C₄²

C₂×C₂²⋊D₇₉

yes

characteristic

164

C₂³

—

165

C₂³

#14579148

yes

characteristic

166

Q₈

—

167

D₄

—

168

D₄

—

169

D₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

C₂×C₂²⋊Dic₇₉

yes

characteristic

175

C₂×C₄

D₄×Dic₇₉

yes

—

176

C₂×C₄

#14579132

yes

—

177

C₂×C₄

#14579157

yes

characteristic

178

C₂×C₄

C₂×#11929609

yes

—

179

C₂×C₄

C₂×#11929612

yes

—

180

C₈

—

632

—

181

C₈

—

158

—

182

C₈

#14579082

yes

—

183

C₈

#14579082

yes

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

#19098466

yes

characteristic

188

C₂²

#19098909

yes

—

189

C₄

—

190

C₄

—

191

C₄

C₂×#14579082

yes

—

192

C₄

#19098680

yes

—

193

C₄

#19098935

yes

characteristic

194

C₄

#19098939

yes

characteristic

195

C₂

—

196

C₂

—

197

C₂

#23764207

yes

characteristic

198

C₂

#23764289

yes

—

	subgroup	G/H
center	C₂×C₄	#14579157
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting, prime radical	C₇₉×#3147	C₂
socle	C₂²×C₇₉	#390
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#3147	D₇₉
hypocenter	C₇₉	#9856
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3147

subgroup

G/H

center

C₂×C₄

#14579157

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting, prime radical

C₇₉×#3147

C₂

socle

C₂²×C₇₉

#390

cosocle

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3147

N	H ≅ G/N	type
1. #23763572	C₂	semidirect
2. #23761851	C₂	semidirect
3. C₂×C₈×C₇₉⋊C₈	C₂	semidirect
104. C₇₉	#9856	semidirect

H ≅ G/N

type

1. #23763572

C₂

semidirect

2. #23761851

C₂

semidirect

3. C₂×C₈×C₇₉⋊C₈

C₂

semidirect

104. C₇₉

#9856

semidirect

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

subgroup	index
#23763572	2
#23761851	2
C₂×C₈×C₇₉⋊C₈	2
C₇₉×#3147	2
#9856	79

subgroup

index

#23763572

#23761851

C₂×C₈×C₇₉⋊C₈

C₇₉×#3147

#9856

Subgroups — #34188212

Conjugacy classes of subgroups

Complements

Maximal subgroups