Group #34208100 — subgroups

Subgroups — #34208100

Order 20224.

302 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 93 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×#19098378	C₂	1	2	yes	maximal
2	#23764271	C₂	1	1	yes	maximal, characteristic
3	#23761500	C₂	1	2	yes	maximal
4	#23761484	C₂	1	1	yes	maximal, characteristic
5	C₇₉×#1849	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₁₆	C₂²	1	1	yes	characteristic
7	C₂×#14579101	C₂²	1	2	yes	—
8	C₂×#14579093	C₂²	1	1	yes	characteristic
9	C₂×#14579086	—	2	1	no	—
10	#19098469	—	2	1	no	—
11	#19098378	—	2	4	no	—
12	C₂×#108×C₇₉	—	2	1	no	—
13	C₂×#108×C₇₉	C₂²	1	2	yes	—
14	#423×C₇₉	—	2	1	no	—
15	C₂×#97×C₇₉	C₂²	1	1	yes	characteristic
16	C₂²×Dic₁₅₈	—	2	1	no	—
17	C₂×Dic₃₁₆	—	2	2	no	—
18	C₂×Dic₃₁₆	—	2	2	no	—
19	C₂×Dic₃₁₆	—	2	1	no	—
20	C₂×Dic₃₁₆	D₄	1	2	yes	—
21	C₂×C₄⋊Dic₇₉	—	2	1	no	—
22	#14579101	—	2	2	no	—
23	#14579101	—	2	2	no	—
24	C₂×C₇₉⋊C₁₆	—	2	2	no	—
25	#14579093	—	2	2	no	—
26	#14579086	—	4	2	no	—
27	C₇₉⋊#95	—	4	2	no	—
28	C₂×M₁₆×C₇₉	—	2	2	no	—
29	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
32	#108×C₇₉	—	2	2	no	—
33	#108×C₇₉	—	2	2	no	—
34	#108×C₇₉	—	4	2	no	—
35	C₄⋊C₈×C₇₉	—	4	2	no	—
36	#97×C₇₉	—	2	2	no	—
37	C₂×Dic₁₅₈	—	2	2	no	—
38	C₂×Dic₁₅₈	—	4	1	no	—
39	C₂×Dic₁₅₈	—	4	1	no	—
40	C₂×Dic₁₅₈	—	4	1	no	—
41	C₂×Dic₁₅₈	—	4	1	no	—
42	C₂×Dic₁₅₈	—	4	1	no	—
43	C₂²×Dic₇₉	—	4	1	no	—
44	C₂²×Dic₇₉	—	4	1	no	—
45	Dic₃₁₆	—	2	4	no	—
46	Dic₃₁₆	—	4	2	no	—
47	C₄⋊Dic₇₉	—	4	1	no	—
48	C₄⋊Dic₇₉	—	4	1	no	—
49	C₇₉⋊C₁₆	—	4	2	no	—
50	C₂²×C₄×C₇₉	SD₁₆	1	2	yes	—
51	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
52	M₁₆×C₇₉	—	4	2	no	—
53	M₁₆×C₇₉	—	4	2	no	—
54	C₂×C₈×C₇₉	—	2	1	no	—
55	C₂×C₈×C₇₉	—	2	1	no	—
56	C₂×C₈×C₇₉	—	4	2	no	—
57	C₂×C₈×C₇₉	D₈	1	2	yes	—
58	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
60	C₂×C₈×C₇₉	#41	1	2	yes	—
61	C₂×C₈×C₇₉	#41	1	2	yes	—
62	C₄²×C₇₉	—	2	1	no	—
63	C₄²×C₇₉	—	2	1	no	—
64	C₂³×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×C₄×C₇₉	—	2	2	no	—
66	C₂×C₄×C₇₉	—	4	2	no	—
67	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
68	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
69	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
70	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
72	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
73	C₂×Dic₇₉	—	4	2	no	—
74	C₂×Dic₇₉	—	8	1	no	—
75	C₂×Dic₇₉	—	8	1	no	—
76	C₂×Dic₇₉	—	8	2	no	—
77	C₂×Dic₇₉	—	8	1	no	—
78	Dic₁₅₈	—	4	2	no	—
79	Dic₁₅₈	—	4	2	no	—
80	Dic₁₅₈	—	8	1	no	—
81	Dic₁₅₈	—	8	1	no	—
82	C₈×C₇₉	—	2	2	no	—
83	C₈×C₇₉	—	8	2	no	—
84	C₈×C₇₉	#123	1	2	yes	—
85	C₈×C₇₉	#126	1	2	yes	—
86	C₂²×C₇₉	—	2	2	no	—
87	C₂²×C₇₉	#461	1	1	yes	characteristic
88	C₂²×C₇₉	#481	1	1	yes	characteristic
89	C₂²×C₇₉	#486	1	1	yes	characteristic
90	C₄×C₇₉	—	4	2	no	—
91	C₄×C₇₉	#397	1	1	yes	characteristic
92	C₄×C₇₉	#459	1	1	yes	characteristic
93	C₄×C₇₉	#480	1	1	yes	characteristic
94	C₄×C₇₉	#489	1	1	yes	characteristic
95	Dic₇₉	—	8	2	no	—
96	Dic₇₉	—	16	1	no	—
97	#13488	—	79	1	no	maximal, Sylow
98	C₂×C₇₉	—	2	1	no	—
99	C₂×C₇₉	—	2	1	no	—
100	C₂×C₇₉	#2071	1	1	yes	characteristic
101	C₂×C₇₉	#2239	1	2	yes	—
102	C₂×#362	—	79	2	no	—
103	#1971	—	79	1	no	—
104	#1849	D₇₉	1	1	yes	characteristic
105	#1383	—	79	2	no	—
106	#1377	—	79	1	no	—
107	C₇₉	#13488	1	1	yes	characteristic, complemented, Sylow
108	C₂²×Q₁₆	—	79	1	no	—
109	C₂×M₃₂	—	79	2	no	—
110	C₂×#108	—	2	1	no	—
111	C₂×#108	C₂×D₇₉	1	2	yes	—
112	C₂×#106	—	79	1	no	—
113	#423	—	2	1	no	—
114	C₂×#103	—	158	1	no	—
115	C₂×#97	C₂×D₇₉	1	1	yes	characteristic
116	#391	—	158	1	no	—
117	#362	—	158	4	no	—
118	C₂²×Q₈	—	158	1	no	—
119	C₂×Q₁₆	—	79	2	no	—
120	C₂×Q₁₆	—	158	1	no	—
121	C₂×Q₁₆	—	158	2	no	—
122	C₂×Q₁₆	—	158	2	no	—
123	C₂×M₁₆	—	2	2	no	—
124	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
125	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
126	C₂×C₄⋊C₄	—	158	1	no	—
127	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
128	M₃₂	—	158	2	no	—
129	M₃₂	—	158	2	no	—
130	C₂×C₁₆	—	158	2	no	—
131	#108	—	2	2	no	—
132	#108	—	2	2	no	—
133	#108	—	4	2	no	—
134	#106	—	158	2	no	—
135	C₄⋊C₈	—	4	2	no	—
136	#103	—	316	2	no	—
137	#97	—	2	2	no	—
138	#95	—	316	2	no	—
139	C₂×Q₈	—	316	1	no	—
140	C₂×Q₈	—	316	1	no	—
141	C₂×Q₈	—	316	1	no	—
142	C₂×Q₈	—	316	1	no	—
143	C₂×Q₈	—	316	1	no	—
144	C₂×Q₈	—	158	2	no	—
145	C₂²×C₄	—	316	1	no	—
146	C₂²×C₄	—	316	1	no	—
147	C₂²×C₄	#11929611	1	2	yes	—
148	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
149	Q₁₆	—	316	2	no	—
150	Q₁₆	—	158	4	no	—
151	M₁₆	—	4	2	no	—
152	M₁₆	—	4	2	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	#11929609	1	2	yes	—
157	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
158	C₂×C₈	#11929623	1	1	yes	characteristic
159	C₂×C₈	#11929624	1	2	yes	—
160	C₂×C₈	#11929625	1	2	yes	—
161	C₄⋊C₄	—	316	1	no	—
162	C₄⋊C₄	—	316	1	no	—
163	C₄²	—	2	1	no	—
164	C₄²	—	2	1	no	—
165	C₁₆	—	316	2	no	—
166	C₂³	C₂×#11929611	1	1	yes	characteristic
167	Q₈	—	316	2	no	—
168	Q₈	—	316	2	no	—
169	Q₈	—	632	1	no	—
170	Q₈	—	632	1	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	316	2	no	—
174	C₂×C₄	—	632	1	no	—
175	C₂×C₄	—	632	1	no	—
176	C₂×C₄	—	632	2	no	—
177	C₂×C₄	—	632	1	no	—
178	C₂×C₄	#14579132	1	1	yes	characteristic
179	C₂×C₄	#14579144	1	1	yes	characteristic
180	C₂×C₄	#14579150	1	1	yes	characteristic
181	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
182	C₂×C₄	#14579187	1	1	yes	characteristic
183	C₂×C₄	#14579192	1	1	yes	characteristic
184	C₈	—	2	2	no	—
185	C₈	—	8	2	no	—
186	C₈	#14579146	1	2	yes	—
187	C₈	#14579156	1	2	yes	—
188	C₂²	—	2	2	no	—
189	C₂²	#19098765	1	1	yes	characteristic
190	C₂²	#19098881	1	1	yes	characteristic
191	C₂²	#19098911	1	1	yes	characteristic
192	C₄	—	4	2	no	—
193	C₄	—	632	2	no	—
194	C₄	—	1264	1	no	—
195	C₄	#19098489	1	1	yes	characteristic
196	C₄	#19098755	1	1	yes	characteristic
197	C₄	#19098876	1	1	yes	characteristic
198	C₄	#19098925	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#23764804	1	1	yes	characteristic
202	C₂	#23765619	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19098378

C₂

yes

maximal

#23764271

C₂

yes

maximal, characteristic

#23761500

C₂

yes

maximal

#23761484

C₂

yes

maximal, characteristic

C₇₉×#1849

C₂

yes

maximal, characteristic

C₂²×Dic₃₁₆

C₂²

yes

characteristic

C₂×#14579101

C₂²

yes

—

C₂×#14579093

C₂²

yes

—

—

—

C₂×#108×C₇₉

—

C₂×#108×C₇₉

C₂²

yes

—

#423×C₇₉

—

C₂×#97×C₇₉

C₂²

yes

characteristic

C₂²×Dic₁₅₈

—

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

D₄

yes

—

C₂×C₄⋊Dic₇₉

—

#14579101

—

#14579101

—

C₂×C₇₉⋊C₁₆

—

#14579093

—

#14579086

—

C₇₉⋊#95

—

C₂×M₁₆×C₇₉

—

C₂²×C₈×C₇₉

D₄

yes

characteristic

C₂²×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

characteristic

#108×C₇₉

—

#108×C₇₉

—

#108×C₇₉

—

C₄⋊C₈×C₇₉

—

#97×C₇₉

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

Dic₃₁₆

—

Dic₃₁₆

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

SD₁₆

yes

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

D₈

yes

—

C₂×C₈×C₇₉

C₂×D₄

yes

C₂×C₈×C₇₉

yes

C₂×C₈×C₇₉

yes

—

C₂×C₈×C₇₉

#41

yes

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₂³×C₇₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

C₂×D₈

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

#123

yes

—

C₈×C₇₉

#126

yes

—

C₂²×C₇₉

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

#13488

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

100

C₂×C₇₉

#2071

yes

characteristic

101

C₂×C₇₉

#2239

yes

—

102

C₂×#362

—

103

#1971

—

104

#1849

D₇₉

yes

characteristic

105

#1383

—

106

#1377

—

107

C₇₉

#13488

yes

characteristic, complemented, Sylow

108

C₂²×Q₁₆

—

109

C₂×M₃₂

—

110

C₂×#108

—

111

C₂×#108

C₂×D₇₉

yes

—

112

C₂×#106

—

113

#423

—

114

C₂×#103

—

158

—

115

C₂×#97

C₂×D₇₉

yes

characteristic

116

#391

—

158

—

117

#362

—

158

—

118

C₂²×Q₈

—

158

—

119

C₂×Q₁₆

—

120

C₂×Q₁₆

—

158

—

121

C₂×Q₁₆

—

158

—

122

C₂×Q₁₆

—

158

—

123

C₂×M₁₆

—

124

C₂²×C₈

C₂²⋊D₇₉

yes

characteristic

125

C₂²×C₈

C₂²×D₇₉

yes

characteristic

126

C₂×C₄⋊C₄

—

158

—

127

C₂×C₄²

C₂²⋊D₇₉

yes

characteristic

128

M₃₂

—

158

—

129

M₃₂

—

158

—

130

C₂×C₁₆

—

158

—

131

#108

—

132

#108

—

133

#108

—

134

#106

—

158

—

135

C₄⋊C₈

—

136

#103

—

316

—

137

#97

—

138

#95

—

316

—

139

C₂×Q₈

—

316

—

140

C₂×Q₈

—

316

—

141

C₂×Q₈

—

316

—

142

C₂×Q₈

—

316

—

143

C₂×Q₈

—

316

—

144

C₂×Q₈

—

158

—

145

C₂²×C₄

—

316

—

146

C₂²×C₄

—

316

—

147

C₂²×C₄

#11929611

yes

—

148

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

149

Q₁₆

—

316

—

150

Q₁₆

—

158

—

151

M₁₆

—

152

M₁₆

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

#11929609

yes

—

157

C₂×C₈

D₄×D₇₉

yes

characteristic

158

C₂×C₈

#11929623

yes

characteristic

159

C₂×C₈

#11929624

yes

—

160

C₂×C₈

#11929625

yes

—

161

C₄⋊C₄

—

316

—

162

C₄⋊C₄

—

316

—

163

C₄²

—

164

C₄²

—

165

C₁₆

—

316

—

166

C₂³

C₂×#11929611

yes

characteristic

167

Q₈

—

316

—

168

Q₈

—

316

—

169

Q₈

—

632

—

170

Q₈

—

632

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

316

—

174

C₂×C₄

—

632

—

175

C₂×C₄

—

632

—

176

C₂×C₄

—

632

—

177

C₂×C₄

—

632

—

178

C₂×C₄

#14579132

yes

characteristic

179

C₂×C₄

#14579144

yes

characteristic

180

C₂×C₄

#14579150

yes

characteristic

181

C₂×C₄

C₂×#11929609

yes

characteristic

182

C₂×C₄

#14579187

yes

characteristic

183

C₂×C₄

#14579192

yes

characteristic

184

C₈

—

185

C₈

—

186

C₈

#14579146

yes

—

187

C₈

#14579156

yes

—

188

C₂²

—

189

C₂²

#19098765

yes

characteristic

190

C₂²

#19098881

yes

characteristic

191

C₂²

#19098911

yes

characteristic

192

C₄

—

193

C₄

—

632

—

194

C₄

—

1264

—

195

C₄

#19098489

yes

characteristic

196

C₄

#19098755

yes

characteristic

197

C₄

#19098876

yes

characteristic

198

C₄

#19098925

yes

characteristic

199

C₂

—

200

C₂

—

201

C₂

#23764804

yes

characteristic

202

C₂

#23765619

yes

—

	subgroup	G/H
center	C₂²	#19098881
commutator, cosocle	C₂²×C₈×C₇₉	C₂³
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1849	C₂
socle	C₂²×C₇₉	#481
prime radical	C₂³×C₇₉	C₂×SD₁₆
hypercenter	#1849	D₇₉
hypocenter	C₇₉	#13488
LCS	G ▷ C₂²×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1849

subgroup

G/H

center

C₂²

#19098881

commutator, cosocle

C₂²×C₈×C₇₉

C₂³

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₇₉×#1849

C₂

socle

C₂²×C₇₉

#481

prime radical

C₂³×C₇₉

C₂×SD₁₆

D₇₉

C₇₉

G ▷ C₂²×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1849

N	H ≅ G/N	type
107. C₇₉	#13488	semidirect

H ≅ G/N

type

107. C₇₉

#13488

semidirect

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

subgroup	index
C₂×#19098378	2
#23764271	2
#23761500	2
#23761484	2
C₇₉×#1849	2
#13488	79

subgroup

index

C₇₉×#1849

Subgroups — #34208100

Conjugacy classes of subgroups

Complements

Maximal subgroups