Group #34186976 — subgroups

Subgroups — #34186976

Order 20224.

290 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 119 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	#23764173	C₂	1	2	yes	maximal
2	C₂×#19098316	C₂	1	1	yes	maximal, characteristic
3	#23761689	C₂	1	1	yes	maximal, characteristic
4	#23761295	C₂	1	2	yes	maximal
5	C₇₉×#1566	C₂	1	1	yes	maximal, characteristic
6	#19098616	C₂²	1	1	yes	characteristic
7	C₂×#14579096	—	2	2	no	—
8	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
9	C₄×#11929608	C₂²	1	1	yes	characteristic
10	#19098327	—	2	2	no	—
11	#19098316	C₄	1	2	yes	—
12	#19098316	C₄	1	2	yes	—
13	#19098285	—	2	1	no	—
14	#19098284	—	2	1	no	—
15	#423×C₇₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
17	#334×C₇₉	—	2	1	no	—
18	#334×C₇₉	—	2	1	no	—
19	C₂×#11929608	D₄	1	2	yes	—
20	C₂²×C₇₉⋊C₈	—	2	2	no	—
21	#14579096	—	4	4	no	—
22	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
23	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
24	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	4	yes	—
25	C₂²⋊(C₇₉⋊C₈)	Q₈	1	2	yes	—
26	#14579072	—	2	1	no	—
27	C₄×C₇₉⋊C₈	—	2	1	no	—
28	C₄×C₇₉⋊C₈	—	2	1	no	—
29	C₂×M₁₆×C₇₉	—	2	2	no	—
30	C₂²×C₈×C₇₉	Q₈	1	1	yes	characteristic
31	C₂²×C₈×C₇₉	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₇₉	—	2	2	no	—
34	C₄⋊C₈×C₇₉	—	2	2	no	—
35	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
38	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
39	#11929608	—	2	2	no	—
40	#11929608	—	2	2	no	—
41	C₂×C₇₉⋊C₈	—	2	4	no	—
42	C₂×C₇₉⋊C₈	—	2	2	no	—
43	C₂×C₇₉⋊C₈	—	2	2	no	—
44	C₂×C₇₉⋊C₈	—	4	2	no	—
45	C₂×C₇₉⋊C₈	—	4	2	no	—
46	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
49	M₁₆×C₇₉	—	4	2	no	—
50	M₁₆×C₇₉	—	4	2	no	—
51	C₂×C₈×C₇₉	—	2	2	no	—
52	C₂×C₈×C₇₉	—	4	1	no	—
53	C₂×C₈×C₇₉	—	4	2	no	—
54	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
58	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₇₉	#41	1	1	yes	characteristic
61	C₄²×C₇₉	#41	1	1	yes	characteristic
62	C₂³×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×C₇₉	—	2	1	no	—
64	C₂×C₄×C₇₉	—	2	1	no	—
65	C₂×C₄×C₇₉	—	4	1	no	—
66	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₇₉	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
70	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₇₉	#125	1	1	yes	characteristic
72	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
73	C₂×C₄×C₇₉	#137	1	2	yes	—
74	C₈×C₇₉	—	2	1	no	—
75	C₈×C₇₉	—	2	1	no	—
76	C₈×C₇₉	—	2	1	no	—
77	C₈×C₇₉	—	8	2	no	—
78	C₈×C₇₉	#108	1	2	yes	complemented
79	C₇₉⋊C₈	—	4	2	no	—
80	C₇₉⋊C₈	—	4	4	no	—
81	C₇₉⋊C₈	—	4	2	no	—
82	C₂²×C₇₉	—	2	1	no	—
83	C₂²×C₇₉	—	2	1	no	—
84	C₂²×C₇₉	#389	1	1	yes	characteristic
85	C₂²×C₇₉	#501	1	2	yes	—
86	C₄×C₇₉	—	2	1	no	—
87	C₄×C₇₉	—	2	1	no	—
88	C₄×C₇₉	—	2	1	no	—
89	C₄×C₇₉	C₂×#108	1	1	yes	characteristic
90	C₄×C₇₉	#430	1	1	yes	characteristic
91	C₄×C₇₉	#441	1	1	yes	characteristic
92	C₄×C₇₉	#442	1	1	yes	characteristic
93	C₄×C₇₉	#468	1	2	yes	—
94	#9625	—	79	1	no	maximal, Sylow
95	C₂×C₇₉	—	2	2	no	—
96	C₂×C₇₉	#1943	1	1	yes	characteristic
97	C₂×C₇₉	#1950	1	1	yes	characteristic
98	C₂×C₇₉	#1953	1	1	yes	characteristic
99	#1950	—	79	2	no	—
100	#1566	D₇₉	1	1	yes	characteristic
101	C₂×#335	—	79	1	no	—
102	#1451	—	79	1	no	—
103	#1301	—	79	2	no	—
104	C₇₉	#9625	1	1	yes	characteristic, complemented, Sylow
105	#431	—	79	1	no	—
106	C₂×#108	—	158	2	no	—
107	#423	C₂×D₇₉	1	2	yes	—
108	C₂×C₄⋊C₈	—	79	2	no	—
109	C₄×M₁₆	—	79	1	no	—
110	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
111	#341	—	158	2	no	—
112	#335	—	79	2	no	—
113	#335	—	79	2	no	—
114	#334	—	2	1	no	—
115	#334	—	2	1	no	—
116	#322	—	158	1	no	—
117	#322	—	158	1	no	—
118	C₂×M₁₆	—	2	2	no	—
119	C₂×M₁₆	—	79	2	no	—
120	C₂²×C₈	—	158	2	no	—
121	C₂²×C₈	Dic₁₅₈	1	1	yes	characteristic
122	C₂²×C₈	Dic₁₅₈	1	1	yes	characteristic
123	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
124	#108	—	316	4	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	79	4	no	—
128	C₄⋊C₈	—	158	2	no	—
129	C₄⋊C₈	—	158	1	no	—
130	C₂²⋊C₈	—	79	2	no	—
131	#97	—	158	1	no	—
132	C₄×C₈	—	158	1	no	—
133	C₄×C₈	—	158	1	no	—
134	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
135	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
136	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
137	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
138	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
139	C₂²×C₄	Q₈×D₇₉	1	1	yes	characteristic
140	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
141	M₁₆	—	4	2	no	—
142	M₁₆	—	4	2	no	—
143	M₁₆	—	158	2	no	—
144	M₁₆	—	158	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	4	1	no	—
147	C₂×C₈	—	4	2	no	—
148	C₂×C₈	—	316	2	no	—
149	C₂×C₈	—	316	2	no	—
150	C₂×C₈	—	158	4	no	—
151	C₂×C₈	—	158	2	no	—
152	C₂×C₈	—	158	2	no	—
153	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
154	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
155	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
156	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
157	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
158	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
159	C₄²	#11929623	1	1	yes	characteristic
160	C₄²	#11929625	1	1	yes	characteristic
161	C₂³	#14579160	1	1	yes	characteristic
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	4	1	no	—
165	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
166	C₂×C₄	#14579123	1	1	yes	characteristic
167	C₂×C₄	#14579126	1	1	yes	characteristic
168	C₂×C₄	#14579132	1	1	yes	characteristic
169	C₂×C₄	#14579137	1	1	yes	characteristic
170	C₂×C₄	#14579152	1	1	yes	characteristic
171	C₂×C₄	#14579188	1	2	yes	—
172	C₂×C₄	#14579190	1	2	yes	—
173	C₈	—	2	1	no	—
174	C₈	—	2	1	no	—
175	C₈	—	2	1	no	—
176	C₈	—	8	2	no	—
177	C₈	—	316	2	no	—
178	C₈	—	316	4	no	—
179	C₈	—	316	2	no	—
180	C₈	#14579096	1	2	yes	complemented
181	C₂²	—	2	1	no	—
182	C₂²	—	2	1	no	—
183	C₂²	#19098459	1	1	yes	characteristic
184	C₂²	#19098970	1	2	yes	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	C₂×#14579096	1	1	yes	characteristic
189	C₄	#19098611	1	1	yes	characteristic
190	C₄	#19098669	1	1	yes	characteristic
191	C₄	#19098674	1	1	yes	characteristic
192	C₄	#19098806	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	#23764136	1	1	yes	characteristic
195	C₂	#23764173	1	1	yes	characteristic
196	C₂	#23764188	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23764173

C₂

yes

maximal

C₂×#19098316

C₂

yes

maximal, characteristic

#23761689

C₂

yes

maximal, characteristic

#23761295

C₂

yes

maximal

C₇₉×#1566

C₂

yes

maximal, characteristic

#19098616

C₂²

yes

characteristic

C₂×#14579096

—

C₂×C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

C₄×#11929608

C₂²

yes

characteristic

#19098327

—

#19098316

C₄

yes

—

#19098316

C₄

yes

—

#19098285

—

#19098284

—

#423×C₇₉

C₂²

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#334×C₇₉

—

#334×C₇₉

—

C₂×#11929608

D₄

yes

—

C₂²×C₇₉⋊C₈

—

#14579096

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

C₂×C₄

yes

—

C₂²⋊(C₇₉⋊C₈)

Q₈

yes

—

#14579072

—

C₄×C₇₉⋊C₈

—

C₄×C₇₉⋊C₈

—

C₂×M₁₆×C₇₉

—

C₂²×C₈×C₇₉

Q₈

yes

characteristic

C₂²×C₈×C₇₉

Q₈

yes

characteristic

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×Q₈

yes

characteristic

C₂²×C₄×C₇₉

C₂×Q₈

yes

characteristic

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₉

C₄⋊C₄

yes

characteristic

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

C₄²×C₇₉

yes

C₄²×C₇₉

yes

C₂³×C₇₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄×D₄

yes

characteristic

C₂×C₄×C₇₉

C₄×Q₈

yes

characteristic

C₂×C₄×C₇₉

#121

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

—

C₂×C₄×C₇₉

#137

yes

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

#108

yes

complemented

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#389

yes

characteristic

C₂²×C₇₉

#501

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

#9625

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

100

#1566

D₇₉

yes

characteristic

101

C₂×#335

—

102

#1451

—

103

#1301

—

104

C₇₉

#9625

yes

characteristic, complemented, Sylow

105

#431

—

106

C₂×#108

—

158

—

107

#423

C₂×D₇₉

yes

—

108

C₂×C₄⋊C₈

—

109

C₄×M₁₆

—

110

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

111

#341

—

158

—

112

#335

—

113

#335

—

114

#334

—

115

#334

—

116

#322

—

158

—

117

#322

—

158

—

118

C₂×M₁₆

—

119

C₂×M₁₆

—

120

C₂²×C₈

—

158

—

121

C₂²×C₈

Dic₁₅₈

yes

characteristic

122

C₂²×C₈

Dic₁₅₈

yes

characteristic

123

C₂×C₄²

C₂²×D₇₉

yes

characteristic

124

#108

—

316

—

125

C₄⋊C₈

—

126

C₄⋊C₈

—

127

C₄⋊C₈

—

128

C₄⋊C₈

—

158

—

129

C₄⋊C₈

—

158

—

130

C₂²⋊C₈

—

131

#97

—

158

—

132

C₄×C₈

—

158

—

133

C₄×C₈

—

158

—

134

C₄×C₈

C₄×D₇₉

yes

characteristic

135

C₄×C₈

C₄×D₇₉

yes

characteristic

136

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

137

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

138

C₂²×C₄

D₄×D₇₉

yes

characteristic

139

C₂²×C₄

Q₈×D₇₉

yes

characteristic

140

C₂²×C₄

C₂×Dic₁₅₈

yes

characteristic

141

M₁₆

—

142

M₁₆

—

143

M₁₆

—

158

—

144

M₁₆

—

158

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

316

—

149

C₂×C₈

—

316

—

150

C₂×C₈

—

158

—

151

C₂×C₈

—

158

—

152

C₂×C₈

—

158

—

153

C₂×C₈

Dic₇₉⋊C₄

yes

characteristic

154

C₂×C₈

Dic₇₉⋊C₄

yes

characteristic

155

C₂×C₈

Dic₇₉⋊C₄

yes

characteristic

156

C₂×C₈

Dic₇₉⋊C₄

yes

characteristic

157

C₄²

C₂×C₄×D₇₉

yes

characteristic

158

C₄²

C₂×C₂²⋊D₇₉

yes

characteristic

159

C₄²

#11929623

yes

characteristic

160

C₄²

#11929625

yes

characteristic

161

C₂³

#14579160

yes

characteristic

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

C₂×Dic₇₉⋊C₄

yes

characteristic

166

C₂×C₄

#14579123

yes

characteristic

167

C₂×C₄

#14579126

yes

characteristic

168

C₂×C₄

#14579132

yes

characteristic

169

C₂×C₄

#14579137

yes

characteristic

170

C₂×C₄

#14579152

yes

characteristic

171

C₂×C₄

#14579188

yes

—

172

C₂×C₄

#14579190

yes

—

173

C₈

—

174

C₈

—

175

C₈

—

176

C₈

—

177

C₈

—

316

—

178

C₈

—

316

—

179

C₈

—

316

—

180

C₈

#14579096

yes

complemented

181

C₂²

—

182

C₂²

—

183

C₂²

#19098459

yes

characteristic

184

C₂²

#19098970

yes

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

C₂×#14579096

yes

characteristic

189

C₄

#19098611

yes

characteristic

190

C₄

#19098669

yes

characteristic

191

C₄

#19098674

yes

characteristic

192

C₄

#19098806

yes

—

193

C₂

—

194

C₂

#23764136

yes

characteristic

195

C₂

#23764173

yes

characteristic

196

C₂

#23764188

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14579132
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1566	C₂
socle	C₂²×C₇₉	#389
prime radical	C₂³×C₇₉	C₄⋊Q₈
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1566	D₇₉
hypocenter	C₇₉	#9625
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

subgroup

G/H

center

C₂×C₄

#14579132

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1566

C₂

socle

C₂²×C₇₉

#389

prime radical

C₂³×C₇₉

C₄⋊Q₈

cosocle

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

N	H ≅ G/N	type
78. C₈×C₇₉	#108	semidirect
104. C₇₉	#9625	semidirect
180. C₈	#14579096	semidirect

H ≅ G/N

type

78. C₈×C₇₉

#108

semidirect

104. C₇₉

#9625

semidirect

180. C₈

#14579096

semidirect

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

subgroup	index
#23764173	2
C₂×#19098316	2
#23761689	2
#23761295	2
C₇₉×#1566	2
#9625	79

subgroup

index

C₇₉×#1566

Subgroups — #34186976

Conjugacy classes of subgroups

Complements

Maximal subgroups