Group #34186981 — subgroups

Subgroups — #34186981

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×#19098314	C₂	1	1	yes	maximal, characteristic
3	#23761689	C₂	1	1	yes	maximal, characteristic
4	#23761295	C₂	1	2	yes	maximal
5	C₇₉×#1567	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	#19098314	C₄	1	2	yes	—
12	#19098314	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	#335×C₇₉	—	2	1	no	—
18	#335×C₇₉	—	2	1	no	—
19	C₂×#11929608	D₄	1	1	yes	characteristic
20	C₂×#11929608	D₄	1	1	yes	characteristic
21	C₂²×C₇₉⋊C₈	—	2	2	no	—
22	#14579096	—	4	4	no	—
23	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
24	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
25	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	4	yes	—
26	C₂²⋊(C₇₉⋊C₈)	Q₈	1	2	yes	—
27	#14579072	—	2	1	no	—
28	C₄×C₇₉⋊C₈	—	2	1	no	—
29	C₄×C₇₉⋊C₈	—	2	1	no	—
30	C₂×M₁₆×C₇₉	—	2	2	no	—
31	C₂²×C₈×C₇₉	Q₈	1	1	yes	characteristic
32	C₂²×C₈×C₇₉	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₇₉	—	2	2	no	—
35	C₄⋊C₈×C₇₉	—	2	2	no	—
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	1	yes	characteristic
39	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
40	#11929608	—	2	2	no	—
41	#11929608	—	2	2	no	—
42	C₂×C₇₉⋊C₈	—	2	4	no	—
43	C₂×C₇₉⋊C₈	—	2	1	no	—
44	C₂×C₇₉⋊C₈	—	2	1	no	—
45	C₂×C₇₉⋊C₈	—	2	2	no	—
46	C₂×C₇₉⋊C₈	—	4	2	no	—
47	C₂×C₇₉⋊C₈	—	4	2	no	—
48	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
51	M₁₆×C₇₉	—	4	2	no	—
52	M₁₆×C₇₉	—	4	2	no	—
53	C₂×C₈×C₇₉	—	2	2	no	—
54	C₂×C₈×C₇₉	—	4	1	no	—
55	C₂×C₈×C₇₉	—	4	2	no	—
56	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
60	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
61	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₇₉	#41	1	1	yes	characteristic
63	C₄²×C₇₉	#41	1	1	yes	characteristic
64	C₂³×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₇₉	—	2	1	no	—
66	C₂×C₄×C₇₉	—	2	1	no	—
67	C₂×C₄×C₇₉	—	4	1	no	—
68	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₇₉	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
72	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₇₉	#125	1	1	yes	characteristic
74	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
75	C₂×C₄×C₇₉	#137	1	2	yes	—
76	C₈×C₇₉	—	2	1	no	—
77	C₈×C₇₉	—	2	1	no	—
78	C₈×C₇₉	—	2	1	no	—
79	C₈×C₇₉	—	8	2	no	—
80	C₈×C₇₉	#108	1	2	yes	complemented
81	C₇₉⋊C₈	—	4	2	no	—
82	C₇₉⋊C₈	—	4	4	no	—
83	C₇₉⋊C₈	—	4	1	no	—
84	C₇₉⋊C₈	—	4	1	no	—
85	C₂²×C₇₉	—	2	1	no	—
86	C₂²×C₇₉	—	2	1	no	—
87	C₂²×C₇₉	#389	1	1	yes	characteristic
88	C₂²×C₇₉	#501	1	2	yes	—
89	C₄×C₇₉	—	2	1	no	—
90	C₄×C₇₉	—	2	1	no	—
91	C₄×C₇₉	—	2	1	no	—
92	C₄×C₇₉	C₂×#108	1	1	yes	characteristic
93	C₄×C₇₉	#430	1	1	yes	characteristic
94	C₄×C₇₉	#440	1	2	yes	—
95	C₄×C₇₉	#469	1	1	yes	characteristic
96	C₄×C₇₉	#470	1	1	yes	characteristic
97	#9626	—	79	1	no	maximal, Sylow
98	C₂×C₇₉	—	2	2	no	—
99	C₂×C₇₉	#1944	1	1	yes	characteristic
100	C₂×C₇₉	#1950	1	1	yes	characteristic
101	C₂×C₇₉	#1953	1	1	yes	characteristic
102	#1950	—	79	2	no	—
103	#1567	D₇₉	1	1	yes	characteristic
104	C₂×#334	—	79	1	no	—
105	#1451	—	79	1	no	—
106	#1301	—	79	2	no	—
107	C₇₉	#9626	1	1	yes	characteristic, complemented, Sylow
108	#431	—	79	1	no	—
109	C₂×#108	—	158	2	no	—
110	#423	C₂×D₇₉	1	2	yes	—
111	C₂×C₄⋊C₈	—	79	2	no	—
112	C₄×M₁₆	—	79	1	no	—
113	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
114	#341	—	158	2	no	—
115	#335	—	2	1	no	—
116	#335	—	2	1	no	—
117	#334	—	79	2	no	—
118	#334	—	79	2	no	—
119	#322	—	158	1	no	—
120	#322	—	158	1	no	—
121	C₂×M₁₆	—	2	2	no	—
122	C₂×M₁₆	—	79	1	no	—
123	C₂×M₁₆	—	79	1	no	—
124	C₂²×C₈	—	158	2	no	—
125	C₂²×C₈	Dic₁₅₈	1	1	yes	characteristic
126	C₂²×C₈	Dic₁₅₈	1	1	yes	characteristic
127	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
128	#108	—	316	4	no	—
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	2	2	no	—
131	C₄⋊C₈	—	79	4	no	—
132	C₄⋊C₈	—	158	2	no	—
133	C₄⋊C₈	—	158	1	no	—
134	C₂²⋊C₈	—	79	2	no	—
135	#97	—	158	1	no	—
136	C₄×C₈	—	158	1	no	—
137	C₄×C₈	—	158	1	no	—
138	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
139	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
140	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
141	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
142	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
143	C₂²×C₄	Q₈×D₇₉	1	1	yes	characteristic
144	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
145	M₁₆	—	4	2	no	—
146	M₁₆	—	4	2	no	—
147	M₁₆	—	158	2	no	—
148	M₁₆	—	158	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	4	1	no	—
151	C₂×C₈	—	4	2	no	—
152	C₂×C₈	—	316	2	no	—
153	C₂×C₈	—	316	2	no	—
154	C₂×C₈	—	158	4	no	—
155	C₂×C₈	—	158	1	no	—
156	C₂×C₈	—	158	1	no	—
157	C₂×C₈	—	158	2	no	—
158	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
159	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
160	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
161	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
162	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
163	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
164	C₄²	#11929623	1	1	yes	characteristic
165	C₄²	#11929625	1	1	yes	characteristic
166	C₂³	#14579160	1	1	yes	characteristic
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
171	C₂×C₄	#14579123	1	1	yes	characteristic
172	C₂×C₄	#14579126	1	1	yes	characteristic
173	C₂×C₄	#14579132	1	1	yes	characteristic
174	C₂×C₄	#14579137	1	1	yes	characteristic
175	C₂×C₄	#14579152	1	1	yes	characteristic
176	C₂×C₄	#14579185	1	2	yes	—
177	C₂×C₄	#14579193	1	2	yes	—
178	C₈	—	2	1	no	—
179	C₈	—	2	1	no	—
180	C₈	—	2	1	no	—
181	C₈	—	8	2	no	—
182	C₈	—	316	2	no	—
183	C₈	—	316	4	no	—
184	C₈	—	316	1	no	—
185	C₈	—	316	1	no	—
186	C₈	#14579096	1	2	yes	complemented
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	#19098459	1	1	yes	characteristic
190	C₂²	#19098971	1	2	yes	—
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	C₂×#14579096	1	1	yes	characteristic
195	C₄	#19098611	1	1	yes	characteristic
196	C₄	#19098662	1	2	yes	—
197	C₄	#19098813	1	1	yes	characteristic
198	C₄	#19098818	1	1	yes	characteristic
199	C₂	—	2	2	no	—
200	C₂	#23764141	1	1	yes	characteristic
201	C₂	#23764173	1	1	yes	characteristic
202	C₂	#23764188	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23764173

C₂

yes

maximal

C₂×#19098314

C₂

yes

maximal, characteristic

#23761689

C₂

yes

maximal, characteristic

#23761295

C₂

yes

maximal

C₇₉×#1567

C₂

yes

maximal, characteristic

#19098616

C₂²

yes

characteristic

C₂×#14579096

—

C₂×C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

C₄×#11929608

C₂²

yes

characteristic

#19098327

—

#19098314

C₄

yes

—

#19098314

C₄

yes

—

#19098285

—

#19098284

—

#423×C₇₉

C₂²

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#335×C₇₉

—

#335×C₇₉

—

C₂×#11929608

D₄

yes

characteristic

C₂×#11929608

D₄

yes

characteristic

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

—

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

—

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

—

C₂×C₇₉

—

C₂×C₇₉

#1944

yes

characteristic

100

C₂×C₇₉

#1950

yes

characteristic

101

C₂×C₇₉

#1953

yes

characteristic

102

#1950

—

103

#1567

D₇₉

yes

characteristic

104

C₂×#334

—

105

#1451

—

106

#1301

—

107

C₇₉

#9626

yes

characteristic, complemented, Sylow

108

#431

—

109

C₂×#108

—

158

—

110

#423

C₂×D₇₉

yes

—

111

C₂×C₄⋊C₈

—

112

C₄×M₁₆

—

113

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

114

#341

—

158

—

115

#335

—

116

#335

—

117

#334

—

118

#334

—

119

#322

—

158

—

120

#322

—

158

—

121

C₂×M₁₆

—

122

C₂×M₁₆

—

123

C₂×M₁₆

—

124

C₂²×C₈

—

158

—

125

C₂²×C₈

Dic₁₅₈

yes

characteristic

126

C₂²×C₈

Dic₁₅₈

yes

characteristic

127

C₂×C₄²

C₂²×D₇₉

yes

characteristic

128

#108

—

316

—

129

C₄⋊C₈

—

130

C₄⋊C₈

—

131

C₄⋊C₈

—

132

C₄⋊C₈

—

158

—

133

C₄⋊C₈

—

158

—

134

C₂²⋊C₈

—

135

#97

—

158

—

136

C₄×C₈

—

158

—

137

C₄×C₈

—

158

—

138

C₄×C₈

C₄×D₇₉

yes

characteristic

139

C₄×C₈

C₄×D₇₉

yes

characteristic

140

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

141

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

142

C₂²×C₄

D₄×D₇₉

yes

characteristic

143

C₂²×C₄

Q₈×D₇₉

yes

characteristic

144

C₂²×C₄

C₂×Dic₁₅₈

yes

characteristic

145

M₁₆

—

146

M₁₆

—

147

M₁₆

—

158

—

148

M₁₆

—

158

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

316

—

153

C₂×C₈

—

316

—

154

C₂×C₈

—

158

—

155

C₂×C₈

—

158

—

156

C₂×C₈

—

158

—

157

C₂×C₈

—

158

—

158

C₂×C₈

Dic₇₉⋊C₄

yes

characteristic

159

C₂×C₈

Dic₇₉⋊C₄

yes

characteristic

160

C₂×C₈

Dic₇₉⋊C₄

yes

characteristic

161

C₂×C₈

Dic₇₉⋊C₄

yes

characteristic

162

C₄²

C₂×C₄×D₇₉

yes

characteristic

163

C₄²

C₂×C₂²⋊D₇₉

yes

characteristic

164

C₄²

#11929623

yes

characteristic

165

C₄²

#11929625

yes

characteristic

166

C₂³

#14579160

yes

characteristic

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

C₂×Dic₇₉⋊C₄

yes

characteristic

171

C₂×C₄

#14579123

yes

characteristic

172

C₂×C₄

#14579126

yes

characteristic

173

C₂×C₄

#14579132

yes

characteristic

174

C₂×C₄

#14579137

yes

characteristic

175

C₂×C₄

#14579152

yes

characteristic

176

C₂×C₄

#14579185

yes

—

177

C₂×C₄

#14579193

yes

—

178

C₈

—

179

C₈

—

180

C₈

—

181

C₈

—

182

C₈

—

316

—

183

C₈

—

316

—

184

C₈

—

316

—

185

C₈

—

316

—

186

C₈

#14579096

yes

complemented

187

C₂²

—

188

C₂²

—

189

C₂²

#19098459

yes

characteristic

190

C₂²

#19098971

yes

—

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

C₂×#14579096

yes

characteristic

195

C₄

#19098611

yes

characteristic

196

C₄

#19098662

yes

—

197

C₄

#19098813

yes

characteristic

198

C₄

#19098818

yes

characteristic

199

C₂

—

200

C₂

#23764141

yes

characteristic

201

C₂

#23764173

yes

characteristic

202

C₂

#23764188

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14579132
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1567	C₂
socle	C₂²×C₇₉	#389
prime radical	C₂³×C₇₉	C₄⋊Q₈
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1567	D₇₉
hypocenter	C₇₉	#9626
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

subgroup

G/H

center

C₂×C₄

#14579132

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1567

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 ◁ #1567

N	H ≅ G/N	type
80. C₈×C₇₉	#108	semidirect
107. C₇₉	#9626	semidirect
186. C₈	#14579096	semidirect

H ≅ G/N

type

80. C₈×C₇₉

#108

semidirect

107. C₇₉

#9626

semidirect

186. 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×#19098314	2
#23761689	2
#23761295	2
C₇₉×#1567	2
#9626	79

subgroup

index

C₇₉×#1567

Subgroups — #34186981

Conjugacy classes of subgroups

Complements

Maximal subgroups