Group #34208358 — subgroups

Subgroups — #34208358

Order 20224.

358 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 125 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×#19098391	C₂	1	2	yes	maximal
2	C₂×#19098368	C₂	1	2	yes	maximal
3	#23764261	C₂	1	1	yes	maximal, characteristic
4	#23763560	C₂	1	1	yes	maximal, characteristic
5	C₇₉×#1374	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₁₆	C₂²	1	1	yes	characteristic
7	C₂×#14579095	—	2	1	no	—
8	C₂×#14579095	C₂²	1	1	yes	characteristic
9	C₂×#14579095	C₂²	1	2	yes	—
10	C₂×#14579086	—	2	1	no	—
11	#19098469	—	2	1	no	—
12	#19098433	—	2	1	no	—
13	#19098391	—	2	4	no	—
14	#19098368	—	2	4	no	—
15	C₂²×C₁₆×C₇₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₅₈	—	2	1	no	—
18	C₂×Dic₃₁₆	—	2	2	no	—
19	C₂×Dic₃₁₆	—	2	2	no	—
20	C₂×Dic₃₁₆	—	2	1	no	—
21	C₂×Dic₃₁₆	Q₈	1	2	yes	—
22	C₂×C₄⋊Dic₇₉	—	2	2	no	—
23	C₂×C₄⋊Dic₇₉	—	2	1	no	—
24	C₂×C₄⋊Dic₇₉	—	4	2	no	—
25	#14579095	—	2	2	no	—
26	#14579095	—	2	2	no	—
27	#14579095	—	2	2	no	—
28	#14579095	—	4	2	no	—
29	#14579086	—	4	2	no	—
30	C₇₉⋊#95	—	4	2	no	—
31	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
34	C₂×C₁₆×C₇₉	—	2	2	no	—
35	C₂×C₁₆×C₇₉	—	2	2	no	—
36	C₂×C₁₆×C₇₉	D₄	1	4	yes	—
37	C₄×C₈×C₇₉	—	2	2	no	—
38	C₂×Dic₁₅₈	—	2	2	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	C₂×Dic₁₅₈	—	4	1	no	—
44	C₂²×Dic₇₉	—	4	1	no	—
45	C₂²×Dic₇₉	—	4	1	no	—
46	C₂²×Dic₇₉	—	4	2	no	—
47	Dic₃₁₆	—	2	4	no	—
48	Dic₃₁₆	—	4	2	no	—
49	C₄⋊Dic₇₉	—	4	2	no	—
50	C₄⋊Dic₇₉	—	4	1	no	—
51	C₄⋊Dic₇₉	—	4	1	no	—
52	C₄⋊Dic₇₉	—	4	2	no	—
53	C₄⋊Dic₇₉	—	8	4	no	—
54	C₂²×C₄×C₇₉	D₈	1	2	yes	—
55	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₇₉	—	2	1	no	—
57	C₂×C₈×C₇₉	—	2	1	no	—
58	C₂×C₈×C₇₉	D₈	1	2	yes	—
59	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
62	C₂×C₈×C₇₉	#41	1	2	yes	—
63	C₄²×C₇₉	—	2	1	no	—
64	C₄²×C₇₉	—	2	1	no	—
65	C₁₆×C₇₉	—	2	4	no	—
66	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₇₉	—	2	2	no	—
68	C₂×C₄×C₇₉	—	2	2	no	—
69	C₂×C₄×C₇₉	SD₃₂	1	4	yes	—
70	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
72	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
73	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
75	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
76	C₂×Dic₇₉	—	4	2	no	—
77	C₂×Dic₇₉	—	8	1	no	—
78	C₂×Dic₇₉	—	8	1	no	—
79	C₂×Dic₇₉	—	8	2	no	—
80	C₂×Dic₇₉	—	8	1	no	—
81	C₂×Dic₇₉	—	8	4	no	—
82	C₂×Dic₇₉	—	8	2	no	—
83	Dic₁₅₈	—	4	2	no	—
84	Dic₁₅₈	—	4	2	no	—
85	Dic₁₅₈	—	8	1	no	—
86	Dic₁₅₈	—	8	1	no	—
87	C₈×C₇₉	—	2	2	no	—
88	C₈×C₇₉	C₂²⋊Q₈	1	2	yes	—
89	C₈×C₇₉	#123	1	2	yes	—
90	C₂²×C₇₉	#474	1	1	yes	characteristic
91	C₂²×C₇₉	#480	1	1	yes	characteristic
92	C₂²×C₇₉	#493	1	1	yes	characteristic
93	C₂²×C₇₉	C₂×SD₃₂	1	2	yes	—
94	C₂²×C₇₉	#510	1	2	yes	—
95	C₄×C₇₉	—	2	4	no	—
96	C₄×C₇₉	#399	1	1	yes	characteristic
97	C₄×C₇₉	#474	1	1	yes	characteristic
98	C₄×C₇₉	#480	1	1	yes	characteristic
99	C₄×C₇₉	#497	1	1	yes	characteristic
100	Dic₇₉	—	8	2	no	—
101	Dic₇₉	—	16	2	no	—
102	Dic₇₉	—	16	1	no	—
103	#13531	—	79	1	no	maximal, Sylow
104	C₂×C₇₉	#2093	1	1	yes	characteristic
105	C₂×C₇₉	#2225	1	2	yes	—
106	C₂×C₇₉	#2235	1	2	yes	—
107	C₂×C₇₉	#2248	1	1	yes	characteristic
108	C₂×C₇₉	#2251	1	1	yes	characteristic
109	C₂×#367	—	79	2	no	—
110	C₂×#358	—	79	2	no	—
111	#1969	—	79	1	no	—
112	#1845	—	79	1	no	—
113	#1374	D₇₉	1	1	yes	characteristic
114	C₇₉	#13531	1	1	yes	characteristic, complemented, Sylow
115	C₂²×Q₁₆	—	79	1	no	—
116	C₂²×C₁₆	C₂×D₇₉	1	2	yes	—
117	C₂×#107	—	79	2	no	—
118	C₂×#107	—	79	1	no	—
119	C₂×#107	—	158	1	no	—
120	C₂×#103	—	158	1	no	—
121	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
122	#391	—	158	1	no	—
123	#384	—	158	1	no	—
124	#367	—	158	4	no	—
125	#358	—	158	4	no	—
126	C₂²×Q₈	—	158	1	no	—
127	C₂×Q₁₆	—	79	2	no	—
128	C₂×Q₁₆	—	158	2	no	—
129	C₂×Q₁₆	—	158	1	no	—
130	C₂×Q₁₆	—	158	2	no	—
131	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
132	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
133	C₂×C₄⋊C₄	—	316	2	no	—
134	C₂×C₄⋊C₄	—	158	2	no	—
135	C₂×C₄⋊C₄	—	158	1	no	—
136	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
137	C₂×C₁₆	—	2	2	no	—
138	C₂×C₁₆	—	2	2	no	—
139	C₂×C₁₆	D₃₁₆	1	4	yes	—
140	#107	—	316	2	no	—
141	#107	—	158	2	no	—
142	#107	—	158	2	no	—
143	#107	—	158	2	no	—
144	#103	—	316	2	no	—
145	C₄×C₈	—	2	2	no	—
146	#95	—	316	2	no	—
147	C₂×Q₈	—	316	1	no	—
148	C₂×Q₈	—	316	1	no	—
149	C₂×Q₈	—	316	1	no	—
150	C₂×Q₈	—	316	1	no	—
151	C₂×Q₈	—	316	1	no	—
152	C₂×Q₈	—	158	2	no	—
153	C₂²×C₄	—	316	1	no	—
154	C₂²×C₄	—	316	1	no	—
155	C₂²×C₄	—	316	2	no	—
156	C₂²×C₄	D₆₃₂	1	2	yes	—
157	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
158	Q₁₆	—	316	2	no	—
159	Q₁₆	—	158	4	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	D₆₃₂	1	2	yes	—
163	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
164	C₂×C₈	Q₈×D₇₉	1	1	yes	characteristic
165	C₂×C₈	#11929623	1	2	yes	—
166	C₂×C₈	#11929623	1	1	yes	characteristic
167	C₄⋊C₄	—	316	2	no	—
168	C₄⋊C₄	—	316	1	no	—
169	C₄⋊C₄	—	316	1	no	—
170	C₄⋊C₄	—	316	2	no	—
171	C₄⋊C₄	—	632	4	no	—
172	C₄²	—	2	1	no	—
173	C₄²	—	2	1	no	—
174	C₁₆	—	2	4	no	—
175	C₂³	C₂×D₆₃₂	1	1	yes	characteristic
176	Q₈	—	316	2	no	—
177	Q₈	—	316	2	no	—
178	Q₈	—	632	1	no	—
179	Q₈	—	632	1	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	316	2	no	—
183	C₂×C₄	—	632	1	no	—
184	C₂×C₄	—	632	1	no	—
185	C₂×C₄	—	632	2	no	—
186	C₂×C₄	—	632	1	no	—
187	C₂×C₄	—	632	4	no	—
188	C₂×C₄	—	632	2	no	—
189	C₂×C₄	#14579106	1	4	yes	—
190	C₂×C₄	#14579141	1	1	yes	characteristic
191	C₂×C₄	#14579145	1	1	yes	characteristic
192	C₂×C₄	#14579158	1	1	yes	characteristic
193	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
194	C₂×C₄	#14579194	1	1	yes	characteristic
195	C₂×C₄	#14579194	1	1	yes	characteristic
196	C₈	—	2	2	no	—
197	C₈	#14579141	1	2	yes	—
198	C₈	#14579145	1	2	yes	—
199	C₂²	#19098844	1	1	yes	characteristic
200	C₂²	#19098877	1	1	yes	characteristic
201	C₂²	#19098940	1	1	yes	characteristic
202	C₂²	C₂×#14579106	1	2	yes	—
203	C₂²	#19099006	1	2	yes	—
204	C₄	—	2	4	no	—
205	C₄	—	632	2	no	—
206	C₄	—	1264	2	no	—
207	C₄	—	1264	1	no	—
208	C₄	#19098500	1	1	yes	characteristic
209	C₄	#19098844	1	1	yes	characteristic
210	C₄	#19098877	1	1	yes	characteristic
211	C₄	#19098957	1	1	yes	characteristic
212	C₂	#23764927	1	1	yes	characteristic
213	C₂	#23765543	1	2	yes	—
214	C₂	#23765598	1	2	yes	—
215	C₂	#23765656	1	1	yes	characteristic
216	C₂	#23765671	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19098391

C₂

yes

maximal

C₂×#19098368

C₂

yes

maximal

#23764261

C₂

yes

maximal, characteristic

#23763560

C₂

yes

maximal, characteristic

C₇₉×#1374

C₂

yes

maximal, characteristic

C₂²×Dic₃₁₆

C₂²

yes

characteristic

C₂×#14579095

—

C₂×#14579095

C₂²

yes

characteristic

C₂×#14579095

C₂²

yes

—

—

—

—

—

—

C₂²×C₁₆×C₇₉

C₂²

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂²×Dic₁₅₈

—

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

Q₈

yes

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

—

—

—

—

—

C₇₉⋊#95

—

C₂²×C₈×C₇₉

D₄

yes

characteristic

C₂²×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

D₄

yes

—

C₄×C₈×C₇₉

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

Dic₃₁₆

—

Dic₃₁₆

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂²×C₄×C₇₉

D₈

yes

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

D₈

yes

—

C₂×C₈×C₇₉

C₂×D₄

yes

—

C₂×C₈×C₇₉

C₂×Q₈

yes

C₂×C₈×C₇₉

yes

C₂×C₈×C₇₉

yes

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₁₆×C₇₉

—

C₂³×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

SD₃₂

yes

—

C₂×C₄×C₇₉

C₂²⋊Q₈

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

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

C₂²⋊Q₈

yes

—

C₈×C₇₉

#123

yes

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

C₂×SD₃₂

yes

—

C₂²×C₇₉

#510

yes

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

100

Dic₇₉

—

101

Dic₇₉

—

102

Dic₇₉

—

103

#13531

—

maximal, Sylow

104

C₂×C₇₉

#2093

yes

characteristic

105

C₂×C₇₉

#2225

yes

—

106

C₂×C₇₉

#2235

yes

—

107

C₂×C₇₉

#2248

yes

characteristic

108

C₂×C₇₉

#2251

yes

characteristic

109

C₂×#367

—

110

C₂×#358

—

111

#1969

—

112

#1845

—

113

#1374

D₇₉

yes

characteristic

114

C₇₉

#13531

yes

characteristic, complemented, Sylow

115

C₂²×Q₁₆

—

116

C₂²×C₁₆

C₂×D₇₉

yes

—

117

C₂×#107

—

118

C₂×#107

—

119

C₂×#107

—

158

—

120

C₂×#103

—

158

—

121

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

122

#391

—

158

—

123

#384

—

158

—

124

#367

—

158

—

125

#358

—

158

—

126

C₂²×Q₈

—

158

—

127

C₂×Q₁₆

—

128

C₂×Q₁₆

—

158

—

129

C₂×Q₁₆

—

158

—

130

C₂×Q₁₆

—

158

—

131

C₂²×C₈

D₃₁₆

yes

characteristic

132

C₂²×C₈

C₂²×D₇₉

yes

characteristic

133

C₂×C₄⋊C₄

—

316

—

134

C₂×C₄⋊C₄

—

158

—

135

C₂×C₄⋊C₄

—

158

—

136

C₂×C₄²

D₃₁₆

yes

characteristic

137

C₂×C₁₆

—

138

C₂×C₁₆

—

139

C₂×C₁₆

D₃₁₆

yes

—

140

#107

—

316

—

141

#107

—

158

—

142

#107

—

158

—

143

#107

—

158

—

144

#103

—

316

—

145

C₄×C₈

—

146

#95

—

316

—

147

C₂×Q₈

—

316

—

148

C₂×Q₈

—

316

—

149

C₂×Q₈

—

316

—

150

C₂×Q₈

—

316

—

151

C₂×Q₈

—

316

—

152

C₂×Q₈

—

158

—

153

C₂²×C₄

—

316

—

154

C₂²×C₄

—

316

—

155

C₂²×C₄

—

316

—

156

C₂²×C₄

D₆₃₂

yes

—

157

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

158

Q₁₆

—

316

—

159

Q₁₆

—

158

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

D₆₃₂

yes

—

163

C₂×C₈

C₂×D₃₁₆

yes

—

164

C₂×C₈

Q₈×D₇₉

yes

characteristic

165

C₂×C₈

#11929623

yes

—

166

C₂×C₈

#11929623

yes

characteristic

167

C₄⋊C₄

—

316

—

168

C₄⋊C₄

—

316

—

169

C₄⋊C₄

—

316

—

170

C₄⋊C₄

—

316

—

171

C₄⋊C₄

—

632

—

172

C₄²

—

173

C₄²

—

174

C₁₆

—

175

C₂³

C₂×D₆₃₂

yes

characteristic

176

Q₈

—

316

—

177

Q₈

—

316

—

178

Q₈

—

632

—

179

Q₈

—

632

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

316

—

183

C₂×C₄

—

632

—

184

C₂×C₄

—

632

—

185

C₂×C₄

—

632

—

186

C₂×C₄

—

632

—

187

C₂×C₄

—

632

—

188

C₂×C₄

—

632

—

189

C₂×C₄

#14579106

yes

—

190

C₂×C₄

#14579141

yes

characteristic

191

C₂×C₄

#14579145

yes

characteristic

192

C₂×C₄

#14579158

yes

characteristic

193

C₂×C₄

C₂×D₆₃₂

yes

characteristic

194

C₂×C₄

#14579194

yes

characteristic

195

C₂×C₄

#14579194

yes

characteristic

196

C₈

—

197

C₈

#14579141

yes

—

198

C₈

#14579145

yes

—

199

C₂²

#19098844

yes

characteristic

200

C₂²

#19098877

yes

characteristic

201

C₂²

#19098940

yes

characteristic

202

C₂²

C₂×#14579106

yes

—

203

C₂²

#19099006

yes

—

204

C₄

—

205

C₄

—

632

—

206

C₄

—

1264

—

207

C₄

—

1264

—

208

C₄

#19098500

yes

characteristic

209

C₄

#19098844

yes

characteristic

210

C₄

#19098877

yes

characteristic

211

C₄

#19098957

yes

characteristic

212

C₂

#23764927

yes

characteristic

213

C₂

#23765543

yes

—

214

C₂

#23765598

yes

—

215

C₂

#23765656

yes

characteristic

216

C₂

#23765671

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₆₃₂
commutator, cosocle	C₂²×C₈×C₇₉	C₂³
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1374	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₈
hypercenter	#1374	D₇₉
hypocenter	C₇₉	#13531
LCS	G ▷ C₂²×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1374

subgroup

G/H

center

C₂³

C₂×D₆₃₂

commutator, cosocle

C₂²×C₈×C₇₉

C₂³

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₇₉×#1374

C₂

socle, prime radical

C₂³×C₇₉

C₂×D₈

D₇₉

C₇₉

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

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1374

N	H ≅ G/N	type
114. C₇₉	#13531	semidirect

H ≅ G/N

type

114. C₇₉

#13531

semidirect

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

subgroup	index
C₂×#19098391	2
C₂×#19098368	2
#23764261	2
#23763560	2
C₇₉×#1374	2
#13531	79

subgroup

index

C₇₉×#1374

Subgroups — #34208358

Conjugacy classes of subgroups

Complements

Maximal subgroups