Group #34206505 — subgroups

Subgroups — #34206505

Order 20224.

486 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 163 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×#19098943	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098368	C₂	1	4	yes	maximal
3	#23763564	C₂	1	1	yes	maximal, characteristic
4	C₂×#346×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₁₆	—	2	1	no	—
6	C₂²×Dic₃₁₆	C₂²	1	2	yes	—
7	C₂×#14579161	—	2	1	no	—
8	#19098943	—	2	2	no	—
9	#19098943	C₄	1	4	yes	complemented
10	C₂×#14579095	C₂²	1	2	yes	—
11	C₂×#14579093	—	2	1	no	—
12	#19098429	—	2	1	no	—
13	#19098368	—	2	8	no	—
14	C₂²×C₁₆×C₇₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
16	#346×C₇₉	—	2	2	no	—
17	C₂²×Dic₁₅₈	—	2	2	no	—
18	C₂×Dic₃₁₆	—	2	4	no	—
19	C₂×Dic₃₁₆	—	2	4	no	—
20	C₂×Dic₃₁₆	—	2	4	no	—
21	C₂×Dic₃₁₆	—	2	2	no	—
22	C₂×Dic₃₁₆	—	4	2	no	—
23	C₂×Dic₃₁₆	—	4	2	no	—
24	C₂×Dic₃₁₆	C₂×C₄	1	4	yes	—
25	#14579161	—	2	4	no	—
26	#14579161	—	4	2	no	—
27	C₂×C₄⋊Dic₇₉	—	2	2	no	—
28	C₂×C₄⋊Dic₇₉	—	4	2	no	—
29	#14579095	—	2	4	no	—
30	#14579093	—	4	2	no	—
31	C₇₉⋊#95	—	4	2	no	—
32	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₇₉	—	2	2	no	—
36	C₂×C₁₆×C₇₉	—	2	2	no	—
37	C₂×C₁₆×C₇₉	D₄	1	4	yes	—
38	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₇₉	D₄	1	2	yes	—
40	C₂×Dic₁₅₈	—	2	4	no	—
41	C₂×Dic₁₅₈	—	4	2	no	—
42	C₂×Dic₁₅₈	—	4	2	no	—
43	C₂×Dic₁₅₈	—	4	2	no	—
44	C₂×Dic₁₅₈	—	4	2	no	—
45	C₂×Dic₁₅₈	—	4	2	no	—
46	C₂²×Dic₇₉	—	4	2	no	—
47	C₂²×Dic₇₉	—	4	2	no	—
48	Dic₃₁₆	—	2	8	no	—
49	Dic₃₁₆	—	4	8	no	—
50	Dic₃₁₆	—	4	4	no	—
51	C₄⋊Dic₇₉	—	4	8	no	—
52	C₄⋊Dic₇₉	—	4	2	no	—
53	C₄⋊Dic₇₉	—	4	2	no	—
54	C₂²×C₄×C₇₉	D₈	1	2	yes	—
55	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₇₉	D₈	1	2	yes	—
58	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
59	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₇₉	#41	1	2	yes	—
63	C₄²×C₇₉	C₂²⋊C₄	1	2	yes	—
64	C₄²×C₇₉	SD₁₆	1	2	yes	—
65	C₁₆×C₇₉	—	2	4	no	—
66	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₇₉	—	2	4	no	—
68	C₂×C₄×C₇₉	#102	1	4	yes	—
69	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
71	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
72	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×Dic₇₉	—	4	4	no	—
76	C₂×Dic₇₉	—	8	2	no	—
77	C₂×Dic₇₉	—	8	2	no	—
78	C₂×Dic₇₉	—	8	2	no	—
79	C₂×Dic₇₉	—	8	4	no	—
80	Dic₁₅₈	—	4	4	no	—
81	Dic₁₅₈	—	4	4	no	—
82	Dic₁₅₈	—	8	2	no	—
83	Dic₁₅₈	—	8	2	no	—
84	C₈×C₇₉	#102	1	4	yes	—
85	C₈×C₇₉	C₄×D₄	1	2	yes	—
86	C₈×C₇₉	#121	1	2	yes	—
87	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
88	C₂²×C₇₉	#486	1	1	yes	characteristic
89	C₂²×C₇₉	#493	1	1	yes	characteristic
90	C₂²×C₇₉	#510	1	4	yes	—
91	C₄×C₇₉	—	2	4	no	—
92	C₄×C₇₉	#390	1	1	yes	characteristic
93	C₄×C₇₉	C₂×#102	1	1	yes	characteristic
94	C₄×C₇₉	#486	1	1	yes	characteristic
95	C₄×C₇₉	#492	1	1	yes	characteristic
96	Dic₇₉	—	8	4	no	—
97	Dic₇₉	—	16	2	no	—
98	#13207	—	79	1	no	maximal, Sylow
99	C₂×C₇₉	#1957	1	1	yes	characteristic
100	C₂×C₇₉	#2141	1	2	yes	—
101	C₂×C₇₉	#2245	1	2	yes	—
102	C₂×C₇₉	#2251	1	2	yes	—
103	C₂×#494	—	79	1	no	—
104	C₂×#358	—	79	4	no	—
105	C₂×#346	D₇₉	1	1	yes	characteristic
106	#1846	—	79	1	no	—
107	C₇₉	#13207	1	1	yes	characteristic, complemented, Sylow
108	C₂²×Q₁₆	—	79	2	no	—
109	C₂²×Q₁₆	—	158	1	no	—
110	C₂×C₄⋊Q₈	—	158	1	no	—
111	C₂²×C₁₆	C₂×D₇₉	1	2	yes	—
112	#494	—	79	4	no	—
113	#494	—	158	2	no	—
114	C₂×#107	—	79	2	no	—
115	C₂×#106	—	158	1	no	—
116	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
117	#382	—	158	1	no	—
118	#358	—	158	8	no	—
119	#346	—	2	2	no	—
120	C₂²×Q₈	—	158	2	no	—
121	C₂×Q₁₆	—	316	2	no	—
122	C₂×Q₁₆	—	316	2	no	—
123	C₂×Q₁₆	—	79	4	no	—
124	C₂×Q₁₆	—	158	4	no	—
125	C₂×Q₁₆	—	158	2	no	—
126	C₂×Q₁₆	—	158	4	no	—
127	C₂×Q₁₆	—	158	4	no	—
128	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
129	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
130	C₄⋊Q₈	—	316	2	no	—
131	C₄⋊Q₈	—	158	4	no	—
132	C₂×C₄⋊C₄	—	316	2	no	—
133	C₂×C₄⋊C₄	—	158	2	no	—
134	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	—	2	2	no	—
137	C₂×C₁₆	D₃₁₆	1	4	yes	—
138	#107	—	158	4	no	—
139	#106	—	316	2	no	—
140	C₄×C₈	C₄×D₇₉	1	2	yes	—
141	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
142	#95	—	316	2	no	—
143	C₂×Q₈	—	316	2	no	—
144	C₂×Q₈	—	316	2	no	—
145	C₂×Q₈	—	316	2	no	—
146	C₂×Q₈	—	316	2	no	—
147	C₂×Q₈	—	316	2	no	—
148	C₂×Q₈	—	158	4	no	—
149	C₂²×C₄	—	316	2	no	—
150	C₂²×C₄	—	316	2	no	—
151	C₂²×C₄	D₆₃₂	1	2	yes	—
152	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
153	Q₁₆	—	316	8	no	—
154	Q₁₆	—	316	4	no	—
155	Q₁₆	—	158	8	no	—
156	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
157	C₂×C₈	D₆₃₂	1	2	yes	—
158	C₂×C₈	#11929613	1	2	yes	—
159	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
160	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
161	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
162	C₂×C₈	#11929624	1	2	yes	—
163	C₄⋊C₄	—	316	8	no	—
164	C₄⋊C₄	—	316	2	no	—
165	C₄⋊C₄	—	316	2	no	—
166	C₄²	(C₂×D₇₉)⋊C₄	1	2	yes	—
167	C₄²	#11929613	1	2	yes	—
168	C₁₆	—	2	4	no	—
169	C₂³	C₂×D₆₃₂	1	1	yes	characteristic
170	Q₈	—	316	4	no	—
171	Q₈	—	316	4	no	—
172	Q₈	—	632	2	no	—
173	Q₈	—	632	2	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	316	4	no	—
176	C₂×C₄	—	632	2	no	—
177	C₂×C₄	—	632	2	no	—
178	C₂×C₄	—	632	2	no	—
179	C₂×C₄	—	632	4	no	—
180	C₂×C₄	#14579083	1	4	yes	—
181	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
182	C₂×C₄	#14579149	1	1	yes	characteristic
183	C₂×C₄	#14579158	1	1	yes	characteristic
184	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
185	C₂×C₄	C₂×#11929613	1	1	yes	characteristic
186	C₂×C₄	C₂×#11929613	1	1	yes	characteristic
187	C₈	#14579083	1	4	yes	—
188	C₈	C₄×D₃₁₆	1	2	yes	—
189	C₈	#14579135	1	2	yes	—
190	C₂²	C₂×#14579083	1	1	yes	characteristic
191	C₂²	#19098910	1	1	yes	characteristic
192	C₂²	#19098940	1	1	yes	characteristic
193	C₂²	#19099006	1	4	yes	—
194	C₄	—	2	4	no	—
195	C₄	—	632	4	no	—
196	C₄	—	1264	2	no	—
197	C₄	#19098465	1	1	yes	characteristic
198	C₄	C₂×#14579083	1	1	yes	characteristic
199	C₄	#19098910	1	1	yes	characteristic
200	C₄	#19098936	1	1	yes	characteristic
201	C₂	#23764206	1	1	yes	characteristic
202	C₂	#23765121	1	2	yes	—
203	C₂	#23765646	1	2	yes	—
204	C₂	#23765671	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19098943

C₂

yes

maximal, characteristic

C₂×#19098368

C₂

yes

maximal

#23763564

C₂

yes

maximal, characteristic

C₂×#346×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₃₁₆

—

C₂²×Dic₃₁₆

C₂²

yes

—

C₂×#14579161

—

#19098943

—

#19098943

C₄

yes

complemented

C₂×#14579095

C₂²

yes

—

C₂×#14579093

—

#19098429

—

#19098368

—

C₂²×C₁₆×C₇₉

C₂²

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#346×C₇₉

—

C₂²×Dic₁₅₈

—

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

C₂×C₄

yes

—

#14579161

—

#14579161

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

#14579095

—

#14579093

—

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×C₄

yes

—

C₄×C₈×C₇₉

D₄

yes

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

Dic₃₁₆

—

Dic₃₁₆

—

Dic₃₁₆

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂²×C₄×C₇₉

D₈

yes

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

D₈

yes

—

C₂×C₈×C₇₉

SD₁₆

yes

—

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

—

C₂×C₈×C₇₉

#41

yes

—

C₄²×C₇₉

C₂²⋊C₄

yes

—

C₄²×C₇₉

SD₁₆

yes

—

C₁₆×C₇₉

—

C₂³×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#102

yes

—

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₉

C₂×SD₁₆

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

#102

yes

—

C₈×C₇₉

C₄×D₄

yes

—

C₈×C₇₉

#121

yes

—

C₂²×C₇₉

yes

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

—

—

C₂×C₇₉

yes

100

C₂×C₇₉

#2141

yes

—

101

C₂×C₇₉

#2245

yes

—

102

C₂×C₇₉

#2251

yes

—

103

C₂×#494

—

104

C₂×#358

—

105

C₂×#346

D₇₉

yes

characteristic

106

#1846

—

107

C₇₉

#13207

yes

characteristic, complemented, Sylow

108

C₂²×Q₁₆

—

109

C₂²×Q₁₆

—

158

—

110

C₂×C₄⋊Q₈

—

158

—

111

C₂²×C₁₆

C₂×D₇₉

yes

—

112

#494

—

113

#494

—

158

—

114

C₂×#107

—

115

C₂×#106

—

158

—

116

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

117

#382

—

158

—

118

#358

—

158

—

119

#346

—

120

C₂²×Q₈

—

158

—

121

C₂×Q₁₆

—

316

—

122

C₂×Q₁₆

—

316

—

123

C₂×Q₁₆

—

124

C₂×Q₁₆

—

158

—

125

C₂×Q₁₆

—

158

—

126

C₂×Q₁₆

—

158

—

127

C₂×Q₁₆

—

158

—

128

C₂²×C₈

D₃₁₆

yes

characteristic

129

C₂²×C₈

C₂²×D₇₉

yes

characteristic

130

C₄⋊Q₈

—

316

—

131

C₄⋊Q₈

—

158

—

132

C₂×C₄⋊C₄

—

316

—

133

C₂×C₄⋊C₄

—

158

—

134

C₂×C₄²

D₃₁₆

yes

characteristic

135

C₂×C₁₆

—

136

C₂×C₁₆

—

137

C₂×C₁₆

D₃₁₆

yes

—

138

#107

—

158

—

139

#106

—

316

—

140

C₄×C₈

C₄×D₇₉

yes

—

141

C₄×C₈

C₂²⋊D₇₉

yes

—

142

#95

—

316

—

143

C₂×Q₈

—

316

—

144

C₂×Q₈

—

316

—

145

C₂×Q₈

—

316

—

146

C₂×Q₈

—

316

—

147

C₂×Q₈

—

316

—

148

C₂×Q₈

—

158

—

149

C₂²×C₄

—

316

—

150

C₂²×C₄

—

316

—

151

C₂²×C₄

D₆₃₂

yes

—

152

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

153

Q₁₆

—

316

—

154

Q₁₆

—

316

—

155

Q₁₆

—

158

—

156

C₂×C₈

(C₂×D₇₉)⋊C₄

yes

—

157

C₂×C₈

D₆₃₂

yes

—

158

C₂×C₈

#11929613

yes

—

159

C₂×C₈

C₂×C₄×D₇₉

yes

characteristic

160

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

161

C₂×C₈

C₂×D₃₁₆

yes

—

162

C₂×C₈

#11929624

yes

—

163

C₄⋊C₄

—

316

—

164

C₄⋊C₄

—

316

—

165

C₄⋊C₄

—

316

—

166

C₄²

(C₂×D₇₉)⋊C₄

yes

—

167

C₄²

#11929613

yes

—

168

C₁₆

—

169

C₂³

C₂×D₆₃₂

yes

characteristic

170

Q₈

—

316

—

171

Q₈

—

316

—

172

Q₈

—

632

—

173

Q₈

—

632

—

174

C₂×C₄

—

175

C₂×C₄

—

316

—

176

C₂×C₄

—

632

—

177

C₂×C₄

—

632

—

178

C₂×C₄

—

632

—

179

C₂×C₄

—

632

—

180

C₂×C₄

#14579083

yes

—

181

C₂×C₄

C₂×(C₂×D₇₉)⋊C₄

yes

characteristic

182

C₂×C₄

#14579149

yes

characteristic

183

C₂×C₄

#14579158

yes

characteristic

184

C₂×C₄

C₂×D₆₃₂

yes

characteristic

185

C₂×C₄

C₂×#11929613

yes

characteristic

186

C₂×C₄

C₂×#11929613

yes

characteristic

187

C₈

#14579083

yes

—

188

C₈

C₄×D₃₁₆

yes

—

189

C₈

#14579135

yes

—

190

C₂²

C₂×#14579083

yes

characteristic

191

C₂²

#19098910

yes

characteristic

192

C₂²

#19098940

yes

characteristic

193

C₂²

#19099006

yes

—

194

C₄

—

195

C₄

—

632

—

196

C₄

—

1264

—

197

C₄

#19098465

yes

characteristic

198

C₄

C₂×#14579083

yes

characteristic

199

C₄

#19098910

yes

characteristic

200

C₄

#19098936

yes

characteristic

201

C₂

#23764206

yes

characteristic

202

C₂

#23765121

yes

—

203

C₂

#23765646

yes

—

204

C₂

#23765671

yes

—

	subgroup	G/H
center	C₂³	C₂×D₆₃₂
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#346×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#346	D₇₉
hypocenter	C₇₉	#13207
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂³

C₂×D₆₃₂

commutator

C₂×C₈×C₇₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₂×#346×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂×D₈

cosocle

C₂²×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
9. #19098943	C₄	semidirect
107. C₇₉	#13207	semidirect

H ≅ G/N

type

9. #19098943

C₄

semidirect

107. C₇₉

#13207

semidirect

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

subgroup	index
C₂×#19098943	2
C₂×#19098368	2
#23763564	2
C₂×#346×C₇₉	2
#13207	79

subgroup

index

C₂×#346×C₇₉

Subgroups — #34206505

Conjugacy classes of subgroups

Complements

Maximal subgroups