Group #34188003 — subgroups

Subgroups — #34188003

Order 20224.

526 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 211 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×#19098961	C₂	1	1	yes	maximal, characteristic
2	#23764361	C₂	1	2	yes	maximal
3	#23764202	C₂	1	2	yes	maximal
4	#23763557	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈²×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579161	C₂²	1	2	yes	—
7	#19098961	—	2	2	no	—
8	#19098961	C₄	1	4	yes	complemented
9	C₂×#14579093	—	2	2	no	—
10	C₂×#14579093	—	2	2	no	—
11	C₂×#14579086	—	2	4	no	—
12	#19098433	C₂²	1	1	yes	characteristic
13	#19098433	C₂²	1	1	yes	characteristic
14	#19098326	—	2	4	no	—
15	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
17	C₈²×C₇₉	—	2	2	no	—
18	C₂²×Dic₁₅₈	—	2	2	no	—
19	#14579161	—	2	2	no	—
20	#14579161	—	2	2	no	—
21	#14579161	—	2	2	no	—
22	#14579161	C₂×C₄	1	4	yes	—
23	C₂×C₄⋊Dic₇₉	—	2	2	no	—
24	C₂×C₄⋊Dic₇₉	—	2	2	no	—
25	C₂×C₄⋊Dic₇₉	—	2	2	no	—
26	C₂×C₄⋊Dic₇₉	—	2	2	no	—
27	C₂×C₄⋊Dic₇₉	—	2	1	no	—
28	C₂×C₄⋊Dic₇₉	—	2	1	no	—
29	#14579093	—	2	8	no	—
30	#14579093	—	4	4	no	—
31	#14579086	—	4	8	no	—
32	C₂²×C₈×C₇₉	D₄	1	4	yes	—
33	C₂²×C₈×C₇₉	D₄	1	2	yes	—
34	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
35	C₄×C₈×C₇₉	—	2	2	no	—
36	C₄×C₈×C₇₉	—	2	2	no	—
37	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₇₉	D₄	1	2	yes	—
39	C₂×Dic₁₅₈	—	2	4	no	—
40	C₂×Dic₁₅₈	—	4	2	no	—
41	C₂×Dic₁₅₈	—	4	4	no	—
42	C₂×Dic₁₅₈	—	4	4	no	—
43	C₂²×Dic₇₉	—	4	2	no	—
44	C₂²×Dic₇₉	—	4	1	no	—
45	C₂²×Dic₇₉	—	4	1	no	—
46	C₄⋊Dic₇₉	—	2	4	no	—
47	C₄⋊Dic₇₉	—	2	4	no	—
48	C₄⋊Dic₇₉	—	4	2	no	—
49	C₄⋊Dic₇₉	—	4	2	no	—
50	C₄⋊Dic₇₉	—	4	4	no	—
51	C₄⋊Dic₇₉	—	4	2	no	—
52	C₄⋊Dic₇₉	—	4	2	no	—
53	C₄⋊Dic₇₉	—	4	4	no	—
54	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₇₉	—	2	4	no	—
57	C₂×C₈×C₇₉	—	2	4	no	—
58	C₂×C₈×C₇₉	C₂²⋊C₄	1	4	yes	—
59	C₂×C₈×C₇₉	SD₁₆	1	8	yes	—
60	C₂×C₈×C₇₉	SD₁₆	1	4	yes	—
61	C₂×C₈×C₇₉	Q₁₆	1	4	yes	—
62	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₇₉	#41	1	2	yes	—
65	C₂³×C₇₉	#127	1	1	yes	characteristic
66	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
68	C₂×C₄×C₇₉	#121	1	2	yes	—
69	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
70	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
71	C₂×C₄×C₇₉	C₂×SD₁₆	1	4	yes	—
72	C₂×C₄×C₇₉	C₂×Q₁₆	1	2	yes	—
73	C₂×C₄×C₇₉	#135	1	4	yes	—
74	C₂×Dic₇₉	—	4	4	no	—
75	C₂×Dic₇₉	—	8	2	no	—
76	C₂×Dic₇₉	—	8	2	no	—
77	C₂×Dic₇₉	—	8	1	no	—
78	C₂×Dic₇₉	—	8	2	no	—
79	C₂×Dic₇₉	—	8	1	no	—
80	C₂×Dic₇₉	—	8	2	no	—
81	Dic₁₅₈	—	4	8	no	—
82	Dic₁₅₈	—	8	4	no	—
83	C₈×C₇₉	—	2	8	no	—
84	C₈×C₇₉	#103	1	8	yes	complemented
85	C₂²×C₇₉	#390	1	1	yes	characteristic
86	C₂²×C₇₉	#492	1	1	yes	characteristic
87	C₂²×C₇₉	#492	1	2	yes	—
88	C₂²×C₇₉	#494	1	1	yes	characteristic
89	C₂²×C₇₉	#495	1	2	yes	—
90	C₄×C₇₉	C₂×#103	1	2	yes	—
91	C₄×C₇₉	C₄×SD₁₆	1	4	yes	—
92	C₄×C₇₉	#465	1	4	yes	—
93	C₄×C₇₉	#487	1	2	yes	—
94	Dic₇₉	—	8	4	no	—
95	Dic₇₉	—	16	1	no	—
96	Dic₇₉	—	16	1	no	—
97	#9819	—	79	1	no	maximal, Sylow
98	C₂×C₇₉	#1705	1	1	yes	characteristic
99	C₂×C₇₉	#1707	1	2	yes	—
100	C₂×C₇₉	#1708	1	1	yes	characteristic
101	C₂×C₇₉	#1956	1	1	yes	characteristic
102	C₂×C₇₉	#1968	1	2	yes	—
103	C₂×#498	—	79	1	no	—
104	#1990	—	79	2	no	—
105	#1956	—	79	2	no	—
106	#1844	—	79	1	no	—
107	C₂×C₈²	D₇₉	1	1	yes	characteristic
108	C₇₉	#9819	1	1	yes	characteristic, complemented, Sylow
109	C₂×C₄⋊Q₈	—	79	2	no	—
110	#498	—	79	4	no	—
111	#498	—	158	2	no	—
112	C₂×#106	—	158	2	no	—
113	C₂×#106	—	158	2	no	—
114	C₂×#103	—	158	4	no	—
115	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
116	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
117	#384	—	79	1	no	—
118	#384	—	79	1	no	—
119	#340	—	158	4	no	—
120	C₈²	—	2	2	no	—
121	C₂²×Q₈	—	158	2	no	—
122	C₂²×C₈	D₃₁₆	1	4	yes	—
123	C₂²×C₈	D₃₁₆	1	2	yes	—
124	C₄⋊Q₈	—	79	4	no	—
125	C₄⋊Q₈	—	158	2	no	—
126	C₄⋊Q₈	—	158	2	no	—
127	C₄⋊Q₈	—	158	2	no	—
128	C₂×C₄⋊C₄	—	158	2	no	—
129	C₂×C₄⋊C₄	—	158	2	no	—
130	C₂×C₄⋊C₄	—	158	1	no	—
131	C₂×C₄⋊C₄	—	158	1	no	—
132	C₂×C₄⋊C₄	—	158	2	no	—
133	C₂×C₄⋊C₄	—	158	2	no	—
134	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
135	#106	—	316	4	no	—
136	#106	—	158	8	no	—
137	#103	—	316	8	no	—
138	C₄×C₈	—	2	2	no	—
139	C₄×C₈	—	2	2	no	—
140	C₄×C₈	C₄×D₇₉	1	2	yes	—
141	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
142	C₂×Q₈	—	316	2	no	—
143	C₂×Q₈	—	316	4	no	—
144	C₂×Q₈	—	316	4	no	—
145	C₂×Q₈	—	158	4	no	—
146	C₂²×C₄	—	316	2	no	—
147	C₂²×C₄	—	316	1	no	—
148	C₂²×C₄	—	316	1	no	—
149	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
150	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	(C₂×D₇₉)⋊C₄	1	4	yes	—
154	C₂×C₈	#11929613	1	8	yes	—
155	C₂×C₈	#11929613	1	4	yes	—
156	C₂×C₈	Dic₃₁₆	1	4	yes	—
157	C₄⋊C₄	—	316	2	no	—
158	C₄⋊C₄	—	316	2	no	—
159	C₄⋊C₄	—	316	4	no	—
160	C₄⋊C₄	—	316	2	no	—
161	C₄⋊C₄	—	316	2	no	—
162	C₄⋊C₄	—	316	4	no	—
163	C₄⋊C₄	—	158	4	no	—
164	C₄⋊C₄	—	158	4	no	—
165	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
166	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
167	C₄²	#11929624	1	2	yes	—
168	C₂³	#14579158	1	1	yes	characteristic
169	Q₈	—	316	8	no	—
170	Q₈	—	632	4	no	—
171	C₂×C₄	—	316	4	no	—
172	C₂×C₄	—	632	2	no	—
173	C₂×C₄	—	632	2	no	—
174	C₂×C₄	—	632	2	no	—
175	C₂×C₄	—	632	1	no	—
176	C₂×C₄	—	632	1	no	—
177	C₂×C₄	—	632	2	no	—
178	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
179	C₂×C₄	C₄×D₃₁₆	1	2	yes	—
180	C₂×C₄	#14579135	1	2	yes	—
181	C₂×C₄	#14579149	1	1	yes	characteristic
182	C₂×C₄	C₂×#11929613	1	4	yes	—
183	C₂×C₄	C₂×#11929613	1	2	yes	—
184	C₂×C₄	C₂×Dic₃₁₆	1	2	yes	—
185	C₂×C₄	#14579183	1	4	yes	—
186	C₈	—	2	8	no	—
187	C₈	#14579086	1	8	yes	complemented
188	C₂²	#19098465	1	1	yes	characteristic
189	C₂²	#19098936	1	2	yes	—
190	C₂²	#19098936	1	1	yes	characteristic
191	C₂²	#19098943	1	1	yes	characteristic
192	C₂²	#19098947	1	2	yes	—
193	C₄	—	632	4	no	—
194	C₄	—	1264	1	no	—
195	C₄	—	1264	1	no	—
196	C₄	C₂×#14579086	1	2	yes	—
197	C₄	C₄×#11929613	1	4	yes	—
198	C₄	#19098792	1	4	yes	—
199	C₄	#19098914	1	2	yes	—
200	C₂	#23762987	1	1	yes	characteristic
201	C₂	#23762997	1	2	yes	—
202	C₂	#23763001	1	1	yes	characteristic
203	C₂	#23764202	1	1	yes	characteristic
204	C₂	#23764254	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19098961

C₂

yes

maximal, characteristic

#23764361

C₂

yes

maximal

#23764202

C₂

yes

maximal

#23763557

C₂

yes

maximal, characteristic

C₂×C₈²×C₇₉

C₂

yes

maximal, characteristic

C₂×#14579161

C₂²

yes

—

#19098961

—

#19098961

C₄

yes

—

—

—

C₂²

yes

characteristic

#19098433

C₂²

yes

characteristic

#19098326

—

C₂×C₄×C₈×C₇₉

C₂²

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₈²×C₇₉

—

C₂²×Dic₁₅₈

—

—

—

—

C₂×C₄

yes

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

#14579093

—

#14579093

—

#14579086

—

C₂²×C₈×C₇₉

D₄

yes

—

C₂²×C₈×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄×C₈×C₇₉

—

C₄×C₈×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

D₄

yes

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

SD₁₆

yes

—

C₂×C₈×C₇₉

SD₁₆

yes

—

C₂×C₈×C₇₉

Q₁₆

yes

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

#41

yes

—

C₂³×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄×D₄

yes

—

C₂×C₄×C₇₉

#121

yes

—

C₂×C₄×C₇₉

#124

yes

characteristic

C₂×C₄×C₇₉

C₂×SD₁₆

yes

—

C₂×C₄×C₇₉

C₂×SD₁₆

yes

—

C₂×C₄×C₇₉

C₂×Q₁₆

yes

—

C₂×C₄×C₇₉

#135

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

—

C₂²×C₇₉

#494

yes

characteristic

C₂²×C₇₉

#495

yes

—

C₄×C₇₉

C₂×#103

yes

—

C₄×C₇₉

C₄×SD₁₆

yes

—

C₄×C₇₉

#465

yes

—

C₄×C₇₉

#487

yes

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

100

C₂×C₇₉

#1708

yes

characteristic

101

C₂×C₇₉

#1956

yes

characteristic

102

C₂×C₇₉

#1968

yes

—

103

C₂×#498

—

104

#1990

—

105

#1956

—

106

#1844

—

107

C₂×C₈²

D₇₉

yes

characteristic

108

C₇₉

#9819

yes

characteristic, complemented, Sylow

109

C₂×C₄⋊Q₈

—

110

#498

—

111

#498

—

158

—

112

C₂×#106

—

158

—

113

C₂×#106

—

158

—

114

C₂×#103

—

158

—

115

C₂×C₄×C₈

C₂×D₇₉

yes

—

116

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

117

#384

—

118

#384

—

119

#340

—

158

—

120

C₈²

—

121

C₂²×Q₈

—

158

—

122

C₂²×C₈

D₃₁₆

yes

—

123

C₂²×C₈

D₃₁₆

yes

—

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

158

—

126

C₄⋊Q₈

—

158

—

127

C₄⋊Q₈

—

158

—

128

C₂×C₄⋊C₄

—

158

—

129

C₂×C₄⋊C₄

—

158

—

130

C₂×C₄⋊C₄

—

158

—

131

C₂×C₄⋊C₄

—

158

—

132

C₂×C₄⋊C₄

—

158

—

133

C₂×C₄⋊C₄

—

158

—

134

C₂×C₄²

C₂²×D₇₉

yes

characteristic

135

#106

—

316

—

136

#106

—

158

—

137

#103

—

316

—

138

C₄×C₈

—

139

C₄×C₈

—

140

C₄×C₈

C₄×D₇₉

yes

—

141

C₄×C₈

C₂²⋊D₇₉

yes

—

142

C₂×Q₈

—

316

—

143

C₂×Q₈

—

316

—

144

C₂×Q₈

—

316

—

145

C₂×Q₈

—

158

—

146

C₂²×C₄

—

316

—

147

C₂²×C₄

—

316

—

148

C₂²×C₄

—

316

—

149

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

150

C₂²×C₄

C₂×D₃₁₆

yes

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

(C₂×D₇₉)⋊C₄

yes

—

154

C₂×C₈

#11929613

yes

—

155

C₂×C₈

#11929613

yes

—

156

C₂×C₈

Dic₃₁₆

yes

—

157

C₄⋊C₄

—

316

—

158

C₄⋊C₄

—

316

—

159

C₄⋊C₄

—

316

—

160

C₄⋊C₄

—

316

—

161

C₄⋊C₄

—

316

—

162

C₄⋊C₄

—

316

—

163

C₄⋊C₄

—

158

—

164

C₄⋊C₄

—

158

—

165

C₄²

C₂×C₄×D₇₉

yes

characteristic

166

C₄²

C₂×C₂²⋊D₇₉

yes

characteristic

167

C₄²

#11929624

yes

—

168

C₂³

#14579158

yes

characteristic

169

Q₈

—

316

—

170

Q₈

—

632

—

171

C₂×C₄

—

316

—

172

C₂×C₄

—

632

—

173

C₂×C₄

—

632

—

174

C₂×C₄

—

632

—

175

C₂×C₄

—

632

—

176

C₂×C₄

—

632

—

177

C₂×C₄

—

632

—

178

C₂×C₄

C₂×(C₂×D₇₉)⋊C₄

yes

characteristic

179

C₂×C₄

C₄×D₃₁₆

yes

—

180

C₂×C₄

#14579135

yes

—

181

C₂×C₄

#14579149

yes

characteristic

182

C₂×C₄

C₂×#11929613

yes

—

183

C₂×C₄

C₂×#11929613

yes

—

184

C₂×C₄

C₂×Dic₃₁₆

yes

—

185

C₂×C₄

#14579183

yes

—

186

C₈

—

187

C₈

#14579086

yes

complemented

188

C₂²

#19098465

yes

characteristic

189

C₂²

#19098936

yes

—

190

C₂²

#19098936

yes

characteristic

191

C₂²

#19098943

yes

characteristic

192

C₂²

#19098947

yes

—

193

C₄

—

632

—

194

C₄

—

1264

—

195

C₄

—

1264

—

196

C₄

C₂×#14579086

yes

—

197

C₄

C₄×#11929613

yes

—

198

C₄

#19098792

yes

—

199

C₄

#19098914

yes

—

200

C₂

#23762987

yes

characteristic

201

C₂

#23762997

yes

—

202

C₂

#23763001

yes

characteristic

203

C₂

#23764202

yes

characteristic

204

C₂

#23764254

yes

—

	subgroup	G/H
center	C₂³	#14579158
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×C₈²×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	#127
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×C₈²	D₇₉
hypocenter	C₇₉	#9819
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂³

#14579158

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₂×C₈²×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

#127

cosocle

C₂×C₄²×C₇₉

C₂³

hypercenter

C₂×C₈²

D₇₉

hypocenter

C₇₉

#9819

LCS

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
8. #19098961	C₄	semidirect
84. C₈×C₇₉	#103	semidirect
108. C₇₉	#9819	semidirect
187. C₈	#14579086	semidirect

H ≅ G/N

type

8. #19098961

C₄

semidirect

84. C₈×C₇₉

#103

semidirect

108. C₇₉

#9819

semidirect

187. C₈

#14579086

semidirect

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

subgroup	index
C₂×#19098961	2
#23764361	2
#23764202	2
#23763557	2
C₂×C₈²×C₇₉	2
#9819	79

subgroup

index

C₂×C₈²×C₇₉

Subgroups — #34188003

Conjugacy classes of subgroups

Complements

Maximal subgroups