Group #34110590 — subgroups

Subgroups — #34110590

Order 20096.

552 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 145 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×#19097800	C₂	1	4	yes	maximal, complemented
2	#23757821	C₂	1	1	yes	maximal, characteristic
3	C₄⋊Q₈×D₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#23757664	C₂	1	4	yes	maximal, complemented
5	#23757653	C₂	1	1	yes	maximal, characteristic, complemented
6	#23757652	C₂	1	1	yes	maximal, characteristic, complemented
7	#23757645	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#14578937	C₂	1	1	yes	maximal, characteristic, complemented
9	#492×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14578953	C₂²	1	2	yes	—
11	C₂×Q₈×D₁₅₇	C₂²	1	2	yes	—
12	#19097800	—	2	16	no	—
13	C₂×Dic₆₂₈	C₂²	1	4	yes	—
14	C₂×#14578944	C₂²	1	4	yes	—
15	C₂×#14578942	C₂²	1	4	yes	—
16	C₂×#14578937	C₂²	1	2	yes	—
17	#19097771	C₂²	1	1	yes	characteristic
18	#19097771	C₂²	1	1	yes	characteristic
19	#19097770	—	2	2	no	—
20	#19097769	C₂²	1	2	yes	—
21	#19097758	C₂²	1	2	yes	—
22	#19097742	—	2	2	no	—
23	D₄×Dic₁₅₇	—	2	2	no	—
24	C₄⋊C₄×D₁₅₇	—	2	2	no	—
25	C₄²×D₁₅₇	C₂²	1	1	yes	characteristic
26	#19097681	C₂²	1	2	yes	—
27	C₄×#11929434	C₂²	1	1	yes	characteristic
28	C₂×SD₁₆×C₁₅₇	C₂²	1	4	yes	—
29	C₄⋊Q₈×C₁₅₇	C₂²	1	1	yes	characteristic
30	#127×C₁₅₇	C₂²	1	1	yes	characteristic
31	C₄×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
32	#14578953	—	2	8	no	—
33	C₂×Dic₃₁₄	—	2	2	no	—
34	C₂×Dic₃₁₄	C₂³	1	2	yes	—
35	C₂×Dic₃₁₄	C₂³	1	2	yes	—
36	Q₈×D₁₅₇	—	2	8	no	—
37	C₂×C₂²⋊D₁₅₇	—	2	2	no	—
38	C₂²×Dic₁₅₇	—	2	2	no	—
39	C₂×C₄×D₁₅₇	—	2	2	no	—
40	C₂×C₄×D₁₅₇	D₄	1	2	yes	—
41	C₂×C₄×D₁₅₇	C₂³	1	1	yes	characteristic
42	Dic₆₂₈	—	2	8	no	—
43	#14578944	—	2	8	no	—
44	#14578942	—	2	8	no	—
45	#14578937	D₄	1	8	yes	complemented
46	C₂×#11929434	C₂³	1	2	yes	—
47	C₄⋊Dic₁₅₇	—	2	2	no	—
48	C₄⋊Dic₁₅₇	—	2	2	no	—
49	#14578933	—	2	4	no	—
50	C₂²⋊Dic₁₅₇	—	2	4	no	—
51	C₄×Dic₁₅₇	D₄	1	2	yes	—
52	C₄×Dic₁₅₇	C₂³	1	1	yes	characteristic
53	C₂×Q₈×C₁₅₇	C₂³	1	2	yes	—
54	C₂×D₄×C₁₅₇	—	2	2	no	—
55	C₂×D₄×C₁₅₇	C₂³	1	2	yes	—
56	SD₁₆×C₁₅₇	—	2	8	no	—
57	C₂×C₈×C₁₅₇	C₂³	1	2	yes	—
58	C₄⋊C₄×C₁₅₇	—	2	2	no	—
59	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
60	C₂²×D₁₅₇	C₂×D₄	1	1	yes	characteristic
61	Dic₃₁₄	—	2	4	no	—
62	Dic₃₁₄	—	2	4	no	—
63	Dic₃₁₄	—	4	4	no	—
64	C₂²⋊D₁₅₇	—	4	4	no	—
65	C₂×Dic₁₅₇	—	2	2	no	—
66	C₂×Dic₁₅₇	—	2	2	no	—
67	C₂×Dic₁₅₇	—	4	4	no	—
68	C₂×Dic₁₅₇	C₂×D₄	1	2	yes	—
69	C₂×Dic₁₅₇	C₂×D₄	1	1	yes	characteristic
70	C₄×D₁₅₇	—	2	4	no	—
71	C₄×D₁₅₇	—	4	4	no	—
72	C₄×D₁₅₇	C₂×D₄	1	4	yes	—
73	#11929434	C₂×D₄	1	4	yes	—
74	C₂³×C₁₅₇	—	2	2	no	—
75	Q₈×C₁₅₇	—	2	4	no	—
76	D₄×C₁₅₇	—	2	4	no	—
77	D₄×C₁₅₇	—	4	4	no	—
78	C₂×C₄×C₁₅₇	—	2	2	no	—
79	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
81	C₈×C₁₅₇	C₂×D₄	1	4	yes	—
82	C₂²×C₁₅₇	—	4	2	no	—
83	C₂²×C₁₅₇	—	4	4	no	—
84	C₂²×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
85	C₂×D₁₅₇	—	2	2	no	—
86	C₂×D₁₅₇	#127	1	2	yes	—
87	C₄×C₁₅₇	—	4	2	no	—
88	C₄×C₁₅₇	#137	1	4	yes	complemented
89	C₄×C₁₅₇	C₂²×D₄	1	2	yes	—
90	Dic₁₅₇	—	2	2	no	—
91	Dic₁₅₇	—	4	2	no	—
92	Dic₁₅₇	—	4	2	no	—
93	Dic₁₅₇	#127	1	2	yes	—
94	C₂×C₁₅₇	—	8	2	no	—
95	C₂×C₁₅₇	C₂×#127	1	1	yes	characteristic
96	C₂×C₁₅₇	C₂×#137	1	2	yes	—
97	D₁₅₇	—	2	2	no	—
98	C₁₅₇	#3152	1	1	yes	characteristic, complemented, Sylow
99	#3152	—	157	1	no	maximal, Sylow
100	C₂×#137	—	157	4	no	—
101	#532	—	157	1	no	—
102	C₂×C₄⋊Q₈	—	157	1	no	—
103	#497	—	157	4	no	—
104	#494	—	157	1	no	—
105	#494	—	157	1	no	—
106	#492	—	157	1	no	—
107	#492	D₁₅₇	1	1	yes	characteristic, complemented
108	C₄×M₁₆	—	157	1	no	—
109	C₂×#41	—	157	2	no	—
110	C₂²×Q₈	—	157	2	no	—
111	#137	—	314	16	no	—
112	C₂×Q₁₆	—	157	4	no	—
113	C₂×Q₁₆	—	157	4	no	—
114	C₂×SD₁₆	—	157	4	no	—
115	C₂×SD₁₆	C₂×D₁₅₇	1	4	yes	—
116	C₂×M₁₆	—	157	2	no	—
117	C₄⋊Q₈	—	314	2	no	—
118	C₄⋊Q₈	—	157	2	no	—
119	C₄⋊Q₈	—	157	1	no	—
120	C₄⋊Q₈	—	157	1	no	—
121	C₄⋊Q₈	C₂×D₁₅₇	1	1	yes	characteristic
122	#127	C₂×D₁₅₇	1	1	yes	characteristic
123	#124	—	157	2	no	—
124	#121	—	314	2	no	—
125	C₄×D₄	—	314	2	no	—
126	C₂×C₄⋊C₄	—	314	2	no	—
127	C₂×C₄²	—	157	1	no	—
128	#97	—	157	2	no	—
129	C₄×C₈	—	157	1	no	—
130	C₄×C₈	C₂×D₁₅₇	1	1	yes	characteristic
131	#41	—	314	8	no	—
132	C₂×Q₈	—	314	2	no	—
133	C₂×Q₈	—	314	8	no	—
134	C₂×Q₈	—	157	2	no	—
135	C₂×Q₈	—	157	2	no	—
136	C₂×Q₈	C₂²×D₁₅₇	1	2	yes	—
137	C₂×D₄	—	2	2	no	—
138	C₂×D₄	—	314	2	no	—
139	C₂×D₄	C₂²×D₁₅₇	1	2	yes	—
140	C₂²×C₄	—	314	2	no	—
141	C₂²×C₄	—	314	2	no	—
142	C₂²×C₄	—	157	2	no	—
143	C₂²×C₄	—	157	1	no	—
144	Q₁₆	—	314	8	no	—
145	Q₁₆	—	314	8	no	—
146	SD₁₆	—	2	8	no	—
147	SD₁₆	—	314	8	no	—
148	M₁₆	—	157	8	no	—
149	C₂×C₈	—	157	2	no	—
150	C₂×C₈	C₂²×D₁₅₇	1	2	yes	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	314	4	no	—
153	C₄⋊C₄	—	314	2	no	—
154	C₄⋊C₄	—	314	2	no	—
155	C₂²⋊C₄	—	314	4	no	—
156	C₄²	—	157	2	no	—
157	C₄²	—	157	1	no	—
158	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
159	C₂³	—	2	2	no	—
160	C₂³	—	157	1	no	—
161	Q₈	—	2	4	no	—
162	Q₈	—	314	4	no	—
163	Q₈	—	314	4	no	—
164	Q₈	—	628	4	no	—
165	D₄	—	2	4	no	—
166	D₄	—	4	4	no	—
167	D₄	—	628	4	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	314	4	no	—
170	C₂×C₄	—	314	2	no	—
171	C₂×C₄	—	314	2	no	—
172	C₂×C₄	—	628	4	no	—
173	C₂×C₄	—	628	4	no	—
174	C₂×C₄	—	157	2	no	—
175	C₂×C₄	—	157	1	no	—
176	C₂×C₄	—	157	4	no	—
177	C₂×C₄	D₄×D₁₅₇	1	2	yes	—
178	C₂×C₄	C₂³×D₁₅₇	1	1	yes	characteristic
179	C₈	—	157	4	no	—
180	C₈	D₄×D₁₅₇	1	4	yes	—
181	C₂²	—	4	4	no	—
182	C₂²	—	4	2	no	—
183	C₂²	—	314	2	no	—
184	C₂²	—	157	2	no	—
185	C₂²	C₂×D₄×D₁₅₇	1	1	yes	characteristic
186	C₄	—	4	2	no	—
187	C₄	—	314	2	no	—
188	C₄	—	628	2	no	—
189	C₄	—	628	2	no	—
190	C₄	—	157	2	no	—
191	C₄	#19097800	1	4	yes	complemented
192	C₄	C₂×D₄×D₁₅₇	1	2	yes	—
193	C₂	—	8	2	no	—
194	C₂	—	314	2	no	—
195	C₂	#127×D₁₅₇	1	1	yes	characteristic
196	C₂	C₂×#19097800	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19097800

C₂

yes

maximal, complemented

#23757821

C₂

yes

maximal, characteristic

C₄⋊Q₈×D₁₅₇

C₂

yes

maximal, characteristic, complemented

#23757664

C₂

yes

maximal, complemented

#23757653

C₂

yes

maximal, characteristic, complemented

#23757652

C₂

yes

maximal, characteristic, complemented

#23757645

C₂

yes

maximal, characteristic, complemented

C₄×#14578937

C₂

yes

maximal, characteristic, complemented

#492×C₁₅₇

C₂

yes

maximal, characteristic, complemented

C₂×#14578953

C₂²

yes

—

C₂×Q₈×D₁₅₇

C₂²

yes

—

#19097800

—

C₂×Dic₆₂₈

C₂²

yes

—

C₂×#14578944

C₂²

yes

—

C₂×#14578942

C₂²

yes

—

C₂×#14578937

C₂²

yes

—

#19097771

C₂²

yes

characteristic

#19097771

C₂²

yes

characteristic

#19097770

—

#19097769

C₂²

yes

—

#19097758

C₂²

yes

—

#19097742

—

D₄×Dic₁₅₇

—

C₄⋊C₄×D₁₅₇

—

C₄²×D₁₅₇

C₂²

yes

characteristic

#19097681

C₂²

yes

—

C₄×#11929434

C₂²

yes

characteristic

C₂×SD₁₆×C₁₅₇

C₂²

yes

—

C₄⋊Q₈×C₁₅₇

C₂²

yes

characteristic

#127×C₁₅₇

C₂²

yes

characteristic

C₄×C₈×C₁₅₇

C₂²

yes

characteristic

#14578953

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

C₂³

yes

—

C₂×Dic₃₁₄

C₂³

yes

—

Q₈×D₁₅₇

—

C₂×C₂²⋊D₁₅₇

—

C₂²×Dic₁₅₇

—

C₂×C₄×D₁₅₇

—

C₂×C₄×D₁₅₇

D₄

yes

—

C₂×C₄×D₁₅₇

C₂³

yes

characteristic

Dic₆₂₈

—

#14578944

—

#14578942

—

#14578937

D₄

yes

complemented

C₂×#11929434

C₂³

yes

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

#14578933

—

C₂²⋊Dic₁₅₇

—

C₄×Dic₁₅₇

D₄

yes

—

C₄×Dic₁₅₇

C₂³

yes

characteristic

C₂×Q₈×C₁₅₇

C₂³

yes

—

C₂×D₄×C₁₅₇

—

C₂×D₄×C₁₅₇

C₂³

yes

—

SD₁₆×C₁₅₇

—

C₂×C₈×C₁₅₇

C₂³

yes

—

C₄⋊C₄×C₁₅₇

—

C₄²×C₁₅₇

C₂³

yes

characteristic

C₂²×D₁₅₇

C₂×D₄

yes

characteristic

Dic₃₁₄

—

Dic₃₁₄

—

Dic₃₁₄

—

C₂²⋊D₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

C₂×D₄

yes

—

C₂×Dic₁₅₇

C₂×D₄

yes

characteristic

C₄×D₁₅₇

—

C₄×D₁₅₇

—

C₄×D₁₅₇

C₂×D₄

yes

—

#11929434

C₂×D₄

yes

—

C₂³×C₁₅₇

—

Q₈×C₁₅₇

—

D₄×C₁₅₇

—

D₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂⁴

yes

characteristic

C₈×C₁₅₇

C₂×D₄

yes

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂²×D₄

yes

characteristic

C₂×D₁₅₇

—

C₂×D₁₅₇

#127

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

#137

yes

complemented

C₄×C₁₅₇

C₂²×D₄

yes

—

Dic₁₅₇

—

Dic₁₅₇

—

Dic₁₅₇

—

Dic₁₅₇

#127

yes

—

C₂×C₁₅₇

—

C₂×C₁₅₇

C₂×#127

yes

characteristic

C₂×C₁₅₇

C₂×#137

yes

—

D₁₅₇

—

C₁₅₇

#3152

yes

characteristic, complemented, Sylow

#3152

—

157

maximal, Sylow

100

C₂×#137

—

157

—

101

#532

—

157

—

102

C₂×C₄⋊Q₈

—

157

—

103

#497

—

157

—

104

#494

—

157

—

105

#494

—

157

—

106

#492

—

157

—

107

#492

D₁₅₇

yes

characteristic, complemented

108

C₄×M₁₆

—

157

—

109

C₂×#41

—

157

—

110

C₂²×Q₈

—

157

—

111

#137

—

314

—

112

C₂×Q₁₆

—

157

—

113

C₂×Q₁₆

—

157

—

114

C₂×SD₁₆

—

157

—

115

C₂×SD₁₆

C₂×D₁₅₇

yes

—

116

C₂×M₁₆

—

157

—

117

C₄⋊Q₈

—

314

—

118

C₄⋊Q₈

—

157

—

119

C₄⋊Q₈

—

157

—

120

C₄⋊Q₈

—

157

—

121

C₄⋊Q₈

C₂×D₁₅₇

yes

characteristic

122

#127

C₂×D₁₅₇

yes

characteristic

123

#124

—

157

—

124

#121

—

314

—

125

C₄×D₄

—

314

—

126

C₂×C₄⋊C₄

—

314

—

127

C₂×C₄²

—

157

—

128

#97

—

157

—

129

C₄×C₈

—

157

—

130

C₄×C₈

C₂×D₁₅₇

yes

characteristic

131

#41

—

314

—

132

C₂×Q₈

—

314

—

133

C₂×Q₈

—

314

—

134

C₂×Q₈

—

157

—

135

C₂×Q₈

—

157

—

136

C₂×Q₈

C₂²×D₁₅₇

yes

—

137

C₂×D₄

—

138

C₂×D₄

—

314

—

139

C₂×D₄

C₂²×D₁₅₇

yes

—

140

C₂²×C₄

—

314

—

141

C₂²×C₄

—

314

—

142

C₂²×C₄

—

157

—

143

C₂²×C₄

—

157

—

144

Q₁₆

—

314

—

145

Q₁₆

—

314

—

146

SD₁₆

—

147

SD₁₆

—

314

—

148

M₁₆

—

157

—

149

C₂×C₈

—

157

—

150

C₂×C₈

C₂²×D₁₅₇

yes

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

314

—

153

C₄⋊C₄

—

314

—

154

C₄⋊C₄

—

314

—

155

C₂²⋊C₄

—

314

—

156

C₄²

—

157

—

157

C₄²

—

157

—

158

C₄²

C₂²×D₁₅₇

yes

characteristic

159

C₂³

—

160

C₂³

—

157

—

161

Q₈

—

162

Q₈

—

314

—

163

Q₈

—

314

—

164

Q₈

—

628

—

165

D₄

—

166

D₄

—

167

D₄

—

628

—

168

C₂×C₄

—

169

C₂×C₄

—

314

—

170

C₂×C₄

—

314

—

171

C₂×C₄

—

314

—

172

C₂×C₄

—

628

—

173

C₂×C₄

—

628

—

174

C₂×C₄

—

157

—

175

C₂×C₄

—

157

—

176

C₂×C₄

—

157

—

177

C₂×C₄

D₄×D₁₅₇

yes

—

178

C₂×C₄

C₂³×D₁₅₇

yes

characteristic

179

C₈

—

157

—

180

C₈

D₄×D₁₅₇

yes

—

181

C₂²

—

182

C₂²

—

183

C₂²

—

314

—

184

C₂²

—

157

—

185

C₂²

C₂×D₄×D₁₅₇

yes

characteristic

186

C₄

—

187

C₄

—

314

—

188

C₄

—

628

—

189

C₄

—

628

—

190

C₄

—

157

—

191

C₄

#19097800

yes

complemented

192

C₄

C₂×D₄×D₁₅₇

yes

—

193

C₂

—

194

C₂

—

314

—

195

C₂

#127×D₁₅₇

yes

characteristic

196

C₂

C₂×#19097800

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₅₇
commutator, cosocle	C₂×C₄×C₁₅₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅₇
Fitting	#492×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₂²×D₄
prime radical	#23757821	C₂
hypercenter	#492	D₁₅₇
hypocenter	C₁₅₇	#3152
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

C₂×D₄×D₁₅₇

commutator, cosocle

C₂×C₄×C₁₅₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₅₇

Fitting

#492×C₁₅₇

C₂

socle

C₂²×C₁₅₇

C₂²×D₄

C₂

D₁₅₇

C₁₅₇

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. C₂×#19097800	C₂	semidirect
3. C₄⋊Q₈×D₁₅₇	C₂	semidirect
4. #23757664	C₂	semidirect
5. #23757653	C₂	semidirect
6. #23757652	C₂	semidirect
7. #23757645	C₂	semidirect
8. C₄×#14578937	C₂	semidirect
9. #492×C₁₅₇	C₂	semidirect
45. #14578937	D₄	semidirect
88. C₄×C₁₅₇	#137	semidirect
98. C₁₅₇	#3152	semidirect
107. #492	D₁₅₇	semidirect
191. C₄	#19097800	semidirect

H ≅ G/N

type

1. C₂×#19097800

C₂

semidirect

3. C₄⋊Q₈×D₁₅₇

C₂

semidirect

4. #23757664

C₂

semidirect

5. #23757653

C₂

semidirect

6. #23757652

C₂

semidirect

7. #23757645

C₂

semidirect

8. C₄×#14578937

C₂

semidirect

9. #492×C₁₅₇

C₂

semidirect

45. #14578937

D₄

semidirect

88. C₄×C₁₅₇

#137

semidirect

98. C₁₅₇

#3152

semidirect

107. #492

D₁₅₇

semidirect

191. C₄

#19097800

semidirect

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

subgroup	index
C₂×#19097800	2
#23757821	2
C₄⋊Q₈×D₁₅₇	2
#23757664	2
#23757653	2
#23757652	2
#23757645	2
C₄×#14578937	2
#492×C₁₅₇	2
#3152	157

subgroup

index

C₄⋊Q₈×D₁₅₇

#492×C₁₅₇

157

Subgroups — #34110590

Conjugacy classes of subgroups

Complements

Maximal subgroups