Group #34000191 — subgroups

Subgroups — #34000191

Order 18688.

262 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 102 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	#22626261	C₂	1	1	yes	maximal, characteristic
2	C₈×#11924036	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556664	—	2	1	no	—
5	C₄×#11924036	C₄	1	1	yes	characteristic
6	C₄×#11924036	C₄	1	1	yes	characteristic
7	#17989508	—	2	1	no	—
8	C₄⋊#11924031	C₄	1	2	yes	—
9	C₈×#10596833	—	2	1	no	—
10	C₈×#10596833	—	2	1	no	—
11	C₂×C₁₆×D₇₃	—	2	1	no	—
12	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
13	C₄⋊#11924003	C₄	1	1	yes	characteristic
14	#17988405	—	2	1	no	—
15	C₄⋊C₁₆×C₇₃	C₄	1	1	yes	characteristic
16	C₂×#11924036	—	2	1	no	—
17	C₂×#11924036	C₈	1	2	yes	—
18	#14556664	—	4	1	no	—
19	#14556664	—	4	1	no	—
20	C₂×#11924031	—	2	2	no	—
21	C₄×#10596833	—	2	1	no	—
22	C₄×#10596833	—	2	1	no	—
23	C₄×#10596833	—	2	1	no	—
24	C₄×#10596833	C₈	1	2	yes	—
25	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
26	C₂×C₈×D₇₃	Q₈	1	1	yes	characteristic
27	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂×#11924003	—	2	1	no	—
29	C₁₆×D₇₃	—	4	1	no	—
30	C₁₆×D₇₃	—	4	1	no	—
31	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
32	C₈×Dic₇₃	D₄	1	1	yes	characteristic
33	C₈×Dic₇₃	D₄	1	1	yes	characteristic
34	C₂×C₁₆×C₇₃	—	2	1	no	—
35	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
36	#11924036	—	2	4	no	—
37	#11924036	—	2	2	no	—
38	#11924036	C₁₆	1	4	yes	complemented
39	C₂×#10596833	—	2	2	no	—
40	C₂×#10596833	—	2	2	no	—
41	#11924031	—	4	2	no	—
42	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
44	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
45	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
47	C₈×D₇₃	—	2	1	no	—
48	C₈×D₇₃	—	2	1	no	—
49	C₈×D₇₃	D₈	1	1	yes	characteristic
50	C₈×D₇₃	SD₁₆	1	1	yes	characteristic
51	C₈×D₇₃	SD₁₆	1	1	yes	characteristic
52	C₈×D₇₃	Q₁₆	1	1	yes	characteristic
53	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
55	C₄×Dic₇₃	M₁₆	1	1	yes	characteristic
56	#11924003	—	4	1	no	—
57	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
59	C₄²×C₇₃	C₄²	1	1	yes	characteristic
60	C₁₆×C₇₃	—	4	1	no	—
61	C₂²×D₇₃	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
63	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
64	C₂×C₄×C₇₃	#97	1	1	yes	characteristic
65	C₂×Dic₇₃	C₂²⋊C₈	1	1	yes	characteristic
66	C₂×Dic₇₃	C₂²⋊C₈	1	1	yes	characteristic
67	C₂×Dic₇₃	C₄⋊C₈	1	1	yes	characteristic
68	C₄×D₇₃	—	2	2	no	—
69	C₄×D₇₃	#106	1	1	yes	characteristic
70	C₄×D₇₃	#107	1	1	yes	characteristic
71	C₄×D₇₃	#108	1	1	yes	characteristic
72	C₄×D₇₃	#108	1	1	yes	characteristic
73	C₄×D₇₃	C₂×C₁₆	1	2	yes	—
74	C₄×D₇₃	M₃₂	1	2	yes	—
75	#10596833	—	2	4	no	—
76	#10596833	—	2	4	no	—
77	C₈×C₇₃	—	2	1	no	—
78	C₈×C₇₃	#102	1	1	yes	characteristic
79	C₈×C₇₃	#103	1	1	yes	characteristic
80	#10596831	—	2	1	no	—
81	#10596831	#102	1	1	yes	characteristic
82	#10596831	#103	1	1	yes	characteristic
83	C₂²×C₇₃	#336	1	1	yes	characteristic
84	C₂×D₇₃	#334	1	1	yes	characteristic
85	C₂×D₇₃	#335	1	1	yes	characteristic
86	C₂×D₇₃	C₄⋊C₁₆	1	2	yes	—
87	C₂×D₇₃	#364	1	2	yes	—
88	C₄×C₇₃	—	2	1	no	—
89	C₄×C₇₃	#340	1	1	yes	characteristic
90	C₄×C₇₃	#341	1	1	yes	characteristic
91	C₄×C₇₃	C₄×C₁₆	1	1	yes	characteristic
92	C₄×C₇₃	#346	1	1	yes	characteristic
93	Dic₇₃	—	2	1	no	—
94	Dic₇₃	#334	1	1	yes	characteristic
95	Dic₇₃	#335	1	1	yes	characteristic
96	Dic₇₃	C₂²⋊C₁₆	1	2	yes	—
97	#7151	—	73	1	no	maximal, Sylow
98	C₂×C₇₃	#1277	1	1	yes	characteristic
99	C₂×C₇₃	#1374	1	1	yes	characteristic
100	C₂×C₇₃	#1376	1	1	yes	characteristic
101	D₇₃	#1367	1	2	yes	—
102	D₇₃	#1370	1	2	yes	—
103	C₂×C₄⋊C₁₆	—	73	1	no	—
104	C₂×C₄⋊C₁₆	—	73	1	no	—
105	C₂×C₈²	—	73	1	no	—
106	C₇₃	#7151	1	1	yes	characteristic, complemented, Sylow
107	C₂²×C₁₆	—	146	1	no	—
108	C₂²×C₁₆	—	146	1	no	—
109	C₂×C₄×C₈	—	73	1	no	—
110	C₂×C₄×C₈	—	73	1	no	—
111	C₂×C₄×C₈	—	73	1	no	—
112	C₄⋊C₁₆	—	73	2	no	—
113	C₄⋊C₁₆	—	73	1	no	—
114	C₄⋊C₁₆	—	146	1	no	—
115	C₄⋊C₁₆	—	146	1	no	—
116	C₄⋊C₁₆	#64408	1	1	yes	characteristic
117	C₈²	—	146	1	no	—
118	C₈²	—	146	1	no	—
119	C₂²×C₈	—	73	1	no	—
120	C₂²×C₈	—	73	2	no	—
121	C₂²×C₈	—	73	1	no	—
122	C₂²×C₈	—	146	1	no	—
123	C₂×C₄²	—	73	1	no	—
124	C₂×C₁₆	—	2	1	no	—
125	C₂×C₁₆	—	292	1	no	—
126	C₂×C₁₆	—	292	1	no	—
127	C₂×C₁₆	—	292	1	no	—
128	C₂×C₁₆	—	292	1	no	—
129	C₂×C₁₆	—	146	2	no	—
130	C₂×C₁₆	—	146	1	no	—
131	C₄×C₈	—	73	1	no	—
132	C₄×C₈	—	73	1	no	—
133	C₄×C₈	—	73	2	no	—
134	C₄×C₈	—	73	1	no	—
135	C₄×C₈	—	146	1	no	—
136	C₄×C₈	—	146	1	no	—
137	C₄×C₈	—	146	1	no	—
138	C₄×C₈	C₂×#64408	1	1	yes	characteristic
139	C₂²×C₄	—	73	1	no	—
140	C₂²×C₄	—	73	1	no	—
141	C₂²×C₄	—	73	1	no	—
142	C₂×C₈	—	73	1	no	—
143	C₂×C₈	—	73	1	no	—
144	C₂×C₈	—	73	1	no	—
145	C₂×C₈	—	73	1	no	—
146	C₂×C₈	—	73	1	no	—
147	C₂×C₈	—	73	4	no	—
148	C₂×C₈	—	73	1	no	—
149	C₂×C₈	—	146	1	no	—
150	C₂×C₈	—	146	1	no	—
151	C₂×C₈	—	146	2	no	—
152	C₂×C₈	—	146	2	no	—
153	C₂×C₈	—	146	2	no	—
154	C₂×C₈	—	146	4	no	—
155	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
156	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
157	C₄²	—	73	1	no	—
158	C₄²	—	73	1	no	—
159	C₄²	—	73	1	no	—
160	C₄²	C₄×#64408	1	1	yes	characteristic
161	C₁₆	—	4	1	no	—
162	C₁₆	—	292	2	no	—
163	C₁₆	—	292	1	no	—
164	C₂³	—	73	1	no	—
165	C₂×C₄	—	73	1	no	—
166	C₂×C₄	—	73	1	no	—
167	C₂×C₄	—	73	1	no	—
168	C₂×C₄	—	73	1	no	—
169	C₂×C₄	—	73	1	no	—
170	C₂×C₄	—	73	1	no	—
171	C₂×C₄	—	73	2	no	—
172	C₂×C₄	—	73	2	no	—
173	C₂×C₄	—	73	1	no	—
174	C₂×C₄	—	146	2	no	—
175	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
176	C₂×C₄	C₈×#64408	1	1	yes	characteristic
177	C₂×C₄	#14556645	1	1	yes	characteristic
178	C₈	—	2	1	no	—
179	C₈	—	73	1	no	—
180	C₈	—	73	1	no	—
181	C₈	—	146	4	no	—
182	C₈	—	146	4	no	—
183	C₈	—	146	1	no	—
184	C₈	#14556653	1	1	yes	characteristic
185	C₈	#14556654	1	1	yes	characteristic
186	C₂²	—	73	2	no	—
187	C₂²	—	73	2	no	—
188	C₂²	—	73	1	no	—
189	C₂²	—	73	1	no	—
190	C₂²	#17989452	1	1	yes	characteristic
191	C₄	—	2	1	no	—
192	C₄	—	73	2	no	—
193	C₄	—	73	1	no	—
194	C₄	—	73	1	no	—
195	C₄	—	146	1	no	—
196	C₄	#17989460	1	1	yes	characteristic
197	C₄	#17989462	1	1	yes	characteristic
198	C₄	C₁₆×#64408	1	1	yes	characteristic
199	C₄	#17989474	1	1	yes	characteristic
200	C₂	—	73	2	no	—
201	C₂	—	73	2	no	—
202	C₂	#22624760	1	1	yes	characteristic
203	C₂	#22624968	1	1	yes	characteristic
204	C₂	#22624974	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626261

C₂

yes

maximal, characteristic

C₈×#11924036

C₂

yes

maximal, characteristic

C₄⋊C₁₆×D₇₃

C₂

yes

maximal, characteristic

C₂×#14556664

—

C₄×#11924036

C₄

yes

characteristic

C₄×#11924036

C₄

yes

characteristic

#17989508

—

C₄⋊#11924031

C₄

yes

—

C₈×#10596833

—

C₈×#10596833

—

C₂×C₁₆×D₇₃

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

C₄⋊#11924003

C₄

yes

characteristic

#17988405

—

C₄⋊C₁₆×C₇₃

C₄

yes

characteristic

C₂×#11924036

—

C₂×#11924036

C₈

yes

—

—

—

—

—

—

—

C₈

yes

—

C₂×C₈×D₇₃

D₄

yes

characteristic

C₂×C₈×D₇₃

Q₈

yes

characteristic

C₄²×D₇₃

C₂×C₄

yes

characteristic

C₂×#11924003

—

C₁₆×D₇₃

—

C₁₆×D₇₃

—

C₄×#10596831

C₂×C₄

yes

characteristic

C₈×Dic₇₃

D₄

yes

characteristic

C₈×Dic₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

—

C₁₆

yes

—

—

—

C₂×C₄×D₇₃

C₄⋊C₄

yes

characteristic

C₂×C₄×D₇₃

C₂×C₈

yes

characteristic

C₂×C₄×D₇₃

M₁₆

yes

characteristic

C₂×#10596831

C₂²⋊C₄

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

C₈×D₇₃

D₈

yes

characteristic

C₈×D₇₃

SD₁₆

yes

characteristic

C₈×D₇₃

SD₁₆

yes

characteristic

C₈×D₇₃

Q₁₆

yes

characteristic

C₄×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₇₃

C₂×C₈

yes

characteristic

C₄×Dic₇₃

M₁₆

yes

characteristic

#11924003

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₄²×C₇₃

C₄²

yes

characteristic

C₁₆×C₇₃

—

C₂²×D₇₃

C₄⋊C₈

yes

characteristic

C₂×C₄×C₇₃

#95

yes

characteristic

C₂×C₄×C₇₃

C₄×C₈

yes

characteristic

C₂×C₄×C₇₃

#97

yes

characteristic

C₂×Dic₇₃

C₂²⋊C₈

yes

characteristic

C₂×Dic₇₃

C₂²⋊C₈

yes

characteristic

C₂×Dic₇₃

C₄⋊C₈

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

yes

C₄×D₇₃

yes

C₄×D₇₃

yes

C₄×D₇₃

yes

C₄×D₇₃

C₂×C₁₆

yes

—

C₄×D₇₃

M₃₂

yes

—

#10596833

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

yes

C₈×C₇₃

yes

—

yes

yes

C₂²×C₇₃

yes

C₂×D₇₃

yes

C₂×D₇₃

yes

C₂×D₇₃

C₄⋊C₁₆

yes

—

C₂×D₇₃

#364

yes

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

C₄×C₁₆

yes

characteristic

C₄×C₇₃

#346

yes

characteristic

Dic₇₃

—

Dic₇₃

yes

Dic₇₃

yes

Dic₇₃

C₂²⋊C₁₆

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

100

C₂×C₇₃

#1376

yes

characteristic

101

D₇₃

#1367

yes

—

102

D₇₃

#1370

yes

—

103

C₂×C₄⋊C₁₆

—

104

C₂×C₄⋊C₁₆

—

105

C₂×C₈²

—

106

C₇₃

#7151

yes

characteristic, complemented, Sylow

107

C₂²×C₁₆

—

146

—

108

C₂²×C₁₆

—

146

—

109

C₂×C₄×C₈

—

110

C₂×C₄×C₈

—

111

C₂×C₄×C₈

—

112

C₄⋊C₁₆

—

113

C₄⋊C₁₆

—

114

C₄⋊C₁₆

—

146

—

115

C₄⋊C₁₆

—

146

—

116

C₄⋊C₁₆

#64408

yes

characteristic

117

C₈²

—

146

—

118

C₈²

—

146

—

119

C₂²×C₈

—

120

C₂²×C₈

—

121

C₂²×C₈

—

122

C₂²×C₈

—

146

—

123

C₂×C₄²

—

124

C₂×C₁₆

—

125

C₂×C₁₆

—

292

—

126

C₂×C₁₆

—

292

—

127

C₂×C₁₆

—

292

—

128

C₂×C₁₆

—

292

—

129

C₂×C₁₆

—

146

—

130

C₂×C₁₆

—

146

—

131

C₄×C₈

—

132

C₄×C₈

—

133

C₄×C₈

—

134

C₄×C₈

—

135

C₄×C₈

—

146

—

136

C₄×C₈

—

146

—

137

C₄×C₈

—

146

—

138

C₄×C₈

C₂×#64408

yes

characteristic

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

146

—

150

C₂×C₈

—

146

—

151

C₂×C₈

—

146

—

152

C₂×C₈

—

146

—

153

C₂×C₈

—

146

—

154

C₂×C₈

—

146

—

155

C₂×C₈

C₂²⋊#64408

yes

characteristic

156

C₂×C₈

C₄⋊#64408

yes

characteristic

157

C₄²

—

158

C₄²

—

159

C₄²

—

160

C₄²

C₄×#64408

yes

characteristic

161

C₁₆

—

162

C₁₆

—

292

—

163

C₁₆

—

292

—

164

C₂³

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

146

—

175

C₂×C₄

(C₂×#64408)⋊C₄

yes

characteristic

176

C₂×C₄

C₈×#64408

yes

characteristic

177

C₂×C₄

#14556645

yes

characteristic

178

C₈

—

179

C₈

—

180

C₈

—

181

C₈

—

146

—

182

C₈

—

146

—

183

C₈

—

146

—

184

C₈

#14556653

yes

characteristic

185

C₈

#14556654

yes

characteristic

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

—

190

C₂²

#17989452

yes

characteristic

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

146

—

196

C₄

#17989460

yes

characteristic

197

C₄

#17989462

yes

characteristic

198

C₄

C₁₆×#64408

yes

characteristic

199

C₄

#17989474

yes

characteristic

200

C₂

—

201

C₂

—

202

C₂

#22624760

yes

characteristic

203

C₂

#22624968

yes

characteristic

204

C₂

#22624974

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊#64408
commutator	C₄×C₇₃	C₄×C₁₆
Frattini	C₄×C₈	C₂×#64408
Fitting	C₄⋊C₁₆×C₇₃	C₄
socle	C₂²×C₇₃	#336
prime radical	C₂²×D₇₃	C₄⋊C₈
cosocle	C₄×C₈×D₇₃	C₂²
hypercenter	C₄⋊C₁₆	#64408
hypocenter	C₇₃	#7151
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center

C₂×C₈

C₂²⋊#64408

commutator

C₄×C₇₃

C₄×C₁₆

Frattini

C₄×C₈

C₂×#64408

Fitting

C₄⋊C₁₆×C₇₃

C₄

socle

C₂²×C₇₃

#336

prime radical

C₂²×D₇₃

C₄⋊C₈

cosocle

C₄×C₈×D₇₃

C₂²

C₄⋊C₁₆

C₇₃

G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
38. #11924036	C₁₆	semidirect
106. C₇₃	#7151	semidirect

H ≅ G/N

type

38. #11924036

C₁₆

semidirect

106. C₇₃

#7151

semidirect

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

subgroup	index
#22626261	2
C₈×#11924036	2
C₄⋊C₁₆×D₇₃	2
#7151	73

subgroup

index

#22626261

C₈×#11924036

C₄⋊C₁₆×D₇₃

#7151

Subgroups — #34000191

Conjugacy classes of subgroups

Complements

Maximal subgroups