Group #34007490 — subgroups

Subgroups — #34007490

Order 18688.

378 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 124 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	#22626028	C₂	1	2	yes	maximal
2	C₂×#17989448	C₂	1	1	yes	maximal, characteristic
3	C₂×#17989429	C₂	1	1	yes	maximal, characteristic
4	#22624767	C₂	1	2	yes	maximal
5	#22609823	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#10596833	—	2	2	no	—
7	C₂×C₄⋊#10596833	C₂²	1	2	yes	—
8	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
9	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
10	#17989453	—	2	2	no	—
11	#17989448	—	2	2	no	—
12	#17989429	—	2	2	no	—
13	C₂×#14556524	C₄	1	2	yes	—
14	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×Dic₇₃	C₂²	1	2	yes	—
16	#17988350	C₄	1	1	yes	characteristic
17	#340×C₇₃	C₄	1	1	yes	characteristic
18	C₂²×#10596833	—	2	2	no	—
19	C₂²×#10596833	—	2	1	no	—
20	C₂²×#10596833	D₄	1	2	yes	—
21	C₄⋊#10596833	—	2	2	no	—
22	C₄⋊#10596833	—	2	2	no	—
23	C₄⋊#10596833	—	4	4	no	—
24	C₄×#10596833	—	2	1	no	—
25	C₄×#10596833	—	2	1	no	—
26	C₄×#10596833	—	2	1	no	—
27	C₄×#10596833	Q₈	1	2	yes	—
28	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
30	C₂×C₄×Dic₇₃	—	2	2	no	—
31	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
32	#14556524	—	2	4	no	—
33	C₈×Dic₇₃	D₄	1	2	yes	—
34	C₈×Dic₇₃	Q₈	1	2	yes	—
35	#14556498	—	2	2	no	—
36	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
38	C₂×#10596833	—	2	4	no	—
39	C₂×#10596833	—	2	2	no	—
40	C₂×#10596833	—	2	2	no	—
41	C₂×#10596833	—	2	2	no	—
42	C₂×#10596833	—	2	2	no	—
43	C₂×#10596833	—	4	2	no	—
44	C₂×#10596833	—	4	2	no	—
45	C₂×#10596833	—	4	2	no	—
46	C₂²×Dic₇₃	—	2	2	no	—
47	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×Dic₇₃	#41	1	1	yes	characteristic
49	C₂×#10596831	—	2	2	no	—
50	C₂×#10596831	C₄⋊C₄	1	2	yes	—
51	C₄⋊Dic₇₃	—	2	2	no	—
52	C₄⋊Dic₇₃	—	2	2	no	—
53	C₄×Dic₇₃	—	4	2	no	—
54	C₄×Dic₇₃	—	4	2	no	—
55	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
56	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
57	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
58	C₄×Dic₇₃	#41	1	1	yes	characteristic
59	C₂²×C₄×C₇₃	—	2	2	no	—
60	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₇₃	—	2	2	no	—
62	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
63	C₄⋊C₄×C₇₃	—	2	2	no	—
64	C₄⋊C₄×C₇₃	M₁₆	1	4	yes	—
65	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₇₃	—	2	4	no	—
67	C₂×C₄×C₇₃	—	4	2	no	—
68	C₂×C₄×C₇₃	—	4	2	no	—
69	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
72	C₂×C₄×C₇₃	#131	1	2	yes	—
73	C₂×Dic₇₃	—	2	2	no	—
74	C₂×Dic₇₃	—	4	2	no	—
75	C₂×Dic₇₃	—	4	2	no	—
76	C₂×Dic₇₃	—	4	2	no	—
77	C₂×Dic₇₃	#121	1	1	yes	characteristic
78	C₂×Dic₇₃	C₂²⋊Q₈	1	1	yes	characteristic
79	C₂×Dic₇₃	#125	1	1	yes	characteristic
80	C₂×Dic₇₃	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×Dic₇₃	AGL(1,ℤ/8)	1	2	yes	—
82	C₂×Dic₇₃	#137	1	2	yes	—
83	#10596833	—	2	4	no	—
84	#10596833	—	4	2	no	—
85	#10596833	—	8	2	no	—
86	C₈×C₇₃	—	2	2	no	—
87	#10596831	—	2	2	no	—
88	C₂²×C₇₃	#389	1	1	yes	characteristic
89	C₂²×C₇₃	#435	1	2	yes	—
90	C₂²×C₇₃	#441	1	1	yes	characteristic
91	C₂²×C₇₃	#442	1	1	yes	characteristic
92	C₂²×C₇₃	#443	1	2	yes	—
93	C₄×C₇₃	—	4	4	no	—
94	C₄×C₇₃	#423	1	1	yes	characteristic
95	C₄×C₇₃	#424	1	1	yes	characteristic
96	C₄×C₇₃	#446	1	2	yes	—
97	Dic₇₃	—	2	1	no	—
98	Dic₇₃	—	2	1	no	—
99	Dic₇₃	—	8	2	no	—
100	Dic₇₃	#468	1	2	yes	—
101	Dic₇₃	#501	1	2	yes	—
102	#9599	—	73	1	no	maximal, Sylow
103	C₂×C₇₃	#1586	1	2	yes	—
104	C₂×C₇₃	#1587	1	2	yes	—
105	C₂×C₇₃	#1940	1	1	yes	characteristic
106	C₂×C₇₃	#1943	1	1	yes	characteristic
107	C₂×C₇₃	#1945	1	1	yes	characteristic
108	#1939	—	73	2	no	—
109	C₄×#107	—	73	1	no	—
110	C₂×#334	—	73	1	no	—
111	C₂×#322	—	73	1	no	—
112	#1279	—	73	2	no	—
113	C₇₃	#9599	1	1	yes	characteristic, complemented, Sylow
114	C₂×#107	—	73	2	no	—
115	C₂×C₄⋊C₈	—	73	2	no	—
116	C₂×C₄⋊C₈	—	146	2	no	—
117	C₂×C₄×C₈	—	73	1	no	—
118	C₂×C₄×C₈	—	73	1	no	—
119	C₂×C₄×C₈	—	73	1	no	—
120	C₄×C₄⋊C₄	—	73	2	no	—
121	#340	—	73	1	no	—
122	#340	#64408	1	1	yes	characteristic
123	#336	—	146	2	no	—
124	#334	—	146	2	no	—
125	#322	—	146	2	no	—
126	C₂²×C₈	—	73	1	no	—
127	C₂²×C₈	—	73	2	no	—
128	C₂²×C₈	—	146	2	no	—
129	C₂²×C₈	—	146	1	no	—
130	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
131	C₂×C₄⋊C₄	—	73	2	no	—
132	C₂×C₄⋊C₄	C₂×#64408	1	2	yes	—
133	C₂×C₄²	—	73	1	no	—
134	C₂×C₄²	—	146	2	no	—
135	#107	—	146	4	no	—
136	C₄⋊C₈	—	292	4	no	—
137	C₄⋊C₈	—	146	2	no	—
138	C₄⋊C₈	—	146	2	no	—
139	C₄×C₈	—	73	2	no	—
140	C₄×C₈	—	73	2	no	—
141	C₄×C₈	—	73	2	no	—
142	C₄×C₈	—	146	1	no	—
143	C₄×C₈	—	146	1	no	—
144	C₄×C₈	—	146	1	no	—
145	#95	—	146	2	no	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	73	1	no	—
148	C₂²×C₄	—	73	1	no	—
149	C₂²×C₄	—	146	2	no	—
150	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	292	2	no	—
153	C₂×C₈	—	292	2	no	—
154	C₂×C₈	—	292	2	no	—
155	C₂×C₈	—	73	2	no	—
156	C₂×C₈	—	146	2	no	—
157	C₂×C₈	—	146	2	no	—
158	C₂×C₈	—	146	2	no	—
159	C₂×C₈	—	146	2	no	—
160	C₂×C₈	—	146	4	no	—
161	C₂×C₈	—	146	2	no	—
162	C₂×C₈	C₄⋊#64408	1	2	yes	—
163	C₄⋊C₄	—	2	2	no	—
164	C₄⋊C₄	—	146	2	no	—
165	C₄⋊C₄	—	146	2	no	—
166	C₄⋊C₄	#11924038	1	4	yes	—
167	C₄²	—	292	2	no	—
168	C₄²	—	292	2	no	—
169	C₄²	—	73	1	no	—
170	C₄²	—	73	1	no	—
171	C₄²	—	73	1	no	—
172	C₄²	—	73	1	no	—
173	C₂³	D₄×#64408	1	1	yes	characteristic
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	292	2	no	—
178	C₂×C₄	—	292	2	no	—
179	C₂×C₄	—	292	2	no	—
180	C₂×C₄	—	73	1	no	—
181	C₂×C₄	—	73	1	no	—
182	C₂×C₄	—	73	2	no	—
183	C₂×C₄	—	73	2	no	—
184	C₂×C₄	—	73	1	no	—
185	C₂×C₄	—	73	1	no	—
186	C₂×C₄	—	146	2	no	—
187	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
188	C₂×C₄	Q₈×#64408	1	1	yes	characteristic
189	C₂×C₄	C₂×#11924038	1	2	yes	—
190	C₂×C₄	#14556678	1	2	yes	—
191	C₈	—	2	2	no	—
192	C₈	—	292	2	no	—
193	C₈	—	584	2	no	—
194	C₈	—	146	4	no	—
195	C₈	—	146	2	no	—
196	C₂²	#17989538	1	1	yes	characteristic
197	C₂²	#17989616	1	2	yes	—
198	C₂²	#17989627	1	1	yes	characteristic
199	C₂²	#17989628	1	1	yes	characteristic
200	C₂²	#17989630	1	2	yes	—
201	C₄	—	4	4	no	—
202	C₄	—	584	2	no	—
203	C₄	—	73	2	no	—
204	C₄	—	73	2	no	—
205	C₄	—	146	1	no	—
206	C₄	—	146	1	no	—
207	C₄	#17989591	1	1	yes	characteristic
208	C₄	#17989593	1	1	yes	characteristic
209	C₄	#17989637	1	2	yes	—
210	C₂	#22625550	1	2	yes	—
211	C₂	#22625554	1	2	yes	—
212	C₂	#22626031	1	1	yes	characteristic
213	C₂	#22626036	1	1	yes	characteristic
214	C₂	#22626038	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626028

C₂

yes

maximal

C₂×#17989448

C₂

yes

maximal, characteristic

C₂×#17989429

C₂

yes

maximal, characteristic

#22624767

C₂

yes

maximal

#22609823

C₂

yes

maximal, characteristic

C₂×C₄⋊#10596833

—

C₂×C₄⋊#10596833

C₂²

yes

—

C₂×C₄×#10596833

C₂²

yes

characteristic

C₂×C₄×#10596833

C₂²

yes

—

—

—

C₄

yes

—

C₂×C₈×Dic₇₃

C₂²

yes

characteristic

C₄⋊C₄×Dic₇₃

C₂²

yes

—

#17988350

C₄

yes

characteristic

#340×C₇₃

C₄

yes

—

—

D₄

yes

—

—

—

—

—

—

—

Q₈

yes

—

C₂²×#10596831

C₂×C₄

yes

characteristic

C₂×C₄⋊Dic₇₃

C₂×C₄

yes

—

C₂×C₄×Dic₇₃

—

C₂×C₄×Dic₇₃

C₂³

yes

characteristic

#14556524

—

C₈×Dic₇₃

D₄

yes

—

C₈×Dic₇₃

Q₈

yes

—

#14556498

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×C₄⋊C₄×C₇₃

C₂×C₄

yes

—

—

—

—

—

—

—

—

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

C₂×D₄

yes

C₂²×Dic₇₃

yes

—

C₄⋊C₄

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

C₂×D₄

yes

characteristic

C₄×Dic₇₃

C₂×Q₈

yes

characteristic

C₄×Dic₇₃

C₂×Q₈

yes

characteristic

C₄×Dic₇₃

#41

yes

characteristic

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

M₁₆

yes

—

C₂³×C₇₃

C₄×D₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₄×Q₈

yes

characteristic

C₂×C₄×C₇₃

C₂×M₁₆

yes

—

C₂×C₄×C₇₃

#131

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

#121

yes

characteristic

C₂×Dic₇₃

C₂²⋊Q₈

yes

characteristic

C₂×Dic₇₃

#125

yes

characteristic

C₂×Dic₇₃

C₄⋊Q₈

yes

characteristic

C₂×Dic₇₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₇₃

#137

yes

—

#10596833

—

#10596833

—

#10596833

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

#389

yes

characteristic

C₂²×C₇₃

#435

yes

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

100

Dic₇₃

#468

yes

—

101

Dic₇₃

#501

yes

—

102

#9599

—

maximal, Sylow

103

C₂×C₇₃

#1586

yes

—

104

C₂×C₇₃

#1587

yes

—

105

C₂×C₇₃

#1940

yes

characteristic

106

C₂×C₇₃

#1943

yes

characteristic

107

C₂×C₇₃

#1945

yes

characteristic

108

#1939

—

109

C₄×#107

—

110

C₂×#334

—

111

C₂×#322

—

112

#1279

—

113

C₇₃

#9599

yes

characteristic, complemented, Sylow

114

C₂×#107

—

115

C₂×C₄⋊C₈

—

116

C₂×C₄⋊C₈

—

146

—

117

C₂×C₄×C₈

—

118

C₂×C₄×C₈

—

119

C₂×C₄×C₈

—

120

C₄×C₄⋊C₄

—

121

#340

—

122

yes

123

—

146

—

124

#334

—

146

—

125

#322

—

146

—

126

C₂²×C₈

—

127

C₂²×C₈

—

128

C₂²×C₈

—

146

—

129

C₂²×C₈

—

146

—

130

C₂²×C₈

C₂×#64408

yes

characteristic

131

C₂×C₄⋊C₄

—

132

C₂×C₄⋊C₄

C₂×#64408

yes

—

133

C₂×C₄²

—

134

C₂×C₄²

—

146

—

135

#107

—

146

—

136

C₄⋊C₈

—

292

—

137

C₄⋊C₈

—

146

—

138

C₄⋊C₈

—

146

—

139

C₄×C₈

—

140

C₄×C₈

—

141

C₄×C₈

—

142

C₄×C₈

—

146

—

143

C₄×C₈

—

146

—

144

C₄×C₈

—

146

—

145

#95

—

146

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

—

146

—

150

C₂²×C₄

C₂²×#64408

yes

characteristic

151

C₂×C₈

—

152

C₂×C₈

—

292

—

153

C₂×C₈

—

292

—

154

C₂×C₈

—

292

—

155

C₂×C₈

—

156

C₂×C₈

—

146

—

157

C₂×C₈

—

146

—

158

C₂×C₈

—

146

—

159

C₂×C₈

—

146

—

160

C₂×C₈

—

146

—

161

C₂×C₈

—

146

—

162

C₂×C₈

C₄⋊#64408

yes

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

146

—

165

C₄⋊C₄

—

146

—

166

C₄⋊C₄

#11924038

yes

—

167

C₄²

—

292

—

168

C₄²

—

292

—

169

C₄²

—

170

C₄²

—

171

C₄²

—

172

C₄²

—

173

C₂³

D₄×#64408

yes

characteristic

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

292

—

178

C₂×C₄

—

292

—

179

C₂×C₄

—

292

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

146

—

187

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

188

C₂×C₄

Q₈×#64408

yes

characteristic

189

C₂×C₄

C₂×#11924038

yes

—

190

C₂×C₄

#14556678

yes

—

191

C₈

—

192

C₈

—

292

—

193

C₈

—

584

—

194

C₈

—

146

—

195

C₈

—

146

—

196

C₂²

#17989538

yes

characteristic

197

C₂²

#17989616

yes

—

198

C₂²

#17989627

yes

characteristic

199

C₂²

#17989628

yes

characteristic

200

C₂²

#17989630

yes

—

201

C₄

—

202

C₄

—

584

—

203

C₄

—

204

C₄

—

205

C₄

—

146

—

206

C₄

—

146

—

207

C₄

#17989591

yes

characteristic

208

C₄

#17989593

yes

characteristic

209

C₄

#17989637

yes

—

210

C₂

#22625550

yes

—

211

C₂

#22625554

yes

—

212

C₂

#22626031

yes

characteristic

213

C₂

#22626036

yes

characteristic

214

C₂

#22626038

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×#64408
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×#64408
Fitting	#340×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₄×D₄
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	#340	#64408
hypocenter	C₇₃	#9599
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×#64408

commutator

C₂²×C₄×C₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×#64408

Fitting

#340×C₇₃

C₄

socle, prime radical

C₂³×C₇₃

C₄×D₄

cosocle

C₂×C₄×Dic₇₃

C₂³

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
113. C₇₃	#9599	semidirect

H ≅ G/N

type

113. C₇₃

#9599

semidirect

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

subgroup	index
#22626028	2
C₂×#17989448	2
C₂×#17989429	2
#22624767	2
#22609823	2
#9599	73

subgroup

index

Subgroups — #34007490

Conjugacy classes of subgroups

Complements

Maximal subgroups