Group #34004058 — subgroups

Subgroups — #34004058

Order 18688.

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

normal subgroups only

This group has 350 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	C₄×#14556661	C₂	1	1	yes	maximal, characteristic
2	#22625143	C₂	1	2	yes	maximal
3	C₈×#11924036	C₂	1	1	yes	maximal, characteristic
4	#22624762	C₂	1	2	yes	maximal
5	#22608676	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556661	C₄	1	2	yes	—
7	C₄×#11924038	C₂²	1	2	yes	—
8	C₄×#11924036	C₂²	1	1	yes	characteristic
9	C₄×#11924036	C₂²	1	1	yes	characteristic
10	#17989463	C₄	1	2	yes	—
11	#17989429	—	2	2	no	—
12	#17989428	—	2	2	no	—
13	C₈×#10596833	C₄	1	2	yes	—
14	C₈×#10596833	C₄	1	2	yes	—
15	#17988639	C₂²	1	2	yes	—
16	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
17	#17988339	C₄	1	2	yes	—
18	#17988338	C₄	1	1	yes	characteristic
19	#334×C₇₃	C₄	1	1	yes	characteristic
20	C₂×#11924038	C₂×C₄	1	4	yes	—
21	C₂×#11924036	—	2	1	no	—
22	C₂×#11924036	Q₈	1	2	yes	—
23	#14556661	—	2	4	no	—
24	#14556646	—	2	2	no	—
25	C₄×#10596833	—	2	2	no	—
26	C₄×#10596833	—	2	1	no	—
27	C₄×#10596833	C₂×C₄	1	2	yes	—
28	C₄×#10596833	C₂×C₄	1	2	yes	—
29	C₄×#10596833	D₄	1	1	yes	characteristic
30	C₄×#10596833	D₄	1	1	yes	characteristic
31	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
33	C₄²×D₇₃	C₂³	1	1	yes	characteristic
34	C₄⋊#10596831	C₂×C₄	1	2	yes	—
35	#14556503	—	2	2	no	—
36	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
37	C₈×Dic₇₃	—	2	2	no	—
38	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
39	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
40	C₄⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
42	#11924038	—	2	4	no	—
43	#11924038	—	2	4	no	—
44	#11924036	—	2	4	no	—
45	#11924036	—	2	4	no	—
46	C₂×#10596833	—	2	2	no	—
47	C₂×#10596833	—	2	4	no	—
48	C₂×#10596833	C₄⋊C₄	1	2	yes	—
49	C₂×#10596833	C₄⋊C₄	1	2	yes	—
50	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×D₇₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×D₇₃	#41	1	1	yes	characteristic
53	C₂×#10596831	—	2	2	no	—
54	C₂×#10596831	C₄²	1	1	yes	characteristic
55	C₂×#10596831	C₄²	1	1	yes	characteristic
56	C₈×D₇₃	—	2	2	no	—
57	C₈×D₇₃	M₁₆	1	4	yes	—
58	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
59	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
60	C₄×Dic₇₃	#41	1	1	yes	characteristic
61	C₂×C₈×C₇₃	—	2	2	no	—
62	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
63	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
64	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×D₇₃	C₄×Q₈	1	1	yes	characteristic
66	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
67	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
69	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×Dic₇₃	#117	1	1	yes	characteristic
71	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
72	C₄×D₇₃	—	2	2	no	—
73	C₄×D₇₃	—	2	2	no	—
74	C₄×D₇₃	C₂×M₁₆	1	2	yes	—
75	C₄×D₇₃	#131	1	2	yes	—
76	#10596833	—	2	4	no	—
77	#10596833	—	2	2	no	—
78	#10596833	—	4	4	no	—
79	#10596833	#108	1	4	yes	complemented
80	C₈×C₇₃	—	2	1	no	—
81	C₈×C₇₃	—	4	2	no	—
82	C₈×C₇₃	#97	1	2	yes	—
83	#10596831	—	2	1	no	—
84	#10596831	—	4	2	no	—
85	#10596831	#97	1	2	yes	—
86	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
87	C₂×D₇₃	—	2	1	no	—
88	C₂×D₇₃	—	2	1	no	—
89	C₂×D₇₃	#446	1	2	yes	—
90	C₄×C₇₃	—	2	1	no	—
91	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
92	C₄×C₇₃	#405	1	1	yes	characteristic
93	C₄×C₇₃	#443	1	2	yes	—
94	Dic₇₃	—	2	1	no	—
95	Dic₇₃	C₂×#108	1	2	yes	—
96	Dic₇₃	#436	1	2	yes	—
97	#8293	—	73	1	no	maximal, Sylow
98	C₂×C₇₃	#1770	1	1	yes	characteristic
99	C₂×C₇₃	#1772	1	1	yes	characteristic
100	C₂×C₇₃	C₄×#108	1	1	yes	characteristic
101	D₇₃	—	2	2	no	—
102	C₄×#108	—	73	1	no	—
103	#1564	—	73	1	no	—
104	#1451	—	73	2	no	—
105	C₂×C₈²	—	73	1	no	—
106	#1278	—	73	2	no	—
107	C₇₃	#8293	1	1	yes	characteristic, complemented, Sylow
108	#431	—	73	2	no	—
109	C₂×#108	—	73	2	no	—
110	C₄×M₁₆	—	73	2	no	—
111	C₂×C₄×C₈	—	73	1	no	—
112	C₂×C₄×C₈	—	73	1	no	—
113	C₂×C₄×C₈	—	73	1	no	—
114	#341	—	73	2	no	—
115	#335	—	73	2	no	—
116	#334	—	73	1	no	—
117	#334	#64408	1	1	yes	characteristic
118	#322	—	146	2	no	—
119	#322	—	146	2	no	—
120	C₈²	—	73	2	no	—
121	C₈²	—	73	2	no	—
122	C₂×M₁₆	—	73	4	no	—
123	C₂²×C₈	—	73	1	no	—
124	C₂²×C₈	—	73	2	no	—
125	C₂²×C₈	—	73	1	no	—
126	C₂²×C₈	—	146	1	no	—
127	C₂×C₄²	—	73	1	no	—
128	#108	—	146	4	no	—
129	C₄⋊C₈	—	73	2	no	—
130	C₄⋊C₈	C₂×#64408	1	2	yes	—
131	C₂²⋊C₈	—	146	2	no	—
132	#97	—	146	2	no	—
133	C₄×C₈	—	73	2	no	—
134	C₄×C₈	—	73	1	no	—
135	C₄×C₈	—	73	1	no	—
136	C₄×C₈	—	73	1	no	—
137	C₄×C₈	—	73	1	no	—
138	C₄×C₈	—	73	2	no	—
139	C₄×C₈	—	73	1	no	—
140	C₄×C₈	—	146	2	no	—
141	C₄×C₈	—	146	2	no	—
142	C₄×C₈	—	146	1	no	—
143	C₄×C₈	C₂×#64408	1	1	yes	characteristic
144	C₂²×C₄	—	73	1	no	—
145	C₂²×C₄	—	73	1	no	—
146	C₂²×C₄	—	73	1	no	—
147	M₁₆	—	146	4	no	—
148	M₁₆	—	146	4	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	73	2	no	—
151	C₂×C₈	—	73	1	no	—
152	C₂×C₈	—	73	4	no	—
153	C₂×C₈	—	73	2	no	—
154	C₂×C₈	—	73	1	no	—
155	C₂×C₈	—	146	2	no	—
156	C₂×C₈	—	146	4	no	—
157	C₂×C₈	—	146	2	no	—
158	C₂×C₈	—	146	4	no	—
159	C₂×C₈	—	146	2	no	—
160	C₂×C₈	—	146	4	no	—
161	C₂×C₈	C₄×#64408	1	1	yes	characteristic
162	C₂×C₈	C₄×#64408	1	1	yes	characteristic
163	C₄²	—	73	1	no	—
164	C₄²	—	73	1	no	—
165	C₄²	—	73	1	no	—
166	C₄²	C₂²×#64408	1	1	yes	characteristic
167	C₂³	—	73	1	no	—
168	C₂×C₄	—	73	1	no	—
169	C₂×C₄	—	73	1	no	—
170	C₂×C₄	—	73	2	no	—
171	C₂×C₄	—	73	2	no	—
172	C₂×C₄	—	73	1	no	—
173	C₂×C₄	—	146	2	no	—
174	C₂×C₄	—	146	2	no	—
175	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
176	C₂×C₄	D₄×#64408	1	1	yes	characteristic
177	C₂×C₄	Q₈×#64408	1	1	yes	characteristic
178	C₈	—	2	1	no	—
179	C₈	—	4	2	no	—
180	C₈	—	292	4	no	—
181	C₈	—	292	2	no	—
182	C₈	—	73	4	no	—
183	C₈	—	73	2	no	—
184	C₈	—	146	2	no	—
185	C₈	—	146	4	no	—
186	C₈	—	146	1	no	—
187	C₈	#14556645	1	2	yes	—
188	C₂²	—	73	2	no	—
189	C₂²	—	146	1	no	—
190	C₂²	—	146	1	no	—
191	C₂²	C₄⋊C₄×#64408	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	73	2	no	—
194	C₄	—	73	2	no	—
195	C₄	—	146	1	no	—
196	C₄	C₂×#14556645	1	1	yes	characteristic
197	C₄	#17989553	1	1	yes	characteristic
198	C₄	#17989630	1	2	yes	—
199	C₂	—	146	2	no	—
200	C₂	#22625728	1	1	yes	characteristic
201	C₂	#22625734	1	1	yes	characteristic
202	C₂	#108×#64408	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14556661

C₂

yes

maximal, characteristic

#22625143

C₂

yes

maximal

C₈×#11924036

C₂

yes

maximal, characteristic

#22624762

C₂

yes

maximal

#22608676

C₂

yes

maximal, characteristic

C₂×#14556661

C₄

yes

—

C₄×#11924038

C₂²

yes

—

C₄×#11924036

C₂²

yes

characteristic

C₄×#11924036

C₂²

yes

characteristic

#17989463

C₄

yes

—

#17989429

—

#17989428

—

C₈×#10596833

C₄

yes

—

C₈×#10596833

C₄

yes

—

#17988639

C₂²

yes

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

#17988339

C₄

yes

—

#17988338

C₄

yes

characteristic

#334×C₇₃

C₄

yes

characteristic

C₂×#11924038

C₂×C₄

yes

—

C₂×#11924036

—

C₂×#11924036

Q₈

yes

—

—

—

—

—

C₂×C₄

yes

—

C₄×#10596833

C₂×C₄

yes

—

C₄×#10596833

D₄

yes

characteristic

C₄×#10596833

D₄

yes

characteristic

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

C₄²×D₇₃

C₂³

yes

characteristic

C₄⋊#10596831

C₂×C₄

yes

—

#14556503

—

C₄×#10596831

C₂×C₄

yes

characteristic

C₈×Dic₇₃

—

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

—

—

—

—

—

C₄⋊C₄

yes

—

C₂×#10596833

C₄⋊C₄

yes

—

C₂×C₄×D₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×D₇₃

C₂×Q₈

yes

C₂×C₄×D₇₃

yes

—

C₄²

yes

characteristic

C₂×#10596831

C₄²

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

M₁₆

yes

—

C₄×Dic₇₃

C₂²×C₄

yes

characteristic

C₄×Dic₇₃

C₂×D₄

yes

characteristic

C₄×Dic₇₃

#41

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₂²×D₇₃

C₄×Q₈

yes

characteristic

C₂×C₄×C₇₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₃

C₄×D₄

yes

characteristic

C₂×C₄×C₇₃

C₄×Q₈

yes

characteristic

C₂×Dic₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₇₃

#117

yes

characteristic

C₂×Dic₇₃

C₄×D₄

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

C₂×M₁₆

yes

—

C₄×D₇₃

#131

yes

—

—

—

—

yes

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

#97

yes

—

—

—

yes

—

C₂²×C₇₃

C₄×C₄⋊C₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

#446

yes

—

C₄×C₇₃

—

C₄×C₇₃

C₂×#97

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

Dic₇₃

—

Dic₇₃

C₂×#108

yes

—

Dic₇₃

#436

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

100

C₂×C₇₃

C₄×#108

yes

characteristic

101

D₇₃

—

102

C₄×#108

—

103

#1564

—

104

#1451

—

105

C₂×C₈²

—

106

#1278

—

107

C₇₃

#8293

yes

characteristic, complemented, Sylow

108

#431

—

109

C₂×#108

—

110

C₄×M₁₆

—

111

C₂×C₄×C₈

—

112

C₂×C₄×C₈

—

113

C₂×C₄×C₈

—

114

#341

—

115

#335

—

116

#334

—

117

yes

118

—

146

—

119

#322

—

146

—

120

C₈²

—

121

C₈²

—

122

C₂×M₁₆

—

123

C₂²×C₈

—

124

C₂²×C₈

—

125

C₂²×C₈

—

126

C₂²×C₈

—

146

—

127

C₂×C₄²

—

128

#108

—

146

—

129

C₄⋊C₈

—

130

C₄⋊C₈

C₂×#64408

yes

—

131

C₂²⋊C₈

—

146

—

132

#97

—

146

—

133

C₄×C₈

—

134

C₄×C₈

—

135

C₄×C₈

—

136

C₄×C₈

—

137

C₄×C₈

—

138

C₄×C₈

—

139

C₄×C₈

—

140

C₄×C₈

—

146

—

141

C₄×C₈

—

146

—

142

C₄×C₈

—

146

—

143

C₄×C₈

C₂×#64408

yes

characteristic

144

C₂²×C₄

—

145

C₂²×C₄

—

146

C₂²×C₄

—

147

M₁₆

—

146

—

148

M₁₆

—

146

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

146

—

156

C₂×C₈

—

146

—

157

C₂×C₈

—

146

—

158

C₂×C₈

—

146

—

159

C₂×C₈

—

146

—

160

C₂×C₈

—

146

—

161

C₂×C₈

C₄×#64408

yes

characteristic

162

C₂×C₈

C₄×#64408

yes

characteristic

163

C₄²

—

164

C₄²

—

165

C₄²

—

166

C₄²

C₂²×#64408

yes

characteristic

167

C₂³

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

146

—

174

C₂×C₄

—

146

—

175

C₂×C₄

C₂×C₄×#64408

yes

characteristic

176

C₂×C₄

D₄×#64408

yes

characteristic

177

C₂×C₄

Q₈×#64408

yes

characteristic

178

C₈

—

179

C₈

—

180

C₈

—

292

—

181

C₈

—

292

—

182

C₈

—

183

C₈

—

184

C₈

—

146

—

185

C₈

—

146

—

186

C₈

—

146

—

187

C₈

#14556645

yes

—

188

C₂²

—

189

C₂²

—

146

—

190

C₂²

—

146

—

191

C₂²

C₄⋊C₄×#64408

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

146

—

196

C₄

C₂×#14556645

yes

characteristic

197

C₄

#17989553

yes

characteristic

198

C₄

#17989630

yes

—

199

C₂

—

146

—

200

C₂

#22625728

yes

characteristic

201

C₂

#22625734

yes

characteristic

202

C₂

#108×#64408

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#64408
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₄²	C₂²×#64408
Fitting	#334×C₇₃	C₄
socle	C₂²×C₇₃	C₄×C₄⋊C₄
prime radical	C₂²×D₇₃	C₄×Q₈
cosocle	C₄²×D₇₃	C₂³
hypercenter	#334	#64408
hypocenter	C₇₃	#8293
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

subgroup

G/H

center

C₂×C₄

D₄×#64408

commutator

C₂×C₄×C₇₃

C₂×C₄²

Frattini

C₄²

C₂²×#64408

Fitting

#334×C₇₃

C₄

socle

C₂²×C₇₃

C₄×C₄⋊C₄

prime radical

C₂²×D₇₃

C₄×Q₈

cosocle

C₄²×D₇₃

C₂³

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

N	H ≅ G/N	type
79. #10596833	#108	semidirect
107. C₇₃	#8293	semidirect

H ≅ G/N

type

79. #10596833

#108

semidirect

107. C₇₃

#8293

semidirect

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

subgroup	index
C₄×#14556661	2
#22625143	2
C₈×#11924036	2
#22624762	2
#22608676	2
#8293	73

subgroup

index

Subgroups — #34004058

Conjugacy classes of subgroups

Complements

Maximal subgroups