Group #34004067 — subgroups

Subgroups — #34004067

Order 18688.

358 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 136 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	#22625133	C₂	1	2	yes	maximal
3	#22625125	C₂	1	1	yes	maximal, characteristic
4	#22624762	C₂	1	2	yes	maximal
5	#22608678	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	C₄	1	2	yes	—
13	#17989428	C₄	1	2	yes	—
14	C₈×#10596833	—	2	2	no	—
15	#17988639	C₂²	1	2	yes	—
16	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
17	#17988340	C₄	1	1	yes	characteristic
18	#17988337	C₄	1	2	yes	—
19	#335×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	C₈	1	8	yes	complemented
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	2	yes	—
30	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
32	C₄²×D₇₃	C₂³	1	1	yes	characteristic
33	C₄⋊#10596831	C₂×C₄	1	2	yes	—
34	#14556503	—	2	2	no	—
35	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
36	C₈×Dic₇₃	—	2	2	no	—
37	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
38	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
39	C₄⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
41	#11924038	—	2	4	no	—
42	#11924038	C₂×C₈	1	8	yes	—
43	#11924036	—	4	2	no	—
44	#11924036	—	4	1	no	—
45	#11924036	—	4	1	no	—
46	C₂×#10596833	—	2	4	no	—
47	C₂×#10596833	—	2	2	no	—
48	C₂×#10596833	C₄⋊C₄	1	4	yes	—
49	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×D₇₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×D₇₃	#41	1	1	yes	characteristic
52	C₂×#10596831	—	2	2	no	—
53	C₂×#10596831	C₄²	1	1	yes	characteristic
54	C₂×#10596831	C₄²	1	1	yes	characteristic
55	C₈×D₇₃	—	2	2	no	—
56	C₈×D₇₃	C₂×C₈	1	4	yes	—
57	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
58	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
59	C₄×Dic₇₃	#41	1	1	yes	characteristic
60	C₂×C₈×C₇₃	—	2	2	no	—
61	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
62	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
63	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×D₇₃	C₄×Q₈	1	1	yes	characteristic
65	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
68	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×Dic₇₃	#117	1	1	yes	characteristic
70	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
71	C₄×D₇₃	—	2	1	no	—
72	C₄×D₇₃	—	2	1	no	—
73	C₄×D₇₃	—	2	2	no	—
74	C₄×D₇₃	C₂²×C₈	1	2	yes	—
75	C₄×D₇₃	#131	1	2	yes	—
76	#10596833	—	2	4	no	—
77	#10596833	—	2	8	no	—
78	#10596833	—	4	2	no	—
79	C₈×C₇₃	—	2	1	no	—
80	C₈×C₇₃	—	4	2	no	—
81	C₈×C₇₃	C₄×C₈	1	2	yes	—
82	#10596831	—	2	1	no	—
83	#10596831	—	4	2	no	—
84	#10596831	C₄×C₈	1	2	yes	—
85	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
86	C₂×D₇₃	—	2	1	no	—
87	C₂×D₇₃	—	2	1	no	—
88	C₂×D₇₃	C₈×Q₈	1	2	yes	—
89	C₄×C₇₃	—	2	1	no	—
90	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
91	C₄×C₇₃	#405	1	1	yes	characteristic
92	C₄×C₇₃	#444	1	1	yes	characteristic
93	C₄×C₇₃	#444	1	1	yes	characteristic
94	Dic₇₃	—	2	1	no	—
95	Dic₇₃	#430	1	2	yes	—
96	Dic₇₃	C₈×D₄	1	2	yes	—
97	#8294	—	73	1	no	maximal, Sylow
98	C₂×C₇₃	C₈×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₇₃	#1773	1	1	yes	characteristic
100	C₂×C₇₃	#1778	1	1	yes	characteristic
101	D₇₃	—	2	2	no	—
102	C₄×#108	—	73	1	no	—
103	#1564	—	73	1	no	—
104	C₈×M₁₆	—	73	2	no	—
105	C₂×#322	—	73	1	no	—
106	#1278	—	73	2	no	—
107	C₇₃	#8294	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	1	no	—
116	#335	#64408	1	1	yes	characteristic
117	#334	—	73	2	no	—
118	#322	—	73	2	no	—
119	#322	—	73	2	no	—
120	#322	—	146	2	no	—
121	C₈²	—	146	2	no	—
122	C₂×M₁₆	—	73	4	no	—
123	C₂²×C₈	—	73	2	no	—
124	C₂²×C₈	—	73	1	no	—
125	C₂²×C₈	—	73	1	no	—
126	C₂²×C₈	—	146	1	no	—
127	C₂×C₄²	—	73	1	no	—
128	#108	—	73	8	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	2	no	—
135	C₄×C₈	—	73	1	no	—
136	C₄×C₈	—	73	1	no	—
137	C₄×C₈	—	73	2	no	—
138	C₄×C₈	—	73	1	no	—
139	C₄×C₈	—	146	2	no	—
140	C₄×C₈	—	146	2	no	—
141	C₄×C₈	—	146	1	no	—
142	C₄×C₈	C₂×#64408	1	1	yes	characteristic
143	C₂²×C₄	—	73	1	no	—
144	C₂²×C₄	—	73	1	no	—
145	C₂²×C₄	—	73	1	no	—
146	M₁₆	—	73	8	no	—
147	M₁₆	—	146	4	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	292	1	no	—
150	C₂×C₈	—	292	1	no	—
151	C₂×C₈	—	292	2	no	—
152	C₂×C₈	—	73	1	no	—
153	C₂×C₈	—	73	4	no	—
154	C₂×C₈	—	73	4	no	—
155	C₂×C₈	—	73	1	no	—
156	C₂×C₈	—	146	2	no	—
157	C₂×C₈	—	146	2	no	—
158	C₂×C₈	—	146	2	no	—
159	C₂×C₈	—	146	4	no	—
160	C₂×C₈	C₄×#64408	1	1	yes	characteristic
161	C₂×C₈	C₄×#64408	1	1	yes	characteristic
162	C₄²	—	73	1	no	—
163	C₄²	—	73	1	no	—
164	C₄²	—	73	1	no	—
165	C₄²	C₂²×#64408	1	1	yes	characteristic
166	C₂³	—	73	1	no	—
167	C₂×C₄	—	73	1	no	—
168	C₂×C₄	—	73	1	no	—
169	C₂×C₄	—	73	2	no	—
170	C₂×C₄	—	73	2	no	—
171	C₂×C₄	—	73	1	no	—
172	C₂×C₄	—	146	2	no	—
173	C₂×C₄	—	146	1	no	—
174	C₂×C₄	—	146	1	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	2	no	—
181	C₈	—	292	2	no	—
182	C₈	—	73	2	no	—
183	C₈	—	146	4	no	—
184	C₈	—	146	8	no	—
185	C₈	—	146	1	no	—
186	C₈	C₈×#64408	1	2	yes	—
187	C₂²	—	73	2	no	—
188	C₂²	—	146	1	no	—
189	C₂²	—	146	1	no	—
190	C₂²	C₄⋊C₄×#64408	1	1	yes	characteristic
191	C₄	—	2	1	no	—
192	C₄	—	73	2	no	—
193	C₄	—	73	2	no	—
194	C₄	—	146	1	no	—
195	C₄	C₂×C₈×#64408	1	1	yes	characteristic
196	C₄	#17989553	1	1	yes	characteristic
197	C₄	#17989633	1	1	yes	characteristic
198	C₄	#17989634	1	1	yes	characteristic
199	C₂	—	146	2	no	—
200	C₂	#22625724	1	1	yes	characteristic
201	C₂	#22625738	1	1	yes	characteristic
202	C₂	#22625752	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14556661

C₂

yes

maximal, characteristic

#22625133

C₂

yes

maximal

#22625125

C₂

yes

maximal, characteristic

#22624762

C₂

yes

maximal

#22608678

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

C₄

yes

—

#17989428

C₄

yes

—

C₈×#10596833

—

#17988639

C₂²

yes

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

#17988340

C₄

yes

characteristic

#17988337

C₄

yes

—

#335×C₇₃

C₄

yes

characteristic

C₂×#11924038

C₂×C₄

yes

—

C₂×#11924036

—

C₂×#11924036

Q₈

yes

—

#14556661

C₈

yes

—

—

—

C₂×C₄

yes

—

C₄×#10596833

C₂×C₄

yes

—

C₄×#10596833

D₄

yes

—

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

characteristic

#11924038

—

#11924038

C₂×C₈

yes

—

—

—

—

—

—

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

C₂×C₈

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

—

C₄×D₇₃

C₂²×C₈

yes

—

C₄×D₇₃

#131

yes

—

#10596833

—

#10596833

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

C₄×C₈

yes

—

#10596831

—

#10596831

—

#10596831

C₄×C₈

yes

—

C₂²×C₇₃

C₄×C₄⋊C₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

C₈×Q₈

yes

—

C₄×C₇₃

—

C₄×C₇₃

C₂×C₄×C₈

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

#430

yes

—

Dic₇₃

C₈×D₄

yes

—

#8294

—

maximal, Sylow

C₂×C₇₃

C₈×C₄⋊C₄

yes

characteristic

C₂×C₇₃

#1773

yes

characteristic

100

C₂×C₇₃

#1778

yes

characteristic

101

D₇₃

—

102

C₄×#108

—

103

#1564

—

104

C₈×M₁₆

—

105

C₂×#322

—

106

#1278

—

107

C₇₃

#8294

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

yes

117

—

118

#322

—

119

#322

—

120

#322

—

146

—

121

C₈²

—

146

—

122

C₂×M₁₆

—

123

C₂²×C₈

—

124

C₂²×C₈

—

125

C₂²×C₈

—

126

C₂²×C₈

—

146

—

127

C₂×C₄²

—

128

#108

—

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

—

146

—

140

C₄×C₈

—

146

—

141

C₄×C₈

—

146

—

142

C₄×C₈

C₂×#64408

yes

characteristic

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

—

146

M₁₆

—

147

M₁₆

—

146

—

148

C₂×C₈

—

149

C₂×C₈

—

292

—

150

C₂×C₈

—

292

—

151

C₂×C₈

—

292

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

146

—

157

C₂×C₈

—

146

—

158

C₂×C₈

—

146

—

159

C₂×C₈

—

146

—

160

C₂×C₈

C₄×#64408

yes

characteristic

161

C₂×C₈

C₄×#64408

yes

characteristic

162

C₄²

—

163

C₄²

—

164

C₄²

—

165

C₄²

C₂²×#64408

yes

characteristic

166

C₂³

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

146

—

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

—

146

—

184

C₈

—

146

—

185

C₈

—

146

—

186

C₈

C₈×#64408

yes

—

187

C₂²

—

188

C₂²

—

146

—

189

C₂²

—

146

—

190

C₂²

C₄⋊C₄×#64408

yes

characteristic

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

—

146

—

195

C₄

C₂×C₈×#64408

yes

characteristic

196

C₄

#17989553

yes

characteristic

197

C₄

#17989633

yes

characteristic

198

C₄

#17989634

yes

characteristic

199

C₂

—

146

—

200

C₂

#22625724

yes

characteristic

201

C₂

#22625738

yes

characteristic

202

C₂

#22625752

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#64408
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₄²	C₂²×#64408
Fitting	#335×C₇₃	C₄
socle	C₂²×C₇₃	C₄×C₄⋊C₄
prime radical	C₂²×D₇₃	C₄×Q₈
cosocle	C₄²×D₇₃	C₂³
hypercenter	#335	#64408
hypocenter	C₇₃	#8294
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

subgroup

G/H

center

C₂×C₄

D₄×#64408

commutator

C₄×C₇₃

C₂×C₄×C₈

Frattini

C₄²

C₂²×#64408

Fitting

#335×C₇₃

C₄

socle

C₂²×C₇₃

C₄×C₄⋊C₄

prime radical

C₂²×D₇₃

C₄×Q₈

cosocle

C₄²×D₇₃

C₂³

C₇₃

G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

N	H ≅ G/N	type
23. #14556661	C₈	semidirect
107. C₇₃	#8294	semidirect

H ≅ G/N

type

23. #14556661

C₈

semidirect

107. C₇₃

#8294

semidirect

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

subgroup	index
C₄×#14556661	2
#22625133	2
#22625125	2
#22624762	2
#22608678	2
#8294	73

subgroup

index

Subgroups — #34004067

Conjugacy classes of subgroups

Complements

Maximal subgroups