Group #34004065 — subgroups

Subgroups — #34004065

Order 18688.

358 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 138 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	#22625481	C₂	1	1	yes	maximal, characteristic
3	C₈×#11924038	C₂	1	2	yes	maximal
4	#22624763	C₂	1	2	yes	maximal
5	#322×D₇₃	C₂	1	1	yes	maximal, characteristic
6	#17989606	C₂²	1	2	yes	—
7	C₂×#14556661	C₄	1	2	yes	—
8	C₄×#11924038	C₂²	1	2	yes	—
9	#17989463	C₄	1	2	yes	—
10	#17989450	C₄	1	2	yes	—
11	#17989449	C₄	1	2	yes	—
12	#17989429	—	2	2	no	—
13	C₈×#10596833	—	2	2	no	—
14	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
15	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
16	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
17	#17988309	C₄	1	1	yes	characteristic
18	#17988309	C₄	1	1	yes	characteristic
19	#17988308	C₄	1	1	yes	characteristic
20	#322×C₇₃	C₄	1	1	yes	characteristic
21	C₂×#11924038	C₂×C₄	1	4	yes	—
22	#14556661	C₈	1	8	yes	complemented
23	C₄⋊#10596833	C₂×C₄	1	4	yes	—
24	#14556648	—	2	2	no	—
25	#14556646	—	2	2	no	—
26	C₄×#10596833	—	2	2	no	—
27	C₄×#10596833	—	2	2	no	—
28	C₂×C₈×D₇₃	—	2	1	no	—
29	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×D₇₃	Q₈	1	2	yes	—
32	C₄²×D₇₃	C₂³	1	1	yes	characteristic
33	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
34	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
35	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
36	C₈×Dic₇₃	—	2	1	no	—
37	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
38	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
39	C₈×Dic₇₃	D₄	1	2	yes	—
40	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
43	#11924038	—	2	4	no	—
44	#11924038	C₂×C₈	1	8	yes	—
45	C₂×#10596833	—	2	4	no	—
46	C₂×#10596833	—	2	4	no	—
47	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×D₇₃	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₄×D₇₃	#41	1	1	yes	characteristic
50	C₂×#10596831	—	2	1	no	—
51	C₂×#10596831	C₄²	1	1	yes	characteristic
52	C₂×#10596831	C₄²	1	1	yes	characteristic
53	C₂×#10596831	C₄⋊C₄	1	2	yes	—
54	C₈×D₇₃	—	2	2	no	—
55	C₈×D₇₃	—	4	2	no	—
56	C₈×D₇₃	—	4	2	no	—
57	C₈×D₇₃	C₂×C₈	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	1	no	—
62	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
63	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
64	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
65	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×D₇₃	C₄×Q₈	1	1	yes	characteristic
67	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
68	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
70	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
71	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
72	C₂×Dic₇₃	C₄×Q₈	1	1	yes	characteristic
73	C₄×D₇₃	—	2	2	no	—
74	C₄×D₇₃	—	2	2	no	—
75	C₄×D₇₃	C₂²×C₈	1	2	yes	—
76	C₄×D₇₃	#131	1	2	yes	—
77	#10596833	—	2	8	no	—
78	#10596833	—	4	4	no	—
79	C₈×C₇₃	—	2	1	no	—
80	C₈×C₇₃	—	2	2	no	—
81	C₈×C₇₃	—	4	1	no	—
82	C₈×C₇₃	C₄×C₈	1	2	yes	—
83	#10596831	—	2	1	no	—
84	#10596831	—	2	2	no	—
85	#10596831	—	4	1	no	—
86	#10596831	C₄×C₈	1	2	yes	—
87	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
88	C₂×D₇₃	—	2	1	no	—
89	C₂×D₇₃	—	2	1	no	—
90	C₂×D₇₃	C₈×Q₈	1	2	yes	—
91	C₄×C₇₃	—	2	1	no	—
92	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
93	C₄×C₇₃	#405	1	1	yes	characteristic
94	C₄×C₇₃	#430	1	2	yes	—
95	Dic₇₃	—	2	1	no	—
96	Dic₇₃	C₈×D₄	1	2	yes	—
97	Dic₇₃	#444	1	2	yes	—
98	#8294	—	73	1	no	maximal, Sylow
99	C₂×C₇₃	C₈×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₇₃	#1773	1	1	yes	characteristic
101	C₂×C₇₃	#1778	1	1	yes	characteristic
102	D₇₃	—	2	2	no	—
103	C₄×#108	—	73	1	no	—
104	#1564	—	73	1	no	—
105	C₈×M₁₆	—	73	2	no	—
106	C₂×#322	—	73	1	no	—
107	#1278	—	73	2	no	—
108	C₇₃	#8294	1	1	yes	characteristic, complemented, Sylow
109	#431	—	73	2	no	—
110	C₂×#108	—	73	2	no	—
111	C₄×M₁₆	—	73	2	no	—
112	C₂×C₄×C₈	—	73	1	no	—
113	C₂×C₄×C₈	—	73	1	no	—
114	C₂×C₄×C₈	—	73	1	no	—
115	#341	—	73	2	no	—
116	#335	—	73	2	no	—
117	#334	—	73	2	no	—
118	#322	—	73	1	no	—
119	#322	—	73	1	no	—
120	#322	—	73	1	no	—
121	#322	—	146	2	no	—
122	#322	#64408	1	1	yes	characteristic
123	C₈²	—	146	2	no	—
124	C₂×M₁₆	—	73	4	no	—
125	C₂²×C₈	—	73	2	no	—
126	C₂²×C₈	—	73	1	no	—
127	C₂²×C₈	—	73	1	no	—
128	C₂²×C₈	—	146	1	no	—
129	C₂×C₄²	—	73	1	no	—
130	#108	—	73	8	no	—
131	C₄⋊C₈	—	73	4	no	—
132	C₂²⋊C₈	—	146	2	no	—
133	#97	—	146	2	no	—
134	C₄×C₈	—	73	1	no	—
135	C₄×C₈	—	73	1	no	—
136	C₄×C₈	—	73	2	no	—
137	C₄×C₈	—	73	1	no	—
138	C₄×C₈	—	73	1	no	—
139	C₄×C₈	—	73	1	no	—
140	C₄×C₈	—	146	1	no	—
141	C₄×C₈	—	146	2	no	—
142	C₄×C₈	—	146	2	no	—
143	C₄×C₈	C₂×#64408	1	1	yes	characteristic
144	C₄×C₈	C₂×#64408	1	1	yes	characteristic
145	C₄×C₈	C₂×#64408	1	1	yes	characteristic
146	C₂²×C₄	—	73	1	no	—
147	C₂²×C₄	—	73	1	no	—
148	C₂²×C₄	—	73	1	no	—
149	M₁₆	—	73	8	no	—
150	M₁₆	—	146	4	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	292	2	no	—
153	C₂×C₈	—	292	2	no	—
154	C₂×C₈	—	73	1	no	—
155	C₂×C₈	—	73	4	no	—
156	C₂×C₈	—	73	1	no	—
157	C₂×C₈	—	73	2	no	—
158	C₂×C₈	—	146	2	no	—
159	C₂×C₈	—	146	4	no	—
160	C₂×C₈	—	146	1	no	—
161	C₂×C₈	—	146	4	no	—
162	C₂×C₈	C₄×#64408	1	1	yes	characteristic
163	C₂×C₈	C₄×#64408	1	1	yes	characteristic
164	C₂×C₈	C₄⋊#64408	1	2	yes	—
165	C₄²	—	73	1	no	—
166	C₄²	—	73	1	no	—
167	C₄²	—	73	1	no	—
168	C₄²	C₂²×#64408	1	1	yes	characteristic
169	C₂³	—	73	1	no	—
170	C₂×C₄	—	73	1	no	—
171	C₂×C₄	—	73	1	no	—
172	C₂×C₄	—	73	2	no	—
173	C₂×C₄	—	73	2	no	—
174	C₂×C₄	—	73	1	no	—
175	C₂×C₄	—	146	2	no	—
176	C₂×C₄	—	146	2	no	—
177	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
179	C₂×C₄	#14556670	1	1	yes	characteristic
180	C₈	—	2	1	no	—
181	C₈	—	2	2	no	—
182	C₈	—	4	1	no	—
183	C₈	—	292	1	no	—
184	C₈	—	292	4	no	—
185	C₈	—	73	2	no	—
186	C₈	—	146	8	no	—
187	C₈	—	146	1	no	—
188	C₈	—	146	2	no	—
189	C₈	C₈×#64408	1	2	yes	—
190	C₂²	—	73	2	no	—
191	C₂²	—	146	1	no	—
192	C₂²	—	146	1	no	—
193	C₂²	C₄×C₄⋊#64408	1	1	yes	characteristic
194	C₄	—	2	1	no	—
195	C₄	—	73	2	no	—
196	C₄	—	73	2	no	—
197	C₄	—	146	1	no	—
198	C₄	C₂×C₈×#64408	1	1	yes	characteristic
199	C₄	#17989550	1	1	yes	characteristic
200	C₄	#17989602	1	2	yes	—
201	C₂	—	146	2	no	—
202	C₂	C₈×C₄⋊#64408	1	1	yes	characteristic
203	C₂	#22625737	1	1	yes	characteristic
204	C₂	#22625751	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14556661

C₂

yes

maximal, characteristic

#22625481

C₂

yes

maximal, characteristic

C₈×#11924038

C₂

yes

maximal

#22624763

C₂

yes

maximal

#322×D₇₃

C₂

yes

maximal, characteristic

#17989606

C₂²

yes

—

C₂×#14556661

C₄

yes

—

C₄×#11924038

C₂²

yes

—

#17989463

C₄

yes

—

#17989450

C₄

yes

—

#17989449

C₄

yes

—

#17989429

—

C₈×#10596833

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

C₄×C₈×D₇₃

C₂²

yes

characteristic

C₄×C₈×D₇₃

C₂²

yes

characteristic

#17988309

C₄

yes

characteristic

#17988309

C₄

yes

characteristic

#17988308

C₄

yes

characteristic

#322×C₇₃

C₄

yes

characteristic

C₂×#11924038

C₂×C₄

yes

—

#14556661

C₈

yes

complemented

C₄⋊#10596833

C₂×C₄

yes

—

—

—

—

—

C₂×C₈×D₇₃

—

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

C₂×C₈×D₇₃

Q₈

yes

—

C₄²×D₇₃

C₂³

yes

characteristic

C₄×#10596831

C₂×C₄

yes

characteristic

C₄×#10596831

C₂×C₄

yes

characteristic

C₄×#10596831

C₂×C₄

yes

characteristic

C₈×Dic₇₃

—

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

D₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

#11924038

—

#11924038

C₂×C₈

yes

—

C₂×#10596833

—

C₂×#10596833

—

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

C₂×#10596831

C₄⋊C₄

yes

—

C₈×D₇₃

—

C₈×D₇₃

—

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

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₂²×D₇₃

C₄×Q₈

yes

characteristic

C₂×C₄×C₇₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

#117

yes

characteristic

C₂×Dic₇₃

C₄×D₄

yes

characteristic

C₂×Dic₇₃

C₄×D₄

yes

characteristic

C₂×Dic₇₃

C₄×Q₈

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

C₂²×C₈

yes

—

C₄×D₇₃

#131

yes

—

#10596833

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

C₄×C₈

yes

—

—

—

—

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

—

Dic₇₃

—

Dic₇₃

C₈×D₄

yes

—

Dic₇₃

#444

yes

—

#8294

—

maximal, Sylow

C₂×C₇₃

C₈×C₄⋊C₄

yes

characteristic

100

C₂×C₇₃

#1773

yes

characteristic

101

C₂×C₇₃

#1778

yes

characteristic

102

D₇₃

—

103

C₄×#108

—

104

#1564

—

105

C₈×M₁₆

—

106

C₂×#322

—

107

#1278

—

108

C₇₃

#8294

yes

characteristic, complemented, Sylow

109

#431

—

110

C₂×#108

—

111

C₄×M₁₆

—

112

C₂×C₄×C₈

—

113

C₂×C₄×C₈

—

114

C₂×C₄×C₈

—

115

#341

—

116

#335

—

117

#334

—

118

#322

—

119

#322

—

120

#322

—

121

#322

—

146

—

122

#322

#64408

yes

characteristic

123

C₈²

—

146

—

124

C₂×M₁₆

—

125

C₂²×C₈

—

126

C₂²×C₈

—

127

C₂²×C₈

—

128

C₂²×C₈

—

146

—

129

C₂×C₄²

—

130

#108

—

131

C₄⋊C₈

—

132

C₂²⋊C₈

—

146

—

133

#97

—

146

—

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

C₄×C₈

C₂×#64408

yes

characteristic

145

C₄×C₈

C₂×#64408

yes

characteristic

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

M₁₆

—

150

M₁₆

—

146

—

151

C₂×C₈

—

152

C₂×C₈

—

292

—

153

C₂×C₈

—

292

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

146

—

159

C₂×C₈

—

146

—

160

C₂×C₈

—

146

—

161

C₂×C₈

—

146

—

162

C₂×C₈

C₄×#64408

yes

characteristic

163

C₂×C₈

C₄×#64408

yes

characteristic

164

C₂×C₈

C₄⋊#64408

yes

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

168

C₄²

C₂²×#64408

yes

characteristic

169

C₂³

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

146

—

176

C₂×C₄

—

146

—

177

C₂×C₄

C₂×C₄×#64408

yes

characteristic

178

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

179

C₂×C₄

#14556670

yes

characteristic

180

C₈

—

181

C₈

—

182

C₈

—

183

C₈

—

292

—

184

C₈

—

292

—

185

C₈

—

186

C₈

—

146

—

187

C₈

—

146

—

188

C₈

—

146

—

189

C₈

C₈×#64408

yes

—

190

C₂²

—

191

C₂²

—

146

—

192

C₂²

—

146

—

193

C₂²

C₄×C₄⋊#64408

yes

characteristic

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

—

146

—

198

C₄

C₂×C₈×#64408

yes

characteristic

199

C₄

#17989550

yes

characteristic

200

C₄

#17989602

yes

—

201

C₂

—

146

—

202

C₂

C₈×C₄⋊#64408

yes

characteristic

203

C₂

#22625737

yes

characteristic

204

C₂

#22625751

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#64408
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₄²	C₂²×#64408
Fitting	#322×C₇₃	C₄
socle	C₂²×C₇₃	C₄×C₄⋊C₄
prime radical	C₂²×D₇₃	C₄×Q₈
cosocle	C₄²×D₇₃	C₂³
hypercenter	#322	#64408
hypocenter	C₇₃	#8294
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

subgroup

G/H

center

C₂×C₄

C₂×C₄⋊#64408

commutator

C₄×C₇₃

C₂×C₄×C₈

Frattini

C₄²

C₂²×#64408

Fitting

#322×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 ◁ C₄×C₈ ◁ #322

N	H ≅ G/N	type
22. #14556661	C₈	semidirect
108. C₇₃	#8294	semidirect

H ≅ G/N

type

22. #14556661

C₈

semidirect

108. 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
#22625481	2
C₈×#11924038	2
#22624763	2
#322×D₇₃	2
#8294	73

subgroup

index

#322×D₇₃

Subgroups — #34004065

Conjugacy classes of subgroups

Complements

Maximal subgroups