Group #34013916 — subgroups

Subgroups — #34013916

Order 18688.

310 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 92 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	D₇₃⋊#365	C₂	1	2	yes	maximal
2	#22625866	C₂	1	1	yes	maximal, characteristic
3	#22625733	C₂	1	1	yes	maximal, characteristic
4	#22625003	C₂	1	2	yes	maximal
5	#364×D₇₃	C₂	1	1	yes	maximal, characteristic
6	D₇₃⋊#108	C₂²	1	1	yes	characteristic
7	#17989597	C₂²	1	1	yes	characteristic
8	C₂×#14556659	—	2	1	no	—
9	C₂×#14556659	C₂²	1	2	yes	—
10	#17989550	—	2	1	no	—
11	#17989533	—	2	1	no	—
12	#17989512	—	2	2	no	—
13	#17989511	—	2	2	no	—
14	#17989458	—	2	1	no	—
15	M₃₂×D₇₃	C₂²	1	2	yes	—
16	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
17	#17988408	C₄	1	1	yes	characteristic, complemented
18	#17988407	C₄	1	2	yes	complemented
19	#364×C₇₃	C₄	1	1	yes	characteristic, complemented
20	D₇₃⋊M₁₆	—	2	1	no	—
21	#14556670	—	2	1	no	—
22	C₂×C₄⋊#64408	—	2	2	no	—
23	C₂×C₄⋊#64408	—	4	2	no	—
24	#14556662	—	2	1	no	—
25	#14556661	D₄	1	2	yes	—
26	#14556660	Q₈	1	2	yes	—
27	#14556659	—	2	2	no	—
28	#14556659	—	2	1	no	—
29	#14556659	—	2	2	no	—
30	#14556659	—	4	1	no	—
31	#14556659	—	4	1	no	—
32	#14556645	—	4	1	no	—
33	C₈×#64408	—	4	1	no	—
34	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
35	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
36	C₄²×D₇₃	D₄	1	1	yes	characteristic
37	#14556531	C₂×C₄	1	2	yes	—
38	#14556530	—	2	2	no	—
39	C₁₆×D₇₃	—	2	2	no	—
40	C₄×#10596831	Q₈	1	1	yes	characteristic
41	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
42	C₈×Dic₇₃	Q₈	1	1	yes	characteristic
43	M₃₂×C₇₃	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×#64408	—	4	2	no	—
46	#11924038	—	2	2	no	—
47	#11924037	—	4	1	no	—
48	#11924036	—	4	1	no	—
49	C₄⋊#64408	—	2	2	no	—
50	C₄⋊#64408	—	4	2	no	—
51	C₄⋊#64408	—	4	2	no	—
52	C₄⋊#64408	—	8	4	no	—
53	C₂²⋊#64408	—	4	1	no	—
54	C₄×#64408	—	4	1	no	—
55	C₂×C₄×D₇₃	—	2	1	no	—
56	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
57	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
58	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
59	C₈×D₇₃	SD₁₆	1	1	yes	characteristic
60	C₈×D₇₃	SD₁₆	1	1	yes	characteristic
61	C₈×D₇₃	SD₁₆	1	1	yes	characteristic
62	C₈×D₇₃	SD₁₆	1	1	yes	characteristic
63	C₈×D₇₃	C₂×D₄	1	1	yes	characteristic
64	C₈×D₇₃	C₂×Q₈	1	1	yes	characteristic
65	C₈×D₇₃	#41	1	1	yes	characteristic
66	C₈×D₇₃	#41	1	1	yes	characteristic
67	C₄×Dic₇₃	—	2	1	no	—
68	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
69	#11924003	—	2	2	no	—
70	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
73	C₁₆×C₇₃	—	2	2	no	—
74	C₂²×D₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
75	C₂×#64408	—	4	2	no	—
76	C₂×#64408	—	8	4	no	—
77	C₂×#64408	—	8	2	no	—
78	C₂×C₄×C₇₃	—	2	1	no	—
79	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×Dic₇₃	—	2	1	no	—
81	C₂×Dic₇₃	#137	1	1	yes	characteristic
82	C₄×D₇₃	—	2	4	no	—
83	C₄×D₇₃	#121	1	1	yes	characteristic
84	C₄×D₇₃	C₂²⋊Q₈	1	1	yes	characteristic
85	C₄×D₇₃	C₂×SD₁₆	1	1	yes	characteristic
86	C₄×D₇₃	C₂×SD₁₆	1	1	yes	characteristic
87	#10596833	—	4	2	no	—
88	C₈×C₇₃	#106	1	1	yes	characteristic
89	C₈×C₇₃	#106	1	1	yes	characteristic
90	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
91	C₈×C₇₃	C₄×Q₈	1	1	yes	characteristic
92	#10596831	#106	1	1	yes	characteristic
93	#10596831	#106	1	1	yes	characteristic
94	#10596831	#125	1	1	yes	characteristic
95	#10596831	C₄⋊Q₈	1	1	yes	characteristic
96	C₂²×C₇₃	#428	1	1	yes	characteristic
97	C₂×D₇₃	—	2	2	no	—
98	C₂×D₇₃	#460	1	1	yes	characteristic
99	C₂×D₇₃	#476	1	1	yes	characteristic
100	#64408	—	8	2	no	—
101	#64408	—	16	2	no	—
102	C₄×C₇₃	—	2	2	no	—
103	C₄×C₇₃	#389	1	1	yes	characteristic
104	C₄×C₇₃	C₂×#106	1	1	yes	characteristic
105	Dic₇₃	—	2	2	no	—
106	Dic₇₃	#461	1	1	yes	characteristic
107	Dic₇₃	#475	1	1	yes	characteristic
108	#13186	—	73	1	no	maximal, Sylow
109	C₂×C₇₃	—	2	1	no	—
110	C₂×C₇₃	#1948	1	1	yes	characteristic
111	D₇₃	—	2	1	no	—
112	D₇₃	#2230	1	2	yes	—
113	C₂×#365	—	73	2	no	—
114	C₂×#364	—	73	1	no	—
115	#1844	—	73	1	no	—
116	#1772	—	73	1	no	—
117	#1386	—	73	2	no	—
118	C₇₃	#13186	1	1	yes	characteristic, complemented, Sylow
119	C₂×M₃₂	—	73	2	no	—
120	C₂×#108	—	73	1	no	—
121	#427	—	73	1	no	—
122	C₂×#106	—	73	2	no	—
123	C₂×#106	—	146	1	no	—
124	#405	—	146	1	no	—
125	C₂×C₄×C₈	—	73	1	no	—
126	#384	—	146	1	no	—
127	#365	—	146	2	no	—
128	#365	—	146	2	no	—
129	#364	—	73	2	no	—
130	#364	—	73	1	no	—
131	#364	#64408	1	1	yes	characteristic, complemented
132	#339	—	146	1	no	—
133	C₂×M₁₆	—	146	1	no	—
134	C₂²×C₈	—	73	1	no	—
135	C₂²×C₈	—	73	1	no	—
136	#117	—	146	1	no	—
137	C₂×C₄⋊C₄	—	292	2	no	—
138	C₂×C₄⋊C₄	—	146	2	no	—
139	C₂×C₄²	—	73	1	no	—
140	M₃₂	—	73	2	no	—
141	M₃₂	—	146	2	no	—
142	M₃₂	C₂×#64408	1	2	yes	—
143	C₂×C₁₆	—	146	2	no	—
144	#108	—	73	2	no	—
145	#108	—	146	1	no	—
146	#107	—	73	2	no	—
147	#106	—	292	1	no	—
148	#106	—	292	1	no	—
149	#106	—	146	2	no	—
150	#106	—	146	2	no	—
151	#106	—	146	1	no	—
152	#97	—	292	1	no	—
153	C₄×C₈	—	292	1	no	—
154	C₄×C₈	—	73	1	no	—
155	C₄×C₈	—	73	1	no	—
156	C₄×C₈	—	73	1	no	—
157	C₄×C₈	C₂×#64408	1	1	yes	characteristic
158	C₂²×C₄	—	292	2	no	—
159	C₂²×C₄	—	73	1	no	—
160	C₂²×C₄	—	146	1	no	—
161	M₁₆	—	292	1	no	—
162	M₁₆	—	146	2	no	—
163	C₂×C₈	—	292	1	no	—
164	C₂×C₈	—	73	1	no	—
165	C₂×C₈	—	73	1	no	—
166	C₂×C₈	—	73	1	no	—
167	C₂×C₈	—	73	1	no	—
168	C₂×C₈	—	73	1	no	—
169	C₂×C₈	—	73	1	no	—
170	C₂×C₈	—	73	1	no	—
171	C₂×C₈	—	73	1	no	—
172	C₂×C₈	—	73	1	no	—
173	C₂×C₈	—	73	1	no	—
174	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
175	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
176	C₄⋊C₄	—	292	2	no	—
177	C₄⋊C₄	—	292	2	no	—
178	C₄⋊C₄	—	584	4	no	—
179	C₄⋊C₄	—	146	2	no	—
180	C₂²⋊C₄	—	292	1	no	—
181	C₄²	—	292	1	no	—
182	C₄²	—	73	1	no	—
183	C₄²	—	146	1	no	—
184	C₄²	C₄⋊#64408	1	1	yes	characteristic
185	C₁₆	—	2	2	no	—
186	C₁₆	—	146	2	no	—
187	C₂³	—	73	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	292	2	no	—
190	C₂×C₄	—	584	2	no	—
191	C₂×C₄	—	584	4	no	—
192	C₂×C₄	—	73	1	no	—
193	C₂×C₄	—	73	1	no	—
194	C₂×C₄	—	73	1	no	—
195	C₂×C₄	—	73	1	no	—
196	C₂×C₄	—	73	1	no	—
197	C₂×C₄	—	146	4	no	—
198	C₂×C₄	—	146	1	no	—
199	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
200	C₈	—	292	2	no	—
201	C₈	—	73	1	no	—
202	C₈	—	73	1	no	—
203	C₈	—	73	1	no	—
204	C₈	—	73	1	no	—
205	C₈	#14556659	1	1	yes	characteristic
206	C₈	#14556659	1	1	yes	characteristic
207	C₈	D₄×#64408	1	1	yes	characteristic
208	C₈	Q₈×#64408	1	1	yes	characteristic
209	C₂²	—	73	1	no	—
210	C₂²	—	73	1	no	—
211	C₂²	—	146	2	no	—
212	C₂²	#17989598	1	1	yes	characteristic
213	C₄	—	2	2	no	—
214	C₄	—	584	2	no	—
215	C₄	—	73	1	no	—
216	C₄	—	73	1	no	—
217	C₄	—	1168	2	no	—
218	C₄	—	146	2	no	—
219	C₄	#17989538	1	1	yes	characteristic
220	C₄	C₂×#14556659	1	1	yes	characteristic
221	C₂	—	2	1	no	—
222	C₂	—	73	2	no	—
223	C₂	—	146	1	no	—
224	C₂	#22626043	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₇₃⋊#365

C₂

yes

maximal

#22625866

C₂

yes

maximal, characteristic

#22625733

C₂

yes

maximal, characteristic

#22625003

C₂

yes

maximal

#364×D₇₃

C₂

yes

maximal, characteristic

D₇₃⋊#108

C₂²

yes

characteristic

#17989597

C₂²

yes

characteristic

C₂×#14556659

—

C₂×#14556659

C₂²

yes

—

—

—

—

—

—

M₃₂×D₇₃

C₂²

yes

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

#17988408

C₄

yes

characteristic, complemented

#17988407

C₄

yes

complemented

#364×C₇₃

C₄

yes

characteristic, complemented

D₇₃⋊M₁₆

—

#14556670

—

C₂×C₄⋊#64408

—

C₂×C₄⋊#64408

—

#14556662

—

#14556661

D₄

yes

—

#14556660

Q₈

yes

—

—

—

—

—

—

—

—

C₂×C₈×D₇₃

D₄

yes

characteristic

C₂×C₈×D₇₃

C₂³

yes

characteristic

C₄²×D₇₃

D₄

yes

characteristic

#14556531

C₂×C₄

yes

—

#14556530

—

C₁₆×D₇₃

—

C₄×#10596831

Q₈

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

Q₈

yes

characteristic

M₃₂×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×#64408

—

—

—

—

—

—

—

—

—

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

C₂×D₄

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₂×#10596831

C₂×Q₈

yes

characteristic

C₈×D₇₃

SD₁₆

yes

characteristic

C₈×D₇₃

SD₁₆

yes

characteristic

C₈×D₇₃

SD₁₆

yes

characteristic

C₈×D₇₃

SD₁₆

yes

characteristic

C₈×D₇₃

C₂×D₄

yes

characteristic

C₈×D₇₃

C₂×Q₈

yes

C₈×D₇₃

yes

C₈×D₇₃

yes

C₄×Dic₇₃

—

C₄×Dic₇₃

C₄⋊C₄

yes

characteristic

#11924003

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₄⋊C₄

yes

characteristic

C₁₆×C₇₃

—

C₂²×D₇₃

yes

—

—

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

#137

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

#121

yes

characteristic

C₄×D₇₃

C₂²⋊Q₈

yes

characteristic

C₄×D₇₃

C₂×SD₁₆

yes

characteristic

C₄×D₇₃

C₂×SD₁₆

yes

characteristic

#10596833

—

C₈×C₇₃

yes

C₈×C₇₃

yes

C₈×C₇₃

C₄×D₄

yes

characteristic

C₈×C₇₃

C₄×Q₈

yes

yes

yes

yes

C₄⋊Q₈

yes

characteristic

C₂²×C₇₃

#428

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

yes

C₂×D₇₃

yes

100

—

101

#64408

—

102

C₄×C₇₃

—

103

C₄×C₇₃

#389

yes

characteristic

104

C₄×C₇₃

C₂×#106

yes

characteristic

105

Dic₇₃

—

106

Dic₇₃

#461

yes

characteristic

107

Dic₇₃

yes

108

—

109

C₂×C₇₃

—

110

C₂×C₇₃

#1948

yes

characteristic

111

D₇₃

—

112

D₇₃

#2230

yes

—

113

C₂×#365

—

114

C₂×#364

—

115

#1844

—

116

#1772

—

117

#1386

—

118

C₇₃

#13186

yes

characteristic, complemented, Sylow

119

C₂×M₃₂

—

120

C₂×#108

—

121

#427

—

122

C₂×#106

—

123

C₂×#106

—

146

—

124

#405

—

146

—

125

C₂×C₄×C₈

—

126

#384

—

146

—

127

#365

—

146

—

128

#365

—

146

—

129

#364

—

130

#364

—

131

#364

#64408

yes

characteristic, complemented

132

#339

—

146

—

133

C₂×M₁₆

—

146

—

134

C₂²×C₈

—

135

C₂²×C₈

—

136

#117

—

146

—

137

C₂×C₄⋊C₄

—

292

—

138

C₂×C₄⋊C₄

—

146

—

139

C₂×C₄²

—

140

M₃₂

—

141

M₃₂

—

146

—

142

M₃₂

C₂×#64408

yes

—

143

C₂×C₁₆

—

146

—

144

#108

—

145

#108

—

146

—

146

#107

—

147

#106

—

292

—

148

#106

—

292

—

149

#106

—

146

—

150

#106

—

146

—

151

#106

—

146

—

152

#97

—

292

—

153

C₄×C₈

—

292

—

154

C₄×C₈

—

155

C₄×C₈

—

156

C₄×C₈

—

157

C₄×C₈

C₂×#64408

yes

characteristic

158

C₂²×C₄

—

292

—

159

C₂²×C₄

—

160

C₂²×C₄

—

146

—

161

M₁₆

—

292

—

162

M₁₆

—

146

—

163

C₂×C₈

—

292

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₂×C₈

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

C₄⋊#64408

yes

characteristic

175

C₂×C₈

C₂²×#64408

yes

characteristic

176

C₄⋊C₄

—

292

—

177

C₄⋊C₄

—

292

—

178

C₄⋊C₄

—

584

—

179

C₄⋊C₄

—

146

—

180

C₂²⋊C₄

—

292

—

181

C₄²

—

292

—

182

C₄²

—

183

C₄²

—

146

—

184

C₄²

C₄⋊#64408

yes

characteristic

185

C₁₆

—

186

C₁₆

—

146

—

187

C₂³

—

188

C₂×C₄

—

189

C₂×C₄

—

292

—

190

C₂×C₄

—

584

—

191

C₂×C₄

—

584

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

146

—

198

C₂×C₄

—

146

—

199

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

200

C₈

—

292

—

201

C₈

—

202

C₈

—

203

C₈

—

204

C₈

—

205

C₈

#14556659

yes

characteristic

206

C₈

#14556659

yes

characteristic

207

C₈

D₄×#64408

yes

characteristic

208

C₈

Q₈×#64408

yes

characteristic

209

C₂²

—

210

C₂²

—

211

C₂²

—

146

—

212

C₂²

#17989598

yes

characteristic

213

C₄

—

214

C₄

—

584

—

215

C₄

—

216

C₄

—

217

C₄

—

1168

—

218

C₄

—

146

—

219

C₄

#17989538

yes

characteristic

220

C₄

C₂×#14556659

yes

characteristic

221

C₂

—

222

C₂

—

223

C₂

—

146

—

224

C₂

#22626043

yes

characteristic

	subgroup	G/H
center	C₂	#22626043
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×#64408
Fitting	#364×C₇₃	C₄
socle	C₂×C₇₃	#1948
prime radical	C₂²×D₇₃	AGL(1,ℤ/8)
cosocle	C₂×C₈×D₇₃	C₂³
hypercenter	#364	#64408
hypocenter	C₇₃	#13186
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

subgroup

G/H

center

C₂

#22626043

commutator

C₂×C₈×C₇₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×#64408

Fitting

#364×C₇₃

C₄

C₂×C₇₃

C₂²×D₇₃

C₂×C₈×D₇₃

C₂³

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

N	H ≅ G/N	type
17. #17988408	C₄	semidirect
18. #17988407	C₄	semidirect
19. #364×C₇₃	C₄	semidirect
118. C₇₃	#13186	semidirect
131. #364	#64408	semidirect

H ≅ G/N

type

17. #17988408

C₄

semidirect

18. #17988407

C₄

semidirect

19. #364×C₇₃

C₄

semidirect

118. C₇₃

#13186

semidirect

131. #364

#64408

semidirect

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

subgroup	index
D₇₃⋊#365	2
#22625866	2
#22625733	2
#22625003	2
#364×D₇₃	2
#13186	73

subgroup

index

D₇₃⋊#365

#364×D₇₃

Subgroups — #34013916

Conjugacy classes of subgroups

Complements

Maximal subgroups