Group #34012689 — subgroups

Subgroups — #34012689

Order 18688.

326 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 128 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	#22626143	C₂	1	2	yes	maximal
2	#22626132	C₂	1	1	yes	maximal, characteristic
3	#22625733	C₂	1	1	yes	maximal, characteristic
4	#22624987	C₂	1	2	yes	maximal
5	#364×D₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556664	C₂²	1	1	yes	characteristic
7	C₂×#14556664	C₂²	1	1	yes	characteristic
8	D₇₃⋊#108	C₄	1	1	yes	characteristic
9	#17989597	C₄	1	1	yes	characteristic
10	#17989550	C₂²	1	2	yes	—
11	#17989476	C₄	1	2	yes	complemented
12	#17989475	C₄	1	2	yes	complemented
13	#17989474	—	2	2	no	—
14	C₁₆×#64408	—	2	2	no	—
15	#17989458	C₄	1	2	yes	—
16	M₃₂×D₇₃	C₂²	1	2	yes	—
17	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
18	#17988408	C₄	1	1	yes	characteristic, complemented
19	#17988407	C₄	1	2	yes	complemented
20	#364×C₇₃	C₄	1	1	yes	characteristic, complemented
21	D₇₃⋊M₁₆	C₂×C₄	1	2	yes	—
22	#14556670	C₂×C₄	1	2	yes	—
23	#14556664	—	2	1	no	—
24	#14556664	—	2	1	no	—
25	#14556664	D₄	1	2	yes	—
26	#14556664	Q₈	1	2	yes	—
27	C₂×#11924031	C₂×C₄	1	2	yes	—
28	C₂×#11924031	C₂×C₄	1	2	yes	—
29	#14556662	C₈	1	2	yes	—
30	#14556661	C₈	1	2	yes	—
31	#14556660	C₈	1	2	yes	—
32	#14556659	C₈	1	2	yes	—
33	#14556645	—	2	2	no	—
34	C₈×#64408	—	2	2	no	—
35	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
37	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
38	#14556531	C₂×C₄	1	2	yes	—
39	#14556530	—	2	2	no	—
40	C₁₆×D₇₃	—	2	2	no	—
41	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
42	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
43	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
44	M₃₂×C₇₃	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
46	#11924038	C₂×C₈	1	4	yes	—
47	#11924037	—	2	2	no	—
48	#11924036	—	2	2	no	—
49	C₄⋊#64408	C₂×C₈	1	4	yes	—
50	C₂²⋊#64408	—	2	2	no	—
51	C₄×#64408	—	2	2	no	—
52	#11924031	—	2	2	no	—
53	#11924031	C₄⋊C₄	1	4	yes	—
54	C₂×C₄×D₇₃	—	2	1	no	—
55	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×#10596831	C₄²	1	1	yes	characteristic
57	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
58	C₈×D₇₃	C₂×C₈	1	1	yes	characteristic
59	C₈×D₇₃	C₂×C₈	1	1	yes	characteristic
60	C₈×D₇₃	C₂×C₈	1	1	yes	characteristic
61	C₈×D₇₃	C₂×C₈	1	1	yes	characteristic
62	C₈×D₇₃	C₂×D₄	1	1	yes	characteristic
63	C₈×D₇₃	C₂×Q₈	1	1	yes	characteristic
64	C₈×D₇₃	#41	1	1	yes	characteristic
65	C₈×D₇₃	#41	1	1	yes	characteristic
66	C₄×Dic₇₃	—	2	1	no	—
67	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
68	#11924003	—	2	2	no	—
69	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
70	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
71	C₄²×C₇₃	C₄²	1	1	yes	characteristic
72	C₁₆×C₇₃	—	2	2	no	—
73	C₂²×D₇₃	#131	1	1	yes	characteristic
74	C₂×#64408	—	2	4	no	—
75	C₂×C₄×C₇₃	—	2	1	no	—
76	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
77	C₂×Dic₇₃	—	2	1	no	—
78	C₂×Dic₇₃	#131	1	1	yes	characteristic
79	C₄×D₇₃	—	4	2	no	—
80	C₄×D₇₃	C₄×D₄	1	1	yes	characteristic
81	C₄×D₇₃	C₄×Q₈	1	1	yes	characteristic
82	C₄×D₇₃	C₂²×C₈	1	1	yes	characteristic
83	C₄×D₇₃	C₂²×C₈	1	1	yes	characteristic
84	#10596833	—	2	4	no	—
85	C₈×C₇₃	C₄×C₈	1	1	yes	characteristic
86	C₈×C₇₃	C₄×C₈	1	1	yes	characteristic
87	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
88	C₈×C₇₃	C₄×Q₈	1	1	yes	characteristic
89	#10596831	C₄×C₈	1	1	yes	characteristic
90	#10596831	C₄×C₈	1	1	yes	characteristic
91	#10596831	C₂×C₄⋊C₄	1	1	yes	characteristic
92	#10596831	#117	1	1	yes	characteristic
93	C₂²×C₇₃	#405	1	1	yes	characteristic
94	C₂×D₇₃	—	2	2	no	—
95	C₂×D₇₃	C₈×D₄	1	1	yes	characteristic
96	C₂×D₇₃	C₈×Q₈	1	1	yes	characteristic
97	#64408	—	4	4	no	—
98	C₄×C₇₃	—	4	1	no	—
99	C₄×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
100	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
101	Dic₇₃	—	4	1	no	—
102	Dic₇₃	C₈×D₄	1	1	yes	characteristic
103	Dic₇₃	C₈×Q₈	1	1	yes	characteristic
104	#12665	—	73	1	no	maximal, Sylow
105	C₂×C₇₃	—	2	1	no	—
106	C₂×C₇₃	C₈×C₄⋊C₄	1	1	yes	characteristic
107	D₇₃	—	2	1	no	—
108	D₇₃	#2171	1	2	yes	—
109	C₂×#364	—	73	1	no	—
110	#2108	—	73	2	no	—
111	C₂×#346	—	73	1	no	—
112	#1772	—	73	1	no	—
113	#1379	—	73	2	no	—
114	C₇₃	#12665	1	1	yes	characteristic, complemented, Sylow
115	C₂×M₃₂	—	73	2	no	—
116	C₂²×C₁₆	—	73	1	no	—
117	C₂²×C₁₆	—	73	1	no	—
118	C₂×#108	—	73	1	no	—
119	#427	—	73	1	no	—
120	#405	—	73	2	no	—
121	C₂×C₄×C₈	—	73	1	no	—
122	#364	—	73	2	no	—
123	#364	—	73	1	no	—
124	#364	#64408	1	1	yes	characteristic, complemented
125	#346	—	73	2	no	—
126	#346	—	73	2	no	—
127	#346	—	146	2	no	—
128	C₄×C₁₆	—	146	2	no	—
129	#339	—	73	2	no	—
130	C₂×M₁₆	—	73	2	no	—
131	C₂²×C₈	—	73	1	no	—
132	C₂²×C₈	—	73	1	no	—
133	#117	—	73	2	no	—
134	C₂×C₄²	—	73	1	no	—
135	M₃₂	—	73	2	no	—
136	M₃₂	—	146	2	no	—
137	M₃₂	C₂×#64408	1	2	yes	—
138	C₂×C₁₆	—	73	2	no	—
139	C₂×C₁₆	—	73	2	no	—
140	C₂×C₁₆	—	73	2	no	—
141	C₂×C₁₆	—	73	2	no	—
142	C₂×C₁₆	—	146	2	no	—
143	C₂×C₁₆	—	146	1	no	—
144	C₂×C₁₆	—	146	1	no	—
145	#108	—	73	2	no	—
146	#108	—	73	2	no	—
147	#107	—	73	2	no	—
148	#106	—	73	2	no	—
149	#97	—	146	2	no	—
150	C₄×C₈	—	73	1	no	—
151	C₄×C₈	—	73	1	no	—
152	C₄×C₈	—	73	1	no	—
153	C₄×C₈	—	146	2	no	—
154	C₄×C₈	C₂×#64408	1	1	yes	characteristic
155	C₂²×C₄	—	73	1	no	—
156	C₂²×C₄	—	146	1	no	—
157	M₁₆	—	73	4	no	—
158	M₁₆	—	146	2	no	—
159	C₂×C₈	—	73	1	no	—
160	C₂×C₈	—	73	1	no	—
161	C₂×C₈	—	73	1	no	—
162	C₂×C₈	—	73	1	no	—
163	C₂×C₈	—	73	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	—	146	2	no	—
170	C₂×C₈	C₄×#64408	1	1	yes	characteristic
171	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
172	C₄⋊C₄	—	73	4	no	—
173	C₂²⋊C₄	—	146	2	no	—
174	C₄²	—	73	1	no	—
175	C₄²	—	146	1	no	—
176	C₄²	—	146	2	no	—
177	C₄²	C₄×#64408	1	1	yes	characteristic
178	C₁₆	—	2	2	no	—
179	C₁₆	—	73	4	no	—
180	C₁₆	—	146	2	no	—
181	C₁₆	—	146	2	no	—
182	C₂³	—	73	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	292	2	no	—
185	C₂×C₄	—	73	1	no	—
186	C₂×C₄	—	73	1	no	—
187	C₂×C₄	—	73	1	no	—
188	C₂×C₄	—	73	1	no	—
189	C₂×C₄	—	73	1	no	—
190	C₂×C₄	—	146	4	no	—
191	C₂×C₄	—	146	1	no	—
192	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
193	C₈	—	73	1	no	—
194	C₈	—	73	1	no	—
195	C₈	—	73	1	no	—
196	C₈	—	73	1	no	—
197	C₈	—	146	4	no	—
198	C₈	C₈×#64408	1	1	yes	characteristic
199	C₈	C₈×#64408	1	1	yes	characteristic
200	C₈	D₄×#64408	1	1	yes	characteristic
201	C₈	Q₈×#64408	1	1	yes	characteristic
202	C₂²	—	73	1	no	—
203	C₂²	—	73	1	no	—
204	C₂²	—	146	2	no	—
205	C₂²	#17989553	1	1	yes	characteristic
206	C₄	—	4	1	no	—
207	C₄	—	292	1	no	—
208	C₄	—	292	4	no	—
209	C₄	—	73	1	no	—
210	C₄	—	73	1	no	—
211	C₄	C₄⋊C₄×#64408	1	1	yes	characteristic
212	C₄	C₂×C₈×#64408	1	1	yes	characteristic
213	C₂	—	2	1	no	—
214	C₂	—	73	2	no	—
215	C₂	—	146	1	no	—
216	C₂	#22625724	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626143

C₂

yes

maximal

#22626132

C₂

yes

maximal, characteristic

#22625733

C₂

yes

maximal, characteristic

#22624987

C₂

yes

maximal

#364×D₇₃

C₂

yes

maximal, characteristic

C₂×#14556664

C₂²

yes

characteristic

C₂×#14556664

C₂²

yes

characteristic

D₇₃⋊#108

C₄

yes

characteristic

#17989597

C₄

yes

characteristic

#17989550

C₂²

yes

—

#17989476

C₄

yes

complemented

#17989475

C₄

yes

complemented

#17989474

—

C₁₆×#64408

—

#17989458

C₄

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

C₂×C₄

yes

—

#14556670

C₂×C₄

yes

—

#14556664

—

#14556664

—

#14556664

D₄

yes

—

#14556664

Q₈

yes

—

C₂×#11924031

C₂×C₄

yes

—

C₂×#11924031

C₂×C₄

yes

—

#14556662

C₈

yes

—

#14556661

C₈

yes

—

#14556660

C₈

yes

—

#14556659

C₈

yes

—

#14556645

—

C₈×#64408

—

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

C₂×C₈×D₇₃

C₂³

yes

characteristic

C₄²×D₇₃

C₂×C₄

yes

characteristic

#14556531

C₂×C₄

yes

—

#14556530

—

C₁₆×D₇₃

—

C₄×#10596831

C₂×C₄

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

M₃₂×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

#11924038

C₂×C₈

yes

—

#11924037

—

#11924036

—

C₄⋊#64408

C₂×C₈

yes

—

—

—

—

C₄⋊C₄

yes

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

C₂²×C₄

yes

characteristic

C₂×#10596831

C₄²

yes

characteristic

C₂×#10596831

C₂²×C₄

yes

characteristic

C₈×D₇₃

C₂×C₈

yes

characteristic

C₈×D₇₃

C₂×C₈

yes

characteristic

C₈×D₇₃

C₂×C₈

yes

characteristic

C₈×D₇₃

C₂×C₈

yes

characteristic

C₈×D₇₃

C₂×D₄

yes

characteristic

C₈×D₇₃

C₂×Q₈

yes

C₈×D₇₃

yes

C₈×D₇₃

yes

C₄×Dic₇₃

—

C₄×Dic₇₃

C₄²

yes

characteristic

#11924003

—

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₄²

yes

characteristic

C₁₆×C₇₃

—

C₂²×D₇₃

#131

yes

characteristic

C₂×#64408

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₄²

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

#131

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

C₄×D₄

yes

characteristic

C₄×D₇₃

C₄×Q₈

yes

characteristic

C₄×D₇₃

C₂²×C₈

yes

characteristic

C₄×D₇₃

C₂²×C₈

yes

characteristic

#10596833

—

C₈×C₇₃

C₄×C₈

yes

characteristic

C₈×C₇₃

C₄×C₈

yes

characteristic

C₈×C₇₃

C₄×D₄

yes

characteristic

C₈×C₇₃

C₄×Q₈

yes

characteristic

#10596831

C₄×C₈

yes

characteristic

#10596831

C₄×C₈

yes

characteristic

#10596831

C₂×C₄⋊C₄

yes

yes

C₂²×C₇₃

yes

C₂×D₇₃

—

C₂×D₇₃

C₈×D₄

yes

characteristic

C₂×D₇₃

C₈×Q₈

yes

characteristic

#64408

—

C₄×C₇₃

—

C₄×C₇₃

C₄×C₄⋊C₄

yes

characteristic

100

C₄×C₇₃

C₂×C₄×C₈

yes

characteristic

101

Dic₇₃

—

102

Dic₇₃

C₈×D₄

yes

characteristic

103

Dic₇₃

C₈×Q₈

yes

characteristic

104

#12665

—

maximal, Sylow

105

C₂×C₇₃

—

106

C₂×C₇₃

C₈×C₄⋊C₄

yes

characteristic

107

D₇₃

—

108

D₇₃

#2171

yes

—

109

C₂×#364

—

110

#2108

—

111

C₂×#346

—

112

#1772

—

113

#1379

—

114

C₇₃

#12665

yes

characteristic, complemented, Sylow

115

C₂×M₃₂

—

116

C₂²×C₁₆

—

117

C₂²×C₁₆

—

118

C₂×#108

—

119

#427

—

120

#405

—

121

C₂×C₄×C₈

—

122

#364

—

123

#364

—

124

#364

#64408

yes

characteristic, complemented

125

#346

—

126

#346

—

127

#346

—

146

—

128

C₄×C₁₆

—

146

—

129

#339

—

130

C₂×M₁₆

—

131

C₂²×C₈

—

132

C₂²×C₈

—

133

#117

—

134

C₂×C₄²

—

135

M₃₂

—

136

M₃₂

—

146

—

137

M₃₂

C₂×#64408

yes

—

138

C₂×C₁₆

—

139

C₂×C₁₆

—

140

C₂×C₁₆

—

141

C₂×C₁₆

—

142

C₂×C₁₆

—

146

—

143

C₂×C₁₆

—

146

—

144

C₂×C₁₆

—

146

—

145

#108

—

146

#108

—

147

#107

—

148

#106

—

149

#97

—

146

—

150

C₄×C₈

—

151

C₄×C₈

—

152

C₄×C₈

—

153

C₄×C₈

—

146

—

154

C₄×C₈

C₂×#64408

yes

characteristic

155

C₂²×C₄

—

156

C₂²×C₄

—

146

—

157

M₁₆

—

158

M₁₆

—

146

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

146

—

170

C₂×C₈

C₄×#64408

yes

characteristic

171

C₂×C₈

C₂²×#64408

yes

characteristic

172

C₄⋊C₄

—

173

C₂²⋊C₄

—

146

—

174

C₄²

—

175

C₄²

—

146

—

176

C₄²

—

146

—

177

C₄²

C₄×#64408

yes

characteristic

178

C₁₆

—

179

C₁₆

—

180

C₁₆

—

146

—

181

C₁₆

—

146

—

182

C₂³

—

183

C₂×C₄

—

184

C₂×C₄

—

292

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

146

—

191

C₂×C₄

—

146

—

192

C₂×C₄

C₂×C₄×#64408

yes

characteristic

193

C₈

—

194

C₈

—

195

C₈

—

196

C₈

—

197

C₈

—

146

—

198

C₈

C₈×#64408

yes

characteristic

199

C₈

C₈×#64408

yes

characteristic

200

C₈

D₄×#64408

yes

characteristic

201

C₈

Q₈×#64408

yes

characteristic

202

C₂²

—

203

C₂²

—

204

C₂²

—

146

—

205

C₂²

#17989553

yes

characteristic

206

C₄

—

207

C₄

—

292

—

208

C₄

—

292

—

209

C₄

—

210

C₄

—

211

C₄

C₄⋊C₄×#64408

yes

characteristic

212

C₄

C₂×C₈×#64408

yes

characteristic

213

C₂

—

214

C₂

—

215

C₂

—

146

—

216

C₂

#22625724

yes

characteristic

	subgroup	G/H
center	C₈	D₄×#64408
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₂×C₈	C₂²×#64408
Fitting	#364×C₇₃	C₄
socle	C₂×C₇₃	C₈×C₄⋊C₄
prime radical	C₂²×D₇₃	#131
cosocle	C₂×C₈×D₇₃	C₂³
hypercenter	#364	#64408
hypocenter	C₇₃	#12665
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #364

subgroup

G/H

center

C₈

D₄×#64408

commutator

C₄×C₇₃

C₂×C₄×C₈

Frattini

C₂×C₈

C₂²×#64408

Fitting

#364×C₇₃

C₄

socle

C₂×C₇₃

C₈×C₄⋊C₄

prime radical

C₂²×D₇₃

#131

cosocle

C₂×C₈×D₇₃

C₂³

C₇₃

G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ #364

N	H ≅ G/N	type
11. #17989476	C₄	semidirect
12. #17989475	C₄	semidirect
18. #17988408	C₄	semidirect
19. #17988407	C₄	semidirect
20. #364×C₇₃	C₄	semidirect
114. C₇₃	#12665	semidirect
124. #364	#64408	semidirect

H ≅ G/N

type

11. #17989476

C₄

semidirect

12. #17989475

C₄

semidirect

18. #17988408

C₄

semidirect

19. #17988407

C₄

semidirect

20. #364×C₇₃

C₄

semidirect

114. C₇₃

#12665

semidirect

124. #364

#64408

semidirect

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

subgroup	index
#22626143	2
#22626132	2
#22625733	2
#22624987	2
#364×D₇₃	2
#12665	73

subgroup

index

#364×D₇₃

Subgroups — #34012689

Conjugacy classes of subgroups

Complements

Maximal subgroups