Group #34012684 — subgroups

Subgroups — #34012684

Order 18688.

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

normal subgroups only

This group has 334 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	C₄×#14556664	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	#17989474	—	2	2	no	—
12	#17989473	C₄	1	2	yes	complemented
13	C₄×#11924031	C₄	1	2	yes	complemented
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	1	yes	characteristic
26	#14556664	D₄	1	1	yes	characteristic
27	#14556664	Q₈	1	2	yes	—
28	C₂×#11924031	C₂×C₄	1	2	yes	—
29	C₂×#11924031	C₂×C₄	1	2	yes	—
30	#14556662	C₈	1	2	yes	—
31	#14556661	C₈	1	2	yes	—
32	#14556660	C₈	1	2	yes	—
33	#14556659	C₈	1	2	yes	—
34	#14556645	—	2	2	no	—
35	C₈×#64408	—	2	2	no	—
36	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
38	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
39	#14556531	C₂×C₄	1	2	yes	—
40	#14556530	—	2	2	no	—
41	C₁₆×D₇₃	—	2	2	no	—
42	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
43	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
44	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
45	M₃₂×C₇₃	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
47	#11924038	C₂×C₈	1	4	yes	—
48	#11924037	—	2	2	no	—
49	#11924036	—	2	2	no	—
50	C₄⋊#64408	C₂×C₈	1	4	yes	—
51	C₂²⋊#64408	—	2	2	no	—
52	C₄×#64408	—	2	2	no	—
53	#11924031	—	2	2	no	—
54	#11924031	C₄⋊C₄	1	2	yes	—
55	#11924031	C₄⋊C₄	1	2	yes	—
56	C₂×C₄×D₇₃	—	2	1	no	—
57	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×#10596831	C₄²	1	1	yes	characteristic
59	C₂×#10596831	C₂²×C₄	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×C₈	1	1	yes	characteristic
63	C₈×D₇₃	C₂×C₈	1	1	yes	characteristic
64	C₈×D₇₃	C₂×D₄	1	1	yes	characteristic
65	C₈×D₇₃	C₂×Q₈	1	1	yes	characteristic
66	C₈×D₇₃	#41	1	1	yes	characteristic
67	C₈×D₇₃	#41	1	1	yes	characteristic
68	C₄×Dic₇₃	—	2	1	no	—
69	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
70	#11924003	—	2	2	no	—
71	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
72	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₇₃	C₄²	1	1	yes	characteristic
74	C₁₆×C₇₃	—	2	2	no	—
75	C₂²×D₇₃	#131	1	1	yes	characteristic
76	C₂×#64408	—	2	4	no	—
77	C₂×C₄×C₇₃	—	2	1	no	—
78	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
79	C₂×Dic₇₃	—	2	1	no	—
80	C₂×Dic₇₃	#131	1	1	yes	characteristic
81	C₄×D₇₃	—	2	4	no	—
82	C₄×D₇₃	C₄×D₄	1	1	yes	characteristic
83	C₄×D₇₃	C₄×Q₈	1	1	yes	characteristic
84	C₄×D₇₃	C₂²×C₈	1	1	yes	characteristic
85	C₄×D₇₃	C₂²×C₈	1	1	yes	characteristic
86	#10596833	—	2	4	no	—
87	C₈×C₇₃	C₄×C₈	1	1	yes	characteristic
88	C₈×C₇₃	C₄×C₈	1	1	yes	characteristic
89	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
90	C₈×C₇₃	C₄×Q₈	1	1	yes	characteristic
91	#10596831	C₄×C₈	1	1	yes	characteristic
92	#10596831	C₄×C₈	1	1	yes	characteristic
93	#10596831	C₂×C₄⋊C₄	1	1	yes	characteristic
94	#10596831	#117	1	1	yes	characteristic
95	C₂²×C₇₃	#405	1	1	yes	characteristic
96	C₂×D₇₃	—	2	2	no	—
97	C₂×D₇₃	C₈×D₄	1	1	yes	characteristic
98	C₂×D₇₃	C₈×Q₈	1	1	yes	characteristic
99	#64408	—	4	4	no	—
100	C₄×C₇₃	—	2	2	no	—
101	C₄×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
102	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
103	Dic₇₃	—	2	2	no	—
104	Dic₇₃	C₈×D₄	1	1	yes	characteristic
105	Dic₇₃	C₈×Q₈	1	1	yes	characteristic
106	#12664	—	73	1	no	maximal, Sylow
107	C₂×C₇₃	—	2	1	no	—
108	C₂×C₇₃	C₈×C₄⋊C₄	1	1	yes	characteristic
109	D₇₃	—	2	1	no	—
110	D₇₃	#2170	1	2	yes	—
111	C₂×#364	—	73	1	no	—
112	#2108	—	73	2	no	—
113	C₂×C₄×C₁₆	—	73	1	no	—
114	#1772	—	73	1	no	—
115	#1379	—	73	2	no	—
116	C₇₃	#12664	1	1	yes	characteristic, complemented, Sylow
117	C₂×M₃₂	—	73	2	no	—
118	C₂²×C₁₆	—	73	1	no	—
119	C₂²×C₁₆	—	73	1	no	—
120	C₂×#108	—	73	1	no	—
121	#427	—	73	1	no	—
122	#405	—	73	2	no	—
123	C₂×C₄×C₈	—	73	1	no	—
124	#364	—	73	2	no	—
125	#364	—	73	1	no	—
126	#364	#64408	1	1	yes	characteristic, complemented
127	#346	—	146	2	no	—
128	C₄×C₁₆	—	73	2	no	—
129	C₄×C₁₆	—	73	2	no	—
130	C₄×C₁₆	—	146	2	no	—
131	#339	—	73	2	no	—
132	C₂×M₁₆	—	73	2	no	—
133	C₂²×C₈	—	73	1	no	—
134	C₂²×C₈	—	73	1	no	—
135	#117	—	73	2	no	—
136	C₂×C₄²	—	73	1	no	—
137	M₃₂	—	73	2	no	—
138	M₃₂	—	146	2	no	—
139	M₃₂	C₂×#64408	1	2	yes	—
140	C₂×C₁₆	—	73	2	no	—
141	C₂×C₁₆	—	73	1	no	—
142	C₂×C₁₆	—	73	1	no	—
143	C₂×C₁₆	—	73	2	no	—
144	C₂×C₁₆	—	73	2	no	—
145	C₂×C₁₆	—	146	2	no	—
146	C₂×C₁₆	—	146	1	no	—
147	C₂×C₁₆	—	146	1	no	—
148	#108	—	73	2	no	—
149	#108	—	73	2	no	—
150	#107	—	73	2	no	—
151	#106	—	73	2	no	—
152	#97	—	146	2	no	—
153	C₄×C₈	—	73	1	no	—
154	C₄×C₈	—	73	1	no	—
155	C₄×C₈	—	73	1	no	—
156	C₄×C₈	—	146	2	no	—
157	C₄×C₈	C₂×#64408	1	1	yes	characteristic
158	C₂²×C₄	—	73	1	no	—
159	C₂²×C₄	—	146	1	no	—
160	M₁₆	—	73	4	no	—
161	M₁₆	—	146	2	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	—	73	1	no	—
170	C₂×C₈	—	73	1	no	—
171	C₂×C₈	—	73	1	no	—
172	C₂×C₈	—	146	2	no	—
173	C₂×C₈	C₄×#64408	1	1	yes	characteristic
174	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
175	C₄⋊C₄	—	73	4	no	—
176	C₂²⋊C₄	—	146	2	no	—
177	C₄²	—	73	1	no	—
178	C₄²	—	146	1	no	—
179	C₄²	—	146	2	no	—
180	C₄²	C₄×#64408	1	1	yes	characteristic
181	C₁₆	—	2	2	no	—
182	C₁₆	—	73	2	no	—
183	C₁₆	—	73	2	no	—
184	C₁₆	—	146	2	no	—
185	C₁₆	—	146	2	no	—
186	C₂³	—	73	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	73	1	no	—
189	C₂×C₄	—	73	1	no	—
190	C₂×C₄	—	73	1	no	—
191	C₂×C₄	—	73	1	no	—
192	C₂×C₄	—	73	1	no	—
193	C₂×C₄	—	146	4	no	—
194	C₂×C₄	—	146	4	no	—
195	C₂×C₄	—	146	1	no	—
196	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
197	C₈	—	73	1	no	—
198	C₈	—	73	1	no	—
199	C₈	—	73	1	no	—
200	C₈	—	73	1	no	—
201	C₈	—	146	4	no	—
202	C₈	C₈×#64408	1	1	yes	characteristic
203	C₈	C₈×#64408	1	1	yes	characteristic
204	C₈	D₄×#64408	1	1	yes	characteristic
205	C₈	Q₈×#64408	1	1	yes	characteristic
206	C₂²	—	73	1	no	—
207	C₂²	—	73	1	no	—
208	C₂²	—	146	2	no	—
209	C₂²	#17989553	1	1	yes	characteristic
210	C₄	—	2	2	no	—
211	C₄	—	292	4	no	—
212	C₄	—	73	1	no	—
213	C₄	—	73	1	no	—
214	C₄	—	146	2	no	—
215	C₄	C₄⋊C₄×#64408	1	1	yes	characteristic
216	C₄	C₂×C₈×#64408	1	1	yes	characteristic
217	C₂	—	2	1	no	—
218	C₂	—	73	2	no	—
219	C₂	—	146	1	no	—
220	C₂	#22625724	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626143

C₂

yes

maximal

C₄×#14556664

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

—

#17989474

—

#17989473

C₄

yes

complemented

C₄×#11924031

C₄

yes

complemented

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

characteristic

#14556664

D₄

yes

characteristic

#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

—

#11924031

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

—

100

C₄×C₇₃

—

101

C₄×C₇₃

C₄×C₄⋊C₄

yes

characteristic

102

C₄×C₇₃

C₂×C₄×C₈

yes

characteristic

103

Dic₇₃

—

104

Dic₇₃

C₈×D₄

yes

characteristic

105

Dic₇₃

C₈×Q₈

yes

characteristic

106

#12664

—

maximal, Sylow

107

C₂×C₇₃

—

108

C₂×C₇₃

C₈×C₄⋊C₄

yes

characteristic

109

D₇₃

—

110

D₇₃

#2170

yes

—

111

C₂×#364

—

112

#2108

—

113

C₂×C₄×C₁₆

—

114

#1772

—

115

#1379

—

116

C₇₃

#12664

yes

characteristic, complemented, Sylow

117

C₂×M₃₂

—

118

C₂²×C₁₆

—

119

C₂²×C₁₆

—

120

C₂×#108

—

121

#427

—

122

#405

—

123

C₂×C₄×C₈

—

124

#364

—

125

#364

—

126

#364

#64408

yes

characteristic, complemented

127

#346

—

146

—

128

C₄×C₁₆

—

129

C₄×C₁₆

—

130

C₄×C₁₆

—

146

—

131

#339

—

132

C₂×M₁₆

—

133

C₂²×C₈

—

134

C₂²×C₈

—

135

#117

—

136

C₂×C₄²

—

137

M₃₂

—

138

M₃₂

—

146

—

139

M₃₂

C₂×#64408

yes

—

140

C₂×C₁₆

—

141

C₂×C₁₆

—

142

C₂×C₁₆

—

143

C₂×C₁₆

—

144

C₂×C₁₆

—

145

C₂×C₁₆

—

146

—

146

C₂×C₁₆

—

146

—

147

C₂×C₁₆

—

146

—

148

#108

—

149

#108

—

150

#107

—

151

#106

—

152

#97

—

146

—

153

C₄×C₈

—

154

C₄×C₈

—

155

C₄×C₈

—

156

C₄×C₈

—

146

—

157

C₄×C₈

C₂×#64408

yes

characteristic

158

C₂²×C₄

—

159

C₂²×C₄

—

146

—

160

M₁₆

—

161

M₁₆

—

146

—

162

C₂×C₈

—

163

C₂×C₈

—

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

—

146

—

173

C₂×C₈

C₄×#64408

yes

characteristic

174

C₂×C₈

C₂²×#64408

yes

characteristic

175

C₄⋊C₄

—

176

C₂²⋊C₄

—

146

—

177

C₄²

—

178

C₄²

—

146

—

179

C₄²

—

146

—

180

C₄²

C₄×#64408

yes

characteristic

181

C₁₆

—

182

C₁₆

—

183

C₁₆

—

184

C₁₆

—

146

—

185

C₁₆

—

146

—

186

C₂³

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

146

—

194

C₂×C₄

—

146

—

195

C₂×C₄

—

146

—

196

C₂×C₄

C₂×C₄×#64408

yes

characteristic

197

C₈

—

198

C₈

—

199

C₈

—

200

C₈

—

201

C₈

—

146

—

202

C₈

C₈×#64408

yes

characteristic

203

C₈

C₈×#64408

yes

characteristic

204

C₈

D₄×#64408

yes

characteristic

205

C₈

Q₈×#64408

yes

characteristic

206

C₂²

—

207

C₂²

—

208

C₂²

—

146

—

209

C₂²

#17989553

yes

characteristic

210

C₄

—

211

C₄

—

292

—

212

C₄

—

213

C₄

—

214

C₄

—

146

—

215

C₄

C₄⋊C₄×#64408

yes

characteristic

216

C₄

C₂×C₈×#64408

yes

characteristic

217

C₂

—

218

C₂

—

219

C₂

—

146

—

220

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	#12664
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
12. #17989473	C₄	semidirect
13. C₄×#11924031	C₄	semidirect
18. #17988408	C₄	semidirect
19. #17988407	C₄	semidirect
20. #364×C₇₃	C₄	semidirect
116. C₇₃	#12664	semidirect
126. #364	#64408	semidirect

H ≅ G/N

type

12. #17989473

C₄

semidirect

13. C₄×#11924031

C₄

semidirect

18. #17988408

C₄

semidirect

19. #17988407

C₄

semidirect

20. #364×C₇₃

C₄

semidirect

116. C₇₃

#12664

semidirect

126. #364

#64408

semidirect

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

subgroup	index
#22626143	2
C₄×#14556664	2
#22625733	2
#22624987	2
#364×D₇₃	2
#12664	73

subgroup

index

#364×D₇₃

Subgroups — #34012684

Conjugacy classes of subgroups

Complements

Maximal subgroups