Group #34018161 — subgroups

Subgroups — #34018161

Order 18688.

702 conjugacy classes of subgroups; only the 270 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 270 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	#22626577	C₂	1	2	yes	maximal
2	#22625501	C₂	1	6	yes	maximal, complemented
3	#22625490	C₂	1	3	yes	maximal, complemented
4	#22625476	C₂	1	3	yes	maximal, complemented
5	C₈×#11924029	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556679	—	2	2	no	—
7	#17989594	—	2	6	no	—
8	#17989591	C₂²	1	6	yes	—
9	#17989588	C₂²	1	6	yes	—
10	#17989451	C₂²	1	1	yes	characteristic
11	#17989451	C₂²	1	3	yes	—
12	#17989451	C₂²	1	6	yes	—
13	#17989450	C₂²	1	6	yes	—
14	C₄×#11924029	C₂²	1	1	yes	characteristic
15	Q₈×#10596831	C₄	1	1	yes	characteristic
16	C₈×D₂₉₂	C₄	1	3	yes	—
17	#17988652	C₄	1	3	yes	—
18	#17988639	C₂²	1	3	yes	—
19	C₄×C₈×D₇₃	C₂²	1	3	yes	—
20	C₈×Q₈×C₇₃	C₄	1	1	yes	characteristic
21	#14556679	—	4	8	no	—
22	C₂×#11924038	—	2	6	no	—
23	C₂×#11924036	—	2	6	no	—
24	#14556658	—	2	6	no	—
25	C₄⋊#10596833	—	2	6	no	—
26	C₄⋊#10596833	C₂³	1	2	yes	—
27	C₄⋊#10596833	C₂³	1	6	yes	—
28	C₂×#11924029	D₄	1	1	yes	characteristic
29	C₂×C₈×D₇₃	C₂×C₄	1	3	yes	—
30	C₂×C₈×D₇₃	C₂×C₄	1	3	yes	—
31	Q₈×Dic₇₃	Q₈	1	1	yes	characteristic
32	C₄×D₂₉₂	C₂×C₄	1	3	yes	—
33	#14556553	Q₈	1	3	yes	—
34	#14556549	D₄	1	3	yes	—
35	C₄²×D₇₃	C₂³	1	3	yes	—
36	C₄⋊#10596831	C₂×C₄	1	3	yes	—
37	#14556503	C₂×C₄	1	6	yes	—
38	C₄×#10596831	C₂×C₄	1	3	yes	—
39	C₈×Dic₇₃	C₂³	1	3	yes	—
40	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
41	C₄×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
42	C₄⋊C₈×C₇₃	C₂×C₄	1	3	yes	—
43	C₄×C₈×C₇₃	C₂×C₄	1	3	yes	—
44	#11924038	—	4	12	no	—
45	#11924036	—	4	12	no	—
46	C₂×#10596833	—	2	2	no	—
47	C₂×#10596833	—	2	6	no	—
48	#11924029	D₈	1	4	yes	—
49	#11924029	Q₁₆	1	4	yes	—
50	C₂×D₂₉₂	C₄⋊C₄	1	3	yes	—
51	C₂×C₄×D₇₃	C₂²×C₄	1	3	yes	—
52	C₂×C₄×D₇₃	C₂×D₄	1	3	yes	—
53	C₂×C₄×D₇₃	C₂×Q₈	1	3	yes	—
54	C₂×#10596831	C₂²×C₄	1	3	yes	—
55	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
56	C₈×D₇₃	—	2	6	no	—
57	C₈×D₇₃	—	4	3	no	—
58	C₄⋊Dic₇₃	C₄⋊C₄	1	3	yes	—
59	#11924005	C₄⋊C₄	1	6	yes	—
60	C₄×Dic₇₃	C₂×D₄	1	3	yes	—
61	C₄×Dic₇₃	C₂×Q₈	1	3	yes	—
62	C₄×Dic₇₃	C₂⁴	1	1	yes	characteristic
63	C₂×Q₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₇₃	C₂²×C₄	1	3	yes	—
66	C₄⋊C₄×C₇₃	C₄⋊C₄	1	3	yes	—
67	C₄²×C₇₃	C₂²×C₄	1	3	yes	—
68	C₂²×D₇₃	C₂×C₄⋊C₄	1	3	yes	—
69	Q₈×C₇₃	#107	1	4	yes	—
70	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	3	yes	—
71	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	3	yes	—
72	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
73	C₂×Dic₇₃	C₂×C₄⋊C₄	1	3	yes	—
74	C₂×Dic₇₃	C₂²×D₄	1	1	yes	characteristic
75	C₂×Dic₇₃	C₂²×Q₈	1	1	yes	characteristic
76	D₂₉₂	#107	1	12	yes	—
77	C₄×D₇₃	—	2	3	no	—
78	C₄×D₇₃	—	2	3	no	—
79	C₄×D₇₃	—	4	3	no	—
80	C₄×D₇₃	C₂×D₈	1	6	yes	—
81	C₄×D₇₃	C₂×Q₁₆	1	6	yes	—
82	#10596833	—	4	2	no	—
83	#10596833	—	4	6	no	—
84	C₈×C₇₃	—	2	3	no	—
85	C₈×C₇₃	—	2	1	no	—
86	#10596831	—	2	3	no	—
87	#10596831	#131	1	2	yes	complemented
88	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
89	C₂×D₇₃	—	2	3	no	—
90	C₂×D₇₃	—	2	3	no	—
91	C₂×D₇₃	C₂×#107	1	6	yes	—
92	C₄×C₇₃	—	2	3	no	—
93	C₄×C₇₃	C₂×#107	1	6	yes	—
94	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
95	C₄×C₇₃	#568	1	1	yes	characteristic
96	Dic₇₃	—	2	3	no	—
97	Dic₇₃	—	2	1	no	—
98	Dic₇₃	C₂²×D₈	1	1	yes	characteristic
99	Dic₇₃	C₂²×Q₁₆	1	1	yes	characteristic
100	#17828	—	73	1	no	maximal, Sylow
101	C₂×C₇₃	#2906	1	1	yes	characteristic
102	C₂×C₇₃	C₂²×#107	1	1	yes	characteristic
103	C₂×C₇₃	#2916	1	1	yes	characteristic
104	D₇₃	—	2	6	no	—
105	C₈×#41	—	73	1	no	—
106	#2906	—	73	2	no	—
107	#1571	—	73	6	no	—
108	#1567	—	73	3	no	—
109	C₂×#335	—	73	3	no	—
110	C₇₃	#17828	1	1	yes	characteristic, complemented, Sylow
111	C₂×#131	—	146	2	no	—
112	C₄×#41	—	73	1	no	—
113	C₈×Q₈	—	73	1	no	—
114	C₈×Q₈	#64408	1	1	yes	characteristic
115	C₈×D₄	—	73	3	no	—
116	C₈×D₄	—	73	3	no	—
117	#431	—	73	3	no	—
118	#424	—	146	6	no	—
119	#423	—	73	6	no	—
120	C₂×C₄⋊C₈	—	73	6	no	—
121	C₂×C₄×C₈	—	73	3	no	—
122	#335	—	73	1	no	—
123	#335	—	73	3	no	—
124	#335	—	73	6	no	—
125	#335	—	73	6	no	—
126	C₂×#41	—	73	1	no	—
127	#131	—	292	8	no	—
128	C₂×M₁₆	—	146	6	no	—
129	C₂²×C₈	—	73	3	no	—
130	C₂²×C₈	—	73	3	no	—
131	C₂²×C₈	—	146	6	no	—
132	C₄×Q₈	—	73	1	no	—
133	C₄×Q₈	C₂×#64408	1	1	yes	characteristic
134	C₄×D₄	—	73	3	no	—
135	C₄×D₄	—	73	3	no	—
136	#117	—	73	3	no	—
137	C₂×C₄²	—	73	3	no	—
138	C₄⋊C₈	—	73	2	no	—
139	C₄⋊C₈	—	73	6	no	—
140	C₄⋊C₈	—	73	3	no	—
141	C₄⋊C₈	—	146	6	no	—
142	C₄⋊C₈	—	146	6	no	—
143	C₄⋊C₈	C₂×#64408	1	3	yes	—
144	C₂²⋊C₈	—	73	6	no	—
145	C₄×C₈	—	73	1	no	—
146	C₄×C₈	—	73	3	no	—
147	C₄×C₈	—	73	3	no	—
148	C₄×C₈	C₂×#64408	1	3	yes	—
149	#41	—	73	4	no	—
150	#41	—	73	4	no	—
151	C₂×Q₈	C₄⋊#64408	1	1	yes	characteristic
152	C₂×D₄	—	73	3	no	—
153	C₂²×C₄	—	73	3	no	—
154	C₂²×C₄	—	73	3	no	—
155	C₂²×C₄	—	73	3	no	—
156	M₁₆	—	292	12	no	—
157	C₂×C₈	—	292	3	no	—
158	C₂×C₈	—	292	12	no	—
159	C₂×C₈	—	73	1	no	—
160	C₂×C₈	—	73	3	no	—
161	C₂×C₈	—	146	6	no	—
162	C₂×C₈	—	146	2	no	—
163	C₂×C₈	—	146	6	no	—
164	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
165	C₂×C₈	C₂²×#64408	1	3	yes	—
166	C₄⋊C₄	—	73	3	no	—
167	C₄⋊C₄	C₄⋊#64408	1	3	yes	—
168	C₂²⋊C₄	—	73	6	no	—
169	C₄²	—	73	1	no	—
170	C₄²	—	73	3	no	—
171	C₄²	—	73	3	no	—
172	C₄²	C₂²×#64408	1	3	yes	—
173	C₂³	—	73	3	no	—
174	Q₈	#14556660	1	4	yes	—
175	D₄	—	73	12	no	—
176	C₂×C₄	—	292	3	no	—
177	C₂×C₄	—	73	1	no	—
178	C₂×C₄	—	73	6	no	—
179	C₂×C₄	—	73	6	no	—
180	C₂×C₄	—	73	1	no	—
181	C₂×C₄	—	73	3	no	—
182	C₂×C₄	—	146	3	no	—
183	C₂×C₄	—	146	3	no	—
184	C₂×C₄	C₂×C₄⋊#64408	1	3	yes	—
185	C₂×C₄	C₂×C₄⋊#64408	1	3	yes	—
186	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
187	C₈	—	2	3	no	—
188	C₈	—	2	1	no	—
189	C₈	—	292	2	no	—
190	C₈	—	292	6	no	—
191	C₈	—	73	2	no	—
192	C₈	—	146	3	no	—
193	C₂²	—	73	6	no	—
194	C₂²	—	146	3	no	—
195	C₂²	—	146	3	no	—
196	C₂²	C₂²×C₄⋊#64408	1	1	yes	characteristic
197	C₄	—	2	3	no	—
198	C₄	—	73	1	no	—
199	C₄	—	73	1	no	—
200	C₄	—	146	3	no	—
201	C₄	—	146	1	no	—
202	C₄	C₂×#14556660	1	6	yes	—
203	C₄	D₇₃⋊#131	1	1	yes	characteristic
204	C₄	#17989667	1	1	yes	characteristic
205	C₂	—	146	6	no	—
206	C₂	#22626579	1	1	yes	characteristic
207	C₂	C₂²×#14556660	1	1	yes	characteristic
208	C₂	#22626592	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626577

C₂

yes

maximal

#22625501

C₂

yes

maximal, complemented

#22625490

C₂

yes

maximal, complemented

#22625476

C₂

yes

maximal, complemented

C₈×#11924029

C₂

yes

maximal, characteristic

C₂×#14556679

—

#17989594

—

#17989591

C₂²

yes

—

#17989588

C₂²

yes

—

#17989451

C₂²

yes

characteristic

#17989451

C₂²

yes

—

#17989451

C₂²

yes

—

#17989450

C₂²

yes

—

C₄×#11924029

C₂²

yes

characteristic

Q₈×#10596831

C₄

yes

characteristic

C₈×D₂₉₂

C₄

yes

—

#17988652

C₄

yes

—

#17988639

C₂²

yes

—

C₄×C₈×D₇₃

C₂²

yes

—

C₈×Q₈×C₇₃

C₄

yes

—

—

—

—

—

C₂³

yes

—

C₄⋊#10596833

C₂³

yes

—

C₂×#11924029

D₄

yes

characteristic

C₂×C₈×D₇₃

C₂×C₄

yes

—

C₂×C₈×D₇₃

C₂×C₄

yes

—

Q₈×Dic₇₃

Q₈

yes

characteristic

C₄×D₂₉₂

C₂×C₄

yes

—

#14556553

Q₈

yes

—

#14556549

D₄

yes

—

C₄²×D₇₃

C₂³

yes

—

C₄⋊#10596831

C₂×C₄

yes

—

#14556503

C₂×C₄

yes

—

C₄×#10596831

C₂×C₄

yes

—

C₈×Dic₇₃

C₂³

yes

—

C₈×Dic₇₃

C₂³

yes

characteristic

C₄×Q₈×C₇₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

—

—

—

—

D₈

yes

—

#11924029

Q₁₆

yes

—

C₂×D₂₉₂

C₄⋊C₄

yes

—

C₂×C₄×D₇₃

C₂²×C₄

yes

—

C₂×C₄×D₇₃

C₂×D₄

yes

—

C₂×C₄×D₇₃

C₂×Q₈

yes

—

C₂×#10596831

C₂²×C₄

yes

—

C₂×#10596831

C₂²×C₄

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

C₄⋊Dic₇₃

C₄⋊C₄

yes

—

#11924005

C₄⋊C₄

yes

—

C₄×Dic₇₃

C₂×D₄

yes

—

C₄×Dic₇₃

C₂×Q₈

yes

—

C₄×Dic₇₃

C₂⁴

yes

characteristic

C₂×Q₈×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

—

C₄⋊C₄×C₇₃

C₄⋊C₄

yes

—

C₄²×C₇₃

C₂²×C₄

yes

—

C₂²×D₇₃

C₂×C₄⋊C₄

yes

—

Q₈×C₇₃

#107

yes

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₃

C₂³×C₄

yes

characteristic

C₂×Dic₇₃

C₂×C₄⋊C₄

yes

—

C₂×Dic₇₃

C₂²×D₄

yes

characteristic

C₂×Dic₇₃

C₂²×Q₈

yes

characteristic

D₂₉₂

#107

yes

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

C₂×D₈

yes

—

C₄×D₇₃

C₂×Q₁₆

yes

—

#10596833

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

—

—

yes

C₂²×C₇₃

C₂²×C₄⋊C₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

C₂×#107

yes

—

C₄×C₇₃

—

C₄×C₇₃

C₂×#107

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

C₂²×D₈

yes

characteristic

Dic₇₃

C₂²×Q₁₆

yes

characteristic

100

#17828

—

maximal, Sylow

101

C₂×C₇₃

#2906

yes

characteristic

102

C₂×C₇₃

C₂²×#107

yes

characteristic

103

C₂×C₇₃

#2916

yes

characteristic

104

D₇₃

—

105

C₈×#41

—

106

#2906

—

107

#1571

—

108

#1567

—

109

C₂×#335

—

110

C₇₃

#17828

yes

characteristic, complemented, Sylow

111

C₂×#131

—

146

—

112

C₄×#41

—

113

C₈×Q₈

—

114

C₈×Q₈

#64408

yes

characteristic

115

C₈×D₄

—

116

C₈×D₄

—

117

#431

—

118

#424

—

146

—

119

#423

—

120

C₂×C₄⋊C₈

—

121

C₂×C₄×C₈

—

122

#335

—

123

#335

—

124

#335

—

125

#335

—

126

C₂×#41

—

127

#131

—

292

—

128

C₂×M₁₆

—

146

—

129

C₂²×C₈

—

130

C₂²×C₈

—

131

C₂²×C₈

—

146

—

132

C₄×Q₈

—

133

C₄×Q₈

C₂×#64408

yes

characteristic

134

C₄×D₄

—

135

C₄×D₄

—

136

#117

—

137

C₂×C₄²

—

138

C₄⋊C₈

—

139

C₄⋊C₈

—

140

C₄⋊C₈

—

141

C₄⋊C₈

—

146

—

142

C₄⋊C₈

—

146

—

143

C₄⋊C₈

C₂×#64408

yes

—

144

C₂²⋊C₈

—

145

C₄×C₈

—

146

C₄×C₈

—

147

C₄×C₈

—

148

C₄×C₈

C₂×#64408

yes

—

149

#41

—

150

#41

—

151

C₂×Q₈

C₄⋊#64408

yes

characteristic

152

C₂×D₄

—

153

C₂²×C₄

—

154

C₂²×C₄

—

155

C₂²×C₄

—

156

M₁₆

—

292

—

157

C₂×C₈

—

292

—

158

C₂×C₈

—

292

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

146

—

162

C₂×C₈

—

146

—

163

C₂×C₈

—

146

—

164

C₂×C₈

C₂²×#64408

yes

characteristic

165

C₂×C₈

C₂²×#64408

yes

—

166

C₄⋊C₄

—

167

C₄⋊C₄

C₄⋊#64408

yes

—

168

C₂²⋊C₄

—

169

C₄²

—

170

C₄²

—

171

C₄²

—

172

C₄²

C₂²×#64408

yes

—

173

C₂³

—

174

Q₈

#14556660

yes

—

175

D₄

—

176

C₂×C₄

—

292

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

146

—

183

C₂×C₄

—

146

—

184

C₂×C₄

C₂×C₄⋊#64408

yes

—

185

C₂×C₄

C₂×C₄⋊#64408

yes

—

186

C₂×C₄

C₂³×#64408

yes

characteristic

187

C₈

—

188

C₈

—

189

C₈

—

292

—

190

C₈

—

292

—

191

C₈

—

192

C₈

—

146

—

193

C₂²

—

194

C₂²

—

146

—

195

C₂²

—

146

—

196

C₂²

C₂²×C₄⋊#64408

yes

characteristic

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

146

—

201

C₄

—

146

—

202

C₄

C₂×#14556660

yes

—

203

C₄

D₇₃⋊#131

yes

characteristic

204

C₄

#17989667

yes

characteristic

205

C₂

—

146

—

206

C₂

#22626579

yes

characteristic

207

C₂

C₂²×#14556660

yes

characteristic

208

C₂

#22626592

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₄⋊#64408
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#64408
Fitting	C₈×Q₈×C₇₃	C₄
socle	C₂²×C₇₃	C₂²×C₄⋊C₄
prime radical	C₂×#11924029	D₄
cosocle	C₄×Dic₇₃	C₂⁴
hypercenter	C₈×Q₈	#64408
hypocenter	C₇₃	#17828
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×C₄⋊#64408

commutator

C₂×C₄×C₇₃

C₂³×C₄

Frattini

C₂×C₄

C₂³×#64408

Fitting

C₈×Q₈×C₇₃

C₄

socle

C₂²×C₇₃

C₂²×C₄⋊C₄

prime radical

C₂×#11924029

D₄

cosocle

C₄×Dic₇₃

C₂⁴

C₈×Q₈

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
2. #22625501	C₂	semidirect
3. #22625490	C₂	semidirect
4. #22625476	C₂	semidirect
87. #10596831	#131	semidirect
110. C₇₃	#17828	semidirect

H ≅ G/N

type

2. #22625501

C₂

semidirect

3. #22625490

C₂

semidirect

4. #22625476

C₂

semidirect

87. #10596831

#131

semidirect

110. C₇₃

#17828

semidirect

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

subgroup	index
#22626577	2
#22625501	2
#22625490	2
#22625476	2
C₈×#11924029	2
#17828	73

subgroup

index

Subgroups — #34018161

Conjugacy classes of subgroups

Complements

Maximal subgroups