Group #34013288 — subgroups

Subgroups — #34013288

Order 18688.

234 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 84 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	#22626274	C₂	1	1	yes	maximal, characteristic
2	#22626266	C₂	1	1	yes	maximal, characteristic
3	#22624986	C₂	1	1	yes	maximal, characteristic
4	#22624985	C₂	1	1	yes	maximal, characteristic
5	#22624982	C₂	1	1	yes	maximal, characteristic
6	#22624981	C₂	1	1	yes	maximal, characteristic
7	#22609685	C₂	1	1	yes	maximal, characteristic
8	C₂×#14556665	C₂²	1	1	yes	characteristic
9	C₂×#14556665	C₂²	1	1	yes	characteristic
10	C₂×#14556665	C₂²	1	1	yes	characteristic
11	C₂²×#11924031	C₂²	1	1	yes	characteristic
12	#17989510	—	2	2	no	—
13	C₄⋊#11924031	—	2	2	no	—
14	#17988545	C₂²	1	1	yes	characteristic
15	#17988543	C₂²	1	1	yes	characteristic
16	#17988543	C₂²	1	1	yes	characteristic
17	#17988345	C₄	1	1	yes	characteristic
18	#17988343	C₄	1	1	yes	characteristic
19	#17988343	C₄	1	1	yes	characteristic
20	H(3,ℤ/4)×C₇₃	C₄	1	1	yes	characteristic
21	#14556665	—	2	1	no	—
22	#14556665	—	2	1	no	—
23	#14556665	—	2	1	no	—
24	#14556665	—	2	1	no	—
25	#14556665	—	2	1	no	—
26	#14556665	—	2	1	no	—
27	C₂×#11924031	—	2	1	no	—
28	C₂×#11924031	—	2	1	no	—
29	C₂×#11924031	—	2	1	no	—
30	C₂×#11924031	—	2	1	no	—
31	C₂×#11924031	—	2	1	no	—
32	C₂×#11924031	—	2	1	no	—
33	C₂×#11924012	C₂×C₄	1	1	yes	characteristic
34	C₂×#11924012	C₂×C₄	1	1	yes	characteristic
35	C₂²×#10596831	C₂³	1	1	yes	characteristic
36	#14556550	C₂×C₄	1	1	yes	characteristic
37	#14556502	—	2	1	no	—
38	#14556502	Q₈	1	2	yes	—
39	#14556501	—	2	1	no	—
40	C₄×#10596831	—	2	1	no	—
41	C₄×#10596831	D₄	1	2	yes	—
42	C₈×Dic₇₃	—	2	1	no	—
43	C₂×M₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
44	#117×C₇₃	C₂×C₄	1	1	yes	characteristic
45	#117×C₇₃	C₂×C₄	1	1	yes	characteristic
46	#11924031	—	4	1	no	—
47	#11924031	—	4	1	no	—
48	#11924031	—	4	1	no	—
49	#11924031	—	4	1	no	—
50	#11924012	—	2	1	no	—
51	#11924012	—	2	1	no	—
52	#11924012	—	2	1	no	—
53	#11924012	—	2	1	no	—
54	C₂×#10596831	—	2	1	no	—
55	C₂×#10596831	C₄⋊C₄	1	2	yes	—
56	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
57	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
58	C₂×#10596831	#41	1	1	yes	characteristic
59	C₂×#10596831	#41	1	1	yes	characteristic
60	C₂×#10596831	#41	1	1	yes	characteristic
61	C₂×#10596831	#41	1	1	yes	characteristic
62	C₄⋊Dic₇₃	—	2	1	no	—
63	C₂²⋊Dic₇₃	—	2	1	no	—
64	C₄×Dic₇₃	—	2	1	no	—
65	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
66	M₁₆×C₇₃	—	2	1	no	—
67	M₁₆×C₇₃	—	2	1	no	—
68	C₂×C₈×C₇₃	—	2	1	no	—
69	C₄⋊C₄×C₇₃	—	2	1	no	—
70	C₄⋊C₄×C₇₃	—	2	1	no	—
71	C₂²⋊C₄×C₇₃	—	2	1	no	—
72	C₂²⋊C₄×C₇₃	—	2	1	no	—
73	C₄²×C₇₃	—	2	1	no	—
74	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
75	C₂³×C₇₃	#131	1	1	yes	characteristic
76	C₂×C₄×C₇₃	—	2	2	no	—
77	C₂×C₄×C₇₃	—	4	1	no	—
78	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
80	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
81	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
82	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
83	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
84	C₂×Dic₇₃	—	4	1	no	—
85	C₈×C₇₃	—	4	1	no	—
86	#10596831	—	2	2	no	—
87	#10596831	—	4	1	no	—
88	#10596831	#124	1	1	yes	characteristic
89	#10596831	#124	1	1	yes	characteristic
90	#10596831	#125	1	1	yes	characteristic
91	#10596831	#125	1	1	yes	characteristic
92	C₂²×C₇₃	—	4	1	no	—
93	C₂²×C₇₃	#424	1	1	yes	characteristic
94	C₂²×C₇₃	#433	1	1	yes	characteristic
95	C₂²×C₇₃	#433	1	1	yes	characteristic
96	C₄×C₇₃	—	4	2	no	—
97	C₄×C₇₃	—	8	1	no	—
98	C₄×C₇₃	#382	1	1	yes	characteristic
99	C₄×C₇₃	#424	1	1	yes	characteristic
100	C₄×C₇₃	#433	1	1	yes	characteristic
101	C₄×C₇₃	#433	1	1	yes	characteristic
102	Dic₇₃	—	8	1	no	—
103	#12885	—	73	1	no	maximal, Sylow
104	C₂×C₇₃	—	2	1	no	—
105	C₂×C₇₃	—	2	1	no	—
106	C₂×C₇₃	—	2	1	no	—
107	C₂×C₇₃	#1832	1	1	yes	characteristic
108	#2153	—	73	1	no	—
109	#2151	—	73	1	no	—
110	#1747	—	73	1	no	—
111	#1379	—	73	1	no	—
112	#1379	—	73	1	no	—
113	#1378	—	73	1	no	—
114	#1378	—	73	1	no	—
115	C₇₃	#12885	1	1	yes	characteristic, complemented, Sylow
116	C₂×M₃₂	—	73	1	no	—
117	C₂×M₃₂	—	73	1	no	—
118	C₂×M₃₂	—	73	1	no	—
119	C₂²×C₁₆	—	73	1	no	—
120	#405	—	73	1	no	—
121	#405	—	73	1	no	—
122	#405	—	73	1	no	—
123	#364	—	146	2	no	—
124	C₄⋊C₁₆	—	146	2	no	—
125	#338	—	73	1	no	—
126	H(3,ℤ/4)	—	73	1	no	—
127	H(3,ℤ/4)	—	73	1	no	—
128	H(3,ℤ/4)	#64408	1	1	yes	characteristic
129	C₂×M₁₆	—	73	1	no	—
130	C₂×M₁₆	—	73	1	no	—
131	C₂×M₁₆	C₂×#64408	1	1	yes	characteristic
132	C₂²×C₈	—	73	1	no	—
133	#117	—	73	1	no	—
134	#117	C₂×#64408	1	1	yes	characteristic
135	#117	C₂×#64408	1	1	yes	characteristic
136	M₃₂	—	146	1	no	—
137	M₃₂	—	146	1	no	—
138	M₃₂	—	146	1	no	—
139	M₃₂	—	146	1	no	—
140	M₃₂	—	146	1	no	—
141	M₃₂	—	146	1	no	—
142	C₂×C₁₆	—	146	1	no	—
143	C₂×C₁₆	—	146	1	no	—
144	C₂×C₁₆	—	146	1	no	—
145	C₂×C₁₆	—	146	1	no	—
146	C₂×C₁₆	—	146	1	no	—
147	C₂×C₁₆	—	146	1	no	—
148	#97	—	73	2	no	—
149	#97	—	146	1	no	—
150	#97	—	146	1	no	—
151	C₄×C₈	—	73	2	no	—
152	C₄×C₈	—	146	1	no	—
153	C₄×C₈	—	146	1	no	—
154	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
155	M₁₆	—	2	1	no	—
156	M₁₆	—	2	1	no	—
157	M₁₆	—	146	1	no	—
158	M₁₆	—	146	1	no	—
159	M₁₆	—	146	1	no	—
160	M₁₆	—	146	1	no	—
161	C₂×C₈	—	2	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	2	no	—
169	C₂×C₈	—	146	1	no	—
170	C₄⋊C₄	—	2	1	no	—
171	C₄⋊C₄	—	2	1	no	—
172	C₄⋊C₄	—	146	1	no	—
173	C₂²⋊C₄	—	2	1	no	—
174	C₂²⋊C₄	—	2	1	no	—
175	C₂²⋊C₄	—	146	1	no	—
176	C₄²	—	2	1	no	—
177	C₄²	—	146	1	no	—
178	C₄²	C₄⋊#64408	1	2	yes	—
179	C₁₆	—	292	1	no	—
180	C₁₆	—	292	1	no	—
181	C₁₆	—	292	1	no	—
182	C₁₆	—	292	1	no	—
183	C₂³	#14556678	1	1	yes	characteristic
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	4	1	no	—
186	C₂×C₄	—	292	1	no	—
187	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
188	C₂×C₄	#14556670	1	1	yes	characteristic
189	C₂×C₄	#14556670	1	1	yes	characteristic
190	C₂×C₄	#14556678	1	1	yes	characteristic
191	C₂×C₄	#14556678	1	1	yes	characteristic
192	C₂×C₄	#14556679	1	1	yes	characteristic
193	C₈	—	4	1	no	—
194	C₈	—	292	1	no	—
195	C₈	—	73	1	no	—
196	C₈	—	73	1	no	—
197	C₈	—	73	1	no	—
198	C₈	—	73	1	no	—
199	C₈	—	146	2	no	—
200	C₂²	—	4	1	no	—
201	C₂²	#17989593	1	1	yes	characteristic
202	C₂²	#17989611	1	1	yes	characteristic
203	C₂²	#17989612	1	1	yes	characteristic
204	C₄	—	4	2	no	—
205	C₄	—	8	1	no	—
206	C₄	—	584	1	no	—
207	C₄	#17989531	1	1	yes	characteristic
208	C₄	#17989594	1	1	yes	characteristic
209	C₄	#17989611	1	1	yes	characteristic
210	C₄	#17989612	1	1	yes	characteristic
211	C₂	—	2	1	no	—
212	C₂	—	2	1	no	—
213	C₂	—	2	1	no	—
214	C₂	#22625849	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626274

C₂

yes

maximal, characteristic

#22626266

C₂

yes

maximal, characteristic

#22624986

C₂

yes

maximal, characteristic

#22624985

C₂

yes

maximal, characteristic

#22624982

C₂

yes

maximal, characteristic

#22624981

C₂

yes

maximal, characteristic

#22609685

C₂

yes

maximal, characteristic

C₂×#14556665

C₂²

yes

characteristic

C₂×#14556665

C₂²

yes

characteristic

C₂×#14556665

C₂²

yes

characteristic

C₂²×#11924031

C₂²

yes

—

—

C₂²

yes

characteristic

#17988543

C₂²

yes

characteristic

#17988543

C₂²

yes

characteristic

#17988345

C₄

yes

characteristic

#17988343

C₄

yes

characteristic

#17988343

C₄

yes

characteristic

H(3,ℤ/4)×C₇₃

C₄

yes

—

—

—

—

—

—

—

—

—

—

—

—

C₂×C₄

yes

characteristic

C₂×#11924012

C₂×C₄

yes

characteristic

C₂²×#10596831

C₂³

yes

characteristic

#14556550

C₂×C₄

yes

characteristic

#14556502

—

#14556502

Q₈

yes

—

#14556501

—

C₄×#10596831

—

C₄×#10596831

D₄

yes

—

C₈×Dic₇₃

—

C₂×M₁₆×C₇₃

C₂×C₄

yes

characteristic

#117×C₇₃

C₂×C₄

yes

characteristic

#117×C₇₃

C₂×C₄

yes

—

—

—

—

—

—

—

—

—

C₄⋊C₄

yes

—

C₂×#10596831

C₂×D₄

yes

characteristic

C₂×#10596831

C₂×Q₈

yes

yes

yes

yes

yes

C₄⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₄⋊C₄

yes

—

C₂³×C₇₃

#131

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₈×C₇₃

—

—

—

yes

yes

yes

yes

C₂²×C₇₃

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

100

C₄×C₇₃

#433

yes

characteristic

101

C₄×C₇₃

#433

yes

characteristic

102

Dic₇₃

—

103

#12885

—

maximal, Sylow

104

C₂×C₇₃

—

105

C₂×C₇₃

—

106

C₂×C₇₃

—

107

C₂×C₇₃

#1832

yes

characteristic

108

#2153

—

109

#2151

—

110

#1747

—

111

#1379

—

112

#1379

—

113

#1378

—

114

#1378

—

115

C₇₃

#12885

yes

characteristic, complemented, Sylow

116

C₂×M₃₂

—

117

C₂×M₃₂

—

118

C₂×M₃₂

—

119

C₂²×C₁₆

—

120

#405

—

121

#405

—

122

#405

—

123

#364

—

146

—

124

C₄⋊C₁₆

—

146

—

125

#338

—

126

H(3,ℤ/4)

—

127

H(3,ℤ/4)

—

128

H(3,ℤ/4)

#64408

yes

characteristic

129

C₂×M₁₆

—

130

C₂×M₁₆

—

131

C₂×M₁₆

C₂×#64408

yes

characteristic

132

C₂²×C₈

—

133

#117

—

134

yes

135

yes

136

M₃₂

—

146

—

137

M₃₂

—

146

—

138

M₃₂

—

146

—

139

M₃₂

—

146

—

140

M₃₂

—

146

—

141

M₃₂

—

146

—

142

C₂×C₁₆

—

146

—

143

C₂×C₁₆

—

146

—

144

C₂×C₁₆

—

146

—

145

C₂×C₁₆

—

146

—

146

C₂×C₁₆

—

146

—

147

C₂×C₁₆

—

146

—

148

#97

—

149

#97

—

146

—

150

#97

—

146

—

151

C₄×C₈

—

152

C₄×C₈

—

146

—

153

C₄×C₈

—

146

—

154

C₂²×C₄

C₂²×#64408

yes

characteristic

155

M₁₆

—

156

M₁₆

—

157

M₁₆

—

146

—

158

M₁₆

—

146

—

159

M₁₆

—

146

—

160

M₁₆

—

146

—

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

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

146

—

173

C₂²⋊C₄

—

174

C₂²⋊C₄

—

175

C₂²⋊C₄

—

146

—

176

C₄²

—

177

C₄²

—

146

—

178

C₄²

C₄⋊#64408

yes

—

179

C₁₆

—

292

—

180

C₁₆

—

292

—

181

C₁₆

—

292

—

182

C₁₆

—

292

—

183

C₂³

#14556678

yes

characteristic

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

292

—

187

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

188

C₂×C₄

#14556670

yes

characteristic

189

C₂×C₄

#14556670

yes

characteristic

190

C₂×C₄

#14556678

yes

characteristic

191

C₂×C₄

#14556678

yes

characteristic

192

C₂×C₄

#14556679

yes

characteristic

193

C₈

—

194

C₈

—

292

—

195

C₈

—

196

C₈

—

197

C₈

—

198

C₈

—

199

C₈

—

146

—

200

C₂²

—

201

C₂²

#17989593

yes

characteristic

202

C₂²

#17989611

yes

characteristic

203

C₂²

#17989612

yes

characteristic

204

C₄

—

205

C₄

—

206

C₄

—

584

—

207

C₄

#17989531

yes

characteristic

208

C₄

#17989594

yes

characteristic

209

C₄

#17989611

yes

characteristic

210

C₄

#17989612

yes

characteristic

211

C₂

—

212

C₂

—

213

C₂

—

214

C₂

#22625849

yes

characteristic

	subgroup	G/H
center	C₄	#17989531
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×#64408
Fitting	H(3,ℤ/4)×C₇₃	C₄
socle	C₂×C₇₃	#1832
prime radical	C₂³×C₇₃	#131
cosocle	C₂²×#10596831	C₂³
hypercenter	H(3,ℤ/4)	#64408
hypocenter	C₇₃	#12885
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#17989531

commutator

C₂²×C₄×C₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×#64408

Fitting

H(3,ℤ/4)×C₇₃

C₄

C₂×C₇₃

C₂³×C₇₃

C₂³

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
115. C₇₃	#12885	semidirect

H ≅ G/N

type

115. C₇₃

#12885

semidirect

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

subgroup	index
#22626274	2
#22626266	2
#22624986	2
#22624985	2
#22624982	2
#22624981	2
#22609685	2
#12885	73

subgroup

index

Subgroups — #34013288

Conjugacy classes of subgroups

Complements

Maximal subgroups