Group #34034329 — subgroups

Subgroups — #34034329

Order 18688.

284 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 93 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	#22627127	C₂	1	2	yes	maximal
2	C₂×#17989691	C₂	1	1	yes	maximal, characteristic
3	#64408⋊M₃₂	C₂	1	1	yes	maximal, characteristic
4	#22627010	C₂	1	2	yes	maximal
5	#22625736	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556688	C₂²	1	2	yes	—
7	#17989706	C₂²	1	1	yes	characteristic
8	#17989706	C₂²	1	1	yes	characteristic
9	#17989692	—	2	2	no	—
10	#17989691	—	2	1	no	—
11	#17989691	—	2	1	no	—
12	#17989691	—	2	2	no	—
13	#17989682	—	2	1	no	—
14	#64408⋊C₁₆	—	2	1	no	—
15	#17989597	C₄	1	1	yes	characteristic
16	#17989553	C₂²	1	2	yes	—
17	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
18	#17989459	C₄	1	2	yes	—
19	#108×D₇₃	C₄	1	1	yes	characteristic
20	#14556688	—	2	2	no	—
21	#14556688	—	2	2	no	—
22	#14556688	—	2	1	no	—
23	#14556688	D₄	1	2	yes	—
24	#14556687	—	2	1	no	—
25	#14556687	Q₈	1	2	yes	—
26	#14556686	—	2	1	no	—
27	#14556686	—	2	1	no	—
28	C₂×#11924040	—	2	2	no	—
29	C₂×#11924036	C₂³	1	1	yes	characteristic
30	#14556670	C₂×C₄	1	2	yes	—
31	C₂×C₄×#64408	C₂×C₄	1	1	yes	characteristic
32	#14556660	—	2	1	no	—
33	#14556659	—	2	1	no	—
34	#14556645	—	2	2	no	—
35	C₈×#64408	—	2	2	no	—
36	C₈×#64408	D₄	1	1	yes	characteristic
37	C₈×#64408	D₄	1	1	yes	characteristic
38	C₈×#64408	Q₈	1	1	yes	characteristic
39	C₈×#64408	Q₈	1	1	yes	characteristic
40	M₁₆×D₇₃	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
42	#14556527	C₈	1	1	yes	characteristic
43	#14556526	C₈	1	2	yes	—
44	#108×C₇₃	C₈	1	1	yes	characteristic
45	#11924040	—	2	2	no	—
46	#11924040	—	2	2	no	—
47	#11924040	—	4	2	no	—
48	C₂²×#64408	—	2	1	no	—
49	#11924036	C₂×D₄	1	1	yes	characteristic
50	#11924036	C₂×D₄	1	1	yes	characteristic
51	#11924036	C₂×Q₈	1	1	yes	characteristic
52	#11924036	#41	1	1	yes	characteristic
53	C₂×#10596833	C₂×Q₈	1	1	yes	characteristic
54	C₂×#10596833	#41	1	1	yes	characteristic
55	C₄⋊#64408	—	2	2	no	—
56	C₂²⋊#64408	—	2	2	no	—
57	C₄×#64408	—	2	2	no	—
58	C₄×#64408	—	2	1	no	—
59	C₄×#64408	C₄⋊C₄	1	1	yes	characteristic
60	C₄×#64408	C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
62	#11924012	C₂×C₈	1	2	yes	—
63	#11924011	—	2	2	no	—
64	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
65	C₈×D₇₃	—	2	2	no	—
66	C₈×D₇₃	C₄⋊C₄	1	1	yes	characteristic
67	C₈×D₇₃	C₄⋊C₄	1	1	yes	characteristic
68	C₈×D₇₃	M₁₆	1	1	yes	characteristic
69	C₈×D₇₃	M₁₆	1	1	yes	characteristic
70	M₁₆×C₇₃	C₂×C₈	1	2	yes	—
71	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
72	C₂²×D₇₃	#131	1	1	yes	characteristic
73	C₂×#64408	—	2	1	no	—
74	C₂×#64408	—	2	1	no	—
75	C₂×#64408	—	4	2	no	—
76	C₂×#64408	—	4	2	no	—
77	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
78	C₂×Dic₇₃	C₄×Q₈	1	1	yes	characteristic
79	C₄×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₄×D₇₃	C₄×D₄	1	1	yes	characteristic
81	C₄×D₇₃	C₂×M₁₆	1	1	yes	characteristic
82	C₄×D₇₃	#131	1	1	yes	characteristic
83	#10596833	#121	1	1	yes	characteristic
84	#10596833	C₂²⋊Q₈	1	1	yes	characteristic
85	#10596833	#125	1	1	yes	characteristic
86	#10596833	C₄⋊Q₈	1	1	yes	characteristic
87	C₈×C₇₃	—	2	2	no	—
88	C₈×C₇₃	C₄⋊C₈	1	1	yes	characteristic
89	C₈×C₇₃	C₄⋊C₈	1	1	yes	characteristic
90	#10596831	—	2	2	no	—
91	#10596831	C₄⋊C₈	1	1	yes	characteristic
92	#10596831	C₄⋊C₈	1	1	yes	characteristic
93	C₂²×C₇₃	C₈×Q₈	1	1	yes	characteristic
94	C₂×D₇₃	—	4	1	no	—
95	C₂×D₇₃	#424	1	1	yes	characteristic
96	C₂×D₇₃	#436	1	1	yes	characteristic
97	#64408	—	2	2	no	—
98	#64408	—	4	1	no	—
99	#64408	—	8	2	no	—
100	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
101	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
102	Dic₇₃	#389	1	1	yes	characteristic
103	Dic₇₃	#446	1	1	yes	characteristic
104	#13164	—	73	1	no	maximal, Sylow
105	C₂×C₇₃	—	2	1	no	—
106	C₂×C₇₃	#1939	1	1	yes	characteristic
107	D₇₃	—	2	1	no	—
108	D₇₃	—	2	1	no	—
109	#2153	—	73	2	no	—
110	C₂×#364	—	73	1	no	—
111	C₄×M₃₂	—	73	1	no	—
112	#1772	—	73	1	no	—
113	#1378	—	73	2	no	—
114	C₇₃	#13164	1	1	yes	characteristic, complemented, Sylow
115	C₂×M₃₂	—	73	1	no	—
116	C₂×M₃₂	—	73	1	no	—
117	C₂×M₃₂	—	73	2	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	—	146	1	no	—
123	#364	—	146	1	no	—
124	#364	—	146	2	no	—
125	#364	—	146	2	no	—
126	#346	—	146	1	no	—
127	C₄×C₁₆	—	146	1	no	—
128	#339	—	73	2	no	—
129	C₂×M₁₆	—	73	2	no	—
130	C₂²×C₈	—	73	1	no	—
131	C₂²×C₈	—	73	1	no	—
132	#117	—	73	2	no	—
133	C₂×C₄²	—	73	1	no	—
134	M₃₂	—	73	2	no	—
135	M₃₂	—	73	2	no	—
136	M₃₂	—	146	2	no	—
137	M₃₂	—	146	2	no	—
138	M₃₂	—	146	1	no	—
139	M₃₂	—	146	1	no	—
140	C₂×C₁₆	—	146	2	no	—
141	C₂×C₁₆	—	146	1	no	—
142	C₂×C₁₆	—	146	1	no	—
143	#108	—	73	2	no	—
144	#108	—	73	1	no	—
145	#108	#10596842	1	1	yes	characteristic
146	#107	—	146	1	no	—
147	#106	—	146	1	no	—
148	#97	—	146	2	no	—
149	C₄×C₈	—	73	1	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	C₂²×C₄	—	73	1	no	—
155	C₂²×C₄	—	146	1	no	—
156	M₁₆	—	73	2	no	—
157	M₁₆	—	146	2	no	—
158	M₁₆	C₂×#10596842	1	2	yes	—
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	—	73	1	no	—
170	C₂×C₈	—	146	2	no	—
171	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
172	C₄⋊C₄	—	146	2	no	—
173	C₂²⋊C₄	—	146	2	no	—
174	C₄²	—	73	1	no	—
175	C₄²	—	73	1	no	—
176	C₄²	—	146	2	no	—
177	C₄²	—	146	1	no	—
178	C₁₆	—	292	2	no	—
179	C₁₆	—	146	2	no	—
180	C₁₆	—	146	2	no	—
181	C₂³	—	73	1	no	—
182	C₂×C₄	—	292	2	no	—
183	C₂×C₄	—	292	2	no	—
184	C₂×C₄	—	73	1	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	—	146	1	no	—
190	C₂×C₄	—	146	1	no	—
191	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
192	C₈	—	2	2	no	—
193	C₈	—	73	1	no	—
194	C₈	—	73	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	C₈	C₄⋊#10596842	1	1	yes	characteristic
201	C₈	C₄⋊#10596842	1	1	yes	characteristic
202	C₂²	—	292	1	no	—
203	C₂²	—	73	1	no	—
204	C₂²	—	73	1	no	—
205	C₂²	Q₈×#10596842	1	1	yes	characteristic
206	C₄	—	292	1	no	—
207	C₄	—	584	2	no	—
208	C₄	—	73	1	no	—
209	C₄	—	73	1	no	—
210	C₄	—	146	2	no	—
211	C₄	C₂×C₄⋊#10596842	1	1	yes	characteristic
212	C₄	D₄×#10596842	1	1	yes	characteristic
213	C₂	—	2	1	no	—
214	C₂	—	146	1	no	—
215	C₂	—	146	1	no	—
216	C₂	#22627080	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22627127

C₂

yes

maximal

C₂×#17989691

C₂

yes

maximal, characteristic

#64408⋊M₃₂

C₂

yes

maximal, characteristic

#22627010

C₂

yes

maximal

#22625736

C₂

yes

maximal, characteristic

C₂×#14556688

C₂²

yes

—

#17989706

C₂²

yes

characteristic

#17989706

C₂²

yes

—

—

—

—

—

#64408⋊C₁₆

—

#17989597

C₄

yes

characteristic

#17989553

C₂²

yes

—

C₂×C₈×#64408

C₂²

yes

characteristic

#17989459

C₄

yes

—

#108×D₇₃

C₄

yes

—

—

—

D₄

yes

—

#14556687

—

#14556687

Q₈

yes

—

—

—

—

C₂³

yes

characteristic

#14556670

C₂×C₄

yes

—

C₂×C₄×#64408

C₂×C₄

yes

—

—

—

—

D₄

yes

characteristic

C₈×#64408

D₄

yes

characteristic

C₈×#64408

Q₈

yes

characteristic

C₈×#64408

Q₈

yes

characteristic

M₁₆×D₇₃

C₂×C₄

yes

—

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

#14556527

C₈

yes

characteristic

#14556526

C₈

yes

—

#108×C₇₃

C₈

yes

—

—

—

C₂²×#64408

—

#11924036

C₂×D₄

yes

characteristic

#11924036

C₂×D₄

yes

characteristic

#11924036

C₂×Q₈

yes

yes

C₂×Q₈

yes

yes

—

—

—

—

C₄⋊C₄

yes

characteristic

C₄×#64408

C₄⋊C₄

yes

characteristic

C₂×C₄×D₇₃

C₂²×C₄

yes

characteristic

#11924012

C₂×C₈

yes

—

#11924011

—

C₂×#10596831

C₂×C₈

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

C₄⋊C₄

yes

characteristic

C₈×D₇₃

C₄⋊C₄

yes

characteristic

C₈×D₇₃

M₁₆

yes

characteristic

C₈×D₇₃

M₁₆

yes

characteristic

M₁₆×C₇₃

C₂×C₈

yes

—

C₂×C₈×C₇₃

C₂×C₈

yes

C₂²×D₇₃

yes

—

—

—

—

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×Dic₇₃

C₄×Q₈

yes

characteristic

C₄×D₇₃

C₂×C₄⋊C₄

yes

characteristic

C₄×D₇₃

C₄×D₄

yes

characteristic

C₄×D₇₃

C₂×M₁₆

yes

C₄×D₇₃

yes

yes

C₂²⋊Q₈

yes

yes

C₄⋊Q₈

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

C₄⋊C₈

yes

characteristic

C₈×C₇₃

C₄⋊C₈

yes

characteristic

#10596831

—

#10596831

C₄⋊C₈

yes

characteristic

#10596831

C₄⋊C₈

yes

characteristic

C₂²×C₇₃

C₈×Q₈

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

yes

C₂×D₇₃

yes

—

—

—

100

C₄×C₇₃

C₂×C₄⋊C₈

yes

characteristic

101

C₄×C₇₃

C₈×D₄

yes

characteristic

102

Dic₇₃

#389

yes

characteristic

103

Dic₇₃

yes

104

—

105

C₂×C₇₃

—

106

C₂×C₇₃

#1939

yes

characteristic

107

D₇₃

—

108

D₇₃

—

109

#2153

—

110

C₂×#364

—

111

C₄×M₃₂

—

112

#1772

—

113

#1378

—

114

C₇₃

#13164

yes

characteristic, complemented, Sylow

115

C₂×M₃₂

—

116

C₂×M₃₂

—

117

C₂×M₃₂

—

118

C₂×#108

—

119

#427

—

120

#405

—

121

C₂×C₄×C₈

—

122

#364

—

146

—

123

#364

—

146

—

124

#364

—

146

—

125

#364

—

146

—

126

#346

—

146

—

127

C₄×C₁₆

—

146

—

128

#339

—

129

C₂×M₁₆

—

130

C₂²×C₈

—

131

C₂²×C₈

—

132

#117

—

133

C₂×C₄²

—

134

M₃₂

—

135

M₃₂

—

136

M₃₂

—

146

—

137

M₃₂

—

146

—

138

M₃₂

—

146

—

139

M₃₂

—

146

—

140

C₂×C₁₆

—

146

—

141

C₂×C₁₆

—

146

—

142

C₂×C₁₆

—

146

—

143

#108

—

144

#108

—

145

yes

146

—

146

—

147

#106

—

146

—

148

#97

—

146

—

149

C₄×C₈

—

150

C₄×C₈

—

151

C₄×C₈

—

152

C₄×C₈

—

153

C₄×C₈

—

146

—

154

C₂²×C₄

—

155

C₂²×C₄

—

146

—

156

M₁₆

—

157

M₁₆

—

146

—

158

M₁₆

C₂×#10596842

yes

—

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

—

170

C₂×C₈

—

146

—

171

C₂×C₈

C₂×#10596842

yes

characteristic

172

C₄⋊C₄

—

146

—

173

C₂²⋊C₄

—

146

—

174

C₄²

—

175

C₄²

—

176

C₄²

—

146

—

177

C₄²

—

146

—

178

C₁₆

—

292

—

179

C₁₆

—

146

—

180

C₁₆

—

146

—

181

C₂³

—

182

C₂×C₄

—

292

—

183

C₂×C₄

—

292

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

146

—

190

C₂×C₄

—

146

—

191

C₂×C₄

C₂²×#10596842

yes

characteristic

192

C₈

—

193

C₈

—

194

C₈

—

195

C₈

—

196

C₈

—

197

C₈

—

198

C₈

—

199

C₈

—

146

—

200

C₈

C₄⋊#10596842

yes

characteristic

201

C₈

C₄⋊#10596842

yes

characteristic

202

C₂²

—

292

—

203

C₂²

—

204

C₂²

—

205

C₂²

Q₈×#10596842

yes

characteristic

206

C₄

—

292

—

207

C₄

—

584

—

208

C₄

—

209

C₄

—

210

C₄

—

146

—

211

C₄

C₂×C₄⋊#10596842

yes

characteristic

212

C₄

D₄×#10596842

yes

characteristic

213

C₂

—

214

C₂

—

146

—

215

C₂

—

146

—

216

C₂

#22627080

yes

characteristic

	subgroup	G/H
center	C₂	#22627080
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂×C₄	C₂²×#10596842
Fitting	#108×C₇₃	C₈
socle	C₂×C₇₃	#1939
prime radical	C₂²×D₇₃	#131
cosocle	C₂×#11924036	C₂³
hypercenter	#108	#10596842
hypocenter	C₇₃	#13164
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #108

subgroup

G/H

center

C₂

#22627080

commutator

C₂×C₄×C₇₃

C₂²×C₈

Frattini

C₂×C₄

C₂²×#10596842

Fitting

#108×C₇₃

C₈

C₂×C₇₃

C₂²×D₇₃

C₂³

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #108

N	H ≅ G/N	type
114. C₇₃	#13164	semidirect

H ≅ G/N

type

114. C₇₃

#13164

semidirect

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

subgroup	index
#22627127	2
C₂×#17989691	2
#64408⋊M₃₂	2
#22627010	2
#22625736	2
#13164	73

subgroup

index

#64408⋊M₃₂

Subgroups — #34034329

Conjugacy classes of subgroups

Complements

Maximal subgroups