Group #34007488 — subgroups

Subgroups — #34007488

Order 18688.

378 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 124 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	#22626029	C₂	1	2	yes	maximal
2	C₂×#17989450	C₂	1	1	yes	maximal, characteristic
3	C₂×#17989429	C₂	1	1	yes	maximal, characteristic
4	#22624767	C₂	1	2	yes	maximal
5	#22609823	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556658	—	2	2	no	—
7	C₂×C₄⋊#10596833	C₂²	1	2	yes	—
8	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
9	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
10	#17989453	—	2	2	no	—
11	#17989450	—	2	2	no	—
12	#17989429	—	2	2	no	—
13	C₂×#14556524	C₄	1	2	yes	—
14	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×Dic₇₃	C₂²	1	2	yes	—
16	#17988350	C₄	1	1	yes	characteristic
17	#340×C₇₃	C₄	1	1	yes	characteristic
18	C₂²×#10596833	—	2	2	no	—
19	C₂²×#10596833	—	2	1	no	—
20	C₂²×#10596833	D₄	1	1	yes	characteristic
21	C₂²×#10596833	D₄	1	1	yes	characteristic
22	#14556658	—	4	4	no	—
23	C₄⋊#10596833	—	2	2	no	—
24	C₄⋊#10596833	—	2	2	no	—
25	C₄×#10596833	—	2	1	no	—
26	C₄×#10596833	—	2	1	no	—
27	C₄×#10596833	—	2	1	no	—
28	C₄×#10596833	Q₈	1	2	yes	—
29	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
31	C₂×C₄×Dic₇₃	—	2	2	no	—
32	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
33	#14556524	—	2	4	no	—
34	C₈×Dic₇₃	D₄	1	2	yes	—
35	C₈×Dic₇₃	Q₈	1	2	yes	—
36	#14556498	—	2	2	no	—
37	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
39	C₂×#10596833	—	2	1	no	—
40	C₂×#10596833	—	2	1	no	—
41	C₂×#10596833	—	2	1	no	—
42	C₂×#10596833	—	2	1	no	—
43	C₂×#10596833	—	2	2	no	—
44	C₂×#10596833	—	4	2	no	—
45	C₂×#10596833	—	4	2	no	—
46	C₂×#10596833	—	4	2	no	—
47	C₂×#10596833	—	4	2	no	—
48	C₂×#10596833	D₈	1	2	yes	—
49	C₂×#10596833	Q₁₆	1	2	yes	—
50	C₂²×Dic₇₃	—	2	2	no	—
51	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×Dic₇₃	#41	1	1	yes	characteristic
53	C₂×#10596831	—	2	2	no	—
54	C₂×#10596831	C₄⋊C₄	1	2	yes	—
55	C₄⋊Dic₇₃	—	2	2	no	—
56	C₄⋊Dic₇₃	M₁₆	1	4	yes	—
57	C₄×Dic₇₃	—	4	2	no	—
58	C₄×Dic₇₃	—	4	2	no	—
59	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
60	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
61	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
62	C₄×Dic₇₃	#41	1	1	yes	characteristic
63	C₂²×C₄×C₇₃	—	2	2	no	—
64	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₇₃	—	2	2	no	—
66	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
67	C₄⋊C₄×C₇₃	—	2	2	no	—
68	C₄⋊C₄×C₇₃	—	2	2	no	—
69	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₇₃	—	4	2	no	—
71	C₂×C₄×C₇₃	—	4	2	no	—
72	C₂×C₄×C₇₃	—	4	2	no	—
73	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
76	C₂×C₄×C₇₃	#131	1	2	yes	—
77	C₂×Dic₇₃	—	2	2	no	—
78	C₂×Dic₇₃	—	2	4	no	—
79	C₂×Dic₇₃	—	4	2	no	—
80	C₂×Dic₇₃	—	4	2	no	—
81	C₂×Dic₇₃	#121	1	1	yes	characteristic
82	C₂×Dic₇₃	C₂²⋊Q₈	1	1	yes	characteristic
83	C₂×Dic₇₃	#125	1	1	yes	characteristic
84	C₂×Dic₇₃	C₄⋊Q₈	1	1	yes	characteristic
85	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
86	C₂×Dic₇₃	C₂×Q₁₆	1	1	yes	characteristic
87	C₂×Dic₇₃	#135	1	1	yes	characteristic
88	C₂×Dic₇₃	#135	1	1	yes	characteristic
89	#10596833	—	2	2	no	—
90	#10596833	—	2	2	no	—
91	#10596833	—	4	2	no	—
92	#10596833	—	8	2	no	—
93	C₈×C₇₃	—	2	2	no	—
94	#10596831	—	2	2	no	—
95	C₂²×C₇₃	#389	1	1	yes	characteristic
96	C₂²×C₇₃	#435	1	2	yes	—
97	C₂²×C₇₃	C₄×D₈	1	1	yes	characteristic
98	C₂²×C₇₃	C₄×Q₁₆	1	1	yes	characteristic
99	C₂²×C₇₃	#444	1	2	yes	—
100	C₄×C₇₃	—	8	2	no	—
101	C₄×C₇₃	#423	1	1	yes	characteristic
102	C₄×C₇₃	#424	1	1	yes	characteristic
103	C₄×C₇₃	#446	1	2	yes	—
104	Dic₇₃	—	2	1	no	—
105	Dic₇₃	—	2	1	no	—
106	Dic₇₃	—	4	4	no	—
107	Dic₇₃	#466	1	1	yes	characteristic
108	Dic₇₃	#467	1	1	yes	characteristic
109	Dic₇₃	#499	1	1	yes	characteristic
110	Dic₇₃	#500	1	1	yes	characteristic
111	#9598	—	73	1	no	maximal, Sylow
112	C₂×C₇₃	#1581	1	2	yes	—
113	C₂×C₇₃	#1584	1	2	yes	—
114	C₂×C₇₃	#1940	1	1	yes	characteristic
115	C₂×C₇₃	#1942	1	1	yes	characteristic
116	C₂×C₇₃	#1946	1	1	yes	characteristic
117	#1939	—	73	2	no	—
118	C₄×#107	—	73	1	no	—
119	C₂×#335	—	73	1	no	—
120	C₂×#322	—	73	1	no	—
121	#1279	—	73	2	no	—
122	C₇₃	#9598	1	1	yes	characteristic, complemented, Sylow
123	C₂×#107	—	73	2	no	—
124	C₂×C₄⋊C₈	—	73	2	no	—
125	C₂×C₄⋊C₈	—	146	2	no	—
126	C₂×C₄×C₈	—	73	1	no	—
127	C₂×C₄×C₈	—	73	1	no	—
128	C₂×C₄×C₈	—	73	1	no	—
129	C₄×C₄⋊C₄	—	73	2	no	—
130	#340	—	73	1	no	—
131	#340	#64408	1	1	yes	characteristic
132	#336	—	146	2	no	—
133	#335	—	146	2	no	—
134	#322	—	146	2	no	—
135	C₂²×C₈	—	73	1	no	—
136	C₂²×C₈	—	73	1	no	—
137	C₂²×C₈	—	73	1	no	—
138	C₂²×C₈	—	146	2	no	—
139	C₂²×C₈	—	146	1	no	—
140	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
141	C₂×C₄⋊C₄	—	73	2	no	—
142	C₂×C₄⋊C₄	C₂×#64408	1	2	yes	—
143	C₂×C₄²	—	73	1	no	—
144	C₂×C₄²	—	146	2	no	—
145	#107	—	146	4	no	—
146	C₄⋊C₈	—	292	4	no	—
147	C₄⋊C₈	—	146	2	no	—
148	C₄⋊C₈	—	146	2	no	—
149	C₄×C₈	—	73	2	no	—
150	C₄×C₈	—	73	2	no	—
151	C₄×C₈	—	73	2	no	—
152	C₄×C₈	—	146	1	no	—
153	C₄×C₈	—	146	1	no	—
154	C₄×C₈	—	146	1	no	—
155	#95	—	146	2	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	73	1	no	—
158	C₂²×C₄	—	73	1	no	—
159	C₂²×C₄	—	146	2	no	—
160	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	292	2	no	—
163	C₂×C₈	—	292	2	no	—
164	C₂×C₈	—	292	2	no	—
165	C₂×C₈	—	292	2	no	—
166	C₂×C₈	—	73	2	no	—
167	C₂×C₈	—	73	2	no	—
168	C₂×C₈	—	73	2	no	—
169	C₂×C₈	—	146	2	no	—
170	C₂×C₈	—	146	1	no	—
171	C₂×C₈	—	146	1	no	—
172	C₂×C₈	—	146	1	no	—
173	C₂×C₈	—	146	1	no	—
174	C₂×C₈	—	146	2	no	—
175	C₂×C₈	C₄⋊#64408	1	2	yes	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	73	4	no	—
179	C₄⋊C₄	—	146	2	no	—
180	C₄²	—	292	2	no	—
181	C₄²	—	292	2	no	—
182	C₄²	—	73	1	no	—
183	C₄²	—	73	1	no	—
184	C₄²	—	73	1	no	—
185	C₄²	—	73	1	no	—
186	C₂³	D₄×#64408	1	1	yes	characteristic
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	292	2	no	—
191	C₂×C₄	—	292	2	no	—
192	C₂×C₄	—	73	1	no	—
193	C₂×C₄	—	73	1	no	—
194	C₂×C₄	—	73	1	no	—
195	C₂×C₄	—	73	1	no	—
196	C₂×C₄	—	73	1	no	—
197	C₂×C₄	—	73	1	no	—
198	C₂×C₄	—	73	1	no	—
199	C₂×C₄	—	73	1	no	—
200	C₂×C₄	—	146	4	no	—
201	C₂×C₄	—	146	2	no	—
202	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
203	C₂×C₄	Q₈×#64408	1	1	yes	characteristic
204	C₂×C₄	D₇₃⋊M₁₆	1	2	yes	—
205	C₂×C₄	#14556678	1	2	yes	—
206	C₈	—	2	2	no	—
207	C₈	—	292	2	no	—
208	C₈	—	584	2	no	—
209	C₈	—	146	2	no	—
210	C₈	—	146	2	no	—
211	C₈	—	146	2	no	—
212	C₂²	#17989538	1	1	yes	characteristic
213	C₂²	#17989619	1	2	yes	—
214	C₂²	D₈×#64408	1	1	yes	characteristic
215	C₂²	Q₁₆×#64408	1	1	yes	characteristic
216	C₂²	#17989632	1	2	yes	—
217	C₄	—	8	2	no	—
218	C₄	—	292	4	no	—
219	C₄	—	73	1	no	—
220	C₄	—	73	1	no	—
221	C₄	—	73	1	no	—
222	C₄	—	73	1	no	—
223	C₄	—	146	1	no	—
224	C₄	—	146	1	no	—
225	C₄	#17989592	1	1	yes	characteristic
226	C₄	#17989593	1	1	yes	characteristic
227	C₄	#17989639	1	2	yes	—
228	C₂	#22625533	1	2	yes	—
229	C₂	#22625545	1	2	yes	—
230	C₂	#22626032	1	1	yes	characteristic
231	C₂	#22626035	1	1	yes	characteristic
232	C₂	#22626041	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626029

C₂

yes

maximal

C₂×#17989450

C₂

yes

maximal, characteristic

C₂×#17989429

C₂

yes

maximal, characteristic

#22624767

C₂

yes

maximal

#22609823

C₂

yes

maximal, characteristic

C₂×#14556658

—

C₂×C₄⋊#10596833

C₂²

yes

—

C₂×C₄×#10596833

C₂²

yes

characteristic

C₂×C₄×#10596833

C₂²

yes

—

—

—

C₄

yes

—

C₂×C₈×Dic₇₃

C₂²

yes

characteristic

C₄⋊C₄×Dic₇₃

C₂²

yes

—

#17988350

C₄

yes

characteristic

#340×C₇₃

C₄

yes

—

—

D₄

yes

characteristic

C₂²×#10596833

D₄

yes

—

—

—

—

—

—

Q₈

yes

—

C₂²×#10596831

C₂×C₄

yes

characteristic

C₂×C₄⋊Dic₇₃

C₂×C₄

yes

—

C₂×C₄×Dic₇₃

—

C₂×C₄×Dic₇₃

C₂³

yes

characteristic

#14556524

—

C₈×Dic₇₃

D₄

yes

—

C₈×Dic₇₃

Q₈

yes

—

#14556498

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×C₄⋊C₄×C₇₃

C₂×C₄

yes

—

—

—

—

—

—

—

—

—

—

D₈

yes

—

C₂×#10596833

Q₁₆

yes

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

C₂×D₄

yes

C₂²×Dic₇₃

yes

—

C₄⋊C₄

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

M₁₆

yes

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

C₂×D₄

yes

characteristic

C₄×Dic₇₃

C₂×Q₈

yes

characteristic

C₄×Dic₇₃

C₂×Q₈

yes

characteristic

C₄×Dic₇₃

#41

yes

characteristic

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₂³×C₇₃

C₄×D₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₄×Q₈

yes

characteristic

C₂×C₄×C₇₃

C₂×M₁₆

yes

—

C₂×C₄×C₇₃

#131

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

#121

yes

characteristic

C₂×Dic₇₃

C₂²⋊Q₈

yes

characteristic

C₂×Dic₇₃

#125

yes

characteristic

C₂×Dic₇₃

C₄⋊Q₈

yes

characteristic

C₂×Dic₇₃

C₂×D₈

yes

characteristic

C₂×Dic₇₃

C₂×Q₁₆

yes

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

—

—

—

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

#389

yes

characteristic

C₂²×C₇₃

#435

yes

—

C₂²×C₇₃

C₄×D₈

yes

characteristic

C₂²×C₇₃

C₄×Q₁₆

yes

characteristic

C₂²×C₇₃

#444

yes

—

100

C₄×C₇₃

—

101

C₄×C₇₃

#423

yes

characteristic

102

C₄×C₇₃

#424

yes

characteristic

103

C₄×C₇₃

#446

yes

—

104

Dic₇₃

—

105

Dic₇₃

—

106

Dic₇₃

—

107

Dic₇₃

#466

yes

characteristic

108

Dic₇₃

#467

yes

characteristic

109

Dic₇₃

#499

yes

characteristic

110

Dic₇₃

yes

111

—

112

C₂×C₇₃

#1581

yes

—

113

C₂×C₇₃

#1584

yes

—

114

C₂×C₇₃

#1940

yes

characteristic

115

C₂×C₇₃

#1942

yes

characteristic

116

C₂×C₇₃

#1946

yes

characteristic

117

#1939

—

118

C₄×#107

—

119

C₂×#335

—

120

C₂×#322

—

121

#1279

—

122

C₇₃

#9598

yes

characteristic, complemented, Sylow

123

C₂×#107

—

124

C₂×C₄⋊C₈

—

125

C₂×C₄⋊C₈

—

146

—

126

C₂×C₄×C₈

—

127

C₂×C₄×C₈

—

128

C₂×C₄×C₈

—

129

C₄×C₄⋊C₄

—

130

#340

—

131

yes

132

—

146

—

133

#335

—

146

—

134

#322

—

146

—

135

C₂²×C₈

—

136

C₂²×C₈

—

137

C₂²×C₈

—

138

C₂²×C₈

—

146

—

139

C₂²×C₈

—

146

—

140

C₂²×C₈

C₂×#64408

yes

characteristic

141

C₂×C₄⋊C₄

—

142

C₂×C₄⋊C₄

C₂×#64408

yes

—

143

C₂×C₄²

—

144

C₂×C₄²

—

146

—

145

#107

—

146

—

146

C₄⋊C₈

—

292

—

147

C₄⋊C₈

—

146

—

148

C₄⋊C₈

—

146

—

149

C₄×C₈

—

150

C₄×C₈

—

151

C₄×C₈

—

152

C₄×C₈

—

146

—

153

C₄×C₈

—

146

—

154

C₄×C₈

—

146

—

155

#95

—

146

—

156

C₂²×C₄

—

157

C₂²×C₄

—

158

C₂²×C₄

—

159

C₂²×C₄

—

146

—

160

C₂²×C₄

C₂²×#64408

yes

characteristic

161

C₂×C₈

—

162

C₂×C₈

—

292

—

163

C₂×C₈

—

292

—

164

C₂×C₈

—

292

—

165

C₂×C₈

—

292

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

146

—

170

C₂×C₈

—

146

—

171

C₂×C₈

—

146

—

172

C₂×C₈

—

146

—

173

C₂×C₈

—

146

—

174

C₂×C₈

—

146

—

175

C₂×C₈

C₄⋊#64408

yes

—

176

C₄⋊C₄

—

177

C₄⋊C₄

—

178

C₄⋊C₄

—

179

C₄⋊C₄

—

146

—

180

C₄²

—

292

—

181

C₄²

—

292

—

182

C₄²

—

183

C₄²

—

184

C₄²

—

185

C₄²

—

186

C₂³

D₄×#64408

yes

characteristic

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

292

—

191

C₂×C₄

—

292

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

146

—

201

C₂×C₄

—

146

—

202

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

203

C₂×C₄

Q₈×#64408

yes

characteristic

204

C₂×C₄

D₇₃⋊M₁₆

yes

—

205

C₂×C₄

#14556678

yes

—

206

C₈

—

207

C₈

—

292

—

208

C₈

—

584

—

209

C₈

—

146

—

210

C₈

—

146

—

211

C₈

—

146

—

212

C₂²

#17989538

yes

characteristic

213

C₂²

#17989619

yes

—

214

C₂²

D₈×#64408

yes

characteristic

215

C₂²

Q₁₆×#64408

yes

characteristic

216

C₂²

#17989632

yes

—

217

C₄

—

218

C₄

—

292

—

219

C₄

—

220

C₄

—

221

C₄

—

222

C₄

—

223

C₄

—

146

—

224

C₄

—

146

—

225

C₄

#17989592

yes

characteristic

226

C₄

#17989593

yes

characteristic

227

C₄

#17989639

yes

—

228

C₂

#22625533

yes

—

229

C₂

#22625545

yes

—

230

C₂

#22626032

yes

characteristic

231

C₂

#22626035

yes

characteristic

232

C₂

#22626041

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×#64408
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×#64408
Fitting	#340×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₄×D₄
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	#340	#64408
hypocenter	C₇₃	#9598
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×#64408

commutator

C₂²×C₄×C₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×#64408

Fitting

#340×C₇₃

C₄

socle, prime radical

C₂³×C₇₃

C₄×D₄

cosocle

C₂×C₄×Dic₇₃

C₂³

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
122. C₇₃	#9598	semidirect

H ≅ G/N

type

122. C₇₃

#9598

semidirect

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

subgroup	index
#22626029	2
C₂×#17989450	2
C₂×#17989429	2
#22624767	2
#22609823	2
#9598	73

subgroup

index

Subgroups — #34007488

Conjugacy classes of subgroups

Complements

Maximal subgroups