Group #32990437 — subgroups

Subgroups — #32990437

Order 18688.

406 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 129 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	#22615173	C₂	1	1	yes	maximal, characteristic
2	#22610695	C₂	1	2	yes	maximal
3	#22610090	C₂	1	1	yes	maximal, characteristic
4	#22607770	C₂	1	2	yes	maximal
5	C₇₃×#1296	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556524	—	2	2	no	—
7	C₂×#14556523	—	2	2	no	—
8	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
9	#17988526	—	2	2	no	—
10	#17988486	—	2	2	no	—
11	#17988457	C₂²	1	1	yes	characteristic
12	#17988452	—	2	2	no	—
13	C₄×C₄⋊Dic₇₃	C₂²	1	2	yes	—
14	#17988350	—	2	2	no	—
15	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
16	#384×C₇₃	C₂²	1	1	yes	characteristic
17	C₂²×#10596831	—	2	1	no	—
18	C₂×C₄⋊Dic₇₃	—	2	2	no	—
19	C₂×C₄⋊Dic₇₃	—	2	2	no	—
20	C₂×C₄⋊Dic₇₃	—	2	1	no	—
21	C₂×C₄⋊Dic₇₃	D₄	1	2	yes	—
22	C₂×C₄⋊Dic₇₃	D₄	1	2	yes	—
23	C₂×C₄×Dic₇₃	—	2	2	no	—
24	#14556524	—	4	4	no	—
25	#14556523	—	4	4	no	—
26	C₄×#10596831	—	2	1	no	—
27	C₄×#10596831	Q₈	1	2	yes	—
28	#14556498	—	2	2	no	—
29	#14556498	—	2	4	no	—
30	#14556498	—	2	2	no	—
31	C₂²×C₈×C₇₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₇₃	D₄	1	2	yes	—
34	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₇₃	—	2	2	no	—
36	C₄⋊C₈×C₇₃	Q₈	1	4	yes	—
37	C₂²×Dic₇₃	—	2	2	no	—
38	C₂²×Dic₇₃	—	2	2	no	—
39	C₂²×Dic₇₃	—	4	1	no	—
40	C₂×#10596831	—	2	2	no	—
41	C₂×#10596831	—	2	2	no	—
42	C₂×#10596831	—	2	2	no	—
43	C₄⋊Dic₇₃	—	2	2	no	—
44	C₄⋊Dic₇₃	—	2	2	no	—
45	C₄⋊Dic₇₃	—	2	2	no	—
46	C₄⋊Dic₇₃	—	2	2	no	—
47	C₄⋊Dic₇₃	—	4	2	no	—
48	C₄⋊Dic₇₃	—	4	2	no	—
49	C₄⋊Dic₇₃	—	4	2	no	—
50	C₄⋊Dic₇₃	—	8	2	no	—
51	C₄×Dic₇₃	—	4	2	no	—
52	C₄×Dic₇₃	—	4	2	no	—
53	C₂²×C₄×C₇₃	—	2	2	no	—
54	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₇₃	—	2	4	no	—
57	C₂×C₈×C₇₃	—	4	2	no	—
58	C₂×C₈×C₇₃	—	4	2	no	—
59	C₄⋊C₄×C₇₃	—	4	2	no	—
60	C₄⋊C₄×C₇₃	—	4	2	no	—
61	C₄⋊C₄×C₇₃	—	4	2	no	—
62	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
63	C₄²×C₇₃	C₂×Q₈	1	2	yes	—
64	C₄²×C₇₃	#41	1	1	yes	characteristic
65	C₂³×C₇₃	#120	1	1	yes	characteristic
66	C₂×C₄×C₇₃	—	2	1	no	—
67	C₂×C₄×C₇₃	—	2	1	no	—
68	C₂×C₄×C₇₃	—	4	4	no	—
69	C₂×C₄×C₇₃	—	4	2	no	—
70	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
71	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
73	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₇₃	#135	1	2	yes	—
75	C₂×C₄×C₇₃	#135	1	2	yes	—
76	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
77	C₂×C₄×C₇₃	#137	1	2	yes	—
78	C₂×Dic₇₃	—	4	2	no	—
79	C₂×Dic₇₃	—	4	2	no	—
80	C₂×Dic₇₃	—	4	2	no	—
81	C₂×Dic₇₃	—	4	2	no	—
82	C₂×Dic₇₃	—	4	2	no	—
83	C₂×Dic₇₃	—	4	2	no	—
84	C₂×Dic₇₃	—	8	2	no	—
85	C₂×Dic₇₃	—	8	1	no	—
86	C₈×C₇₃	—	4	4	no	—
87	#10596831	—	2	4	no	—
88	C₂²×C₇₃	#395	1	1	yes	characteristic
89	C₂²×C₇₃	#449	1	2	yes	—
90	C₂²×C₇₃	#452	1	2	yes	—
91	C₂²×C₇₃	#453	1	1	yes	characteristic
92	C₂²×C₇₃	#456	1	1	yes	characteristic
93	C₄×C₇₃	—	2	1	no	—
94	C₄×C₇₃	—	2	1	no	—
95	C₄×C₇₃	—	8	2	no	—
96	C₄×C₇₃	#478	1	2	yes	—
97	C₄×C₇₃	#479	1	2	yes	—
98	C₄×C₇₃	#499	1	2	yes	—
99	C₄×C₇₃	#501	1	2	yes	—
100	Dic₇₃	—	8	2	no	—
101	Dic₇₃	—	8	2	no	—
102	Dic₇₃	—	16	1	no	—
103	#10576	—	73	1	no	maximal, Sylow
104	C₂×C₇₃	#1620	1	1	yes	characteristic
105	C₂×C₇₃	#1634	1	1	yes	characteristic
106	C₂×C₇₃	#1650	1	2	yes	—
107	C₂×C₇₃	#2059	1	2	yes	—
108	C₂×C₇₃	#2060	1	1	yes	characteristic
109	#2571	—	73	1	no	—
110	#1948	—	73	2	no	—
111	#1845	—	73	1	no	—
112	#1296	D₇₃	1	1	yes	characteristic
113	#1279	—	73	2	no	—
114	C₇₃	#10576	1	1	yes	characteristic, complemented, Sylow
115	C₂×#107	—	146	2	no	—
116	C₂×#106	—	146	2	no	—
117	C₂×C₄⋊C₈	C₂×D₇₃	1	2	yes	—
118	C₂×C₄×C₈	—	73	1	no	—
119	#400	—	146	2	no	—
120	#389	—	146	2	no	—
121	#384	—	73	1	no	—
122	#384	C₂×D₇₃	1	1	yes	characteristic
123	#382	—	146	2	no	—
124	C₄×C₄⋊C₄	—	73	2	no	—
125	#340	—	146	2	no	—
126	C₂²×C₈	—	2	2	no	—
127	C₂²×C₈	—	146	1	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	73	2	no	—
130	C₂×C₄⋊C₄	—	73	2	no	—
131	C₂×C₄⋊C₄	—	146	2	no	—
132	C₂×C₄⋊C₄	—	146	1	no	—
133	C₂×C₄⋊C₄	—	146	2	no	—
134	C₂×C₄⋊C₄	D₂₉₂	1	2	yes	—
135	C₂×C₄²	—	146	2	no	—
136	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
137	#107	—	292	4	no	—
138	#106	—	292	4	no	—
139	C₄⋊C₈	—	2	2	no	—
140	C₄⋊C₈	Dic₁₄₆	1	4	yes	—
141	C₄×C₈	—	73	2	no	—
142	C₄×C₈	—	146	1	no	—
143	#95	—	146	2	no	—
144	#95	—	146	4	no	—
145	#95	—	146	2	no	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	292	1	no	—
148	C₂²×C₄	—	146	2	no	—
149	C₂²×C₄	—	146	2	no	—
150	C₂²×C₄	D₄×D₇₃	1	2	yes	—
151	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	—	146	2	no	—
156	C₂×C₈	—	146	2	no	—
157	C₂×C₈	—	146	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	292	2	no	—
162	C₄⋊C₄	—	292	2	no	—
163	C₄⋊C₄	—	292	2	no	—
164	C₄⋊C₄	—	584	2	no	—
165	C₄⋊C₄	—	146	2	no	—
166	C₄⋊C₄	—	146	2	no	—
167	C₄⋊C₄	—	146	2	no	—
168	C₄⋊C₄	—	146	2	no	—
169	C₄²	—	292	2	no	—
170	C₄²	—	292	2	no	—
171	C₄²	Q₈×D₇₃	1	1	yes	characteristic
172	C₄²	C₂×Dic₁₄₆	1	2	yes	—
173	C₄²	#11924027	1	1	yes	characteristic
174	C₂³	#14556561	1	1	yes	characteristic
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	4	4	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	292	2	no	—
180	C₂×C₄	—	292	2	no	—
181	C₂×C₄	—	292	2	no	—
182	C₂×C₄	—	292	2	no	—
183	C₂×C₄	—	292	2	no	—
184	C₂×C₄	—	292	2	no	—
185	C₂×C₄	—	584	2	no	—
186	C₂×C₄	—	584	1	no	—
187	C₂×C₄	#14556568	1	2	yes	—
188	C₂×C₄	#14556571	1	1	yes	characteristic
189	C₂×C₄	#14556590	1	2	yes	—
190	C₂×C₄	#14556591	1	1	yes	characteristic
191	C₂×C₄	#14556612	1	2	yes	—
192	C₂×C₄	#14556612	1	2	yes	—
193	C₂×C₄	#14556615	1	2	yes	—
194	C₂×C₄	#14556623	1	2	yes	—
195	C₈	—	4	4	no	—
196	C₈	—	146	4	no	—
197	C₂²	#17988508	1	1	yes	characteristic
198	C₂²	#17988733	1	2	yes	—
199	C₂²	#17988752	1	2	yes	—
200	C₂²	#17988757	1	1	yes	characteristic
201	C₂²	#17988772	1	1	yes	characteristic
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	8	2	no	—
205	C₄	—	584	2	no	—
206	C₄	—	584	2	no	—
207	C₄	—	1168	1	no	—
208	C₄	#17988889	1	2	yes	—
209	C₄	#17988892	1	2	yes	—
210	C₄	#17988987	1	2	yes	—
211	C₄	#17988995	1	2	yes	—
212	C₂	#22608991	1	1	yes	characteristic
213	C₂	#22609077	1	1	yes	characteristic
214	C₂	#22609188	1	2	yes	—
215	C₂	#22611268	1	2	yes	—
216	C₂	#22611277	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22615173

C₂

yes

maximal, characteristic

#22610695

C₂

yes

maximal

#22610090

C₂

yes

maximal, characteristic

#22607770

C₂

yes

maximal

C₇₃×#1296

C₂

yes

maximal, characteristic

C₂×#14556524

—

C₂×#14556523

—

C₂×C₄×#10596831

C₂²

yes

—

—

C₂²

yes

characteristic

#17988452

—

C₄×C₄⋊Dic₇₃

C₂²

yes

—

#17988350

—

C₂×C₄⋊C₈×C₇₃

C₂²

yes

—

#384×C₇₃

C₂²

yes

characteristic

C₂²×#10596831

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

D₄

yes

—

C₂×C₄⋊Dic₇₃

D₄

yes

—

C₂×C₄×Dic₇₃

—

—

—

—

Q₈

yes

—

#14556498

—

#14556498

—

#14556498

—

C₂²×C₈×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

D₄

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄⋊C₈×C₇₃

—

C₄⋊C₈×C₇₃

Q₈

yes

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₂×#10596831

—

C₂×#10596831

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×Q₈

yes

characteristic

C₄²×C₇₃

C₂×Q₈

yes

—

C₄²×C₇₃

yes

C₂³×C₇₃

yes

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

C₄⋊Q₈

yes

—

C₂×C₄×C₇₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

#135

yes

—

C₂×C₄×C₇₃

#135

yes

—

C₂×C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇₃

#137

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

#395

yes

characteristic

C₂²×C₇₃

#449

yes

—

C₂²×C₇₃

#452

yes

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#478

yes

—

C₄×C₇₃

#479

yes

—

C₄×C₇₃

#499

yes

—

C₄×C₇₃

#501

yes

—

100

Dic₇₃

—

101

Dic₇₃

—

102

Dic₇₃

—

103

#10576

—

maximal, Sylow

104

C₂×C₇₃

#1620

yes

characteristic

105

C₂×C₇₃

#1634

yes

characteristic

106

C₂×C₇₃

#1650

yes

—

107

C₂×C₇₃

#2059

yes

—

108

C₂×C₇₃

#2060

yes

characteristic

109

#2571

—

110

#1948

—

111

#1845

—

112

#1296

D₇₃

yes

characteristic

113

#1279

—

114

C₇₃

#10576

yes

characteristic, complemented, Sylow

115

C₂×#107

—

146

—

116

C₂×#106

—

146

—

117

C₂×C₄⋊C₈

C₂×D₇₃

yes

—

118

C₂×C₄×C₈

—

119

#400

—

146

—

120

#389

—

146

—

121

#384

—

122

#384

C₂×D₇₃

yes

characteristic

123

#382

—

146

—

124

C₄×C₄⋊C₄

—

125

#340

—

146

—

126

C₂²×C₈

—

127

C₂²×C₈

—

146

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

131

C₂×C₄⋊C₄

—

146

—

132

C₂×C₄⋊C₄

—

146

—

133

C₂×C₄⋊C₄

—

146

—

134

C₂×C₄⋊C₄

D₂₉₂

yes

—

135

C₂×C₄²

—

146

—

136

C₂×C₄²

C₂²×D₇₃

yes

characteristic

137

#107

—

292

—

138

#106

—

292

—

139

C₄⋊C₈

—

140

C₄⋊C₈

Dic₁₄₆

yes

—

141

C₄×C₈

—

142

C₄×C₈

—

146

—

143

#95

—

146

—

144

#95

—

146

—

145

#95

—

146

—

146

C₂²×C₄

—

147

C₂²×C₄

—

292

—

148

C₂²×C₄

—

146

—

149

C₂²×C₄

—

146

—

150

C₂²×C₄

D₄×D₇₃

yes

—

151

C₂²×C₄

C₂×D₂₉₂

yes

characteristic

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

146

—

156

C₂×C₈

—

146

—

157

C₂×C₈

—

146

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

292

—

162

C₄⋊C₄

—

292

—

163

C₄⋊C₄

—

292

—

164

C₄⋊C₄

—

584

—

165

C₄⋊C₄

—

146

—

166

C₄⋊C₄

—

146

—

167

C₄⋊C₄

—

146

—

168

C₄⋊C₄

—

146

—

169

C₄²

—

292

—

170

C₄²

—

292

—

171

C₄²

Q₈×D₇₃

yes

characteristic

172

C₄²

C₂×Dic₁₄₆

yes

—

173

C₄²

#11924027

yes

characteristic

174

C₂³

#14556561

yes

characteristic

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

292

—

180

C₂×C₄

—

292

—

181

C₂×C₄

—

292

—

182

C₂×C₄

—

292

—

183

C₂×C₄

—

292

—

184

C₂×C₄

—

292

—

185

C₂×C₄

—

584

—

186

C₂×C₄

—

584

—

187

C₂×C₄

#14556568

yes

—

188

C₂×C₄

#14556571

yes

characteristic

189

C₂×C₄

#14556590

yes

—

190

C₂×C₄

#14556591

yes

characteristic

191

C₂×C₄

#14556612

yes

—

192

C₂×C₄

#14556612

yes

—

193

C₂×C₄

#14556615

yes

—

194

C₂×C₄

#14556623

yes

—

195

C₈

—

196

C₈

—

146

—

197

C₂²

#17988508

yes

characteristic

198

C₂²

#17988733

yes

—

199

C₂²

#17988752

yes

—

200

C₂²

#17988757

yes

characteristic

201

C₂²

#17988772

yes

characteristic

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

—

584

—

206

C₄

—

584

—

207

C₄

—

1168

—

208

C₄

#17988889

yes

—

209

C₄

#17988892

yes

—

210

C₄

#17988987

yes

—

211

C₄

#17988995

yes

—

212

C₂

#22608991

yes

characteristic

213

C₂

#22609077

yes

characteristic

214

C₂

#22609188

yes

—

215

C₂

#22611268

yes

—

216

C₂

#22611277

yes

characteristic

	subgroup	G/H
center	C₂³	#14556561
commutator, cosocle	C₂×C₄²×C₇₃	C₂³
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1296	C₂
socle, prime radical	C₂³×C₇₃	#120
hypercenter	#1296	D₇₃
hypocenter	C₇₃	#10576
LCS	G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1296

subgroup

G/H

center

C₂³

#14556561

commutator, cosocle

C₂×C₄²×C₇₃

C₂³

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1296

C₂

C₂³×C₇₃

D₇₃

C₇₃

G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1296

N	H ≅ G/N	type
114. C₇₃	#10576	semidirect

H ≅ G/N

type

114. C₇₃

#10576

semidirect

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

subgroup	index
#22615173	2
#22610695	2
#22610090	2
#22607770	2
C₇₃×#1296	2
#10576	73

subgroup

index

C₇₃×#1296

Subgroups — #32990437

Conjugacy classes of subgroups

Complements

Maximal subgroups