Group #32986126 — subgroups

Subgroups — #32986126

Order 18688.

438 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 159 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	#22618294	C₂	1	1	yes	maximal, characteristic
2	#22609784	C₂	1	1	yes	maximal, characteristic
3	#22607815	C₂	1	4	yes	maximal
4	C₇₃×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556583	—	2	1	no	—
6	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
7	C₂×#14556502	—	2	1	no	—
8	#17988485	—	2	2	no	—
9	#17988485	C₄	1	4	yes	—
10	#17988457	C₂²	1	2	yes	—
11	#17988453	—	2	1	no	—
12	#17988342	—	2	1	no	—
13	C₂×C₄⋊C₈×C₇₃	—	2	2	no	—
14	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
15	C₄³×C₇₃	C₂²	1	1	yes	characteristic
16	C₂²×#10596831	—	2	2	no	—
17	#14556583	—	4	4	no	—
18	C₂×C₄⋊Dic₇₃	—	2	2	no	—
19	C₂×C₄⋊Dic₇₃	—	2	2	no	—
20	C₂×C₄⋊Dic₇₃	C₂×C₄	1	4	yes	—
21	C₂×#11924007	—	2	4	no	—
22	C₄⋊#10596831	—	2	2	no	—
23	C₄⋊#10596831	Q₈	1	4	yes	—
24	#14556502	—	4	2	no	—
25	#14556498	—	2	4	no	—
26	C₂²×C₈×C₇₃	—	2	2	no	—
27	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₇₃	D₄	1	2	yes	—
30	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₇₃	—	2	4	no	—
33	C₄⋊C₈×C₇₃	—	4	4	no	—
34	C₂²×Dic₇₃	—	2	4	no	—
35	C₂×#10596831	—	2	4	no	—
36	C₂×#10596831	—	4	2	no	—
37	C₂×#10596831	—	4	2	no	—
38	C₄⋊Dic₇₃	—	4	2	no	—
39	C₄⋊Dic₇₃	—	4	4	no	—
40	C₄⋊Dic₇₃	—	4	2	no	—
41	C₄⋊Dic₇₃	—	4	4	no	—
42	#11924007	—	4	8	no	—
43	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₇₃	—	4	2	no	—
49	C₂×C₈×C₇₃	—	4	4	no	—
50	C₄²×C₇₃	—	2	2	no	—
51	C₄²×C₇₃	—	2	2	no	—
52	C₄²×C₇₃	—	2	2	no	—
53	C₄²×C₇₃	—	4	2	no	—
54	C₄²×C₇₃	SD₁₆	1	2	yes	—
55	C₄²×C₇₃	Q₁₆	1	2	yes	—
56	C₄²×C₇₃	C₂×Q₈	1	2	yes	—
57	C₄²×C₇₃	#41	1	2	yes	—
58	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₇₃	—	2	2	no	—
60	C₂×C₄×C₇₃	—	2	2	no	—
61	C₂×C₄×C₇₃	—	2	2	no	—
62	C₂×C₄×C₇₃	—	2	2	no	—
63	C₂×C₄×C₇₃	—	2	2	no	—
64	C₂×C₄×C₇₃	—	2	2	no	—
65	C₂×C₄×C₇₃	#103	1	4	yes	—
66	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
67	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
68	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
69	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
72	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
73	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
74	C₂×C₄×C₇₃	#137	1	2	yes	—
75	C₂×Dic₇₃	—	4	4	no	—
76	C₂×Dic₇₃	—	4	4	no	—
77	C₂×Dic₇₃	—	4	4	no	—
78	C₈×C₇₃	—	8	2	no	—
79	#10596831	—	4	4	no	—
80	C₂²×C₇₃	#391	1	1	yes	characteristic
81	C₂²×C₇₃	#413	1	2	yes	—
82	C₂²×C₇₃	C₂×#103	1	1	yes	characteristic
83	C₂²×C₇₃	#417	1	2	yes	—
84	C₂²×C₇₃	#418	1	1	yes	characteristic
85	C₄×C₇₃	—	2	4	no	—
86	C₄×C₇₃	—	2	2	no	—
87	C₄×C₇₃	—	4	2	no	—
88	C₄×C₇₃	#476	1	2	yes	—
89	C₄×C₇₃	#477	1	2	yes	—
90	C₄×C₇₃	#490	1	1	yes	characteristic
91	C₄×C₇₃	#491	1	1	yes	characteristic
92	C₄×C₇₃	#501	1	2	yes	—
93	Dic₇₃	—	8	4	no	—
94	#9902	—	73	1	no	maximal, Sylow
95	C₂×C₇₃	#1550	1	2	yes	—
96	C₂×C₇₃	#1557	1	2	yes	—
97	C₂×C₇₃	#1987	1	1	yes	characteristic
98	C₂×C₇₃	#1992	1	1	yes	characteristic
99	C₂×C₇₃	#1998	1	1	yes	characteristic
100	#2859	—	73	1	no	—
101	#1842	D₇₃	1	1	yes	characteristic
102	#1767	—	73	1	no	—
103	#1296	—	73	4	no	—
104	C₇₃	#9902	1	1	yes	characteristic, complemented, Sylow
105	C₂×#125	—	146	1	no	—
106	C₂×C₄⋊C₈	—	2	2	no	—
107	C₂×C₄⋊C₈	—	73	2	no	—
108	C₂×C₄⋊C₈	C₂×D₇₃	1	2	yes	—
109	C₂×#97	—	146	1	no	—
110	#389	—	73	4	no	—
111	#389	—	146	2	no	—
112	#384	—	73	2	no	—
113	#382	—	146	1	no	—
114	C₄³	C₂×D₇₃	1	1	yes	characteristic
115	#336	—	146	1	no	—
116	C₂²×C₈	—	2	2	no	—
117	C₂²×C₈	—	146	2	no	—
118	#125	—	292	4	no	—
119	C₂×C₄⋊C₄	—	73	4	no	—
120	C₂×C₄⋊C₄	—	146	4	no	—
121	C₂×C₄⋊C₄	—	146	2	no	—
122	C₂×C₄⋊C₄	—	146	2	no	—
123	C₂×C₄²	C₄×D₇₃	1	2	yes	—
124	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
125	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
126	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
127	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
128	C₄⋊C₈	—	2	4	no	—
129	C₄⋊C₈	—	4	4	no	—
130	C₄⋊C₈	—	73	4	no	—
131	C₄⋊C₈	—	146	2	no	—
132	#97	—	292	2	no	—
133	#95	—	146	4	no	—
134	C₂²×C₄	—	146	4	no	—
135	C₂²×C₄	#11924005	1	2	yes	—
136	C₂²×C₄	#11924005	1	2	yes	—
137	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
138	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
139	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	4	4	no	—
142	C₂×C₈	—	292	2	no	—
143	C₂×C₈	—	292	2	no	—
144	C₂×C₈	—	146	4	no	—
145	C₄⋊C₄	—	292	4	no	—
146	C₄⋊C₄	—	292	8	no	—
147	C₄⋊C₄	—	292	2	no	—
148	C₄⋊C₄	—	292	4	no	—
149	C₄⋊C₄	—	292	2	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	2	2	no	—
152	C₄²	—	2	2	no	—
153	C₄²	—	4	2	no	—
154	C₄²	#11924017	1	2	yes	—
155	C₄²	Dic₂₉₂	1	2	yes	—
156	C₄²	Q₈×D₇₃	1	2	yes	—
157	C₄²	#11924027	1	2	yes	—
158	C₂³	C₂×#11924005	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	292	4	no	—
166	C₂×C₄	—	292	4	no	—
167	C₂×C₄	—	292	4	no	—
168	C₂×C₄	#14556516	1	4	yes	—
169	C₂×C₄	#14556556	1	2	yes	—
170	C₂×C₄	#14556571	1	2	yes	—
171	C₂×C₄	#14556578	1	1	yes	characteristic
172	C₂×C₄	#14556589	1	1	yes	characteristic
173	C₂×C₄	C₂×#11924017	1	1	yes	characteristic
174	C₂×C₄	C₂×Dic₂₉₂	1	1	yes	characteristic
175	C₂×C₄	#14556617	1	2	yes	—
176	C₂×C₄	#14556619	1	2	yes	—
177	C₂×C₄	#14556622	1	2	yes	—
178	C₈	—	8	2	no	—
179	C₈	—	292	4	no	—
180	C₂²	#17988492	1	1	yes	characteristic
181	C₂²	#17988569	1	2	yes	—
182	C₂²	C₂×#14556516	1	1	yes	characteristic
183	C₂²	#17988586	1	2	yes	—
184	C₂²	#17988593	1	1	yes	characteristic
185	C₄	—	2	2	no	—
186	C₄	—	2	4	no	—
187	C₄	—	4	2	no	—
188	C₄	—	584	4	no	—
189	C₄	#17988878	1	2	yes	—
190	C₄	#17988883	1	2	yes	—
191	C₄	#17988951	1	1	yes	characteristic
192	C₄	#17988955	1	1	yes	characteristic
193	C₄	#17988993	1	2	yes	—
194	C₂	#22608619	1	2	yes	—
195	C₂	#22608651	1	2	yes	—
196	C₂	#22610878	1	1	yes	characteristic
197	C₂	#22610899	1	1	yes	characteristic
198	C₂	#22610923	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22618294

C₂

yes

maximal, characteristic

#22609784

C₂

yes

maximal, characteristic

#22607815

C₂

yes

maximal

C₇₃×#1842

C₂

yes

maximal, characteristic

C₂×#14556583

—

C₂×C₄⋊#10596831

C₂²

yes

—

C₂×#14556502

—

#17988485

—

#17988485

C₄

yes

—

#17988457

C₂²

yes

—

#17988453

—

#17988342

—

C₂×C₄⋊C₈×C₇₃

—

C₂×C₄⋊C₈×C₇₃

C₂²

yes

—

C₄³×C₇₃

C₂²

yes

characteristic

C₂²×#10596831

—

#14556583

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

C₂×C₄

yes

—

C₂×#11924007

—

C₄⋊#10596831

—

C₄⋊#10596831

Q₈

yes

—

#14556502

—

#14556498

—

C₂²×C₈×C₇₃

—

C₂×C₄²×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

D₄

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

—

C₂×C₄²×C₇₃

D₄

yes

characteristic

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄⋊C₈×C₇₃

—

C₄⋊C₈×C₇₃

—

C₂²×Dic₇₃

—

C₂×#10596831

—

C₂×#10596831

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924007

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

SD₁₆

yes

—

C₄²×C₇₃

Q₁₆

yes

—

C₄²×C₇₃

C₂×Q₈

yes

—

C₄²×C₇₃

#41

yes

—

C₂³×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#103

yes

—

C₂×C₄×C₇₃

C₄×Q₈

yes

—

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₃

#124

yes

characteristic

C₂×C₄×C₇₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇₃

#137

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

#391

yes

characteristic

C₂²×C₇₃

#413

yes

—

C₂²×C₇₃

C₂×#103

yes

characteristic

C₂²×C₇₃

#417

yes

—

C₂²×C₇₃

#418

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#476

yes

—

C₄×C₇₃

#477

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

Dic₇₃

—

#9902

—

maximal, Sylow

C₂×C₇₃

#1550

yes

—

C₂×C₇₃

#1557

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

100

—

101

#1842

D₇₃

yes

characteristic

102

#1767

—

103

#1296

—

104

C₇₃

#9902

yes

characteristic, complemented, Sylow

105

C₂×#125

—

146

—

106

C₂×C₄⋊C₈

—

107

C₂×C₄⋊C₈

—

108

C₂×C₄⋊C₈

C₂×D₇₃

yes

—

109

C₂×#97

—

146

—

110

#389

—

111

#389

—

146

—

112

#384

—

113

#382

—

146

—

114

C₄³

C₂×D₇₃

yes

characteristic

115

#336

—

146

—

116

C₂²×C₈

—

117

C₂²×C₈

—

146

—

118

#125

—

292

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

146

—

121

C₂×C₄⋊C₄

—

146

—

122

C₂×C₄⋊C₄

—

146

—

123

C₂×C₄²

C₄×D₇₃

yes

—

124

C₂×C₄²

D₂₉₂

yes

characteristic

125

C₂×C₄²

D₂₉₂

yes

characteristic

126

C₂×C₄²

C₂²⋊D₇₃

yes

—

127

C₂×C₄²

C₂²×D₇₃

yes

characteristic

128

C₄⋊C₈

—

129

C₄⋊C₈

—

130

C₄⋊C₈

—

131

C₄⋊C₈

—

146

—

132

#97

—

292

—

133

#95

—

146

—

134

C₂²×C₄

—

146

—

135

C₂²×C₄

#11924005

yes

—

136

C₂²×C₄

#11924005

yes

—

137

C₂²×C₄

C₂×C₄×D₇₃

yes

characteristic

138

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

139

C₂²×C₄

C₂×D₂₉₂

yes

characteristic

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

292

—

143

C₂×C₈

—

292

—

144

C₂×C₈

—

146

—

145

C₄⋊C₄

—

292

—

146

C₄⋊C₄

—

292

—

147

C₄⋊C₄

—

292

—

148

C₄⋊C₄

—

292

—

149

C₄⋊C₄

—

292

—

150

C₄²

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

154

C₄²

#11924017

yes

—

155

C₄²

Dic₂₉₂

yes

—

156

C₄²

Q₈×D₇₃

yes

—

157

C₄²

#11924027

yes

—

158

C₂³

C₂×#11924005

yes

characteristic

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

292

—

166

C₂×C₄

—

292

—

167

C₂×C₄

—

292

—

168

C₂×C₄

#14556516

yes

—

169

C₂×C₄

#14556556

yes

—

170

C₂×C₄

#14556571

yes

—

171

C₂×C₄

#14556578

yes

characteristic

172

C₂×C₄

#14556589

yes

characteristic

173

C₂×C₄

C₂×#11924017

yes

characteristic

174

C₂×C₄

C₂×Dic₂₉₂

yes

characteristic

175

C₂×C₄

#14556617

yes

—

176

C₂×C₄

#14556619

yes

—

177

C₂×C₄

#14556622

yes

—

178

C₈

—

179

C₈

—

292

—

180

C₂²

#17988492

yes

characteristic

181

C₂²

#17988569

yes

—

182

C₂²

C₂×#14556516

yes

characteristic

183

C₂²

#17988586

yes

—

184

C₂²

#17988593

yes

characteristic

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

584

—

189

C₄

#17988878

yes

—

190

C₄

#17988883

yes

—

191

C₄

#17988951

yes

characteristic

192

C₄

#17988955

yes

characteristic

193

C₄

#17988993

yes

—

194

C₂

#22608619

yes

—

195

C₂

#22608651

yes

—

196

C₂

#22610878

yes

characteristic

197

C₂

#22610899

yes

characteristic

198

C₂

#22610923

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×#11924005
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1842	C₂
socle, prime radical	C₂³×C₇₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1842	D₇₃
hypocenter	C₇₃	#9902
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

subgroup

G/H

center

C₂³

C₂×#11924005

commutator

C₂²×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1842

C₂

socle, prime radical

C₂³×C₇₃

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

N	H ≅ G/N	type
104. C₇₃	#9902	semidirect

H ≅ G/N

type

104. C₇₃

#9902

semidirect

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

subgroup	index
#22618294	2
#22609784	2
#22607815	2
C₇₃×#1842	2
#9902	73

subgroup

index

C₇₃×#1842

Subgroups — #32986126

Conjugacy classes of subgroups

Complements

Maximal subgroups