Group #32997527 — subgroups

Subgroups — #32997527

Order 18688.

270 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 115 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	#22611685	C₂	1	2	yes	maximal
2	C₈²×D₇₃	C₂	1	1	yes	maximal, characteristic
3	#22607995	C₂	1	1	yes	maximal, characteristic, complemented
4	#22607994	C₂	1	2	yes	maximal, complemented
5	C₇₃×#1367	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14556530	—	2	2	no	—
7	C₄×C₈×D₇₃	C₄	1	1	yes	characteristic
8	C₄×C₈×D₇₃	C₄	1	1	yes	characteristic
9	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
10	C₄⋊#11924003	C₂²	1	2	yes	—
11	#17988405	—	2	2	no	—
12	C₈×#10596831	C₄	1	1	yes	characteristic
13	C₈×#10596831	C₄	1	1	yes	characteristic
14	C₈×#10596831	C₂²	1	1	yes	characteristic
15	C₄⋊C₁₆×C₇₃	C₂²	1	2	yes	—
16	C₈²×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₇₃	—	2	1	no	—
18	C₂×C₈×D₇₃	C₈	1	1	yes	characteristic
19	C₂×C₈×D₇₃	C₈	1	1	yes	characteristic
20	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
21	C₂×C₈×D₇₃	Q₈	1	1	yes	characteristic
22	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
23	#14556530	—	4	4	no	—
24	C₂×#11924003	—	2	2	no	—
25	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
26	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
27	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
28	C₈×Dic₇₃	—	2	1	no	—
29	C₈×Dic₇₃	C₈	1	1	yes	characteristic
30	C₈×Dic₇₃	C₈	1	1	yes	characteristic
31	C₈×Dic₇₃	D₄	1	1	yes	characteristic
32	C₈×Dic₇₃	Q₈	1	1	yes	characteristic
33	C₂×C₁₆×C₇₃	—	2	2	no	—
34	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
37	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
39	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
40	C₂×#10596831	—	2	1	no	—
41	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
42	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
43	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
44	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
45	C₈×D₇₃	—	2	4	no	—
46	C₈×D₇₃	—	2	2	no	—
47	C₈×D₇₃	—	2	2	no	—
48	C₈×D₇₃	—	2	2	no	—
49	C₈×D₇₃	SD₁₆	1	4	yes	—
50	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
51	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
52	C₄×Dic₇₃	M₁₆	1	1	yes	characteristic
53	#11924003	—	4	2	no	—
54	C₂×C₈×C₇₃	—	2	1	no	—
55	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
56	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
57	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
60	C₁₆×C₇₃	—	4	2	no	—
61	C₂²×D₇₃	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
65	C₂×Dic₇₃	C₄⋊C₈	1	1	yes	characteristic
66	C₂×Dic₇₃	C₄⋊C₈	1	1	yes	characteristic
67	C₂×Dic₇₃	C₄⋊C₈	1	1	yes	characteristic
68	C₄×D₇₃	—	2	1	no	—
69	C₄×D₇₃	—	2	1	no	—
70	C₄×D₇₃	—	2	2	no	—
71	C₄×D₇₃	#106	1	2	yes	—
72	C₄×D₇₃	#108	1	2	yes	—
73	C₈×C₇₃	—	2	2	no	—
74	C₈×C₇₃	—	2	1	no	—
75	C₈×C₇₃	—	2	1	no	—
76	C₈×C₇₃	M₃₂	1	2	yes	complemented
77	C₈×C₇₃	C₂×SD₁₆	1	2	yes	—
78	#10596831	—	2	2	no	—
79	#10596831	—	2	1	no	—
80	#10596831	—	2	1	no	—
81	#10596831	#106	1	2	yes	—
82	#10596831	M₃₂	1	2	yes	complemented
83	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
84	C₂×D₇₃	—	2	1	no	—
85	C₂×D₇₃	—	2	1	no	—
86	C₂×D₇₃	#334	1	2	yes	—
87	C₄×C₇₃	—	2	1	no	—
88	C₄×C₇₃	C₂×#106	1	1	yes	characteristic
89	C₄×C₇₃	C₂×#108	1	1	yes	characteristic
90	C₄×C₇₃	C₂×M₃₂	1	1	yes	characteristic
91	C₄×C₇₃	C₂×M₃₂	1	1	yes	characteristic
92	Dic₇₃	—	2	1	no	—
93	Dic₇₃	#334	1	2	yes	—
94	Dic₇₃	#364	1	2	yes	—
95	#11996	—	73	1	no	maximal, Sylow
96	C₂×C₇₃	C₂×#334	1	1	yes	characteristic
97	C₂×C₇₃	#2150	1	1	yes	characteristic
98	C₂×C₇₃	C₂×#364	1	1	yes	characteristic
99	D₇₃	—	2	2	no	—
100	#2150	—	73	2	no	—
101	C₂×C₈²	—	73	1	no	—
102	#1367	—	73	2	no	—
103	#1367	—	73	1	no	—
104	#1367	D₇₃	1	1	yes	characteristic, complemented
105	C₇₃	#11996	1	1	yes	characteristic, complemented, Sylow
106	C₂×M₃₂	—	146	2	no	—
107	C₂×C₄×C₈	—	73	1	no	—
108	C₂×C₄×C₈	—	73	1	no	—
109	C₂×C₄×C₈	—	73	1	no	—
110	C₄⋊C₁₆	—	73	2	no	—
111	C₄⋊C₁₆	—	146	2	no	—
112	C₄⋊C₁₆	C₂×D₇₃	1	2	yes	—
113	C₈²	—	73	1	no	—
114	C₈²	—	73	1	no	—
115	C₈²	—	73	1	no	—
116	C₈²	C₂×D₇₃	1	1	yes	characteristic
117	C₂²×C₈	—	73	1	no	—
118	C₂²×C₈	—	73	1	no	—
119	C₂²×C₈	—	73	1	no	—
120	C₂²×C₈	—	73	1	no	—
121	C₂²×C₈	—	146	1	no	—
122	C₂×C₄²	—	73	1	no	—
123	M₃₂	—	292	4	no	—
124	C₂×C₁₆	—	2	2	no	—
125	C₂×C₁₆	—	146	2	no	—
126	C₄×C₈	—	73	1	no	—
127	C₄×C₈	—	73	1	no	—
128	C₄×C₈	—	73	1	no	—
129	C₄×C₈	—	73	1	no	—
130	C₄×C₈	—	73	1	no	—
131	C₄×C₈	—	73	1	no	—
132	C₄×C₈	—	73	1	no	—
133	C₄×C₈	—	146	1	no	—
134	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
135	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
136	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
137	C₂²×C₄	—	73	1	no	—
138	C₂²×C₄	—	73	1	no	—
139	C₂²×C₄	—	73	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	73	1	no	—
142	C₂×C₈	—	73	1	no	—
143	C₂×C₈	—	73	4	no	—
144	C₂×C₈	—	73	1	no	—
145	C₂×C₈	—	73	1	no	—
146	C₂×C₈	—	146	2	no	—
147	C₂×C₈	—	146	2	no	—
148	C₂×C₈	—	146	4	no	—
149	C₂×C₈	—	146	1	no	—
150	C₂×C₈	—	146	2	no	—
151	C₂×C₈	C₈×D₇₃	1	1	yes	characteristic
152	C₂×C₈	C₈×D₇₃	1	1	yes	characteristic
153	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
154	C₂×C₈	Q₈×D₇₃	1	1	yes	characteristic
155	C₄²	—	73	1	no	—
156	C₄²	—	73	1	no	—
157	C₄²	—	73	1	no	—
158	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
159	C₁₆	—	4	2	no	—
160	C₁₆	—	292	2	no	—
161	C₂³	—	73	1	no	—
162	C₂×C₄	—	73	1	no	—
163	C₂×C₄	—	73	1	no	—
164	C₂×C₄	—	73	2	no	—
165	C₂×C₄	—	73	2	no	—
166	C₂×C₄	—	73	1	no	—
167	C₂×C₄	—	146	2	no	—
168	C₂×C₄	—	146	1	no	—
169	C₂×C₄	—	146	1	no	—
170	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
171	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
172	C₂×C₄	M₁₆×D₇₃	1	1	yes	characteristic
173	C₈	—	2	2	no	—
174	C₈	—	2	1	no	—
175	C₈	—	2	1	no	—
176	C₈	—	73	2	no	—
177	C₈	—	73	2	no	—
178	C₈	—	146	2	no	—
179	C₈	—	146	1	no	—
180	C₈	—	146	1	no	—
181	C₈	#14556530	1	2	yes	complemented
182	C₈	SD₁₆×D₇₃	1	2	yes	—
183	C₂²	—	73	2	no	—
184	C₂²	—	146	1	no	—
185	C₂²	—	146	1	no	—
186	C₂²	C₄⋊C₈×D₇₃	1	1	yes	characteristic
187	C₄	—	2	1	no	—
188	C₄	—	73	2	no	—
189	C₄	—	73	2	no	—
190	C₄	—	146	1	no	—
191	C₄	#106×D₇₃	1	1	yes	characteristic
192	C₄	#108×D₇₃	1	1	yes	characteristic
193	C₄	M₃₂×D₇₃	1	1	yes	characteristic
194	C₄	C₂×#14556530	1	1	yes	characteristic
195	C₂	—	146	2	no	—
196	C₂	#334×D₇₃	1	1	yes	characteristic
197	C₂	#22611685	1	1	yes	characteristic
198	C₂	#364×D₇₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22611685

C₂

yes

maximal

C₈²×D₇₃

C₂

yes

maximal, characteristic

#22607995

C₂

yes

maximal, characteristic, complemented

#22607994

C₂

yes

maximal, complemented

C₇₃×#1367

C₂

yes

maximal, characteristic, complemented

C₂×#14556530

—

C₄×C₈×D₇₃

C₄

yes

characteristic

C₄×C₈×D₇₃

C₄

yes

characteristic

C₄×C₈×D₇₃

C₂²

yes

characteristic

C₄⋊#11924003

C₂²

yes

—

#17988405

—

C₈×#10596831

C₄

yes

characteristic

C₈×#10596831

C₄

yes

characteristic

C₈×#10596831

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₃

C₂²

yes

—

C₈²×C₇₃

C₂²

yes

characteristic

C₂×C₈×D₇₃

—

C₂×C₈×D₇₃

C₈

yes

characteristic

C₂×C₈×D₇₃

C₈

yes

characteristic

C₂×C₈×D₇₃

D₄

yes

characteristic

C₂×C₈×D₇₃

Q₈

yes

characteristic

C₄²×D₇₃

C₂×C₄

yes

—

—

C₂×C₄

yes

characteristic

C₄×#10596831

C₂×C₄

yes

characteristic

C₄×#10596831

C₂×C₄

yes

characteristic

C₈×Dic₇₃

—

C₈×Dic₇₃

C₈

yes

characteristic

C₈×Dic₇₃

C₈

yes

characteristic

C₈×Dic₇₃

D₄

yes

characteristic

C₈×Dic₇₃

Q₈

yes

characteristic

C₂×C₁₆×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₄×D₇₃

C₄⋊C₄

yes

characteristic

C₂×C₄×D₇₃

C₂×C₈

yes

characteristic

C₂×C₄×D₇₃

M₁₆

yes

characteristic

C₂×#10596831

—

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₂×#10596831

C₂×C₈

yes

characteristic

C₂×#10596831

C₂×C₈

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

C₈×D₇₃

—

C₈×D₇₃

—

C₈×D₇₃

SD₁₆

yes

—

C₄×Dic₇₃

C₄⋊C₄

yes

characteristic

C₄×Dic₇₃

C₂×C₈

yes

characteristic

C₄×Dic₇₃

M₁₆

yes

characteristic

#11924003

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×Q₈

yes

characteristic

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₁₆×C₇₃

—

C₂²×D₇₃

C₄⋊C₈

yes

characteristic

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂×M₁₆

yes

characteristic

C₂×Dic₇₃

C₄⋊C₈

yes

characteristic

C₂×Dic₇₃

C₄⋊C₈

yes

characteristic

C₂×Dic₇₃

C₄⋊C₈

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

#106

yes

—

C₄×D₇₃

#108

yes

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

M₃₂

yes

complemented

C₈×C₇₃

C₂×SD₁₆

yes

—

—

—

—

yes

—

#10596831

M₃₂

yes

complemented

C₂²×C₇₃

C₂×C₄⋊C₈

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

#334

yes

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

C₂×M₃₂

yes

characteristic

C₄×C₇₃

C₂×M₃₂

yes

characteristic

Dic₇₃

—

Dic₇₃

#334

yes

—

Dic₇₃

#364

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

100

#2150

—

101

C₂×C₈²

—

102

#1367

—

103

#1367

—

104

#1367

D₇₃

yes

characteristic, complemented

105

C₇₃

#11996

yes

characteristic, complemented, Sylow

106

C₂×M₃₂

—

146

—

107

C₂×C₄×C₈

—

108

C₂×C₄×C₈

—

109

C₂×C₄×C₈

—

110

C₄⋊C₁₆

—

111

C₄⋊C₁₆

—

146

—

112

C₄⋊C₁₆

C₂×D₇₃

yes

—

113

C₈²

—

114

C₈²

—

115

C₈²

—

116

C₈²

C₂×D₇₃

yes

characteristic

117

C₂²×C₈

—

118

C₂²×C₈

—

119

C₂²×C₈

—

120

C₂²×C₈

—

121

C₂²×C₈

—

146

—

122

C₂×C₄²

—

123

M₃₂

—

292

—

124

C₂×C₁₆

—

125

C₂×C₁₆

—

146

—

126

C₄×C₈

—

127

C₄×C₈

—

128

C₄×C₈

—

129

C₄×C₈

—

130

C₄×C₈

—

131

C₄×C₈

—

132

C₄×C₈

—

133

C₄×C₈

—

146

—

134

C₄×C₈

C₄×D₇₃

yes

characteristic

135

C₄×C₈

C₄×D₇₃

yes

characteristic

136

C₄×C₈

C₂²×D₇₃

yes

characteristic

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

146

—

147

C₂×C₈

—

146

—

148

C₂×C₈

—

146

—

149

C₂×C₈

—

146

—

150

C₂×C₈

—

146

—

151

C₂×C₈

C₈×D₇₃

yes

characteristic

152

C₂×C₈

C₈×D₇₃

yes

characteristic

153

C₂×C₈

D₄×D₇₃

yes

characteristic

154

C₂×C₈

Q₈×D₇₃

yes

characteristic

155

C₄²

—

156

C₄²

—

157

C₄²

—

158

C₄²

C₂×C₄×D₇₃

yes

characteristic

159

C₁₆

—

160

C₁₆

—

292

—

161

C₂³

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

146

—

168

C₂×C₄

—

146

—

169

C₂×C₄

—

146

—

170

C₂×C₄

C₄⋊C₄×D₇₃

yes

characteristic

171

C₂×C₄

C₂×C₈×D₇₃

yes

characteristic

172

C₂×C₄

M₁₆×D₇₃

yes

characteristic

173

C₈

—

174

C₈

—

175

C₈

—

176

C₈

—

177

C₈

—

178

C₈

—

146

—

179

C₈

—

146

—

180

C₈

—

146

—

181

C₈

#14556530

yes

complemented

182

C₈

SD₁₆×D₇₃

yes

—

183

C₂²

—

184

C₂²

—

146

—

185

C₂²

—

146

—

186

C₂²

C₄⋊C₈×D₇₃

yes

characteristic

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

—

146

—

191

C₄

#106×D₇₃

yes

characteristic

192

C₄

#108×D₇₃

yes

characteristic

193

C₄

M₃₂×D₇₃

yes

characteristic

194

C₄

C₂×#14556530

yes

characteristic

195

C₂

—

146

—

196

C₂

#334×D₇₃

yes

characteristic

197

C₂

#22611685

yes

characteristic

198

C₂

#364×D₇₃

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#1367	C₂
socle	C₂²×C₇₃	C₂×C₄⋊C₈
prime radical	C₂²×D₇₃	C₄⋊C₈
cosocle	C₄×C₈×C₇₃	C₂³
hypercenter	#1367	D₇₃
hypocenter	C₇₃	#11996
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1367

subgroup

G/H

center

C₂×C₈

D₄×D₇₃

commutator

C₂×C₄×C₇₃

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₇₃×#1367

C₂

socle

C₂²×C₇₃

C₂×C₄⋊C₈

prime radical

C₂²×D₇₃

C₄⋊C₈

cosocle

C₄×C₈×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1367

N	H ≅ G/N	type
3. #22607995	C₂	semidirect
4. #22607994	C₂	semidirect
5. C₇₃×#1367	C₂	semidirect
76. C₈×C₇₃	M₃₂	semidirect
82. #10596831	M₃₂	semidirect
104. #1367	D₇₃	semidirect
105. C₇₃	#11996	semidirect
181. C₈	#14556530	semidirect

H ≅ G/N

type

3. #22607995

C₂

semidirect

4. #22607994

C₂

semidirect

5. C₇₃×#1367

C₂

semidirect

76. C₈×C₇₃

M₃₂

semidirect

82. #10596831

M₃₂

semidirect

104. #1367

D₇₃

semidirect

105. C₇₃

#11996

semidirect

181. C₈

#14556530

semidirect

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

subgroup	index
#22611685	2
C₈²×D₇₃	2
#22607995	2
#22607994	2
C₇₃×#1367	2
#11996	73

subgroup

index

C₈²×D₇₃

C₇₃×#1367

Subgroups — #32997527

Conjugacy classes of subgroups

Complements

Maximal subgroups