Group #32921895 — subgroups

Subgroups — C₇₃×#12491 #32921895

Order 18688.

282 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 82 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	C₇₃×#2206	C₂	1	2	yes	maximal, complemented
2	C₇₃×#1712	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₃×#1589	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₃×#1367	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#1361	C₂	1	2	yes	maximal, complemented
6	C₂×D₁₆×C₇₃	—	2	2	no	—
7	#493×C₇₃	—	2	1	no	—
8	#493×C₇₃	—	2	1	no	—
9	#493×C₇₃	C₂²	1	2	yes	—
10	C₄×D₈×C₇₃	C₂²	1	1	yes	characteristic
11	#436×C₇₃	—	2	1	no	—
12	C₄⋊C₁₆×C₇₃	C₂²	1	2	yes	—
13	#357×C₇₃	—	2	2	no	—
14	#335×C₇₃	C₂²	1	1	yes	characteristic
15	D₄⋊C₈×C₇₃	—	2	1	no	—
16	C₈²×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×D₈×C₇₃	—	2	2	no	—
18	C₂×D₈×C₇₃	—	2	2	no	—
19	C₂×D₈×C₇₃	—	4	2	no	—
20	C₂×D₈×C₇₃	—	4	1	no	—
21	C₂×D₈×C₇₃	—	4	1	no	—
22	C₂×D₈×C₇₃	D₄	1	1	yes	characteristic
23	C₂×M₁₆×C₇₃	—	4	1	no	—
24	#127×C₇₃	—	2	2	no	—
25	C₄×D₄×C₇₃	—	2	1	no	—
26	D₁₆×C₇₃	—	4	4	no	—
27	C₂×C₁₆×C₇₃	—	2	2	no	—
28	#107×C₇₃	D₄	1	1	yes	characteristic
29	C₄⋊C₈×C₇₃	—	2	1	no	—
30	#102×C₇₃	—	2	1	no	—
31	C₂²⋊C₈×C₇₃	—	4	1	no	—
32	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
33	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
34	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
35	C₂×D₄×C₇₃	—	2	1	no	—
36	C₂×D₄×C₇₃	—	4	2	no	—
37	C₂×D₄×C₇₃	—	4	2	no	—
38	C₂×D₄×C₇₃	—	4	2	no	—
39	C₂²×C₄×C₇₃	—	4	1	no	—
40	D₈×C₇₃	—	2	2	no	—
41	D₈×C₇₃	—	4	2	no	—
42	D₈×C₇₃	—	4	2	no	—
43	D₈×C₇₃	—	8	2	no	—
44	D₈×C₇₃	—	8	4	no	—
45	D₈×C₇₃	—	8	2	no	—
46	D₈×C₇₃	—	8	2	no	—
47	M₁₆×C₇₃	—	8	2	no	—
48	C₂×C₈×C₇₃	—	2	1	no	—
49	C₂×C₈×C₇₃	—	4	1	no	—
50	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
51	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
52	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
54	C₄⋊C₄×C₇₃	—	2	1	no	—
55	C₂²⋊C₄×C₇₃	—	4	1	no	—
56	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
57	C₁₆×C₇₃	—	4	2	no	—
58	C₂³×C₇₃	—	4	1	no	—
59	C₂³×C₇₃	—	8	2	no	—
60	D₄×C₇₃	—	4	1	no	—
61	D₄×C₇₃	—	4	1	no	—
62	D₄×C₇₃	—	8	4	no	—
63	D₄×C₇₃	—	8	2	no	—
64	D₄×C₇₃	—	8	2	no	—
65	D₄×C₇₃	—	8	2	no	—
66	D₄×C₇₃	—	8	2	no	—
67	C₂×C₄×C₇₃	—	4	1	no	—
68	C₂×C₄×C₇₃	—	8	1	no	—
69	C₂×C₄×C₇₃	—	8	1	no	—
70	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
72	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
73	C₈×C₇₃	—	2	2	no	—
74	C₈×C₇₃	—	2	1	no	—
75	C₈×C₇₃	—	2	1	no	—
76	C₈×C₇₃	—	8	1	no	—
77	C₈×C₇₃	D₁₆	1	2	yes	complemented
78	C₈×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
79	C₂²×C₇₃	—	8	1	no	—
80	C₂²×C₇₃	—	8	1	no	—
81	C₂²×C₇₃	—	8	1	no	—
82	C₂²×C₇₃	—	16	2	no	—
83	C₂²×C₇₃	—	16	2	no	—
84	C₂²×C₇₃	—	16	2	no	—
85	C₂²×C₇₃	#459	1	1	yes	characteristic
86	C₄×C₇₃	—	2	1	no	—
87	C₄×C₇₃	—	8	1	no	—
88	C₄×C₇₃	#469	1	1	yes	characteristic
89	C₄×C₇₃	#472	1	1	yes	characteristic
90	C₄×C₇₃	C₂×D₁₆	1	1	yes	characteristic
91	C₄×C₇₃	#509	1	1	yes	characteristic
92	#12491	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂×C₇₃	—	16	1	no	—
94	C₂×C₇₃	—	32	2	no	—
95	C₂×C₇₃	#1687	1	1	yes	characteristic
96	C₂×C₇₃	#2206	1	1	yes	characteristic
97	C₂×C₇₃	#2213	1	1	yes	characteristic
98	#2206	C₂×C₇₃	1	2	yes	complemented
99	#1712	C₂×C₇₃	1	1	yes	characteristic, complemented
100	#1589	C₂×C₇₃	1	1	yes	characteristic, complemented
101	#1367	C₂×C₇₃	1	1	yes	characteristic, complemented
102	#1361	C₂×C₇₃	1	2	yes	complemented
103	C₇₃	#12491	1	1	yes	characteristic, direct product factor, Sylow
104	C₂×D₁₆	—	2	2	no	—
105	#493	—	2	1	no	—
106	#493	—	2	1	no	—
107	#493	C₂²×C₇₃	1	2	yes	—
108	C₄×D₈	C₂²×C₇₃	1	1	yes	characteristic
109	#436	—	2	1	no	—
110	C₄⋊C₁₆	C₂²×C₇₃	1	2	yes	—
111	#357	—	2	2	no	—
112	#335	C₂²×C₇₃	1	1	yes	characteristic
113	D₄⋊C₈	—	2	1	no	—
114	C₈²	C₂²×C₇₃	1	1	yes	characteristic
115	C₂×D₈	—	2	2	no	—
116	C₂×D₈	—	2	2	no	—
117	C₂×D₈	—	4	2	no	—
118	C₂×D₈	—	4	1	no	—
119	C₂×D₈	—	4	1	no	—
120	C₂×D₈	D₄×C₇₃	1	1	yes	characteristic
121	C₂×M₁₆	—	4	1	no	—
122	#127	—	2	2	no	—
123	C₄×D₄	—	2	1	no	—
124	D₁₆	—	4	4	no	—
125	C₂×C₁₆	—	2	2	no	—
126	#107	D₄×C₇₃	1	1	yes	characteristic
127	C₄⋊C₈	—	2	1	no	—
128	#102	—	2	1	no	—
129	C₂²⋊C₈	—	4	1	no	—
130	C₄×C₈	D₄×C₇₃	1	1	yes	characteristic
131	C₄×C₈	D₄×C₇₃	1	1	yes	characteristic
132	C₄×C₈	C₂³×C₇₃	1	1	yes	characteristic
133	C₂×D₄	—	2	1	no	—
134	C₂×D₄	—	4	2	no	—
135	C₂×D₄	—	4	2	no	—
136	C₂×D₄	—	4	2	no	—
137	C₂²×C₄	—	4	1	no	—
138	D₈	—	2	2	no	—
139	D₈	—	4	2	no	—
140	D₈	—	4	2	no	—
141	D₈	—	8	2	no	—
142	D₈	—	8	4	no	—
143	D₈	—	8	2	no	—
144	D₈	—	8	2	no	—
145	M₁₆	—	8	2	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	4	1	no	—
148	C₂×C₈	D₈×C₇₃	1	1	yes	characteristic
149	C₂×C₈	D₈×C₇₃	1	1	yes	characteristic
150	C₂×C₈	C₂×D₄×C₇₃	1	1	yes	characteristic
151	C₂×C₈	#41×C₇₃	1	1	yes	characteristic
152	C₄⋊C₄	—	2	1	no	—
153	C₂²⋊C₄	—	4	1	no	—
154	C₄²	C₂×D₄×C₇₃	1	1	yes	characteristic
155	C₁₆	—	4	2	no	—
156	C₂³	—	4	1	no	—
157	C₂³	—	8	2	no	—
158	D₄	—	4	1	no	—
159	D₄	—	4	1	no	—
160	D₄	—	8	4	no	—
161	D₄	—	8	2	no	—
162	D₄	—	8	2	no	—
163	D₄	—	8	2	no	—
164	D₄	—	8	2	no	—
165	C₂×C₄	—	4	1	no	—
166	C₂×C₄	—	8	1	no	—
167	C₂×C₄	—	8	1	no	—
168	C₂×C₄	#121×C₇₃	1	1	yes	characteristic
169	C₂×C₄	C₂×D₈×C₇₃	1	1	yes	characteristic
170	C₂×C₄	AGL(1,ℤ/8)×C₇₃	1	1	yes	characteristic
171	C₈	—	2	2	no	—
172	C₈	—	2	1	no	—
173	C₈	—	2	1	no	—
174	C₈	—	8	1	no	—
175	C₈	D₁₆×C₇₃	1	2	yes	complemented
176	C₈	AGL(1,ℤ/8)×C₇₃	1	2	yes	—
177	C₂²	—	8	1	no	—
178	C₂²	—	8	1	no	—
179	C₂²	—	8	1	no	—
180	C₂²	—	16	2	no	—
181	C₂²	—	16	2	no	—
182	C₂²	—	16	2	no	—
183	C₂²	#459×C₇₃	1	1	yes	characteristic
184	C₄	—	2	1	no	—
185	C₄	—	8	1	no	—
186	C₄	#469×C₇₃	1	1	yes	characteristic
187	C₄	#472×C₇₃	1	1	yes	characteristic
188	C₄	C₂×D₁₆×C₇₃	1	1	yes	characteristic
189	C₄	#509×C₇₃	1	1	yes	characteristic
190	C₂	—	16	1	no	—
191	C₂	—	32	2	no	—
192	C₂	C₇₃×#1687	1	1	yes	characteristic
193	C₂	C₇₃×#2206	1	1	yes	characteristic
194	C₂	C₇₃×#2213	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₃×#2206

C₂

yes

maximal, complemented

C₇₃×#1712

C₂

yes

maximal, characteristic, complemented

C₇₃×#1589

C₂

yes

maximal, characteristic, complemented

C₇₃×#1367

C₂

yes

maximal, characteristic, complemented

C₇₃×#1361

C₂

yes

maximal, complemented

C₂×D₁₆×C₇₃

—

#493×C₇₃

—

#493×C₇₃

—

#493×C₇₃

C₂²

yes

—

C₄×D₈×C₇₃

C₂²

yes

characteristic

#436×C₇₃

—

C₄⋊C₁₆×C₇₃

C₂²

yes

—

#357×C₇₃

—

#335×C₇₃

C₂²

yes

characteristic

D₄⋊C₈×C₇₃

—

C₈²×C₇₃

C₂²

yes

characteristic

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

D₄

yes

characteristic

C₂×M₁₆×C₇₃

—

#127×C₇₃

—

C₄×D₄×C₇₃

—

D₁₆×C₇₃

—

C₂×C₁₆×C₇₃

—

#107×C₇₃

D₄

yes

characteristic

C₄⋊C₈×C₇₃

—

#102×C₇₃

—

C₂²⋊C₈×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂²×C₄×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

characteristic

C₂×C₈×C₇₃

D₈

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

#41

yes

characteristic

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#121

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

AGL(1,ℤ/8)

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

D₁₆

yes

complemented

C₈×C₇₃

AGL(1,ℤ/8)

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#459

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

C₂×D₁₆

yes

C₄×C₇₃

yes

C₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

complemented

#1712

C₂×C₇₃

yes

characteristic, complemented

100

#1589

C₂×C₇₃

yes

characteristic, complemented

101

#1367

C₂×C₇₃

yes

characteristic, complemented

102

#1361

C₂×C₇₃

yes

complemented

103

C₇₃

#12491

yes

characteristic, direct product factor, Sylow

104

C₂×D₁₆

—

105

#493

—

106

#493

—

107

#493

C₂²×C₇₃

yes

—

108

C₄×D₈

C₂²×C₇₃

yes

characteristic

109

#436

—

110

C₄⋊C₁₆

C₂²×C₇₃

yes

—

111

#357

—

112

#335

C₂²×C₇₃

yes

characteristic

113

D₄⋊C₈

—

114

C₈²

C₂²×C₇₃

yes

characteristic

115

C₂×D₈

—

116

C₂×D₈

—

117

C₂×D₈

—

118

C₂×D₈

—

119

C₂×D₈

—

120

C₂×D₈

D₄×C₇₃

yes

characteristic

121

C₂×M₁₆

—

122

#127

—

123

C₄×D₄

—

124

D₁₆

—

125

C₂×C₁₆

—

126

#107

D₄×C₇₃

yes

characteristic

127

C₄⋊C₈

—

128

#102

—

129

C₂²⋊C₈

—

130

C₄×C₈

D₄×C₇₃

yes

characteristic

131

C₄×C₈

D₄×C₇₃

yes

characteristic

132

C₄×C₈

C₂³×C₇₃

yes

characteristic

133

C₂×D₄

—

134

C₂×D₄

—

135

C₂×D₄

—

136

C₂×D₄

—

137

C₂²×C₄

—

138

D₈

—

139

D₈

—

140

D₈

—

141

D₈

—

142

D₈

—

143

D₈

—

144

D₈

—

145

M₁₆

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

D₈×C₇₃

yes

characteristic

149

C₂×C₈

D₈×C₇₃

yes

characteristic

150

C₂×C₈

C₂×D₄×C₇₃

yes

characteristic

151

C₂×C₈

#41×C₇₃

yes

characteristic

152

C₄⋊C₄

—

153

C₂²⋊C₄

—

154

C₄²

C₂×D₄×C₇₃

yes

characteristic

155

C₁₆

—

156

C₂³

—

157

C₂³

—

158

D₄

—

159

D₄

—

160

D₄

—

161

D₄

—

162

D₄

—

163

D₄

—

164

D₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

#121×C₇₃

yes

characteristic

169

C₂×C₄

C₂×D₈×C₇₃

yes

characteristic

170

C₂×C₄

AGL(1,ℤ/8)×C₇₃

yes

characteristic

171

C₈

—

172

C₈

—

173

C₈

—

174

C₈

—

175

C₈

D₁₆×C₇₃

yes

complemented

176

C₈

AGL(1,ℤ/8)×C₇₃

yes

—

177

C₂²

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

—

182

C₂²

—

183

C₂²

#459×C₇₃

yes

characteristic

184

C₄

—

185

C₄

—

186

C₄

#469×C₇₃

yes

characteristic

187

C₄

#472×C₇₃

yes

characteristic

188

C₄

C₂×D₁₆×C₇₃

yes

characteristic

189

C₄

#509×C₇₃

yes

characteristic

190

C₂

—

191

C₂

—

192

C₂

C₇₃×#1687

yes

characteristic

193

C₂

C₇₃×#2206

yes

characteristic

194

C₂

C₇₃×#2213

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₇₃	#459
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇₃
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₄²×C₇₃ ◁ C₈²×C₇₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₇₃

#459

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₇₃

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₇₃ ◁ C₄²×C₇₃ ◁ C₈²×C₇₃ ◁ G

N	H ≅ G/N	type
1. C₇₃×#2206	C₂	semidirect
2. C₇₃×#1712	C₂	semidirect
3. C₇₃×#1589	C₂	semidirect
4. C₇₃×#1367	C₂	semidirect
5. C₇₃×#1361	C₂	semidirect
77. C₈×C₇₃	D₁₆	semidirect
92. #12491	C₇₃	direct (N × H)
98. #2206	C₂×C₇₃	semidirect
99. #1712	C₂×C₇₃	semidirect
100. #1589	C₂×C₇₃	semidirect
101. #1367	C₂×C₇₃	semidirect
102. #1361	C₂×C₇₃	semidirect
103. C₇₃	#12491	direct (N × H)
175. C₈	D₁₆×C₇₃	semidirect

H ≅ G/N

type

1. C₇₃×#2206

C₂

semidirect

2. C₇₃×#1712

C₂

semidirect

3. C₇₃×#1589

C₂

semidirect

4. C₇₃×#1367

C₂

semidirect

5. C₇₃×#1361

C₂

semidirect

77. C₈×C₇₃

D₁₆

semidirect

92. #12491

C₇₃

direct (N × H)

98. #2206

C₂×C₇₃

semidirect

99. #1712

C₂×C₇₃

semidirect

100. #1589

C₂×C₇₃

semidirect

101. #1367

C₂×C₇₃

semidirect

102. #1361

C₂×C₇₃

semidirect

103. C₇₃

#12491

direct (N × H)

175. C₈

D₁₆×C₇₃

semidirect

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

subgroup	index
C₇₃×#2206	2
C₇₃×#1712	2
C₇₃×#1589	2
C₇₃×#1367	2
C₇₃×#1361	2
#12491	73

subgroup

index

C₇₃×#2206

C₇₃×#1712

C₇₃×#1589

C₇₃×#1367

C₇₃×#1361

#12491

Subgroups — C₇₃×#12491 #32921895

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#12491 #32921895

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#12491 #32921895