Group #32922945 — subgroups

Subgroups — C₇₃×#13541 #32922945

Order 18688.

254 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 106 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×#1944	C₂	1	1	yes	maximal, characteristic
2	C₇₃×#1849	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#1387	C₂	1	2	yes	maximal
4	C₇₃×#1383	C₂	1	2	yes	maximal
5	C₇₃×#1377	C₂	1	1	yes	maximal, characteristic
6	C₂×M₃₂×C₇₃	C₂²	1	2	yes	—
7	C₂×#108×C₇₃	—	2	1	no	—
8	C₂×#108×C₇₃	C₂²	1	2	yes	—
9	C₂×#107×C₇₃	C₂²	1	1	yes	characteristic
10	C₂×#106×C₇₃	C₂²	1	1	yes	characteristic
11	#423×C₇₃	—	2	1	no	—
12	C₂×#97×C₇₃	C₂²	1	1	yes	characteristic
13	#389×C₇₃	—	2	1	no	—
14	#340×C₇₃	—	2	1	no	—
15	C₂×M₁₆×C₇₃	—	2	2	no	—
16	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
19	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
20	C₂×C₄⋊C₄×C₇₃	—	4	1	no	—
21	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
22	M₃₂×C₇₃	—	2	2	no	—
23	M₃₂×C₇₃	—	2	2	no	—
24	C₂×C₁₆×C₇₃	—	2	2	no	—
25	#108×C₇₃	—	2	4	no	—
26	#108×C₇₃	—	4	2	no	—
27	#107×C₇₃	—	2	2	no	—
28	#106×C₇₃	—	2	1	no	—
29	#106×C₇₃	Q₈	1	2	yes	—
30	C₄⋊C₈×C₇₃	—	4	2	no	—
31	#97×C₇₃	—	2	2	no	—
32	#95×C₇₃	—	4	1	no	—
33	C₂²×C₄×C₇₃	—	4	1	no	—
34	C₂²×C₄×C₇₃	—	4	1	no	—
35	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
36	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
37	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
38	M₁₆×C₇₃	—	4	2	no	—
39	M₁₆×C₇₃	—	4	2	no	—
40	C₂×C₈×C₇₃	—	2	1	no	—
41	C₂×C₈×C₇₃	—	2	1	no	—
42	C₂×C₈×C₇₃	—	4	2	no	—
43	C₂×C₈×C₇₃	Q₁₆	1	2	yes	—
44	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₇₃	#41	1	2	yes	—
46	C₂×C₈×C₇₃	#41	1	2	yes	—
47	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
48	C₄⋊C₄×C₇₃	—	2	2	no	—
49	C₄⋊C₄×C₇₃	—	4	1	no	—
50	C₄⋊C₄×C₇₃	—	4	1	no	—
51	C₄⋊C₄×C₇₃	—	4	1	no	—
52	C₄⋊C₄×C₇₃	—	8	2	no	—
53	C₄²×C₇₃	—	2	1	no	—
54	C₄²×C₇₃	—	2	1	no	—
55	C₁₆×C₇₃	—	4	2	no	—
56	C₂³×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₄×C₇₃	—	2	1	no	—
58	C₂×C₄×C₇₃	—	2	1	no	—
59	C₂×C₄×C₇₃	—	4	1	no	—
60	C₂×C₄×C₇₃	—	4	1	no	—
61	C₂×C₄×C₇₃	—	4	2	no	—
62	C₂×C₄×C₇₃	—	8	1	no	—
63	C₂×C₄×C₇₃	—	8	1	no	—
64	C₂×C₄×C₇₃	—	8	1	no	—
65	C₂×C₄×C₇₃	—	8	1	no	—
66	C₂×C₄×C₇₃	—	8	1	no	—
67	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₇₃	#123	1	1	yes	characteristic
69	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
70	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×C₄×C₇₃	AGL(1,ℤ/8)	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	—	8	2	no	—
75	C₈×C₇₃	#125	1	2	yes	—
76	C₈×C₇₃	#126	1	2	yes	—
77	C₂²×C₇₃	—	2	1	no	—
78	C₂²×C₇₃	—	2	1	no	—
79	C₂²×C₇₃	#476	1	1	yes	characteristic
80	C₂²×C₇₃	#481	1	1	yes	characteristic
81	C₂²×C₇₃	#487	1	1	yes	characteristic
82	C₄×C₇₃	—	4	1	no	—
83	C₄×C₇₃	—	4	1	no	—
84	C₄×C₇₃	—	8	2	no	—
85	C₄×C₇₃	—	16	1	no	—
86	C₄×C₇₃	#400	1	1	yes	characteristic
87	C₄×C₇₃	#477	1	1	yes	characteristic
88	C₄×C₇₃	#484	1	1	yes	characteristic
89	C₄×C₇₃	#490	1	1	yes	characteristic
90	#13541	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂×C₇₃	—	2	1	no	—
92	C₂×C₇₃	—	2	1	no	—
93	C₂×C₇₃	#2100	1	1	yes	characteristic
94	C₂×C₇₃	#2238	1	1	yes	characteristic
95	C₂×C₇₃	#2239	1	1	yes	characteristic
96	#1944	C₂×C₇₃	1	1	yes	characteristic
97	#1849	C₂×C₇₃	1	1	yes	characteristic
98	#1387	C₂×C₇₃	1	2	yes	—
99	#1383	C₂×C₇₃	1	2	yes	—
100	#1377	C₂×C₇₃	1	1	yes	characteristic
101	C₇₃	#13541	1	1	yes	characteristic, direct product factor, Sylow
102	C₂×M₃₂	C₂²×C₇₃	1	2	yes	—
103	C₂×#108	—	2	1	no	—
104	C₂×#108	C₂²×C₇₃	1	2	yes	—
105	C₂×#107	C₂²×C₇₃	1	1	yes	characteristic
106	C₂×#106	C₂²×C₇₃	1	1	yes	characteristic
107	#423	—	2	1	no	—
108	C₂×#97	C₂²×C₇₃	1	1	yes	characteristic
109	#389	—	2	1	no	—
110	#340	—	2	1	no	—
111	C₂×M₁₆	—	2	2	no	—
112	C₂²×C₈	D₄×C₇₃	1	1	yes	characteristic
113	C₂²×C₈	C₂³×C₇₃	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	2	1	no	—
115	C₂×C₄⋊C₄	—	2	1	no	—
116	C₂×C₄⋊C₄	—	4	1	no	—
117	C₂×C₄²	D₄×C₇₃	1	1	yes	characteristic
118	M₃₂	—	2	2	no	—
119	M₃₂	—	2	2	no	—
120	C₂×C₁₆	—	2	2	no	—
121	#108	—	2	4	no	—
122	#108	—	4	2	no	—
123	#107	—	2	2	no	—
124	#106	—	2	1	no	—
125	#106	Q₈×C₇₃	1	2	yes	—
126	C₄⋊C₈	—	4	2	no	—
127	#97	—	2	2	no	—
128	#95	—	4	1	no	—
129	C₂²×C₄	—	4	1	no	—
130	C₂²×C₄	—	4	1	no	—
131	C₂²×C₄	SD₁₆×C₇₃	1	1	yes	characteristic
132	C₂²×C₄	SD₁₆×C₇₃	1	1	yes	characteristic
133	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
134	M₁₆	—	4	2	no	—
135	M₁₆	—	4	2	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	Q₁₆×C₇₃	1	2	yes	—
140	C₂×C₈	C₂×Q₈×C₇₃	1	1	yes	characteristic
141	C₂×C₈	#41×C₇₃	1	2	yes	—
142	C₂×C₈	#41×C₇₃	1	2	yes	—
143	C₂×C₈	#41×C₇₃	1	1	yes	characteristic
144	C₄⋊C₄	—	2	2	no	—
145	C₄⋊C₄	—	4	1	no	—
146	C₄⋊C₄	—	4	1	no	—
147	C₄⋊C₄	—	4	1	no	—
148	C₄⋊C₄	—	8	2	no	—
149	C₄²	—	2	1	no	—
150	C₄²	—	2	1	no	—
151	C₁₆	—	4	2	no	—
152	C₂³	C₂×SD₁₆×C₇₃	1	1	yes	characteristic
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	8	1	no	—
159	C₂×C₄	—	8	1	no	—
160	C₂×C₄	—	8	1	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	—	8	1	no	—
163	C₂×C₄	C₂²⋊Q₈×C₇₃	1	1	yes	characteristic
164	C₂×C₄	#123×C₇₃	1	1	yes	characteristic
165	C₂×C₄	#124×C₇₃	1	1	yes	characteristic
166	C₂×C₄	C₂×Q₁₆×C₇₃	1	1	yes	characteristic
167	C₂×C₄	AGL(1,ℤ/8)×C₇₃	1	1	yes	characteristic
168	C₂×C₄	AGL(1,ℤ/8)×C₇₃	1	1	yes	characteristic
169	C₈	—	2	2	no	—
170	C₈	—	8	2	no	—
171	C₈	#125×C₇₃	1	2	yes	—
172	C₈	#126×C₇₃	1	2	yes	—
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	#476×C₇₃	1	1	yes	characteristic
176	C₂²	#481×C₇₃	1	1	yes	characteristic
177	C₂²	#487×C₇₃	1	1	yes	characteristic
178	C₄	—	4	1	no	—
179	C₄	—	4	1	no	—
180	C₄	—	8	2	no	—
181	C₄	—	16	1	no	—
182	C₄	#400×C₇₃	1	1	yes	characteristic
183	C₄	#477×C₇₃	1	1	yes	characteristic
184	C₄	#484×C₇₃	1	1	yes	characteristic
185	C₄	#490×C₇₃	1	1	yes	characteristic
186	C₂	—	2	1	no	—
187	C₂	—	2	1	no	—
188	C₂	C₇₃×#2100	1	1	yes	characteristic
189	C₂	C₇₃×#2238	1	1	yes	characteristic
190	C₂	C₇₃×#2239	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₃×#1944

C₂

yes

maximal, characteristic

C₇₃×#1849

C₂

yes

maximal, characteristic

C₇₃×#1387

C₂

yes

maximal

C₇₃×#1383

C₂

yes

maximal

C₇₃×#1377

C₂

yes

maximal, characteristic

C₂×M₃₂×C₇₃

C₂²

yes

—

C₂×#108×C₇₃

—

C₂×#108×C₇₃

C₂²

yes

—

C₂×#107×C₇₃

C₂²

yes

characteristic

C₂×#106×C₇₃

C₂²

yes

characteristic

#423×C₇₃

—

C₂×#97×C₇₃

C₂²

yes

characteristic

#389×C₇₃

—

#340×C₇₃

—

C₂×M₁₆×C₇₃

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄²×C₇₃

D₄

yes

characteristic

M₃₂×C₇₃

—

M₃₂×C₇₃

—

C₂×C₁₆×C₇₃

—

#108×C₇₃

—

#108×C₇₃

—

#107×C₇₃

—

#106×C₇₃

—

#106×C₇₃

Q₈

yes

—

C₄⋊C₈×C₇₃

—

#97×C₇₃

—

#95×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

SD₁₆

yes

characteristic

C₂²×C₄×C₇₃

SD₁₆

yes

characteristic

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

Q₁₆

yes

—

C₂×C₈×C₇₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₃

#41

yes

—

C₂×C₈×C₇₃

#41

yes

—

C₂×C₈×C₇₃

#41

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₁₆×C₇₃

—

C₂³×C₇₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

C₂×Q₁₆

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

#125

yes

—

C₈×C₇₃

#126

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

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

characteristic

#1849

C₂×C₇₃

yes

characteristic

#1387

C₂×C₇₃

yes

—

#1383

C₂×C₇₃

yes

—

100

#1377

C₂×C₇₃

yes

characteristic

101

C₇₃

#13541

yes

characteristic, direct product factor, Sylow

102

C₂×M₃₂

C₂²×C₇₃

yes

—

103

C₂×#108

—

104

C₂×#108

C₂²×C₇₃

yes

—

105

C₂×#107

C₂²×C₇₃

yes

characteristic

106

C₂×#106

C₂²×C₇₃

yes

characteristic

107

#423

—

108

C₂×#97

C₂²×C₇₃

yes

characteristic

109

#389

—

110

#340

—

111

C₂×M₁₆

—

112

C₂²×C₈

D₄×C₇₃

yes

characteristic

113

C₂²×C₈

C₂³×C₇₃

yes

characteristic

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄²

D₄×C₇₃

yes

characteristic

118

M₃₂

—

119

M₃₂

—

120

C₂×C₁₆

—

121

#108

—

122

#108

—

123

#107

—

124

#106

—

125

#106

Q₈×C₇₃

yes

—

126

C₄⋊C₈

—

127

#97

—

128

#95

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

SD₁₆×C₇₃

yes

characteristic

132

C₂²×C₄

SD₁₆×C₇₃

yes

characteristic

133

C₂²×C₄

C₂×D₄×C₇₃

yes

characteristic

134

M₁₆

—

135

M₁₆

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

Q₁₆×C₇₃

yes

—

140

C₂×C₈

C₂×Q₈×C₇₃

yes

characteristic

141

C₂×C₈

#41×C₇₃

yes

—

142

C₂×C₈

#41×C₇₃

yes

—

143

C₂×C₈

#41×C₇₃

yes

characteristic

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄²

—

150

C₄²

—

151

C₁₆

—

152

C₂³

C₂×SD₁₆×C₇₃

yes

characteristic

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

C₂²⋊Q₈×C₇₃

yes

characteristic

164

C₂×C₄

#123×C₇₃

yes

characteristic

165

C₂×C₄

#124×C₇₃

yes

characteristic

166

C₂×C₄

C₂×Q₁₆×C₇₃

yes

characteristic

167

C₂×C₄

AGL(1,ℤ/8)×C₇₃

yes

characteristic

168

C₂×C₄

AGL(1,ℤ/8)×C₇₃

yes

characteristic

169

C₈

—

170

C₈

—

171

C₈

#125×C₇₃

yes

—

172

C₈

#126×C₇₃

yes

—

173

C₂²

—

174

C₂²

—

175

C₂²

#476×C₇₃

yes

characteristic

176

C₂²

#481×C₇₃

yes

characteristic

177

C₂²

#487×C₇₃

yes

characteristic

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

#400×C₇₃

yes

characteristic

183

C₄

#477×C₇₃

yes

characteristic

184

C₄

#484×C₇₃

yes

characteristic

185

C₄

#490×C₇₃

yes

characteristic

186

C₂

—

187

C₂

—

188

C₂

C₇₃×#2100

yes

characteristic

189

C₂

C₇₃×#2238

yes

characteristic

190

C₂

C₇₃×#2239

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₇₃	#481
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇₃
prime radical	C₂³×C₇₃	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂²×C₄×C₇₃ ◁ C₂×#97×C₇₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₇₃

#481

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₇₃

prime radical

C₂³×C₇₃

C₂×SD₁₆

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₇₃ ◁ C₂²×C₄×C₇₃ ◁ C₂×#97×C₇₃ ◁ G

N	H ≅ G/N	type
90. #13541	C₇₃	direct (N × H)
101. C₇₃	#13541	direct (N × H)

H ≅ G/N

type

90. #13541

C₇₃

direct (N × H)

101. C₇₃

#13541

direct (N × H)

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

subgroup	index
C₇₃×#1944	2
C₇₃×#1849	2
C₇₃×#1387	2
C₇₃×#1383	2
C₇₃×#1377	2
#13541	73

subgroup

index

C₇₃×#1944

C₇₃×#1849

C₇₃×#1387

C₇₃×#1383

C₇₃×#1377

#13541

Subgroups — C₇₃×#13541 #32922945

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#13541 #32922945

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#13541 #32922945