Group #22606136 — subgroups

Subgroups — C₇₃×#1983 #22606136

Order 9344.

344 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 128 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	C₂×#124×C₇₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₈×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#103×C₇₃	C₂	1	2	yes	maximal, complemented
4	C₂×#102×C₇₃	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₄×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₈×C₇₃	C₂²	1	1	yes	characteristic
7	C₂²×D₄×C₇₃	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₇₃	C₂²	1	2	yes	—
9	#124×C₇₃	—	2	2	no	—
10	#124×C₇₃	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
12	C₂×C₂²⋊C₄×C₇₃	—	2	1	no	—
13	C₂×C₂²⋊C₄×C₇₃	—	2	1	no	—
14	C₂×C₄²×C₇₃	—	2	1	no	—
15	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₇₃	—	2	2	no	—
17	#103×C₇₃	—	2	4	no	—
18	#102×C₇₃	—	2	4	no	—
19	#95×C₇₃	—	2	1	no	—
20	C₂⁴×C₇₃	—	2	1	no	—
21	C₂×Q₈×C₇₃	—	2	1	no	—
22	C₂×Q₈×C₇₃	—	2	2	no	—
23	C₂×Q₈×C₇₃	—	2	2	no	—
24	C₂×Q₈×C₇₃	C₂×C₄	1	2	yes	—
25	C₂×D₄×C₇₃	—	2	2	no	—
26	C₂×D₄×C₇₃	—	2	1	no	—
27	C₂×D₄×C₇₃	—	2	2	no	—
28	C₂×D₄×C₇₃	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₇₃	—	2	2	no	—
30	C₂²×C₄×C₇₃	—	2	1	no	—
31	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₇₃	—	2	2	no	—
35	C₂×C₈×C₇₃	—	2	2	no	—
36	C₂×C₈×C₇₃	—	2	2	no	—
37	C₄⋊C₄×C₇₃	—	2	2	no	—
38	C₄⋊C₄×C₇₃	D₄	1	4	yes	—
39	C₂²⋊C₄×C₇₃	—	2	4	no	—
40	C₂²⋊C₄×C₇₃	—	2	4	no	—
41	C₄²×C₇₃	—	4	2	no	—
42	C₄²×C₇₃	C₂×C₄	1	2	yes	—
43	C₄²×C₇₃	D₄	1	2	yes	—
44	C₂³×C₇₃	—	2	2	no	—
45	C₂³×C₇₃	—	4	2	no	—
46	C₂³×C₇₃	—	4	2	no	—
47	C₂³×C₇₃	—	4	1	no	—
48	C₂³×C₇₃	—	4	1	no	—
49	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
50	Q₈×C₇₃	—	2	4	no	—
51	Q₈×C₇₃	—	4	1	no	—
52	Q₈×C₇₃	—	4	1	no	—
53	D₄×C₇₃	—	2	4	no	—
54	D₄×C₇₃	—	4	1	no	—
55	D₄×C₇₃	—	4	1	no	—
56	C₂×C₄×C₇₃	—	2	2	no	—
57	C₂×C₄×C₇₃	—	2	1	no	—
58	C₂×C₄×C₇₃	—	2	1	no	—
59	C₂×C₄×C₇₃	—	2	2	no	—
60	C₂×C₄×C₇₃	—	2	4	no	—
61	C₂×C₄×C₇₃	—	4	2	no	—
62	C₂×C₄×C₇₃	—	4	2	no	—
63	C₂×C₄×C₇₃	—	4	2	no	—
64	C₂×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
65	C₂×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
66	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
68	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₇₃	#41	1	2	yes	—
70	C₈×C₇₃	—	4	2	no	—
71	C₂²×C₇₃	—	4	2	no	—
72	C₂²×C₇₃	—	4	2	no	—
73	C₂²×C₇₃	—	4	2	no	—
74	C₂²×C₇₃	—	8	2	no	—
75	C₂²×C₇₃	—	8	2	no	—
76	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₇₃	#121	1	2	yes	—
78	C₂²×C₇₃	#135	1	2	yes	—
79	C₂²×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
80	C₂²×C₇₃	#137	1	1	yes	characteristic
81	C₄×C₇₃	—	2	2	no	—
82	C₄×C₇₃	—	2	2	no	—
83	C₄×C₇₃	—	4	2	no	—
84	C₄×C₇₃	—	4	4	no	—
85	C₄×C₇₃	C₄×D₄	1	2	yes	—
86	C₄×C₇₃	#124	1	1	yes	characteristic
87	C₄×C₇₃	#127	1	1	yes	characteristic
88	C₂×C₇₃	—	8	2	no	—
89	C₂×C₇₃	#390	1	1	yes	characteristic
90	C₂×C₇₃	#416	1	1	yes	characteristic
91	C₂×C₇₃	#417	1	1	yes	characteristic
92	C₂×C₇₃	#463	1	2	yes	—
93	C₂×C₇₃	#464	1	2	yes	—
94	#1983	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₇₃	#1983	1	1	yes	characteristic, direct product factor, Sylow
96	C₂×#124	C₂×C₇₃	1	1	yes	characteristic
97	C₂×C₄⋊C₈	C₂×C₇₃	1	1	yes	characteristic, complemented
98	C₂×#103	C₂×C₇₃	1	2	yes	complemented
99	C₂×#102	C₂×C₇₃	1	2	yes	—
100	C₄×C₄⋊C₄	C₂×C₇₃	1	1	yes	characteristic, complemented
101	C₂²×Q₈	C₂²×C₇₃	1	1	yes	characteristic
102	C₂²×D₄	C₂²×C₇₃	1	1	yes	characteristic
103	C₂²×C₈	C₂²×C₇₃	1	2	yes	—
104	#124	—	2	2	no	—
105	#124	C₄×C₇₃	1	4	yes	complemented
106	C₂×C₄⋊C₄	C₂²×C₇₃	1	2	yes	—
107	C₂×C₂²⋊C₄	—	2	1	no	—
108	C₂×C₂²⋊C₄	—	2	1	no	—
109	C₂×C₄²	—	2	1	no	—
110	C₂×C₄²	C₂²×C₇₃	1	1	yes	characteristic
111	C₄⋊C₈	—	2	2	no	—
112	#103	—	2	4	no	—
113	#102	—	2	4	no	—
114	#95	—	2	1	no	—
115	C₂⁴	—	2	1	no	—
116	C₂×Q₈	—	2	1	no	—
117	C₂×Q₈	—	2	2	no	—
118	C₂×Q₈	—	2	2	no	—
119	C₂×Q₈	C₂×C₄×C₇₃	1	2	yes	—
120	C₂×D₄	—	2	2	no	—
121	C₂×D₄	—	2	1	no	—
122	C₂×D₄	—	2	2	no	—
123	C₂×D₄	C₂×C₄×C₇₃	1	2	yes	—
124	C₂²×C₄	—	2	2	no	—
125	C₂²×C₄	—	2	1	no	—
126	C₂²×C₄	D₄×C₇₃	1	1	yes	characteristic
127	C₂²×C₄	D₄×C₇₃	1	1	yes	characteristic
128	C₂²×C₄	C₂³×C₇₃	1	1	yes	characteristic
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	2	2	no	—
132	C₄⋊C₄	—	2	2	no	—
133	C₄⋊C₄	D₄×C₇₃	1	4	yes	—
134	C₂²⋊C₄	—	2	4	no	—
135	C₂²⋊C₄	—	2	4	no	—
136	C₄²	—	4	2	no	—
137	C₄²	C₂×C₄×C₇₃	1	2	yes	—
138	C₄²	D₄×C₇₃	1	2	yes	—
139	C₂³	—	2	2	no	—
140	C₂³	—	4	2	no	—
141	C₂³	—	4	2	no	—
142	C₂³	—	4	1	no	—
143	C₂³	—	4	1	no	—
144	C₂³	C₂×D₄×C₇₃	1	1	yes	characteristic
145	Q₈	—	2	4	no	—
146	Q₈	—	4	1	no	—
147	Q₈	—	4	1	no	—
148	D₄	—	2	4	no	—
149	D₄	—	4	1	no	—
150	D₄	—	4	1	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	C₂²⋊C₄×C₇₃	1	2	yes	—
160	C₂×C₄	C₂²⋊C₄×C₇₃	1	2	yes	—
161	C₂×C₄	C₂²×C₄×C₇₃	1	1	yes	characteristic
162	C₂×C₄	C₂×D₄×C₇₃	1	2	yes	—
163	C₂×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
164	C₂×C₄	#41×C₇₃	1	2	yes	—
165	C₈	—	4	2	no	—
166	C₂²	—	4	2	no	—
167	C₂²	—	4	2	no	—
168	C₂²	—	4	2	no	—
169	C₂²	—	8	2	no	—
170	C₂²	—	8	2	no	—
171	C₂²	C₂×C₂²⋊C₄×C₇₃	1	1	yes	characteristic
172	C₂²	#121×C₇₃	1	2	yes	—
173	C₂²	#135×C₇₃	1	2	yes	—
174	C₂²	AGL(1,ℤ/8)×C₇₃	1	1	yes	characteristic
175	C₂²	#137×C₇₃	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	4	2	no	—
179	C₄	—	4	4	no	—
180	C₄	C₄×D₄×C₇₃	1	2	yes	—
181	C₄	#124×C₇₃	1	1	yes	characteristic
182	C₄	#127×C₇₃	1	1	yes	characteristic
183	C₂	—	8	2	no	—
184	C₂	#390×C₇₃	1	1	yes	characteristic
185	C₂	#416×C₇₃	1	1	yes	characteristic
186	C₂	#417×C₇₃	1	1	yes	characteristic
187	C₂	#463×C₇₃	1	2	yes	—
188	C₂	#464×C₇₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#124×C₇₃

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#103×C₇₃

C₂

yes

maximal, complemented

C₂×#102×C₇₃

C₂

yes

maximal

C₄×C₄⋊C₄×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₇₃

C₂²

yes

characteristic

C₂²×D₄×C₇₃

C₂²

yes

characteristic

C₂²×C₈×C₇₃

C₂²

yes

—

#124×C₇₃

—

#124×C₇₃

C₄

yes

complemented

C₂×C₄⋊C₄×C₇₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₇₃

—

C₂×C₂²⋊C₄×C₇₃

—

C₂×C₄²×C₇₃

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

—

#103×C₇₃

—

#102×C₇₃

—

#95×C₇₃

—

C₂⁴×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

C₂×C₄

yes

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

C₂×C₄

yes

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

D₄

yes

—

C₂²⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂×C₄

yes

—

C₄²×C₇₃

D₄

yes

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×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⋊C₄

yes

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

#41

yes

—

C₈×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇₃

#121

yes

—

C₂²×C₇₃

#135

yes

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₄×D₄

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₂×C₇₃

#464

yes

—

#1983

C₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₇₃

#1983

yes

characteristic, direct product factor, Sylow

C₂×#124

C₂×C₇₃

yes

characteristic

C₂×C₄⋊C₈

C₂×C₇₃

yes

characteristic, complemented

C₂×#103

C₂×C₇₃

yes

complemented

C₂×#102

C₂×C₇₃

yes

—

100

C₄×C₄⋊C₄

C₂×C₇₃

yes

characteristic, complemented

101

C₂²×Q₈

C₂²×C₇₃

yes

characteristic

102

C₂²×D₄

C₂²×C₇₃

yes

characteristic

103

C₂²×C₈

C₂²×C₇₃

yes

—

104

#124

—

105

#124

C₄×C₇₃

yes

complemented

106

C₂×C₄⋊C₄

C₂²×C₇₃

yes

—

107

C₂×C₂²⋊C₄

—

108

C₂×C₂²⋊C₄

—

109

C₂×C₄²

—

110

C₂×C₄²

C₂²×C₇₃

yes

characteristic

111

C₄⋊C₈

—

112

#103

—

113

#102

—

114

#95

—

115

C₂⁴

—

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

C₂×C₄×C₇₃

yes

—

120

C₂×D₄

—

121

C₂×D₄

—

122

C₂×D₄

—

123

C₂×D₄

C₂×C₄×C₇₃

yes

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

D₄×C₇₃

yes

characteristic

127

C₂²×C₄

D₄×C₇₃

yes

characteristic

128

C₂²×C₄

C₂³×C₇₃

yes

characteristic

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₄⋊C₄

—

133

C₄⋊C₄

D₄×C₇₃

yes

—

134

C₂²⋊C₄

—

135

C₂²⋊C₄

—

136

C₄²

—

137

C₄²

C₂×C₄×C₇₃

yes

—

138

C₄²

D₄×C₇₃

yes

—

139

C₂³

—

140

C₂³

—

141

C₂³

—

142

C₂³

—

143

C₂³

—

144

C₂³

C₂×D₄×C₇₃

yes

characteristic

145

Q₈

—

146

Q₈

—

147

Q₈

—

148

D₄

—

149

D₄

—

150

D₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

C₂²⋊C₄×C₇₃

yes

—

160

C₂×C₄

C₂²⋊C₄×C₇₃

yes

—

161

C₂×C₄

C₂²×C₄×C₇₃

yes

characteristic

162

C₂×C₄

C₂×D₄×C₇₃

yes

—

163

C₂×C₄

C₂×D₄×C₇₃

yes

characteristic

164

C₂×C₄

#41×C₇₃

yes

—

165

C₈

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

C₂×C₂²⋊C₄×C₇₃

yes

characteristic

172

C₂²

#121×C₇₃

yes

—

173

C₂²

#135×C₇₃

yes

—

174

C₂²

AGL(1,ℤ/8)×C₇₃

yes

characteristic

175

C₂²

#137×C₇₃

yes

characteristic

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

C₄×D₄×C₇₃

yes

—

181

C₄

#124×C₇₃

yes

characteristic

182

C₄

#127×C₇₃

yes

characteristic

183

C₂

—

184

C₂

#390×C₇₃

yes

characteristic

185

C₂

#416×C₇₃

yes

characteristic

186

C₂

#417×C₇₃

yes

characteristic

187

C₂

#463×C₇₃

yes

—

188

C₂

#464×C₇₃

yes

—

	subgroup	G/H
center, socle	C₂³×C₇₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₇₃
Frattini, cosocle	C₂²×C₄	C₂³×C₇₃
prime radical	C₂²×D₄×C₇₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ G

subgroup

G/H

center, socle

C₂³×C₇₃

C₂×D₄

commutator

C₂×C₄

C₂²×C₄×C₇₃

Frattini, cosocle

C₂²×C₄

C₂³×C₇₃

prime radical

C₂²×D₄×C₇₃

C₂²

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ G

N	H ≅ G/N	type
2. C₂×C₄⋊C₈×C₇₃	C₂	semidirect
3. C₂×#103×C₇₃	C₂	semidirect
5. C₄×C₄⋊C₄×C₇₃	C₂	semidirect
10. #124×C₇₃	C₄	semidirect
94. #1983	C₇₃	direct (N × H)
95. C₇₃	#1983	direct (N × H)
97. C₂×C₄⋊C₈	C₂×C₇₃	semidirect
98. C₂×#103	C₂×C₇₃	semidirect
100. C₄×C₄⋊C₄	C₂×C₇₃	semidirect
105. #124	C₄×C₇₃	semidirect

H ≅ G/N

type

2. C₂×C₄⋊C₈×C₇₃

C₂

semidirect

3. C₂×#103×C₇₃

C₂

semidirect

5. C₄×C₄⋊C₄×C₇₃

C₂

semidirect

10. #124×C₇₃

C₄

semidirect

94. #1983

C₇₃

direct (N × H)

95. C₇₃

#1983

direct (N × H)

97. C₂×C₄⋊C₈

C₂×C₇₃

semidirect

98. C₂×#103

C₂×C₇₃

semidirect

100. C₄×C₄⋊C₄

C₂×C₇₃

semidirect

105. #124

C₄×C₇₃

semidirect

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

subgroup	index
C₂×#124×C₇₃	2
C₂×C₄⋊C₈×C₇₃	2
C₂×#103×C₇₃	2
C₂×#102×C₇₃	2
C₄×C₄⋊C₄×C₇₃	2
#1983	73

subgroup

index

C₂×#124×C₇₃

C₂×C₄⋊C₈×C₇₃

C₂×#103×C₇₃

C₂×#102×C₇₃

C₄×C₄⋊C₄×C₇₃

#1983

Subgroups — C₇₃×#1983 #22606136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#1983 #22606136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#1983 #22606136