Group #22606154 — subgroups

Subgroups — C₇₃×#2001 #22606154

Order 9344.

392 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 112 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	C₂²×Q₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊Q₈×C₇₃	C₂	1	2	yes	maximal
3	C₂×#103×C₇₃	C₂	1	2	yes	maximal, complemented
4	C₂×C₂²⋊C₈×C₇₃	C₂	1	1	yes	maximal, characteristic
5	#340×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₈×C₇₃	C₂²	1	2	yes	—
7	C₂³×C₄×C₇₃	C₂²	1	1	yes	characteristic
8	C₂×Q₁₆×C₇₃	—	2	4	no	—
9	C₂×Q₁₆×C₇₃	—	2	2	no	—
10	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
12	C₂²⋊Q₈×C₇₃	—	2	4	no	—
13	C₂²⋊Q₈×C₇₃	—	2	4	no	—
14	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
15	C₂×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
16	C₂×C₂²⋊C₄×C₇₃	—	2	2	no	—
17	#103×C₇₃	—	2	4	no	—
18	C₂²⋊C₈×C₇₃	—	2	2	no	—
19	C₂⁴×C₇₃	D₄	1	1	yes	characteristic
20	C₂×Q₈×C₇₃	—	2	2	no	—
21	C₂×Q₈×C₇₃	—	2	2	no	—
22	C₂×Q₈×C₇₃	—	2	2	no	—
23	C₂×Q₈×C₇₃	—	2	2	no	—
24	C₂×Q₈×C₇₃	—	2	2	no	—
25	C₂×Q₈×C₇₃	D₄	1	4	yes	—
26	C₂²×C₄×C₇₃	—	2	2	no	—
27	C₂²×C₄×C₇₃	—	2	2	no	—
28	C₂²×C₄×C₇₃	—	2	1	no	—
29	C₂²×C₄×C₇₃	—	2	2	no	—
30	C₂²×C₄×C₇₃	—	2	2	no	—
31	C₂²×C₄×C₇₃	—	2	1	no	—
32	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
34	Q₁₆×C₇₃	—	2	8	no	—
35	C₂×C₈×C₇₃	—	2	2	no	—
36	C₂×C₈×C₇₃	—	2	1	no	—
37	C₂×C₈×C₇₃	—	2	2	no	—
38	C₂×C₈×C₇₃	D₄	1	2	yes	—
39	C₄⋊C₄×C₇₃	—	2	8	no	—
40	C₄⋊C₄×C₇₃	—	2	2	no	—
41	C₄⋊C₄×C₇₃	D₄	1	4	yes	—
42	C₂²⋊C₄×C₇₃	—	2	8	no	—
43	C₂³×C₇₃	—	2	2	no	—
44	C₂³×C₇₃	—	2	1	no	—
45	C₂³×C₇₃	—	2	1	no	—
46	C₂³×C₇₃	—	2	2	no	—
47	C₂³×C₇₃	Q₁₆	1	2	yes	complemented
48	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
49	Q₈×C₇₃	—	2	4	no	—
50	Q₈×C₇₃	—	2	4	no	—
51	Q₈×C₇₃	—	4	2	no	—
52	Q₈×C₇₃	—	4	2	no	—
53	C₂×C₄×C₇₃	—	2	4	no	—
54	C₂×C₄×C₇₃	—	2	4	no	—
55	C₂×C₄×C₇₃	—	2	2	no	—
56	C₂×C₄×C₇₃	—	2	1	no	—
57	C₂×C₄×C₇₃	—	4	2	no	—
58	C₂×C₄×C₇₃	—	4	2	no	—
59	C₂×C₄×C₇₃	—	4	2	no	—
60	C₂×C₄×C₇₃	—	4	2	no	—
61	C₂×C₄×C₇₃	—	4	1	no	—
62	C₂×C₄×C₇₃	—	4	2	no	—
63	C₂×C₄×C₇₃	—	4	1	no	—
64	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
68	C₈×C₇₃	—	2	2	no	—
69	C₈×C₇₃	—	4	1	no	—
70	C₂²×C₇₃	—	2	4	no	—
71	C₂²×C₇₃	—	2	2	no	—
72	C₂²×C₇₃	—	4	1	no	—
73	C₂²×C₇₃	—	4	2	no	—
74	C₂²×C₇₃	—	4	1	no	—
75	C₂²×C₇₃	#120	1	2	yes	—
76	C₂²×C₇₃	#121	1	1	yes	characteristic
77	C₂²×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
78	C₂²×C₇₃	#135	1	1	yes	characteristic
79	C₂²×C₇₃	#137	1	2	yes	—
80	C₄×C₇₃	—	4	1	no	—
81	C₄×C₇₃	—	4	4	no	—
82	C₄×C₇₃	—	4	4	no	—
83	C₄×C₇₃	#120	1	1	yes	characteristic
84	C₄×C₇₃	#121	1	2	yes	—
85	C₄×C₇₃	#127	1	1	yes	characteristic
86	C₂×C₇₃	—	4	2	no	—
87	C₂×C₇₃	#392	1	1	yes	characteristic
88	C₂×C₇₃	#451	1	2	yes	—
89	C₂×C₇₃	#452	1	2	yes	—
90	C₂×C₇₃	#467	1	1	yes	characteristic
91	C₂×C₇₃	#470	1	1	yes	characteristic
92	#2001	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₇₃	#2001	1	1	yes	characteristic, direct product factor, Sylow
94	C₂²×Q₁₆	C₂×C₇₃	1	1	yes	characteristic, complemented
95	C₂×C₂²⋊Q₈	C₂×C₇₃	1	2	yes	—
96	C₂×#103	C₂×C₇₃	1	2	yes	complemented
97	C₂×C₂²⋊C₈	C₂×C₇₃	1	1	yes	characteristic
98	#340	C₂×C₇₃	1	1	yes	characteristic, complemented
99	C₂²×Q₈	C₂²×C₇₃	1	2	yes	—
100	C₂³×C₄	C₂²×C₇₃	1	1	yes	characteristic
101	C₂×Q₁₆	—	2	4	no	—
102	C₂×Q₁₆	—	2	2	no	—
103	C₂²×C₈	C₂²×C₇₃	1	1	yes	characteristic
104	C₂²×C₈	C₂²×C₇₃	1	1	yes	characteristic
105	C₂²⋊Q₈	—	2	4	no	—
106	C₂²⋊Q₈	—	2	4	no	—
107	C₂×C₄⋊C₄	—	2	2	no	—
108	C₂×C₄⋊C₄	C₂²×C₇₃	1	2	yes	—
109	C₂×C₂²⋊C₄	—	2	2	no	—
110	#103	—	2	4	no	—
111	C₂²⋊C₈	—	2	2	no	—
112	C₂⁴	D₄×C₇₃	1	1	yes	characteristic
113	C₂×Q₈	—	2	2	no	—
114	C₂×Q₈	—	2	2	no	—
115	C₂×Q₈	—	2	2	no	—
116	C₂×Q₈	—	2	2	no	—
117	C₂×Q₈	—	2	2	no	—
118	C₂×Q₈	D₄×C₇₃	1	4	yes	—
119	C₂²×C₄	—	2	2	no	—
120	C₂²×C₄	—	2	2	no	—
121	C₂²×C₄	—	2	1	no	—
122	C₂²×C₄	—	2	2	no	—
123	C₂²×C₄	—	2	2	no	—
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	D₄×C₇₃	1	1	yes	characteristic
126	C₂²×C₄	C₂³×C₇₃	1	1	yes	characteristic
127	Q₁₆	—	2	8	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	D₄×C₇₃	1	2	yes	—
132	C₄⋊C₄	—	2	8	no	—
133	C₄⋊C₄	—	2	2	no	—
134	C₄⋊C₄	D₄×C₇₃	1	4	yes	—
135	C₂²⋊C₄	—	2	8	no	—
136	C₂³	—	2	2	no	—
137	C₂³	—	2	1	no	—
138	C₂³	—	2	1	no	—
139	C₂³	—	2	2	no	—
140	C₂³	Q₁₆×C₇₃	1	2	yes	complemented
141	C₂³	C₂×D₄×C₇₃	1	1	yes	characteristic
142	Q₈	—	2	4	no	—
143	Q₈	—	2	4	no	—
144	Q₈	—	4	2	no	—
145	Q₈	—	4	2	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	C₂×D₄×C₇₃	1	2	yes	—
158	C₂×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
159	C₂×C₄	C₂×D₄×C₇₃	1	2	yes	—
160	C₂×C₄	#41×C₇₃	1	1	yes	characteristic
161	C₈	—	2	2	no	—
162	C₈	—	4	1	no	—
163	C₂²	—	2	4	no	—
164	C₂²	—	2	2	no	—
165	C₂²	—	4	1	no	—
166	C₂²	—	4	2	no	—
167	C₂²	—	4	1	no	—
168	C₂²	#120×C₇₃	1	2	yes	—
169	C₂²	#121×C₇₃	1	1	yes	characteristic
170	C₂²	C₂×Q₁₆×C₇₃	1	1	yes	characteristic
171	C₂²	#135×C₇₃	1	1	yes	characteristic
172	C₂²	#137×C₇₃	1	2	yes	—
173	C₄	—	4	1	no	—
174	C₄	—	4	4	no	—
175	C₄	—	4	4	no	—
176	C₄	#120×C₇₃	1	1	yes	characteristic
177	C₄	#121×C₇₃	1	2	yes	—
178	C₄	#127×C₇₃	1	1	yes	characteristic
179	C₂	—	4	2	no	—
180	C₂	#392×C₇₃	1	1	yes	characteristic
181	C₂	#451×C₇₃	1	2	yes	—
182	C₂	#452×C₇₃	1	2	yes	—
183	C₂	#467×C₇₃	1	1	yes	characteristic
184	C₂	#470×C₇₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Q₁₆×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊Q₈×C₇₃

C₂

yes

maximal

C₂×#103×C₇₃

C₂

yes

maximal, complemented

C₂×C₂²⋊C₈×C₇₃

C₂

yes

maximal, characteristic

#340×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₇₃

C₂²

yes

—

C₂³×C₄×C₇₃

C₂²

yes

characteristic

C₂×Q₁₆×C₇₃

—

C₂×Q₁₆×C₇₃

—

C₂²×C₈×C₇₃

C₂²

yes

characteristic

C₂²×C₈×C₇₃

C₂²

yes

characteristic

C₂²⋊Q₈×C₇₃

—

C₂²⋊Q₈×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₇₃

—

#103×C₇₃

—

C₂²⋊C₈×C₇₃

—

C₂⁴×C₇₃

D₄

yes

characteristic

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

D₄

yes

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

characteristic

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₄

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

D₄

yes

—

C₂²⋊C₄×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

Q₁₆

yes

complemented

C₂³×C₇₃

C₂×D₄

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×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

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

#41

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#120

yes

—

C₂²×C₇₃

#121

yes

characteristic

C₂²×C₇₃

C₂×Q₁₆

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#120

yes

characteristic

C₄×C₇₃

#121

yes

—

C₄×C₇₃

#127

yes

characteristic

C₂×C₇₃

—

C₂×C₇₃

#392

yes

characteristic

C₂×C₇₃

#451

yes

—

C₂×C₇₃

#452

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₇₃

#2001

yes

characteristic, direct product factor, Sylow

C₂²×Q₁₆

C₂×C₇₃

yes

characteristic, complemented

C₂×C₂²⋊Q₈

C₂×C₇₃

yes

—

C₂×#103

C₂×C₇₃

yes

complemented

C₂×C₂²⋊C₈

C₂×C₇₃

yes

characteristic

#340

C₂×C₇₃

yes

characteristic, complemented

C₂²×Q₈

C₂²×C₇₃

yes

—

100

C₂³×C₄

C₂²×C₇₃

yes

characteristic

101

C₂×Q₁₆

—

102

C₂×Q₁₆

—

103

C₂²×C₈

C₂²×C₇₃

yes

characteristic

104

C₂²×C₈

C₂²×C₇₃

yes

characteristic

105

C₂²⋊Q₈

—

106

C₂²⋊Q₈

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

C₂²×C₇₃

yes

—

109

C₂×C₂²⋊C₄

—

110

#103

—

111

C₂²⋊C₈

—

112

C₂⁴

D₄×C₇₃

yes

characteristic

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

D₄×C₇₃

yes

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

D₄×C₇₃

yes

characteristic

126

C₂²×C₄

C₂³×C₇₃

yes

characteristic

127

Q₁₆

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

D₄×C₇₃

yes

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

D₄×C₇₃

yes

—

135

C₂²⋊C₄

—

136

C₂³

—

137

C₂³

—

138

C₂³

—

139

C₂³

—

140

C₂³

Q₁₆×C₇₃

yes

complemented

141

C₂³

C₂×D₄×C₇₃

yes

characteristic

142

Q₈

—

143

Q₈

—

144

Q₈

—

145

Q₈

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

C₂×D₄×C₇₃

yes

—

158

C₂×C₄

C₂×D₄×C₇₃

yes

characteristic

159

C₂×C₄

C₂×D₄×C₇₃

yes

—

160

C₂×C₄

#41×C₇₃

yes

characteristic

161

C₈

—

162

C₈

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

#120×C₇₃

yes

—

169

C₂²

#121×C₇₃

yes

characteristic

170

C₂²

C₂×Q₁₆×C₇₃

yes

characteristic

171

C₂²

#135×C₇₃

yes

characteristic

172

C₂²

#137×C₇₃

yes

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

#120×C₇₃

yes

characteristic

177

C₄

#121×C₇₃

yes

—

178

C₄

#127×C₇₃

yes

characteristic

179

C₂

—

180

C₂

#392×C₇₃

yes

characteristic

181

C₂

#451×C₇₃

yes

—

182

C₂

#452×C₇₃

yes

—

183

C₂

#467×C₇₃

yes

characteristic

184

C₂

#470×C₇₃

yes

characteristic

	subgroup	G/H
center, socle	C₂³×C₇₃	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₇₃
prime radical	C₂⁴×C₇₃	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇₃ ◁ C₂³×C₄×C₇₃ ◁ G

subgroup

G/H

center, socle

C₂³×C₇₃

C₂×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₇₃

prime radical

C₂⁴×C₇₃

D₄

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₇₃ ◁ C₂³×C₄×C₇₃ ◁ G

N	H ≅ G/N	type
1. C₂²×Q₁₆×C₇₃	C₂	semidirect
3. C₂×#103×C₇₃	C₂	semidirect
5. #340×C₇₃	C₂	semidirect
47. C₂³×C₇₃	Q₁₆	semidirect
92. #2001	C₇₃	direct (N × H)
93. C₇₃	#2001	direct (N × H)
94. C₂²×Q₁₆	C₂×C₇₃	semidirect
96. C₂×#103	C₂×C₇₃	semidirect
98. #340	C₂×C₇₃	semidirect
140. C₂³	Q₁₆×C₇₃	semidirect

H ≅ G/N

type

1. C₂²×Q₁₆×C₇₃

C₂

semidirect

3. C₂×#103×C₇₃

C₂

semidirect

5. #340×C₇₃

C₂

semidirect

47. C₂³×C₇₃

Q₁₆

semidirect

92. #2001

C₇₃

direct (N × H)

93. C₇₃

#2001

direct (N × H)

94. C₂²×Q₁₆

C₂×C₇₃

semidirect

96. C₂×#103

C₂×C₇₃

semidirect

98. #340

C₂×C₇₃

semidirect

140. C₂³

Q₁₆×C₇₃

semidirect

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

subgroup	index
C₂²×Q₁₆×C₇₃	2
C₂×C₂²⋊Q₈×C₇₃	2
C₂×#103×C₇₃	2
C₂×C₂²⋊C₈×C₇₃	2
#340×C₇₃	2
#2001	73

subgroup

index

C₂²×Q₁₆×C₇₃

C₂×C₂²⋊Q₈×C₇₃

C₂×#103×C₇₃

C₂×C₂²⋊C₈×C₇₃

#340×C₇₃

#2001

Subgroups — C₇₃×#2001 #22606154

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#2001 #22606154

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#2001 #22606154