Group #32943225 — subgroups

Subgroups — C₇₃×#33821 #32943225

Order 18688.

484 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 202 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×#3170	C₂	1	2	yes	maximal
2	C₇₃×#2225	C₂	1	4	yes	maximal, complemented
3	C₇₃×#2219	C₂	1	4	yes	maximal
4	C₇₃×#2209	C₂	1	4	yes	maximal, complemented
5	C₇₃×#2150	C₂	1	1	yes	maximal, characteristic
6	#533×C₇₃	—	2	2	no	—
7	C₂×Q₃₂×C₇₃	C₂²	1	4	yes	—
8	C₂×M₃₂×C₇₃	C₂²	1	2	yes	—
9	#500×C₇₃	C₂²	1	2	yes	—
10	#494×C₇₃	C₂²	1	2	yes	—
11	#487×C₇₃	—	2	2	no	—
12	#467×C₇₃	—	2	2	no	—
13	#467×C₇₃	C₂²	1	4	yes	—
14	C₄×Q₁₆×C₇₃	—	2	2	no	—
15	C₄×Q₁₆×C₇₃	C₂²	1	4	yes	—
16	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
17	#367×C₇₃	C₂²	1	4	yes	—
18	C₄⋊C₁₆×C₇₃	C₂²	1	4	yes	—
19	#358×C₇₃	C₂²	1	8	yes	—
20	C₂×Q₁₆×C₇₃	—	2	2	no	—
21	C₂×Q₁₆×C₇₃	C₂³	1	4	yes	—
22	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
24	C₄⋊Q₈×C₇₃	—	2	2	no	—
25	C₄⋊Q₈×C₇₃	—	2	2	no	—
26	#125×C₇₃	—	4	2	no	—
27	C₂²⋊Q₈×C₇₃	—	2	4	no	—
28	C₂²⋊Q₈×C₇₃	—	4	2	no	—
29	C₄×Q₈×C₇₃	—	2	4	no	—
30	C₄×Q₈×C₇₃	—	4	2	no	—
31	#117×C₇₃	—	4	2	no	—
32	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
33	Q₃₂×C₇₃	—	4	4	no	—
34	M₃₂×C₇₃	—	4	2	no	—
35	C₂×C₁₆×C₇₃	C₂³	1	4	yes	—
36	#107×C₇₃	—	2	2	no	—
37	#107×C₇₃	C₂³	1	4	yes	—
38	#103×C₇₃	—	2	4	no	—
39	#103×C₇₃	—	2	4	no	—
40	C₄×C₈×C₇₃	D₄	1	2	yes	—
41	C₄×C₈×C₇₃	C₂³	1	2	yes	—
42	C₂×Q₈×C₇₃	—	2	4	no	—
43	C₂×Q₈×C₇₃	—	4	2	no	—
44	C₂²×C₄×C₇₃	D₈	1	2	yes	—
45	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
46	Q₁₆×C₇₃	—	2	4	no	—
47	Q₁₆×C₇₃	—	2	4	no	—
48	Q₁₆×C₇₃	—	4	4	no	—
49	C₂×C₈×C₇₃	—	2	1	no	—
50	C₂×C₈×C₇₃	—	2	1	no	—
51	C₂×C₈×C₇₃	—	4	1	no	—
52	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
54	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
55	C₄⋊C₄×C₇₃	—	2	4	no	—
56	C₄⋊C₄×C₇₃	—	4	2	no	—
57	C₄⋊C₄×C₇₃	—	4	2	no	—
58	C₄⋊C₄×C₇₃	—	4	4	no	—
59	C₄⋊C₄×C₇₃	—	4	4	no	—
60	C₄⋊C₄×C₇₃	—	4	2	no	—
61	C₂²⋊C₄×C₇₃	—	4	4	no	—
62	C₄²×C₇₃	—	4	4	no	—
63	C₄²×C₇₃	D₈	1	1	yes	characteristic
64	C₄²×C₇₃	D₈	1	1	yes	characteristic
65	C₄²×C₇₃	C₂×D₄	1	2	yes	—
66	C₁₆×C₇₃	—	2	4	no	—
67	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
68	Q₈×C₇₃	—	4	4	no	—
69	Q₈×C₇₃	—	4	4	no	—
70	Q₈×C₇₃	—	8	4	no	—
71	C₂×C₄×C₇₃	—	2	1	no	—
72	C₂×C₄×C₇₃	—	2	1	no	—
73	C₂×C₄×C₇₃	—	2	4	no	—
74	C₂×C₄×C₇₃	—	4	4	no	—
75	C₂×C₄×C₇₃	—	4	4	no	—
76	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
78	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
79	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
80	C₈×C₇₃	—	2	1	no	—
81	C₈×C₇₃	—	2	2	no	—
82	C₈×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
83	C₈×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
84	C₂²×C₇₃	—	2	2	no	—
85	C₂²×C₇₃	—	2	1	no	—
86	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
87	C₄×C₇₃	—	2	2	no	—
88	C₄×C₇₃	—	2	1	no	—
89	C₄×C₇₃	—	8	4	no	—
90	C₄×C₇₃	—	8	4	no	—
91	C₄×C₇₃	#510	1	4	yes	—
92	C₄×C₇₃	#537	1	1	yes	characteristic
93	C₄×C₇₃	#575	1	1	yes	characteristic
94	#33821	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂×C₇₃	—	4	1	no	—
96	C₂×C₇₃	#3245	1	1	yes	characteristic
97	C₂×C₇₃	C₂×#510	1	2	yes	—
98	#3170	C₂×C₇₃	1	2	yes	—
99	#2225	C₂×C₇₃	1	4	yes	complemented
100	#2219	C₂×C₇₃	1	4	yes	—
101	#2209	C₂×C₇₃	1	4	yes	complemented
102	#2150	C₂×C₇₃	1	1	yes	characteristic
103	C₇₃	#33821	1	1	yes	characteristic, direct product factor, Sylow
104	#533	—	2	2	no	—
105	C₂×Q₃₂	C₂²×C₇₃	1	4	yes	—
106	C₂×M₃₂	C₂²×C₇₃	1	2	yes	—
107	#500	C₂²×C₇₃	1	2	yes	—
108	#494	C₂²×C₇₃	1	2	yes	—
109	#487	—	2	2	no	—
110	#467	—	2	2	no	—
111	#467	C₂²×C₇₃	1	4	yes	—
112	C₄×Q₁₆	—	2	2	no	—
113	C₄×Q₁₆	C₂²×C₇₃	1	4	yes	—
114	C₂×C₄×C₈	C₂²×C₇₃	1	1	yes	characteristic
115	#367	C₂²×C₇₃	1	4	yes	—
116	C₄⋊C₁₆	C₂²×C₇₃	1	4	yes	—
117	#358	C₂²×C₇₃	1	8	yes	—
118	C₂×Q₁₆	—	2	2	no	—
119	C₂×Q₁₆	C₂³×C₇₃	1	4	yes	—
120	C₂²×C₈	D₄×C₇₃	1	1	yes	characteristic
121	C₂²×C₈	C₂³×C₇₃	1	1	yes	characteristic
122	C₄⋊Q₈	—	2	2	no	—
123	C₄⋊Q₈	—	2	2	no	—
124	#125	—	4	2	no	—
125	C₂²⋊Q₈	—	2	4	no	—
126	C₂²⋊Q₈	—	4	2	no	—
127	C₄×Q₈	—	2	4	no	—
128	C₄×Q₈	—	4	2	no	—
129	#117	—	4	2	no	—
130	C₂×C₄²	D₄×C₇₃	1	1	yes	characteristic
131	Q₃₂	—	4	4	no	—
132	M₃₂	—	4	2	no	—
133	C₂×C₁₆	C₂³×C₇₃	1	4	yes	—
134	#107	—	2	2	no	—
135	#107	C₂³×C₇₃	1	4	yes	—
136	#103	—	2	4	no	—
137	#103	—	2	4	no	—
138	C₄×C₈	D₄×C₇₃	1	2	yes	—
139	C₄×C₈	C₂³×C₇₃	1	2	yes	—
140	C₂×Q₈	—	2	4	no	—
141	C₂×Q₈	—	4	2	no	—
142	C₂²×C₄	D₈×C₇₃	1	2	yes	—
143	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
144	Q₁₆	—	2	4	no	—
145	Q₁₆	—	2	4	no	—
146	Q₁₆	—	4	4	no	—
147	C₂×C₈	—	2	1	no	—
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	4	1	no	—
150	C₂×C₈	C₂×D₄×C₇₃	1	1	yes	characteristic
151	C₂×C₈	C₂×D₄×C₇₃	1	2	yes	—
152	C₂×C₈	C₂⁴×C₇₃	1	1	yes	characteristic
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	4	2	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₄⋊C₄	—	4	4	no	—
157	C₄⋊C₄	—	4	4	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₂²⋊C₄	—	4	4	no	—
160	C₄²	—	4	4	no	—
161	C₄²	D₈×C₇₃	1	1	yes	characteristic
162	C₄²	D₈×C₇₃	1	1	yes	characteristic
163	C₄²	C₂×D₄×C₇₃	1	2	yes	—
164	C₁₆	—	2	4	no	—
165	C₂³	C₂×D₈×C₇₃	1	1	yes	characteristic
166	Q₈	—	4	4	no	—
167	Q₈	—	4	4	no	—
168	Q₈	—	8	4	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	4	4	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	C₂×D₈×C₇₃	1	1	yes	characteristic
175	C₂×C₄	C₂×D₈×C₇₃	1	2	yes	—
176	C₂×C₄	C₂×D₈×C₇₃	1	2	yes	—
177	C₂×C₄	C₂²×D₄×C₇₃	1	1	yes	characteristic
178	C₈	—	2	1	no	—
179	C₈	—	2	2	no	—
180	C₈	H(5,𝔽₂)×C₇₃	1	1	yes	characteristic
181	C₈	2₋¹⁺⁴×C₇₃	1	1	yes	characteristic
182	C₂²	—	2	1	no	—
183	C₂²	—	2	2	no	—
184	C₂²	C₂²×D₈×C₇₃	1	1	yes	characteristic
185	C₄	—	2	1	no	—
186	C₄	—	2	2	no	—
187	C₄	—	8	4	no	—
188	C₄	—	8	4	no	—
189	C₄	#510×C₇₃	1	4	yes	—
190	C₄	#537×C₇₃	1	1	yes	characteristic
191	C₄	#575×C₇₃	1	1	yes	characteristic
192	C₂	—	4	1	no	—
193	C₂	C₇₃×#3245	1	1	yes	characteristic
194	C₂	C₂×#510×C₇₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₃×#3170

C₂

yes

maximal

C₇₃×#2225

C₂

yes

maximal, complemented

C₇₃×#2219

C₂

yes

maximal

C₇₃×#2209

C₂

yes

maximal, complemented

C₇₃×#2150

C₂

yes

maximal, characteristic

#533×C₇₃

—

C₂×Q₃₂×C₇₃

C₂²

yes

—

C₂×M₃₂×C₇₃

C₂²

yes

—

#500×C₇₃

C₂²

yes

—

#494×C₇₃

C₂²

yes

—

#487×C₇₃

—

#467×C₇₃

—

#467×C₇₃

C₂²

yes

—

C₄×Q₁₆×C₇₃

—

C₄×Q₁₆×C₇₃

C₂²

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#367×C₇₃

C₂²

yes

—

C₄⋊C₁₆×C₇₃

C₂²

yes

—

#358×C₇₃

C₂²

yes

—

C₂×Q₁₆×C₇₃

—

C₂×Q₁₆×C₇₃

C₂³

yes

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

C₂³

yes

characteristic

C₄⋊Q₈×C₇₃

—

C₄⋊Q₈×C₇₃

—

#125×C₇₃

—

C₂²⋊Q₈×C₇₃

—

C₂²⋊Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

#117×C₇₃

—

C₂×C₄²×C₇₃

D₄

yes

characteristic

Q₃₂×C₇₃

—

M₃₂×C₇₃

—

C₂×C₁₆×C₇₃

C₂³

yes

—

#107×C₇₃

—

#107×C₇₃

C₂³

yes

—

#103×C₇₃

—

#103×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

C₂³

yes

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

D₈

yes

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₂×C₈×C₇₃

C₂⁴

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

D₈

yes

characteristic

C₄²×C₇₃

D₈

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

—

C₁₆×C₇₃

—

C₂³×C₇₃

C₂×D₈

yes

characteristic

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×D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

—

C₂×C₄×C₇₃

C₂×D₈

yes

—

C₂×C₄×C₇₃

C₂²×D₄

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

H(5,𝔽₂)

yes

characteristic

C₈×C₇₃

2₋¹⁺⁴

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂²×D₈

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#510

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₃

—

C₂×C₇₃

#3245

yes

characteristic

C₂×C₇₃

C₂×#510

yes

—

#3170

C₂×C₇₃

yes

—

#2225

C₂×C₇₃

yes

complemented

100

#2219

C₂×C₇₃

yes

—

101

#2209

C₂×C₇₃

yes

complemented

102

#2150

C₂×C₇₃

yes

characteristic

103

C₇₃

#33821

yes

characteristic, direct product factor, Sylow

104

#533

—

105

C₂×Q₃₂

C₂²×C₇₃

yes

—

106

C₂×M₃₂

C₂²×C₇₃

yes

—

107

#500

C₂²×C₇₃

yes

—

108

#494

C₂²×C₇₃

yes

—

109

#487

—

110

#467

—

111

#467

C₂²×C₇₃

yes

—

112

C₄×Q₁₆

—

113

C₄×Q₁₆

C₂²×C₇₃

yes

—

114

C₂×C₄×C₈

C₂²×C₇₃

yes

characteristic

115

#367

C₂²×C₇₃

yes

—

116

C₄⋊C₁₆

C₂²×C₇₃

yes

—

117

#358

C₂²×C₇₃

yes

—

118

C₂×Q₁₆

—

119

C₂×Q₁₆

C₂³×C₇₃

yes

—

120

C₂²×C₈

D₄×C₇₃

yes

characteristic

121

C₂²×C₈

C₂³×C₇₃

yes

characteristic

122

C₄⋊Q₈

—

123

C₄⋊Q₈

—

124

#125

—

125

C₂²⋊Q₈

—

126

C₂²⋊Q₈

—

127

C₄×Q₈

—

128

C₄×Q₈

—

129

#117

—

130

C₂×C₄²

D₄×C₇₃

yes

characteristic

131

Q₃₂

—

132

M₃₂

—

133

C₂×C₁₆

C₂³×C₇₃

yes

—

134

#107

—

135

#107

C₂³×C₇₃

yes

—

136

#103

—

137

#103

—

138

C₄×C₈

D₄×C₇₃

yes

—

139

C₄×C₈

C₂³×C₇₃

yes

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂²×C₄

D₈×C₇₃

yes

—

143

C₂²×C₄

C₂×D₄×C₇₃

yes

characteristic

144

Q₁₆

—

145

Q₁₆

—

146

Q₁₆

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

C₂×D₄×C₇₃

yes

characteristic

151

C₂×C₈

C₂×D₄×C₇₃

yes

—

152

C₂×C₈

C₂⁴×C₇₃

yes

characteristic

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₂²⋊C₄

—

160

C₄²

—

161

C₄²

D₈×C₇₃

yes

characteristic

162

C₄²

D₈×C₇₃

yes

characteristic

163

C₄²

C₂×D₄×C₇₃

yes

—

164

C₁₆

—

165

C₂³

C₂×D₈×C₇₃

yes

characteristic

166

Q₈

—

167

Q₈

—

168

Q₈

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

C₂×D₈×C₇₃

yes

characteristic

175

C₂×C₄

C₂×D₈×C₇₃

yes

—

176

C₂×C₄

C₂×D₈×C₇₃

yes

—

177

C₂×C₄

C₂²×D₄×C₇₃

yes

characteristic

178

C₈

—

179

C₈

—

180

C₈

H(5,𝔽₂)×C₇₃

yes

characteristic

181

C₈

2₋¹⁺⁴×C₇₃

yes

characteristic

182

C₂²

—

183

C₂²

—

184

C₂²

C₂²×D₈×C₇₃

yes

characteristic

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

#510×C₇₃

yes

—

190

C₄

#537×C₇₃

yes

characteristic

191

C₄

#575×C₇₃

yes

characteristic

192

C₂

—

193

C₂

C₇₃×#3245

yes

characteristic

194

C₂

C₂×#510×C₇₃

yes

—

	subgroup	G/H
center, socle	C₂²×C₇₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₇₃
prime radical	C₂³×C₇₃	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ C₂×C₄×C₈×C₇₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₇₃

C₂²×D₈

commutator, Frattini, cosocle

C₂×C₈

C₂⁴×C₇₃

prime radical

C₂³×C₇₃

C₂×D₈

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ C₂×C₄×C₈×C₇₃ ◁ G

N	H ≅ G/N	type
2. C₇₃×#2225	C₂	semidirect
4. C₇₃×#2209	C₂	semidirect
94. #33821	C₇₃	direct (N × H)
99. #2225	C₂×C₇₃	semidirect
101. #2209	C₂×C₇₃	semidirect
103. C₇₃	#33821	direct (N × H)

H ≅ G/N

type

2. C₇₃×#2225

C₂

semidirect

4. C₇₃×#2209

C₂

semidirect

94. #33821

C₇₃

direct (N × H)

99. #2225

C₂×C₇₃

semidirect

101. #2209

C₂×C₇₃

semidirect

103. C₇₃

#33821

direct (N × H)

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

subgroup	index
C₇₃×#3170	2
C₇₃×#2225	2
C₇₃×#2219	2
C₇₃×#2209	2
C₇₃×#2150	2
#33821	73

subgroup

index

C₇₃×#3170

C₇₃×#2225

C₇₃×#2219

C₇₃×#2209

C₇₃×#2150

#33821

Subgroups — C₇₃×#33821 #32943225

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#33821 #32943225

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#33821 #32943225