Group #32943334 — subgroups

Subgroups — C₇₃×#33930 #32943334

Order 18688.

472 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 218 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×#3395	C₂	1	1	yes	maximal, characteristic
2	Q₈×Q₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#2241	C₂	1	3	yes	maximal
4	C₇₃×#2227	C₂	1	3	yes	maximal
5	C₇₃×#2209	C₂	1	3	yes	maximal
6	Q₈×C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
7	C₄×Q₃₂×C₇₃	C₂	1	3	yes	maximal
8	Q₈²×C₇₃	—	2	1	no	—
9	#557×C₇₃	—	2	1	no	—
10	C₂×Q₃₂×C₇₃	C₂²	1	3	yes	—
11	#500×C₇₃	C₂²	1	3	yes	—
12	#494×C₇₃	C₂²	1	3	yes	—
13	#477×C₇₃	—	2	3	no	—
14	#462×C₇₃	—	2	3	no	—
15	C₈×Q₈×C₇₃	C₂²	1	1	yes	characteristic
16	C₄×Q₁₆×C₇₃	C₂²	1	3	yes	—
17	C₄×Q₁₆×C₇₃	C₂²	1	3	yes	—
18	#366×C₇₃	C₂²	1	3	yes	—
19	C₄⋊C₁₆×C₇₃	C₂²	1	3	yes	—
20	#358×C₇₃	C₂²	1	6	yes	—
21	#358×C₇₃	C₂²	1	3	yes	—
22	#358×C₇₃	C₂²	1	1	yes	characteristic
23	C₄×C₁₆×C₇₃	C₂²	1	3	yes	—
24	C₂×Q₁₆×C₇₃	—	2	3	no	—
25	C₂×Q₁₆×C₇₃	C₂³	1	3	yes	—
26	C₂×Q₁₆×C₇₃	C₂³	1	1	yes	characteristic
27	C₄⋊Q₈×C₇₃	—	2	3	no	—
28	C₄⋊Q₈×C₇₃	—	2	3	no	—
29	C₄⋊Q₈×C₇₃	—	4	3	no	—
30	#125×C₇₃	—	4	3	no	—
31	C₄×Q₈×C₇₃	—	2	3	no	—
32	C₄×Q₈×C₇₃	—	2	3	no	—
33	C₄×Q₈×C₇₃	—	4	1	no	—
34	C₄×Q₈×C₇₃	—	4	1	no	—
35	C₄×Q₈×C₇₃	D₄	1	1	yes	characteristic
36	Q₃₂×C₇₃	—	2	6	no	—
37	C₂×C₁₆×C₇₃	C₂³	1	3	yes	—
38	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
39	#107×C₇₃	—	2	3	no	—
40	#107×C₇₃	C₂³	1	1	yes	characteristic
41	#107×C₇₃	C₂³	1	3	yes	—
42	C₄⋊C₈×C₇₃	D₄	1	3	yes	—
43	#103×C₇₃	—	2	3	no	—
44	#103×C₇₃	—	2	3	no	—
45	C₄×C₈×C₇₃	C₂³	1	3	yes	—
46	C₂×Q₈×C₇₃	—	2	1	no	—
47	C₂×Q₈×C₇₃	—	2	3	no	—
48	C₂×Q₈×C₇₃	—	4	3	no	—
49	C₂×Q₈×C₇₃	D₈	1	1	yes	characteristic
50	Q₁₆×C₇₃	—	2	3	no	—
51	Q₁₆×C₇₃	—	2	1	no	—
52	Q₁₆×C₇₃	—	2	3	no	—
53	Q₁₆×C₇₃	—	8	3	no	—
54	Q₁₆×C₇₃	#41	1	2	yes	—
55	C₂×C₈×C₇₃	C₂×D₄	1	3	yes	—
56	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
57	C₄⋊C₄×C₇₃	—	2	1	no	—
58	C₄⋊C₄×C₇₃	—	2	3	no	—
59	C₄⋊C₄×C₇₃	—	4	3	no	—
60	C₄⋊C₄×C₇₃	—	4	3	no	—
61	C₄⋊C₄×C₇₃	—	4	3	no	—
62	C₄⋊C₄×C₇₃	—	4	3	no	—
63	C₄⋊C₄×C₇₃	—	4	3	no	—
64	C₄⋊C₄×C₇₃	D₈	1	3	yes	—
65	C₄²×C₇₃	—	4	3	no	—
66	C₄²×C₇₃	—	4	3	no	—
67	C₄²×C₇₃	C₂×D₄	1	3	yes	—
68	C₁₆×C₇₃	—	2	3	no	—
69	C₁₆×C₇₃	—	2	1	no	—
70	Q₈×C₇₃	—	2	2	no	—
71	Q₈×C₇₃	—	4	1	no	—
72	Q₈×C₇₃	—	4	3	no	—
73	Q₈×C₇₃	—	4	3	no	—
74	Q₈×C₇₃	—	16	3	no	—
75	Q₈×C₇₃	Q₃₂	1	4	yes	complemented
76	C₂×C₄×C₇₃	—	4	1	no	—
77	C₂×C₄×C₇₃	—	4	1	no	—
78	C₂×C₄×C₇₃	—	4	3	no	—
79	C₂×C₄×C₇₃	—	4	3	no	—
80	C₂×C₄×C₇₃	C₂×D₈	1	3	yes	—
81	C₂×C₄×C₇₃	C₂×D₈	1	3	yes	—
82	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
83	C₈×C₇₃	—	2	3	no	—
84	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
85	C₈×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
86	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
87	C₄×C₇₃	—	2	3	no	—
88	C₄×C₇₃	—	4	2	no	—
89	C₄×C₇₃	—	8	3	no	—
90	C₄×C₇₃	—	8	1	no	—
91	C₄×C₇₃	—	8	3	no	—
92	C₄×C₇₃	C₂×Q₃₂	1	6	yes	—
93	C₄×C₇₃	#548	1	1	yes	characteristic
94	C₄×C₇₃	#577	1	1	yes	characteristic
95	#33930	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂×C₇₃	#3358	1	1	yes	characteristic
97	C₂×C₇₃	C₂²×Q₃₂	1	1	yes	characteristic
98	C₂×C₇₃	#3416	1	1	yes	characteristic
99	#3395	C₂×C₇₃	1	1	yes	characteristic
100	Q₈×Q₁₆	C₂×C₇₃	1	1	yes	characteristic
101	#2241	C₂×C₇₃	1	3	yes	—
102	#2227	C₂×C₇₃	1	3	yes	—
103	#2209	C₂×C₇₃	1	3	yes	—
104	Q₈×C₁₆	C₂×C₇₃	1	1	yes	characteristic
105	C₄×Q₃₂	C₂×C₇₃	1	3	yes	—
106	C₇₃	#33930	1	1	yes	characteristic, direct product factor, Sylow
107	Q₈²	—	2	1	no	—
108	#557	—	2	1	no	—
109	C₂×Q₃₂	C₂²×C₇₃	1	3	yes	—
110	#500	C₂²×C₇₃	1	3	yes	—
111	#494	C₂²×C₇₃	1	3	yes	—
112	#477	—	2	3	no	—
113	#462	—	2	3	no	—
114	C₈×Q₈	C₂²×C₇₃	1	1	yes	characteristic
115	C₄×Q₁₆	C₂²×C₇₃	1	3	yes	—
116	C₄×Q₁₆	C₂²×C₇₃	1	3	yes	—
117	#366	C₂²×C₇₃	1	3	yes	—
118	C₄⋊C₁₆	C₂²×C₇₃	1	3	yes	—
119	#358	C₂²×C₇₃	1	1	yes	characteristic
120	#358	C₂²×C₇₃	1	3	yes	—
121	#358	C₂²×C₇₃	1	6	yes	—
122	C₄×C₁₆	C₂²×C₇₃	1	3	yes	—
123	C₂×Q₁₆	—	2	3	no	—
124	C₂×Q₁₆	C₂³×C₇₃	1	3	yes	—
125	C₂×Q₁₆	C₂³×C₇₃	1	1	yes	characteristic
126	C₄⋊Q₈	—	2	3	no	—
127	C₄⋊Q₈	—	2	3	no	—
128	C₄⋊Q₈	—	4	3	no	—
129	#125	—	4	3	no	—
130	C₄×Q₈	—	2	3	no	—
131	C₄×Q₈	—	2	3	no	—
132	C₄×Q₈	—	4	1	no	—
133	C₄×Q₈	—	4	1	no	—
134	C₄×Q₈	D₄×C₇₃	1	1	yes	characteristic
135	Q₃₂	—	2	6	no	—
136	C₂×C₁₆	C₂³×C₇₃	1	1	yes	characteristic
137	C₂×C₁₆	C₂³×C₇₃	1	3	yes	—
138	#107	—	2	3	no	—
139	#107	C₂³×C₇₃	1	3	yes	—
140	#107	C₂³×C₇₃	1	1	yes	characteristic
141	C₄⋊C₈	D₄×C₇₃	1	3	yes	—
142	#103	—	2	3	no	—
143	#103	—	2	3	no	—
144	C₄×C₈	C₂³×C₇₃	1	3	yes	—
145	C₂×Q₈	—	2	1	no	—
146	C₂×Q₈	—	2	3	no	—
147	C₂×Q₈	—	4	3	no	—
148	C₂×Q₈	D₈×C₇₃	1	1	yes	characteristic
149	Q₁₆	—	2	3	no	—
150	Q₁₆	—	2	1	no	—
151	Q₁₆	—	2	3	no	—
152	Q₁₆	—	8	3	no	—
153	Q₁₆	#41×C₇₃	1	2	yes	—
154	C₂×C₈	C₂×D₄×C₇₃	1	3	yes	—
155	C₂×C₈	C₂⁴×C₇₃	1	1	yes	characteristic
156	C₄⋊C₄	—	2	1	no	—
157	C₄⋊C₄	—	2	3	no	—
158	C₄⋊C₄	—	4	3	no	—
159	C₄⋊C₄	—	4	3	no	—
160	C₄⋊C₄	—	4	3	no	—
161	C₄⋊C₄	—	4	3	no	—
162	C₄⋊C₄	—	4	3	no	—
163	C₄⋊C₄	D₈×C₇₃	1	3	yes	—
164	C₄²	—	4	3	no	—
165	C₄²	—	4	3	no	—
166	C₄²	C₂×D₄×C₇₃	1	3	yes	—
167	C₁₆	—	2	3	no	—
168	C₁₆	—	2	1	no	—
169	Q₈	—	2	2	no	—
170	Q₈	—	4	1	no	—
171	Q₈	—	4	3	no	—
172	Q₈	—	4	3	no	—
173	Q₈	—	16	3	no	—
174	Q₈	Q₃₂×C₇₃	1	4	yes	complemented
175	C₂×C₄	—	4	1	no	—
176	C₂×C₄	—	4	1	no	—
177	C₂×C₄	—	4	3	no	—
178	C₂×C₄	—	4	3	no	—
179	C₂×C₄	C₂×D₈×C₇₃	1	3	yes	—
180	C₂×C₄	C₂×D₈×C₇₃	1	3	yes	—
181	C₂×C₄	C₂²×D₄×C₇₃	1	1	yes	characteristic
182	C₈	—	2	3	no	—
183	C₈	C₂×#41×C₇₃	1	1	yes	characteristic
184	C₈	2₋¹⁺⁴×C₇₃	1	1	yes	characteristic
185	C₂²	C₂²×D₈×C₇₃	1	1	yes	characteristic
186	C₄	—	2	3	no	—
187	C₄	—	4	2	no	—
188	C₄	—	8	3	no	—
189	C₄	—	8	3	no	—
190	C₄	—	8	1	no	—
191	C₄	C₂×Q₃₂×C₇₃	1	6	yes	—
192	C₄	#548×C₇₃	1	1	yes	characteristic
193	C₄	#577×C₇₃	1	1	yes	characteristic
194	C₂	C₇₃×#3358	1	1	yes	characteristic
195	C₂	C₂²×Q₃₂×C₇₃	1	1	yes	characteristic
196	C₂	C₇₃×#3416	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₃×#3395

C₂

yes

maximal, characteristic

Q₈×Q₁₆×C₇₃

C₂

yes

maximal, characteristic

C₇₃×#2241

C₂

yes

maximal

C₇₃×#2227

C₂

yes

maximal

C₇₃×#2209

C₂

yes

maximal

Q₈×C₁₆×C₇₃

C₂

yes

maximal, characteristic

C₄×Q₃₂×C₇₃

C₂

yes

maximal

Q₈²×C₇₃

—

#557×C₇₃

—

C₂×Q₃₂×C₇₃

C₂²

yes

—

#500×C₇₃

C₂²

yes

—

#494×C₇₃

C₂²

yes

—

#477×C₇₃

—

#462×C₇₃

—

C₈×Q₈×C₇₃

C₂²

yes

characteristic

C₄×Q₁₆×C₇₃

C₂²

yes

—

C₄×Q₁₆×C₇₃

C₂²

yes

—

#366×C₇₃

C₂²

yes

—

C₄⋊C₁₆×C₇₃

C₂²

yes

—

#358×C₇₃

C₂²

yes

—

#358×C₇₃

C₂²

yes

—

#358×C₇₃

C₂²

yes

characteristic

C₄×C₁₆×C₇₃

C₂²

yes

—

C₂×Q₁₆×C₇₃

—

C₂×Q₁₆×C₇₃

C₂³

yes

—

C₂×Q₁₆×C₇₃

C₂³

yes

characteristic

C₄⋊Q₈×C₇₃

—

C₄⋊Q₈×C₇₃

—

C₄⋊Q₈×C₇₃

—

#125×C₇₃

—

C₄×Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

C₄×Q₈×C₇₃

D₄

yes

characteristic

Q₃₂×C₇₃

—

C₂×C₁₆×C₇₃

C₂³

yes

—

C₂×C₁₆×C₇₃

C₂³

yes

characteristic

#107×C₇₃

—

#107×C₇₃

C₂³

yes

characteristic

#107×C₇₃

C₂³

yes

—

C₄⋊C₈×C₇₃

D₄

yes

—

#103×C₇₃

—

#103×C₇₃

—

C₄×C₈×C₇₃

C₂³

yes

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

D₈

yes

characteristic

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

#41

yes

—

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

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

D₈

yes

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

—

C₁₆×C₇₃

—

C₁₆×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

Q₃₂

yes

complemented

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₈

yes

—

C₂×C₄×C₇₃

C₂×D₈

yes

—

C₂×C₄×C₇₃

C₂²×D₄

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

C₂×#41

yes

characteristic

C₈×C₇₃

2₋¹⁺⁴

yes

characteristic

C₂²×C₇₃

C₂²×D₈

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×Q₃₂

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₃

#3358

yes

characteristic

C₂×C₇₃

C₂²×Q₃₂

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

characteristic

100

Q₈×Q₁₆

C₂×C₇₃

yes

characteristic

101

#2241

C₂×C₇₃

yes

—

102

#2227

C₂×C₇₃

yes

—

103

#2209

C₂×C₇₃

yes

—

104

Q₈×C₁₆

C₂×C₇₃

yes

characteristic

105

C₄×Q₃₂

C₂×C₇₃

yes

—

106

C₇₃

#33930

yes

characteristic, direct product factor, Sylow

107

Q₈²

—

108

#557

—

109

C₂×Q₃₂

C₂²×C₇₃

yes

—

110

#500

C₂²×C₇₃

yes

—

111

#494

C₂²×C₇₃

yes

—

112

#477

—

113

#462

—

114

C₈×Q₈

C₂²×C₇₃

yes

characteristic

115

C₄×Q₁₆

C₂²×C₇₃

yes

—

116

C₄×Q₁₆

C₂²×C₇₃

yes

—

117

#366

C₂²×C₇₃

yes

—

118

C₄⋊C₁₆

C₂²×C₇₃

yes

—

119

#358

C₂²×C₇₃

yes

characteristic

120

#358

C₂²×C₇₃

yes

—

121

#358

C₂²×C₇₃

yes

—

122

C₄×C₁₆

C₂²×C₇₃

yes

—

123

C₂×Q₁₆

—

124

C₂×Q₁₆

C₂³×C₇₃

yes

—

125

C₂×Q₁₆

C₂³×C₇₃

yes

characteristic

126

C₄⋊Q₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

—

129

#125

—

130

C₄×Q₈

—

131

C₄×Q₈

—

132

C₄×Q₈

—

133

C₄×Q₈

—

134

C₄×Q₈

D₄×C₇₃

yes

characteristic

135

Q₃₂

—

136

C₂×C₁₆

C₂³×C₇₃

yes

characteristic

137

C₂×C₁₆

C₂³×C₇₃

yes

—

138

#107

—

139

#107

C₂³×C₇₃

yes

—

140

#107

C₂³×C₇₃

yes

characteristic

141

C₄⋊C₈

D₄×C₇₃

yes

—

142

#103

—

143

#103

—

144

C₄×C₈

C₂³×C₇₃

yes

—

145

C₂×Q₈

—

146

C₂×Q₈

—

147

C₂×Q₈

—

148

C₂×Q₈

D₈×C₇₃

yes

characteristic

149

Q₁₆

—

150

Q₁₆

—

151

Q₁₆

—

152

Q₁₆

—

153

Q₁₆

#41×C₇₃

yes

—

154

C₂×C₈

C₂×D₄×C₇₃

yes

—

155

C₂×C₈

C₂⁴×C₇₃

yes

characteristic

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

D₈×C₇₃

yes

—

164

C₄²

—

165

C₄²

—

166

C₄²

C₂×D₄×C₇₃

yes

—

167

C₁₆

—

168

C₁₆

—

169

Q₈

—

170

Q₈

—

171

Q₈

—

172

Q₈

—

173

Q₈

—

174

Q₈

Q₃₂×C₇₃

yes

complemented

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

C₂×D₈×C₇₃

yes

—

180

C₂×C₄

C₂×D₈×C₇₃

yes

—

181

C₂×C₄

C₂²×D₄×C₇₃

yes

characteristic

182

C₈

—

183

C₈

C₂×#41×C₇₃

yes

characteristic

184

C₈

2₋¹⁺⁴×C₇₃

yes

characteristic

185

C₂²

C₂²×D₈×C₇₃

yes

characteristic

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

C₂×Q₃₂×C₇₃

yes

—

192

C₄

#548×C₇₃

yes

characteristic

193

C₄

#577×C₇₃

yes

characteristic

194

C₂

C₇₃×#3358

yes

characteristic

195

C₂

C₂²×Q₃₂×C₇₃

yes

characteristic

196

C₂

C₇₃×#3416

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₇₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₇₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₄×Q₈×C₇₃ ◁ C₈×Q₈×C₇₃ ◁ G

subgroup

G/H

center, socle, prime radical

C₂²×C₇₃

C₂²×D₈

commutator, Frattini, cosocle

C₂×C₈

C₂⁴×C₇₃

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₇₃ ◁ C₄×Q₈×C₇₃ ◁ C₈×Q₈×C₇₃ ◁ G

N	H ≅ G/N	type
75. Q₈×C₇₃	Q₃₂	semidirect
95. #33930	C₇₃	direct (N × H)
106. C₇₃	#33930	direct (N × H)
174. Q₈	Q₃₂×C₇₃	semidirect

H ≅ G/N

type

75. Q₈×C₇₃

Q₃₂

semidirect

95. #33930

C₇₃

direct (N × H)

106. C₇₃

#33930

direct (N × H)

174. Q₈

Q₃₂×C₇₃

semidirect

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

subgroup	index
C₇₃×#3395	2
Q₈×Q₁₆×C₇₃	2
C₇₃×#2241	2
C₇₃×#2227	2
C₇₃×#2209	2
Q₈×C₁₆×C₇₃	2
C₄×Q₃₂×C₇₃	2
#33930	73

subgroup

index

C₇₃×#3395

Q₈×Q₁₆×C₇₃

C₇₃×#2241

C₇₃×#2227

C₇₃×#2209

Q₈×C₁₆×C₇₃

C₄×Q₃₂×C₇₃

#33930

Subgroups — C₇₃×#33930 #32943334

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#33930 #32943334

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#33930 #32943334