Group #33004220 — subgroups

Subgroups — #33004220

Order 18688.

486 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 163 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×#17988967	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988392	C₂	1	4	yes	maximal
3	#22610095	C₂	1	1	yes	maximal, characteristic
4	C₂×#346×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₉₂	—	2	1	no	—
6	C₂²×Dic₂₉₂	C₂²	1	2	yes	—
7	C₂×#14556591	—	2	1	no	—
8	#17988967	—	2	2	no	—
9	#17988967	C₄	1	4	yes	complemented
10	C₂×#14556525	C₂²	1	2	yes	—
11	C₂×#14556523	—	2	1	no	—
12	#17988453	—	2	1	no	—
13	#17988392	—	2	8	no	—
14	C₂²×C₁₆×C₇₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
16	#346×C₇₃	—	2	2	no	—
17	C₂²×Dic₁₄₆	—	2	2	no	—
18	C₂×Dic₂₉₂	—	2	4	no	—
19	C₂×Dic₂₉₂	—	2	4	no	—
20	C₂×Dic₂₉₂	—	2	2	no	—
21	C₂×Dic₂₉₂	—	2	4	no	—
22	C₂×Dic₂₉₂	—	4	2	no	—
23	C₂×Dic₂₉₂	—	4	2	no	—
24	C₂×Dic₂₉₂	C₂×C₄	1	4	yes	—
25	#14556591	—	2	4	no	—
26	#14556591	—	4	2	no	—
27	C₂×C₄⋊Dic₇₃	—	2	2	no	—
28	C₂×C₄⋊Dic₇₃	—	4	2	no	—
29	#14556525	—	2	4	no	—
30	#14556523	—	4	2	no	—
31	#14556498	—	4	2	no	—
32	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₇₃	—	2	2	no	—
36	C₂×C₁₆×C₇₃	—	2	2	no	—
37	C₂×C₁₆×C₇₃	D₄	1	4	yes	—
38	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₇₃	D₄	1	2	yes	—
40	C₂×Dic₁₄₆	—	2	4	no	—
41	C₂×Dic₁₄₆	—	4	2	no	—
42	C₂×Dic₁₄₆	—	4	2	no	—
43	C₂×Dic₁₄₆	—	4	2	no	—
44	C₂×Dic₁₄₆	—	4	2	no	—
45	C₂×Dic₁₄₆	—	4	2	no	—
46	C₂²×Dic₇₃	—	4	2	no	—
47	C₂²×Dic₇₃	—	4	2	no	—
48	Dic₂₉₂	—	2	8	no	—
49	Dic₂₉₂	—	4	8	no	—
50	Dic₂₉₂	—	4	4	no	—
51	C₄⋊Dic₇₃	—	4	8	no	—
52	C₄⋊Dic₇₃	—	4	2	no	—
53	C₄⋊Dic₇₃	—	4	2	no	—
54	C₂²×C₄×C₇₃	D₈	1	2	yes	—
55	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₇₃	D₈	1	2	yes	—
58	C₂×C₈×C₇₃	SD₁₆	1	2	yes	—
59	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₇₃	#41	1	2	yes	—
63	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
64	C₄²×C₇₃	SD₁₆	1	2	yes	—
65	C₁₆×C₇₃	—	2	4	no	—
66	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₇₃	—	2	4	no	—
68	C₂×C₄×C₇₃	#102	1	4	yes	—
69	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
71	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
72	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×Dic₇₃	—	4	4	no	—
76	C₂×Dic₇₃	—	8	2	no	—
77	C₂×Dic₇₃	—	8	2	no	—
78	C₂×Dic₇₃	—	8	2	no	—
79	C₂×Dic₇₃	—	8	4	no	—
80	Dic₁₄₆	—	4	4	no	—
81	Dic₁₄₆	—	4	4	no	—
82	Dic₁₄₆	—	8	2	no	—
83	Dic₁₄₆	—	8	2	no	—
84	C₈×C₇₃	#102	1	4	yes	—
85	C₈×C₇₃	C₄×D₄	1	2	yes	—
86	C₈×C₇₃	#121	1	2	yes	—
87	C₂²×C₇₃	C₂×#102	1	1	yes	characteristic
88	C₂²×C₇₃	#486	1	1	yes	characteristic
89	C₂²×C₇₃	#493	1	1	yes	characteristic
90	C₂²×C₇₃	#510	1	4	yes	—
91	C₄×C₇₃	—	2	4	no	—
92	C₄×C₇₃	#390	1	1	yes	characteristic
93	C₄×C₇₃	C₂×#102	1	1	yes	characteristic
94	C₄×C₇₃	#486	1	1	yes	characteristic
95	C₄×C₇₃	#492	1	1	yes	characteristic
96	Dic₇₃	—	8	4	no	—
97	Dic₇₃	—	16	2	no	—
98	#13207	—	73	1	no	maximal, Sylow
99	C₂×C₇₃	#1957	1	1	yes	characteristic
100	C₂×C₇₃	#2141	1	2	yes	—
101	C₂×C₇₃	#2245	1	2	yes	—
102	C₂×C₇₃	#2251	1	2	yes	—
103	C₂×#494	—	73	1	no	—
104	C₂×#358	—	73	4	no	—
105	C₂×#346	D₇₃	1	1	yes	characteristic
106	#1846	—	73	1	no	—
107	C₇₃	#13207	1	1	yes	characteristic, complemented, Sylow
108	C₂²×Q₁₆	—	73	2	no	—
109	C₂²×Q₁₆	—	146	1	no	—
110	C₂×C₄⋊Q₈	—	146	1	no	—
111	C₂²×C₁₆	C₂×D₇₃	1	2	yes	—
112	#494	—	73	4	no	—
113	#494	—	146	2	no	—
114	C₂×#107	—	73	2	no	—
115	C₂×#106	—	146	1	no	—
116	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
117	#382	—	146	1	no	—
118	#358	—	146	8	no	—
119	#346	—	2	2	no	—
120	C₂²×Q₈	—	146	2	no	—
121	C₂×Q₁₆	—	292	2	no	—
122	C₂×Q₁₆	—	292	2	no	—
123	C₂×Q₁₆	—	73	4	no	—
124	C₂×Q₁₆	—	146	4	no	—
125	C₂×Q₁₆	—	146	2	no	—
126	C₂×Q₁₆	—	146	4	no	—
127	C₂×Q₁₆	—	146	4	no	—
128	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
129	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
130	C₄⋊Q₈	—	292	2	no	—
131	C₄⋊Q₈	—	146	4	no	—
132	C₂×C₄⋊C₄	—	292	2	no	—
133	C₂×C₄⋊C₄	—	146	2	no	—
134	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	—	2	2	no	—
137	C₂×C₁₆	D₂₉₂	1	4	yes	—
138	#107	—	146	4	no	—
139	#106	—	292	2	no	—
140	C₄×C₈	C₄×D₇₃	1	2	yes	—
141	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
142	#95	—	292	2	no	—
143	C₂×Q₈	—	292	2	no	—
144	C₂×Q₈	—	292	2	no	—
145	C₂×Q₈	—	292	2	no	—
146	C₂×Q₈	—	292	2	no	—
147	C₂×Q₈	—	292	2	no	—
148	C₂×Q₈	—	146	4	no	—
149	C₂²×C₄	—	292	2	no	—
150	C₂²×C₄	—	292	2	no	—
151	C₂²×C₄	D₅₈₄	1	2	yes	—
152	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
153	Q₁₆	—	292	8	no	—
154	Q₁₆	—	292	4	no	—
155	Q₁₆	—	146	8	no	—
156	C₂×C₈	#11924005	1	2	yes	—
157	C₂×C₈	D₅₈₄	1	2	yes	—
158	C₂×C₈	#11924017	1	2	yes	—
159	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
160	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
161	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
162	C₂×C₈	#11924028	1	2	yes	—
163	C₄⋊C₄	—	292	8	no	—
164	C₄⋊C₄	—	292	2	no	—
165	C₄⋊C₄	—	292	2	no	—
166	C₄²	#11924005	1	2	yes	—
167	C₄²	#11924017	1	2	yes	—
168	C₁₆	—	2	4	no	—
169	C₂³	C₂×D₅₈₄	1	1	yes	characteristic
170	Q₈	—	292	4	no	—
171	Q₈	—	292	4	no	—
172	Q₈	—	584	2	no	—
173	Q₈	—	584	2	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	292	4	no	—
176	C₂×C₄	—	584	2	no	—
177	C₂×C₄	—	584	2	no	—
178	C₂×C₄	—	584	2	no	—
179	C₂×C₄	—	584	4	no	—
180	C₂×C₄	#14556513	1	4	yes	—
181	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
182	C₂×C₄	#14556579	1	1	yes	characteristic
183	C₂×C₄	#14556588	1	1	yes	characteristic
184	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
185	C₂×C₄	C₂×#11924017	1	1	yes	characteristic
186	C₂×C₄	C₂×#11924017	1	1	yes	characteristic
187	C₈	#14556513	1	4	yes	—
188	C₈	C₄×D₂₉₂	1	2	yes	—
189	C₈	#14556565	1	2	yes	—
190	C₂²	C₂×#14556513	1	1	yes	characteristic
191	C₂²	#17988934	1	1	yes	characteristic
192	C₂²	#17988964	1	1	yes	characteristic
193	C₂²	#17989030	1	4	yes	—
194	C₄	—	2	4	no	—
195	C₄	—	584	4	no	—
196	C₄	—	1168	2	no	—
197	C₄	#17988489	1	1	yes	characteristic
198	C₄	C₂×#14556513	1	1	yes	characteristic
199	C₄	#17988934	1	1	yes	characteristic
200	C₄	#17988960	1	1	yes	characteristic
201	C₂	#22610737	1	1	yes	characteristic
202	C₂	#22611652	1	2	yes	—
203	C₂	#22612177	1	2	yes	—
204	C₂	#22612202	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988967

C₂

yes

maximal, characteristic

C₂×#17988392

C₂

yes

maximal

#22610095

C₂

yes

maximal, characteristic

C₂×#346×C₇₃

C₂

yes

maximal, characteristic

C₂²×Dic₂₉₂

—

C₂²×Dic₂₉₂

C₂²

yes

—

C₂×#14556591

—

#17988967

—

#17988967

C₄

yes

complemented

C₂×#14556525

C₂²

yes

—

C₂×#14556523

—

#17988453

—

#17988392

—

C₂²×C₁₆×C₇₃

C₂²

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#346×C₇₃

—

C₂²×Dic₁₄₆

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

C₂×C₄

yes

—

#14556591

—

#14556591

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

—

#14556525

—

#14556523

—

#14556498

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

D₄

yes

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

Dic₂₉₂

—

Dic₂₉₂

—

Dic₂₉₂

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂²×C₄×C₇₃

D₈

yes

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂²⋊C₄

yes

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

SD₁₆

yes

—

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₂×C₈×C₇₃

#41

yes

—

C₄²×C₇₃

C₂²⋊C₄

yes

—

C₄²×C₇₃

SD₁₆

yes

—

C₁₆×C₇₃

—

C₂³×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#102

yes

—

C₂×C₄×C₇₃

C₂×C₂²⋊C₄

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂×SD₁₆

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

#102

yes

—

C₈×C₇₃

C₄×D₄

yes

—

C₈×C₇₃

#121

yes

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

—

C₂×C₇₃

yes

100

C₂×C₇₃

#2141

yes

—

101

C₂×C₇₃

#2245

yes

—

102

C₂×C₇₃

#2251

yes

—

103

C₂×#494

—

104

C₂×#358

—

105

C₂×#346

D₇₃

yes

characteristic

106

#1846

—

107

C₇₃

#13207

yes

characteristic, complemented, Sylow

108

C₂²×Q₁₆

—

109

C₂²×Q₁₆

—

146

—

110

C₂×C₄⋊Q₈

—

146

—

111

C₂²×C₁₆

C₂×D₇₃

yes

—

112

#494

—

113

#494

—

146

—

114

C₂×#107

—

115

C₂×#106

—

146

—

116

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

117

#382

—

146

—

118

#358

—

146

—

119

#346

—

120

C₂²×Q₈

—

146

—

121

C₂×Q₁₆

—

292

—

122

C₂×Q₁₆

—

292

—

123

C₂×Q₁₆

—

124

C₂×Q₁₆

—

146

—

125

C₂×Q₁₆

—

146

—

126

C₂×Q₁₆

—

146

—

127

C₂×Q₁₆

—

146

—

128

C₂²×C₈

D₂₉₂

yes

characteristic

129

C₂²×C₈

C₂²×D₇₃

yes

characteristic

130

C₄⋊Q₈

—

292

—

131

C₄⋊Q₈

—

146

—

132

C₂×C₄⋊C₄

—

292

—

133

C₂×C₄⋊C₄

—

146

—

134

C₂×C₄²

D₂₉₂

yes

characteristic

135

C₂×C₁₆

—

136

C₂×C₁₆

—

137

C₂×C₁₆

D₂₉₂

yes

—

138

#107

—

146

—

139

#106

—

292

—

140

C₄×C₈

C₄×D₇₃

yes

—

141

C₄×C₈

C₂²⋊D₇₃

yes

—

142

#95

—

292

—

143

C₂×Q₈

—

292

—

144

C₂×Q₈

—

292

—

145

C₂×Q₈

—

292

—

146

C₂×Q₈

—

292

—

147

C₂×Q₈

—

292

—

148

C₂×Q₈

—

146

—

149

C₂²×C₄

—

292

—

150

C₂²×C₄

—

292

—

151

C₂²×C₄

D₅₈₄

yes

—

152

C₂²×C₄

C₂×D₂₉₂

yes

characteristic

153

Q₁₆

—

292

—

154

Q₁₆

—

292

—

155

Q₁₆

—

146

—

156

C₂×C₈

#11924005

yes

—

157

C₂×C₈

D₅₈₄

yes

—

158

C₂×C₈

#11924017

yes

—

159

C₂×C₈

C₂×C₄×D₇₃

yes

characteristic

160

C₂×C₈

C₂×C₂²⋊D₇₃

yes

characteristic

161

C₂×C₈

C₂×D₂₉₂

yes

—

162

C₂×C₈

#11924028

yes

—

163

C₄⋊C₄

—

292

—

164

C₄⋊C₄

—

292

—

165

C₄⋊C₄

—

292

—

166

C₄²

#11924005

yes

—

167

C₄²

#11924017

yes

—

168

C₁₆

—

169

C₂³

C₂×D₅₈₄

yes

characteristic

170

Q₈

—

292

—

171

Q₈

—

292

—

172

Q₈

—

584

—

173

Q₈

—

584

—

174

C₂×C₄

—

175

C₂×C₄

—

292

—

176

C₂×C₄

—

584

—

177

C₂×C₄

—

584

—

178

C₂×C₄

—

584

—

179

C₂×C₄

—

584

—

180

C₂×C₄

#14556513

yes

—

181

C₂×C₄

C₂×#11924005

yes

characteristic

182

C₂×C₄

#14556579

yes

characteristic

183

C₂×C₄

#14556588

yes

characteristic

184

C₂×C₄

C₂×D₅₈₄

yes

characteristic

185

C₂×C₄

C₂×#11924017

yes

characteristic

186

C₂×C₄

C₂×#11924017

yes

characteristic

187

C₈

#14556513

yes

—

188

C₈

C₄×D₂₉₂

yes

—

189

C₈

#14556565

yes

—

190

C₂²

C₂×#14556513

yes

characteristic

191

C₂²

#17988934

yes

characteristic

192

C₂²

#17988964

yes

characteristic

193

C₂²

#17989030

yes

—

194

C₄

—

195

C₄

—

584

—

196

C₄

—

1168

—

197

C₄

#17988489

yes

characteristic

198

C₄

C₂×#14556513

yes

characteristic

199

C₄

#17988934

yes

characteristic

200

C₄

#17988960

yes

characteristic

201

C₂

#22610737

yes

characteristic

202

C₂

#22611652

yes

—

203

C₂

#22612177

yes

—

204

C₂

#22612202

yes

—

	subgroup	G/H
center	C₂³	C₂×D₅₈₄
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₂×#346×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂×D₈
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₂×#346	D₇₃
hypocenter	C₇₃	#13207
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂³

C₂×D₅₈₄

commutator

C₂×C₈×C₇₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₂×#346×C₇₃

C₂

socle, prime radical

C₂³×C₇₃

C₂×D₈

cosocle

C₂²×C₈×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
9. #17988967	C₄	semidirect
107. C₇₃	#13207	semidirect

H ≅ G/N

type

9. #17988967

C₄

semidirect

107. C₇₃

#13207

semidirect

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

subgroup	index
C₂×#17988967	2
C₂×#17988392	2
#22610095	2
C₂×#346×C₇₃	2
#13207	73

subgroup

index

C₂×#346×C₇₃

Subgroups — #33004220

Conjugacy classes of subgroups

Complements

Maximal subgroups