Group #33004473 — subgroups

Subgroups — #33004473

Order 18688.

566 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 143 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	#22611687	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#17988400	C₂	1	4	yes	maximal, complemented
3	C₄×#14556527	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#493×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556531	C₂²	1	2	yes	—
6	C₂×#14556527	C₂²	1	2	yes	—
7	C₄×#11924012	—	2	1	no	—
8	C₄⋊#11924003	—	2	2	no	—
9	#17988400	—	2	8	no	—
10	#17988352	—	2	1	no	—
11	C₂²×D₈×C₇₃	—	2	1	no	—
12	C₂²×D₈×C₇₃	C₂²	1	2	yes	—
13	C₂×#127×C₇₃	—	2	1	no	—
14	#493×C₇₃	—	2	2	no	—
15	#493×C₇₃	C₄	1	4	yes	—
16	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×#11924012	—	2	2	no	—
18	#14556531	—	2	4	no	—
19	C₂×#11924003	—	2	2	no	—
20	#14556527	—	2	2	no	—
21	#14556527	D₄	1	4	yes	complemented
22	#14556502	—	4	1	no	—
23	C₄×#10596831	—	4	1	no	—
24	C₂²×D₄×C₇₃	—	2	2	no	—
25	C₂²×D₄×C₇₃	—	4	2	no	—
26	C₂×D₈×C₇₃	—	2	4	no	—
27	C₂×D₈×C₇₃	—	2	4	no	—
28	C₂×D₈×C₇₃	—	2	2	no	—
29	C₂×D₈×C₇₃	—	2	4	no	—
30	C₂×D₈×C₇₃	—	4	4	no	—
31	C₂×D₈×C₇₃	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
34	#127×C₇₃	—	2	4	no	—
35	#127×C₇₃	—	4	1	no	—
36	#127×C₇₃	—	4	1	no	—
37	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
38	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₇₃	D₄	1	2	yes	—
40	#11924012	—	2	4	no	—
41	#11924012	—	4	2	no	—
42	C₂×#10596831	—	4	2	no	—
43	#11924003	—	4	2	no	—
44	C₂⁴×C₇₃	—	4	2	no	—
45	C₂×D₄×C₇₃	—	2	4	no	—
46	C₂×D₄×C₇₃	—	4	4	no	—
47	C₂×D₄×C₇₃	—	4	4	no	—
48	C₂×D₄×C₇₃	—	4	4	no	—
49	C₂×D₄×C₇₃	—	4	2	no	—
50	C₂×D₄×C₇₃	—	4	4	no	—
51	C₂×D₄×C₇₃	—	8	8	no	—
52	C₂²×C₄×C₇₃	SD₁₆	1	2	yes	—
53	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
54	D₈×C₇₃	—	2	8	no	—
55	D₈×C₇₃	—	4	8	no	—
56	D₈×C₇₃	—	4	2	no	—
57	D₈×C₇₃	—	4	2	no	—
58	C₂×C₈×C₇₃	—	2	2	no	—
59	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₇₃	#41	1	2	yes	—
64	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₇₃	D₈	1	1	yes	characteristic
66	C₄²×C₇₃	D₈	1	1	yes	characteristic
67	C₂³×C₇₃	—	4	4	no	—
68	C₂³×C₇₃	—	8	8	no	—
69	C₂³×C₇₃	—	8	4	no	—
70	C₂³×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
71	D₄×C₇₃	—	4	4	no	—
72	D₄×C₇₃	—	4	4	no	—
73	D₄×C₇₃	—	8	8	no	—
74	D₄×C₇₃	—	8	8	no	—
75	D₄×C₇₃	—	8	4	no	—
76	C₂×C₄×C₇₃	—	2	4	no	—
77	C₂×C₄×C₇₃	#102	1	2	yes	—
78	C₂×C₄×C₇₃	#102	1	2	yes	—
79	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₇₃	#121	1	2	yes	—
81	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
83	C₈×C₇₃	—	2	2	no	—
84	C₈×C₇₃	C₄×D₄	1	2	yes	—
85	C₈×C₇₃	#124	1	1	yes	characteristic
86	C₈×C₇₃	#127	1	1	yes	characteristic
87	#10596831	—	4	4	no	—
88	C₂²×C₇₃	—	2	2	no	—
89	C₂²×C₇₃	—	8	8	no	—
90	C₂²×C₇₃	—	8	4	no	—
91	C₂²×C₇₃	—	16	8	no	—
92	C₂²×C₇₃	C₂×#102	1	1	yes	characteristic
93	C₂²×C₇₃	#460	1	2	yes	—
94	C₄×C₇₃	—	2	2	no	—
95	C₄×C₇₃	#361	1	4	yes	complemented
96	C₄×C₇₃	#390	1	1	yes	characteristic
97	C₄×C₇₃	#418	1	1	yes	characteristic
98	C₄×C₇₃	#459	1	2	yes	—
99	#13257	—	73	1	no	maximal, Sylow
100	C₂×C₇₃	—	2	2	no	—
101	C₂×C₇₃	—	16	4	no	—
102	C₂×C₇₃	#1982	1	1	yes	characteristic
103	C₂×C₇₃	C₂×#361	1	2	yes	—
104	C₂×#493	D₇₃	1	1	yes	characteristic
105	#2150	—	73	1	no	—
106	C₂×#361	—	73	4	no	—
107	C₄×#108	—	73	1	no	—
108	C₇₃	#13257	1	1	yes	characteristic, complemented, Sylow
109	C₂²×D₈	—	2	1	no	—
110	C₂²×D₈	C₂×D₇₃	1	2	yes	—
111	C₂×#127	—	2	1	no	—
112	C₂×M₃₂	—	73	2	no	—
113	#493	—	2	2	no	—
114	#493	Dic₇₃	1	4	yes	—
115	C₂×#108	—	73	2	no	—
116	C₄×M₁₆	—	146	1	no	—
117	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
118	C₄⋊C₁₆	—	146	2	no	—
119	#361	—	146	8	no	—
120	#341	—	146	1	no	—
121	C₂²×D₄	—	2	2	no	—
122	C₂²×D₄	—	4	2	no	—
123	C₂×D₈	—	2	4	no	—
124	C₂×D₈	—	2	4	no	—
125	C₂×D₈	—	2	2	no	—
126	C₂×D₈	—	2	4	no	—
127	C₂×D₈	—	4	4	no	—
128	C₂×D₈	C₂×Dic₇₃	1	4	yes	—
129	C₂×M₁₆	—	146	2	no	—
130	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
131	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
132	#127	—	2	4	no	—
133	#127	—	4	1	no	—
134	#127	—	4	1	no	—
135	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
136	M₃₂	—	146	4	no	—
137	C₂×C₁₆	—	146	2	no	—
138	#108	—	73	4	no	—
139	#108	—	146	2	no	—
140	#97	—	292	1	no	—
141	C₄×C₈	—	292	1	no	—
142	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
143	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
144	C₂⁴	—	4	2	no	—
145	C₂×D₄	—	2	4	no	—
146	C₂×D₄	—	4	4	no	—
147	C₂×D₄	—	4	4	no	—
148	C₂×D₄	—	4	4	no	—
149	C₂×D₄	—	4	2	no	—
150	C₂×D₄	—	4	4	no	—
151	C₂×D₄	—	8	8	no	—
152	C₂²×C₄	#11924016	1	2	yes	—
153	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
154	D₈	—	2	8	no	—
155	D₈	—	4	8	no	—
156	D₈	—	4	2	no	—
157	D₈	—	4	2	no	—
158	M₁₆	—	292	2	no	—
159	M₁₆	—	146	4	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	292	2	no	—
162	C₂×C₈	C₂²⋊Dic₇₃	1	2	yes	—
163	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
164	C₂×C₈	D₄×D₇₃	1	2	yes	—
165	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
166	C₂×C₈	#11924027	1	2	yes	—
167	C₄²	C₂²⋊Dic₇₃	1	2	yes	—
168	C₄²	#11924013	1	1	yes	characteristic
169	C₄²	#11924013	1	1	yes	characteristic
170	C₁₆	—	292	2	no	—
171	C₂³	—	4	4	no	—
172	C₂³	—	8	8	no	—
173	C₂³	—	8	4	no	—
174	C₂³	C₂×#11924016	1	1	yes	characteristic
175	D₄	—	4	4	no	—
176	D₄	—	4	4	no	—
177	D₄	—	8	8	no	—
178	D₄	—	8	8	no	—
179	D₄	—	8	4	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	#14556512	1	2	yes	—
182	C₂×C₄	#14556512	1	2	yes	—
183	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
184	C₂×C₄	#14556562	1	2	yes	—
185	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
186	C₂×C₄	#14556617	1	2	yes	—
187	C₈	—	2	2	no	—
188	C₈	—	292	4	no	—
189	C₈	D₄×Dic₇₃	1	2	yes	—
190	C₈	#14556578	1	1	yes	characteristic
191	C₈	#14556587	1	1	yes	characteristic
192	C₂²	—	2	2	no	—
193	C₂²	—	8	4	no	—
194	C₂²	—	8	8	no	—
195	C₂²	—	16	8	no	—
196	C₂²	C₂×#14556512	1	1	yes	characteristic
197	C₂²	#17988784	1	2	yes	—
198	C₄	—	2	2	no	—
199	C₄	#17988400	1	4	yes	complemented
200	C₄	#17988490	1	1	yes	characteristic
201	C₄	#17988592	1	1	yes	characteristic
202	C₄	#17988779	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	—	16	4	no	—
205	C₂	#22610859	1	1	yes	characteristic
206	C₂	C₂×#17988400	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22611687

C₂

yes

maximal, characteristic, complemented

C₂×#17988400

C₂

yes

maximal, complemented

C₄×#14556527

C₂

yes

maximal, characteristic, complemented

C₂×#493×C₇₃

C₂

yes

maximal, characteristic

C₂×#14556531

C₂²

yes

—

C₂×#14556527

C₂²

yes

—

—

—

—

—

C₂²×D₈×C₇₃

—

C₂²×D₈×C₇₃

C₂²

yes

—

C₂×#127×C₇₃

—

#493×C₇₃

—

#493×C₇₃

C₄

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

—

—

—

—

D₄

yes

complemented

#14556502

—

C₄×#10596831

—

C₂²×D₄×C₇₃

—

C₂²×D₄×C₇₃

—

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

C₂×C₄

yes

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

C₂³

yes

characteristic

#127×C₇₃

—

#127×C₇₃

—

#127×C₇₃

—

C₂×C₄²×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

D₄

yes

—

—

—

—

—

C₂⁴×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂²×C₄×C₇₃

SD₁₆

yes

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

D₈×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

—

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

#41

yes

—

C₄²×C₇₃

C₂²⋊C₄

yes

—

C₄²×C₇₃

D₈

yes

characteristic

C₄²×C₇₃

D₈

yes

characteristic

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

C₂×SD₁₆

yes

characteristic

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#102

yes

—

C₂×C₄×C₇₃

#102

yes

—

C₂×C₄×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

#121

yes

—

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₈×C₇₃

—

C₈×C₇₃

C₄×D₄

yes

—

C₈×C₇₃

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×#102

yes

characteristic

C₂²×C₇₃

#460

yes

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

#13257

—

maximal, Sylow

100

C₂×C₇₃

—

101

C₂×C₇₃

—

102

C₂×C₇₃

#1982

yes

characteristic

103

C₂×C₇₃

C₂×#361

yes

—

104

C₂×#493

D₇₃

yes

characteristic

105

#2150

—

106

C₂×#361

—

107

C₄×#108

—

108

C₇₃

#13257

yes

characteristic, complemented, Sylow

109

C₂²×D₈

—

110

C₂²×D₈

C₂×D₇₃

yes

—

111

C₂×#127

—

112

C₂×M₃₂

—

113

#493

—

114

#493

Dic₇₃

yes

—

115

C₂×#108

—

116

C₄×M₁₆

—

146

—

117

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

118

C₄⋊C₁₆

—

146

—

119

#361

—

146

—

120

#341

—

146

—

121

C₂²×D₄

—

122

C₂²×D₄

—

123

C₂×D₈

—

124

C₂×D₈

—

125

C₂×D₈

—

126

C₂×D₈

—

127

C₂×D₈

—

128

C₂×D₈

C₂×Dic₇₃

yes

—

129

C₂×M₁₆

—

146

—

130

C₂²×C₈

C₂²⋊D₇₃

yes

characteristic

131

C₂²×C₈

C₂²×D₇₃

yes

characteristic

132

#127

—

133

#127

—

134

#127

—

135

C₂×C₄²

C₂²⋊D₇₃

yes

characteristic

136

M₃₂

—

146

—

137

C₂×C₁₆

—

146

—

138

#108

—

139

#108

—

146

—

140

#97

—

292

—

141

C₄×C₈

—

292

—

142

C₄×C₈

C₂×Dic₇₃

yes

—

143

C₄×C₈

C₂²⋊D₇₃

yes

—

144

C₂⁴

—

145

C₂×D₄

—

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂×D₄

—

150

C₂×D₄

—

151

C₂×D₄

—

152

C₂²×C₄

#11924016

yes

—

153

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

154

D₈

—

155

D₈

—

156

D₈

—

157

D₈

—

158

M₁₆

—

292

—

159

M₁₆

—

146

—

160

C₂×C₈

—

161

C₂×C₈

—

292

—

162

C₂×C₈

C₂²⋊Dic₇₃

yes

—

163

C₂×C₈

C₂²×Dic₇₃

yes

characteristic

164

C₂×C₈

D₄×D₇₃

yes

—

165

C₂×C₈

C₂×C₂²⋊D₇₃

yes

characteristic

166

C₂×C₈

#11924027

yes

—

167

C₄²

C₂²⋊Dic₇₃

yes

—

168

C₄²

#11924013

yes

characteristic

169

C₄²

#11924013

yes

characteristic

170

C₁₆

—

292

—

171

C₂³

—

172

C₂³

—

173

C₂³

—

174

C₂³

C₂×#11924016

yes

characteristic

175

D₄

—

176

D₄

—

177

D₄

—

178

D₄

—

179

D₄

—

180

C₂×C₄

—

181

C₂×C₄

#14556512

yes

—

182

C₂×C₄

#14556512

yes

—

183

C₂×C₄

C₂×C₂²⋊Dic₇₃

yes

characteristic

184

C₂×C₄

#14556562

yes

—

185

C₂×C₄

C₂×#11924013

yes

characteristic

186

C₂×C₄

#14556617

yes

—

187

C₈

—

188

C₈

—

292

—

189

C₈

D₄×Dic₇₃

yes

—

190

C₈

#14556578

yes

characteristic

191

C₈

#14556587

yes

characteristic

192

C₂²

—

193

C₂²

—

194

C₂²

—

195

C₂²

—

196

C₂²

C₂×#14556512

yes

characteristic

197

C₂²

#17988784

yes

—

198

C₄

—

199

C₄

#17988400

yes

complemented

200

C₄

#17988490

yes

characteristic

201

C₄

#17988592

yes

characteristic

202

C₄

#17988779

yes

—

203

C₂

—

204

C₂

—

205

C₂

#22610859

yes

characteristic

206

C₂

C₂×#17988400

yes

—

	subgroup	G/H
center	C₂²	C₂×#14556512
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting, prime radical	C₂×#493×C₇₃	C₂
socle	C₂²×C₇₃	C₂×#102
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₂×#493	D₇₃
hypocenter	C₇₃	#13257
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#493

subgroup

G/H

center

C₂²

C₂×#14556512

commutator

C₂×C₈×C₇₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting, prime radical

C₂×#493×C₇₃

C₂

socle

C₂²×C₇₃

C₂×#102

cosocle

C₂²×C₈×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#493

N	H ≅ G/N	type
1. #22611687	C₂	semidirect
2. C₂×#17988400	C₂	semidirect
3. C₄×#14556527	C₂	semidirect
21. #14556527	D₄	semidirect
95. C₄×C₇₃	#361	semidirect
108. C₇₃	#13257	semidirect
199. C₄	#17988400	semidirect

H ≅ G/N

type

1. #22611687

C₂

semidirect

2. C₂×#17988400

C₂

semidirect

3. C₄×#14556527

C₂

semidirect

21. #14556527

D₄

semidirect

95. C₄×C₇₃

#361

semidirect

108. C₇₃

#13257

semidirect

199. C₄

#17988400

semidirect

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

subgroup	index
#22611687	2
C₂×#17988400	2
C₄×#14556527	2
C₂×#493×C₇₃	2
#13257	73

subgroup

index

C₂×#493×C₇₃

Subgroups — #33004473

Conjugacy classes of subgroups

Complements

Maximal subgroups