Group #33005524 — subgroups

Subgroups — #33005524

Order 18688.

418 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 95 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×D₁₁₆₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988401	C₂	1	2	yes	maximal, complemented
3	#22610841	C₂	1	2	yes	maximal, complemented
4	#22608025	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#2151	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₅₈₄	C₂²	1	2	yes	—
7	C₂×D₁₁₆₈	—	2	4	no	—
8	C₂×D₁₁₆₈	—	2	2	no	—
9	C₂×#14556527	C₂²	1	2	yes	—
10	#17988591	—	2	2	no	—
11	C₂×#14556507	—	2	2	no	—
12	#17988401	—	2	4	no	—
13	C₂×M₃₂×C₇₃	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
15	#405×C₇₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₁₆×C₇₃	—	2	2	no	—
17	C₂²×D₂₉₂	—	2	2	no	—
18	C₂×D₅₈₄	—	2	2	no	—
19	C₂×D₅₈₄	—	2	2	no	—
20	C₂×D₅₈₄	—	2	2	no	—
21	C₂×D₅₈₄	—	2	2	no	—
22	C₂×D₅₈₄	—	2	2	no	—
23	C₂×D₅₈₄	D₄	1	4	yes	—
24	C₂×#11924012	—	2	2	no	—
25	D₁₁₆₈	—	2	8	no	—
26	#14556527	—	2	2	no	—
27	#14556527	—	2	2	no	—
28	#14556511	—	4	4	no	—
29	#14556507	—	4	4	no	—
30	C₂×M₁₆×C₇₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
32	#117×C₇₃	D₄	1	1	yes	characteristic
33	M₃₂×C₇₃	—	2	2	no	—
34	C₂×C₁₆×C₇₃	—	2	2	no	—
35	C₂×C₁₆×C₇₃	—	2	1	no	—
36	C₂×C₁₆×C₇₃	D₄	1	2	yes	—
37	#97×C₇₃	—	2	1	no	—
38	C₄×C₈×C₇₃	—	2	1	no	—
39	C₂³×D₇₃	—	4	2	no	—
40	C₂×D₂₉₂	—	2	4	no	—
41	C₂×D₂₉₂	—	4	2	no	—
42	C₂×D₂₉₂	—	4	2	no	—
43	C₂×D₂₉₂	—	4	2	no	—
44	C₂×D₂₉₂	—	4	2	no	—
45	C₂×D₂₉₂	—	4	2	no	—
46	D₅₈₄	—	2	4	no	—
47	D₅₈₄	—	2	4	no	—
48	D₅₈₄	—	4	2	no	—
49	D₅₈₄	—	4	2	no	—
50	#11924012	—	4	2	no	—
51	#11924012	—	4	2	no	—
52	C₂×#10596831	—	4	2	no	—
53	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
54	M₁₆×C₇₃	—	2	2	no	—
55	C₂×C₈×C₇₃	—	2	1	no	—
56	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
57	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
59	C₂×C₈×C₇₃	#41	1	2	yes	—
60	C₄⋊C₄×C₇₃	D₈	1	2	yes	—
61	C₂²⋊C₄×C₇₃	—	2	1	no	—
62	C₄²×C₇₃	—	2	1	no	—
63	C₁₆×C₇₃	—	2	2	no	—
64	C₁₆×C₇₃	—	4	1	no	—
65	C₂³×C₇₃	#135	1	1	yes	characteristic
66	C₂²×D₇₃	—	4	4	no	—
67	C₂²×D₇₃	—	8	2	no	—
68	C₂²×D₇₃	—	8	2	no	—
69	C₂²×D₇₃	—	16	4	no	—
70	C₂×C₄×C₇₃	—	2	2	no	—
71	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
72	C₂×C₄×C₇₃	#121	1	2	yes	—
73	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
75	D₂₉₂	—	4	4	no	—
76	D₂₉₂	—	4	4	no	—
77	D₂₉₂	—	8	2	no	—
78	D₂₉₂	—	8	2	no	—
79	C₈×C₇₃	—	4	1	no	—
80	C₈×C₇₃	#120	1	1	yes	characteristic
81	C₈×C₇₃	#123	1	2	yes	—
82	C₈×C₇₃	#124	1	1	yes	characteristic
83	#10596831	—	8	2	no	—
84	C₂²×C₇₃	—	2	2	no	—
85	C₂²×C₇₃	#463	1	2	yes	—
86	C₂²×C₇₃	#466	1	1	yes	characteristic
87	C₂×D₇₃	—	8	4	no	—
88	C₂×D₇₃	—	16	4	no	—
89	C₂×D₇₃	—	16	4	no	—
90	C₂×D₇₃	—	16	4	no	—
91	C₄×C₇₃	—	4	2	no	—
92	C₄×C₇₃	#394	1	1	yes	characteristic
93	C₄×C₇₃	#459	1	2	yes	—
94	C₄×C₇₃	#469	1	1	yes	characteristic
95	#13439	—	73	1	no	maximal, Sylow
96	C₂×C₇₃	—	2	2	no	—
97	C₂×C₇₃	#2054	1	1	yes	characteristic
98	C₂×C₇₃	#2221	1	2	yes	—
99	D₇₃	—	16	4	no	—
100	C₂²×D₁₆	—	73	1	no	—
101	#2151	D₇₃	1	1	yes	characteristic, complemented
102	C₂×#361	—	73	2	no	—
103	#1978	—	73	2	no	—
104	#1381	—	73	1	no	—
105	C₇₃	#13439	1	1	yes	characteristic, complemented, Sylow
106	C₂²×D₈	—	73	2	no	—
107	C₂×D₁₆	—	146	4	no	—
108	C₂×D₁₆	—	146	2	no	—
109	C₂×M₃₂	C₂×D₇₃	1	1	yes	characteristic
110	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
111	C₂×#108	—	73	2	no	—
112	#418	—	146	2	no	—
113	C₂×#100	—	146	2	no	—
114	#405	C₂×D₇₃	1	1	yes	characteristic
115	C₄⋊C₁₆	—	2	2	no	—
116	#361	—	146	4	no	—
117	C₂²×D₄	—	146	2	no	—
118	C₂×D₈	—	73	4	no	—
119	C₂×D₈	—	146	2	no	—
120	C₂×D₈	—	146	2	no	—
121	C₂×D₈	—	146	2	no	—
122	C₂×D₈	—	146	2	no	—
123	C₂×D₈	—	146	2	no	—
124	C₂×M₁₆	—	146	2	no	—
125	C₂×M₁₆	D₂₉₂	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
127	#117	D₂₉₂	1	1	yes	characteristic
128	D₁₆	—	146	8	no	—
129	M₃₂	—	2	2	no	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	2	1	no	—
132	C₂×C₁₆	C₂²⋊D₇₃	1	2	yes	—
133	#108	—	146	2	no	—
134	#108	—	146	2	no	—
135	#102	—	292	4	no	—
136	#100	—	292	4	no	—
137	#97	—	2	1	no	—
138	C₄×C₈	—	2	1	no	—
139	C₂⁴	—	292	2	no	—
140	C₂×D₄	—	292	2	no	—
141	C₂×D₄	—	292	2	no	—
142	C₂×D₄	—	292	2	no	—
143	C₂×D₄	—	292	2	no	—
144	C₂×D₄	—	292	2	no	—
145	C₂×D₄	—	146	4	no	—
146	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
147	D₈	—	292	2	no	—
148	D₈	—	292	2	no	—
149	D₈	—	146	4	no	—
150	D₈	—	146	4	no	—
151	M₁₆	—	2	2	no	—
152	M₁₆	—	292	2	no	—
153	M₁₆	—	292	2	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	292	2	no	—
156	C₂×C₈	D₄×D₇₃	1	2	yes	—
157	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
158	C₂×C₈	#11924028	1	1	yes	characteristic
159	C₂×C₈	#11924029	1	2	yes	—
160	C₄⋊C₄	D₅₈₄	1	2	yes	—
161	C₂²⋊C₄	—	2	1	no	—
162	C₄²	—	2	1	no	—
163	C₁₆	—	2	2	no	—
164	C₁₆	—	4	1	no	—
165	C₂³	—	292	4	no	—
166	C₂³	—	584	2	no	—
167	C₂³	—	584	2	no	—
168	C₂³	—	1168	4	no	—
169	C₂³	#14556613	1	1	yes	characteristic
170	D₄	—	292	4	no	—
171	D₄	—	292	4	no	—
172	D₄	—	584	2	no	—
173	D₄	—	584	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	#14556565	1	1	yes	characteristic
176	C₂×C₄	#14556566	1	2	yes	—
177	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
178	C₂×C₄	#14556619	1	2	yes	—
179	C₈	—	4	1	no	—
180	C₈	—	584	2	no	—
181	C₈	#14556559	1	1	yes	characteristic
182	C₈	#14556576	1	2	yes	—
183	C₈	#14556580	1	1	yes	characteristic
184	C₂²	—	2	2	no	—
185	C₂²	—	584	4	no	—
186	C₂²	—	1168	4	no	—
187	C₂²	—	1168	4	no	—
188	C₂²	—	1168	4	no	—
189	C₂²	#17988803	1	2	yes	—
190	C₂²	#17988823	1	1	yes	characteristic
191	C₄	—	4	2	no	—
192	C₄	#17988504	1	1	yes	characteristic
193	C₄	#17988783	1	2	yes	—
194	C₄	#17988840	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	—	1168	4	no	—
197	C₂	#22611242	1	1	yes	characteristic
198	C₂	#22612051	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₁₁₆₈

C₂

yes

maximal, characteristic

C₂×#17988401

C₂

yes

maximal, complemented

#22610841

C₂

yes

maximal, complemented

#22608025

C₂

yes

maximal, characteristic, complemented

C₇₃×#2151

C₂

yes

maximal, characteristic, complemented

C₂²×D₅₈₄

C₂²

yes

—

C₂×D₁₁₆₈

—

C₂×D₁₁₆₈

—

C₂×#14556527

C₂²

yes

—

#17988591

—

C₂×#14556507

—

#17988401

—

C₂×M₃₂×C₇₃

C₂²

yes

characteristic

C₂²×C₁₆×C₇₃

C₂²

yes

characteristic

#405×C₇₃

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₃

—

C₂²×D₂₉₂

—

C₂×D₅₈₄

—

C₂×D₅₈₄

—

C₂×D₅₈₄

—

C₂×D₅₈₄

—

C₂×D₅₈₄

—

C₂×D₅₈₄

D₄

yes

—

C₂×#11924012

—

D₁₁₆₈

—

—

—

—

—

C₂×M₁₆×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

C₂³

yes

characteristic

#117×C₇₃

D₄

yes

characteristic

M₃₂×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

D₄

yes

—

#97×C₇₃

—

C₄×C₈×C₇₃

—

C₂³×D₇₃

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

D₅₈₄

—

D₅₈₄

—

D₅₈₄

—

D₅₈₄

—

#11924012

—

#11924012

—

C₂×#10596831

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

C₂×C₈×C₇₃

yes

C₂×C₈×C₇₃

yes

—

C₄⋊C₄×C₇₃

D₈

yes

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₁₆×C₇₃

—

C₁₆×C₇₃

—

C₂³×C₇₃

#135

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#121

yes

characteristic

C₂×C₄×C₇₃

#121

yes

—

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

AGL(1,ℤ/8)

yes

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₈×C₇₃

—

C₈×C₇₃

#120

yes

characteristic

C₈×C₇₃

#123

yes

—

C₈×C₇₃

#124

yes

characteristic

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

#463

yes

—

C₂²×C₇₃

#466

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#394

yes

characteristic

C₄×C₇₃

#459

yes

—

C₄×C₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

#2054

yes

characteristic

C₂×C₇₃

#2221

yes

—

D₇₃

—

100

C₂²×D₁₆

—

101

#2151

D₇₃

yes

characteristic, complemented

102

C₂×#361

—

103

#1978

—

104

#1381

—

105

C₇₃

#13439

yes

characteristic, complemented, Sylow

106

C₂²×D₈

—

107

C₂×D₁₆

—

146

—

108

C₂×D₁₆

—

146

—

109

C₂×M₃₂

C₂×D₇₃

yes

characteristic

110

C₂²×C₁₆

C₂×D₇₃

yes

characteristic

111

C₂×#108

—

112

#418

—

146

—

113

C₂×#100

—

146

—

114

#405

C₂×D₇₃

yes

characteristic

115

C₄⋊C₁₆

—

116

#361

—

146

—

117

C₂²×D₄

—

146

—

118

C₂×D₈

—

119

C₂×D₈

—

146

—

120

C₂×D₈

—

146

—

121

C₂×D₈

—

146

—

122

C₂×D₈

—

146

—

123

C₂×D₈

—

146

—

124

C₂×M₁₆

—

146

—

125

C₂×M₁₆

D₂₉₂

yes

characteristic

126

C₂²×C₈

C₂²×D₇₃

yes

characteristic

127

#117

D₂₉₂

yes

characteristic

128

D₁₆

—

146

—

129

M₃₂

—

130

C₂×C₁₆

—

131

C₂×C₁₆

—

132

C₂×C₁₆

C₂²⋊D₇₃

yes

—

133

#108

—

146

—

134

#108

—

146

—

135

#102

—

292

—

136

#100

—

292

—

137

#97

—

138

C₄×C₈

—

139

C₂⁴

—

292

—

140

C₂×D₄

—

292

—

141

C₂×D₄

—

292

—

142

C₂×D₄

—

292

—

143

C₂×D₄

—

292

—

144

C₂×D₄

—

292

—

145

C₂×D₄

—

146

—

146

C₂²×C₄

C₂×D₂₉₂

yes

characteristic

147

D₈

—

292

—

148

D₈

—

292

—

149

D₈

—

146

—

150

D₈

—

146

—

151

M₁₆

—

152

M₁₆

—

292

—

153

M₁₆

—

292

—

154

C₂×C₈

—

155

C₂×C₈

—

292

—

156

C₂×C₈

D₄×D₇₃

yes

—

157

C₂×C₈

C₂×C₂²⋊D₇₃

yes

characteristic

158

C₂×C₈

#11924028

yes

characteristic

159

C₂×C₈

#11924029

yes

—

160

C₄⋊C₄

D₅₈₄

yes

—

161

C₂²⋊C₄

—

162

C₄²

—

163

C₁₆

—

164

C₁₆

—

165

C₂³

—

292

—

166

C₂³

—

584

—

167

C₂³

—

584

—

168

C₂³

—

1168

—

169

C₂³

#14556613

yes

characteristic

170

D₄

—

292

—

171

D₄

—

292

—

172

D₄

—

584

—

173

D₄

—

584

—

174

C₂×C₄

—

175

C₂×C₄

#14556565

yes

characteristic

176

C₂×C₄

#14556566

yes

—

177

C₂×C₄

C₂×D₅₈₄

yes

characteristic

178

C₂×C₄

#14556619

yes

—

179

C₈

—

180

C₈

—

584

—

181

C₈

#14556559

yes

characteristic

182

C₈

#14556576

yes

—

183

C₈

#14556580

yes

characteristic

184

C₂²

—

185

C₂²

—

584

—

186

C₂²

—

1168

—

187

C₂²

—

1168

—

188

C₂²

—

1168

—

189

C₂²

#17988803

yes

—

190

C₂²

#17988823

yes

characteristic

191

C₄

—

192

C₄

#17988504

yes

characteristic

193

C₄

#17988783

yes

—

194

C₄

#17988840

yes

characteristic

195

C₂

—

196

C₂

—

1168

—

197

C₂

#22611242

yes

characteristic

198

C₂

#22612051

yes

—

	subgroup	G/H
center	C₂²	#17988823
commutator, cosocle	C₂²×C₈×C₇₃	C₂³
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#2151	C₂
socle	C₂²×C₇₃	#466
prime radical	C₂²×D₁₁₆₈	C₂
hypercenter	#2151	D₇₃
hypocenter	C₇₃	#13439
LCS	G ▷ C₂²×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2151

subgroup

G/H

center

C₂²

#17988823

commutator, cosocle

C₂²×C₈×C₇₃

C₂³

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₇₃×#2151

C₂

socle

C₂²×C₇₃

#466

prime radical

C₂²×D₁₁₆₈

C₂

D₇₃

C₇₃

G ▷ C₂²×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2151

N	H ≅ G/N	type
2. C₂×#17988401	C₂	semidirect
3. #22610841	C₂	semidirect
4. #22608025	C₂	semidirect
5. C₇₃×#2151	C₂	semidirect
101. #2151	D₇₃	semidirect
105. C₇₃	#13439	semidirect

H ≅ G/N

type

2. C₂×#17988401

C₂

semidirect

3. #22610841

C₂

semidirect

4. #22608025

C₂

semidirect

5. C₇₃×#2151

C₂

semidirect

101. #2151

D₇₃

semidirect

105. C₇₃

#13439

semidirect

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

subgroup	index
C₂²×D₁₁₆₈	2
C₂×#17988401	2
#22610841	2
#22608025	2
C₇₃×#2151	2
#13439	73

subgroup

index

C₂²×D₁₁₆₈

C₇₃×#2151

Subgroups — #33005524

Conjugacy classes of subgroups

Complements

Maximal subgroups