Group #32985657 — subgroups

Subgroups — #32985657

Order 18688.

434 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 135 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	#22610776	C₂	1	2	yes	maximal
2	#22608687	C₂	1	1	yes	maximal, characteristic
3	#22608603	C₂	1	2	yes	maximal
4	#22608220	C₂	1	1	yes	maximal, characteristic
5	C₂×#500×C₇₃	C₂	1	1	yes	maximal, characteristic
6	#17988640	C₂²	1	1	yes	characteristic
7	#17988611	C₂²	1	2	yes	—
8	C₂×#14556510	—	2	4	no	—
9	C₄×#11924012	C₂²	1	1	yes	characteristic
10	#17988338	—	2	2	no	—
11	#17988333	—	2	4	no	—
12	#17988309	—	2	1	no	—
13	#17988308	—	2	1	no	—
14	C₂×C₄⋊Q₈×C₇₃	C₂²	1	2	yes	—
15	#500×C₇₃	—	2	1	no	—
16	#500×C₇₃	—	2	1	no	—
17	#500×C₇₃	—	2	2	no	—
18	C₂×#107×C₇₃	C₄	1	4	yes	—
19	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
20	C₂×#11924012	—	2	2	no	—
21	C₂×#11924012	D₄	1	2	yes	—
22	C₄⋊#10596831	—	2	2	no	—
23	C₄⋊#10596831	—	2	2	no	—
24	C₄⋊#10596831	D₄	1	2	yes	—
25	#14556510	—	4	8	no	—
26	C₂²⋊#10596831	—	2	1	no	—
27	#14556502	—	2	1	no	—
28	C₄×#10596831	—	2	1	no	—
29	C₄×#10596831	—	2	1	no	—
30	C₂²×Q₈×C₇₃	—	2	2	no	—
31	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
32	C₄⋊Q₈×C₇₃	—	2	2	no	—
33	C₄⋊Q₈×C₇₃	—	2	2	no	—
34	C₄⋊Q₈×C₇₃	—	2	2	no	—
35	C₄⋊Q₈×C₇₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
39	#107×C₇₃	—	2	4	no	—
40	#107×C₇₃	—	2	4	no	—
41	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
42	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₇₃	D₄	1	2	yes	—
44	#11924012	—	2	4	no	—
45	#11924012	—	4	4	no	—
46	C₂×#10596831	—	2	2	no	—
47	C₂×#10596831	—	2	2	no	—
48	C₂×#10596831	—	4	2	no	—
49	C₂×Q₈×C₇₃	—	4	2	no	—
50	C₂×Q₈×C₇₃	—	4	2	no	—
51	C₂×Q₈×C₇₃	—	4	4	no	—
52	C₂×Q₈×C₇₃	—	4	4	no	—
53	C₂²×C₄×C₇₃	—	2	4	no	—
54	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
57	C₂×C₈×C₇₃	—	2	4	no	—
58	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
59	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
60	C₄⋊C₄×C₇₃	—	2	4	no	—
61	C₄⋊C₄×C₇₃	—	2	4	no	—
62	C₄⋊C₄×C₇₃	—	4	2	no	—
63	C₄⋊C₄×C₇₃	—	4	2	no	—
64	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₇₃	#41	1	1	yes	characteristic
68	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
69	Q₈×C₇₃	—	8	4	no	—
70	Q₈×C₇₃	—	8	2	no	—
71	Q₈×C₇₃	—	8	2	no	—
72	C₂×C₄×C₇₃	—	2	2	no	—
73	C₂×C₄×C₇₃	—	2	2	no	—
74	C₂×C₄×C₇₃	—	4	4	no	—
75	C₂×C₄×C₇₃	—	4	8	no	—
76	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
78	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
79	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
80	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
81	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
82	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	4	yes	—
83	C₈×C₇₃	—	2	2	no	—
84	C₈×C₇₃	#101	1	4	yes	complemented
85	#10596831	—	4	2	no	—
86	#10596831	—	4	2	no	—
87	#10596831	—	8	2	no	—
88	C₂²×C₇₃	—	2	1	no	—
89	C₂²×C₇₃	—	2	1	no	—
90	C₂²×C₇₃	#390	1	1	yes	characteristic
91	C₂²×C₇₃	#442	1	2	yes	—
92	C₄×C₇₃	—	2	1	no	—
93	C₄×C₇₃	—	2	1	no	—
94	C₄×C₇₃	—	8	4	no	—
95	C₄×C₇₃	C₂×#101	1	2	yes	—
96	C₄×C₇₃	#469	1	2	yes	—
97	C₄×C₇₃	#473	1	2	yes	—
98	C₄×C₇₃	#496	1	2	yes	—
99	#9808	—	73	1	no	maximal, Sylow
100	C₂×C₇₃	—	2	2	no	—
101	C₂×C₇₃	#1966	1	1	yes	characteristic
102	C₂×C₇₃	#1972	1	1	yes	characteristic
103	C₂×C₇₃	#1979	1	1	yes	characteristic
104	C₂×#500	D₇₃	1	1	yes	characteristic
105	#1966	—	73	2	no	—
106	#1566	—	73	1	no	—
107	#1546	—	73	2	no	—
108	#1451	—	73	1	no	—
109	C₇₃	#9808	1	1	yes	characteristic, complemented, Sylow
110	C₂×C₄⋊Q₈	C₂×D₇₃	1	2	yes	—
111	#500	—	2	1	no	—
112	#500	—	2	1	no	—
113	#500	—	2	2	no	—
114	#431	—	73	1	no	—
115	C₂×#107	Dic₇₃	1	4	yes	—
116	#423	—	73	2	no	—
117	C₂×#101	—	146	4	no	—
118	C₄×M₁₆	—	73	1	no	—
119	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
120	#334	—	146	2	no	—
121	#332	—	146	4	no	—
122	#322	—	146	1	no	—
123	#322	—	146	1	no	—
124	C₂²×Q₈	—	2	2	no	—
125	C₂×M₁₆	—	73	2	no	—
126	C₂×M₁₆	—	146	2	no	—
127	C₂²×C₈	C₂×Dic₇₃	1	2	yes	—
128	C₄⋊Q₈	—	2	2	no	—
129	C₄⋊Q₈	—	2	2	no	—
130	C₄⋊Q₈	—	2	2	no	—
131	C₄⋊Q₈	—	2	2	no	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄⋊C₄	C₂×Dic₇₃	1	4	yes	—
134	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
135	#107	—	2	4	no	—
136	#107	—	2	4	no	—
137	C₄⋊C₈	—	73	2	no	—
138	C₄⋊C₈	—	146	2	no	—
139	C₄⋊C₈	—	146	2	no	—
140	#101	—	292	8	no	—
141	C₂²⋊C₈	—	146	1	no	—
142	#97	—	146	1	no	—
143	C₄×C₈	—	146	1	no	—
144	C₄×C₈	—	146	1	no	—
145	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
146	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
147	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
148	C₂×Q₈	—	4	2	no	—
149	C₂×Q₈	—	4	2	no	—
150	C₂×Q₈	—	4	4	no	—
151	C₂×Q₈	—	4	4	no	—
152	C₂²×C₄	—	2	4	no	—
153	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
154	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
155	C₂²×C₄	#11924027	1	1	yes	characteristic
156	M₁₆	—	292	4	no	—
157	M₁₆	—	146	4	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	292	2	no	—
160	C₂×C₈	—	146	2	no	—
161	C₂×C₈	—	146	2	no	—
162	C₂×C₈	C₂²⋊Dic₇₃	1	2	yes	—
163	C₂×C₈	C₂²⋊Dic₇₃	1	2	yes	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄⋊C₄	—	4	2	no	—
168	C₄²	D₄×D₇₃	1	1	yes	characteristic
169	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
170	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
171	C₄²	#11924027	1	1	yes	characteristic
172	C₂³	D₄×Dic₇₃	1	1	yes	characteristic
173	Q₈	—	8	2	no	—
174	Q₈	—	8	2	no	—
175	Q₈	—	8	4	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	4	8	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
181	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
182	C₂×C₄	#14556562	1	1	yes	characteristic
183	C₂×C₄	#14556562	1	1	yes	characteristic
184	C₂×C₄	#14556578	1	1	yes	characteristic
185	C₂×C₄	#14556587	1	1	yes	characteristic
186	C₂×C₄	#14556615	1	4	yes	—
187	C₈	—	2	2	no	—
188	C₈	—	292	2	no	—
189	C₈	—	292	2	no	—
190	C₈	—	584	2	no	—
191	C₈	#14556510	1	4	yes	complemented
192	C₂²	—	2	1	no	—
193	C₂²	—	2	1	no	—
194	C₂²	#17988490	1	1	yes	characteristic
195	C₂²	#17988699	1	2	yes	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	8	4	no	—
199	C₄	C₂×#14556510	1	2	yes	—
200	C₄	#17988837	1	2	yes	—
201	C₄	#17988859	1	2	yes	—
202	C₄	#17988975	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	#22610776	1	1	yes	characteristic
205	C₂	#22610808	1	1	yes	characteristic
206	C₂	#22610847	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22610776

C₂

yes

maximal

#22608687

C₂

yes

maximal, characteristic

#22608603

C₂

yes

maximal

#22608220

C₂

yes

maximal, characteristic

C₂×#500×C₇₃

C₂

yes

maximal, characteristic

#17988640

C₂²

yes

characteristic

#17988611

C₂²

yes

—

C₂×#14556510

—

C₄×#11924012

C₂²

yes

—

—

—

—

C₂×C₄⋊Q₈×C₇₃

C₂²

yes

—

#500×C₇₃

—

#500×C₇₃

—

#500×C₇₃

—

C₂×#107×C₇₃

C₄

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

C₂×#11924012

—

C₂×#11924012

D₄

yes

—

C₄⋊#10596831

—

C₄⋊#10596831

—

C₄⋊#10596831

D₄

yes

—

—

—

—

—

—

C₂²×Q₈×C₇₃

—

C₂²×C₈×C₇₃

C₂×C₄

yes

—

C₄⋊Q₈×C₇₃

—

C₄⋊Q₈×C₇₃

—

C₄⋊Q₈×C₇₃

—

C₄⋊Q₈×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

#107×C₇₃

—

#107×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

—

—

—

—

—

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

#41

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₄²×C₇₃

#41

yes

characteristic

C₂³×C₇₃

C₄×D₄

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₄×D₄

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

—

C₈×C₇₃

—

C₈×C₇₃

yes

—

—

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#390

yes

characteristic

C₂²×C₇₃

#442

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×#101

yes

—

C₄×C₇₃

#469

yes

—

C₄×C₇₃

#473

yes

—

C₄×C₇₃

#496

yes

—

#9808

—

maximal, Sylow

100

C₂×C₇₃

—

101

C₂×C₇₃

#1966

yes

characteristic

102

C₂×C₇₃

#1972

yes

characteristic

103

C₂×C₇₃

#1979

yes

characteristic

104

C₂×#500

D₇₃

yes

characteristic

105

#1966

—

106

#1566

—

107

#1546

—

108

#1451

—

109

C₇₃

#9808

yes

characteristic, complemented, Sylow

110

C₂×C₄⋊Q₈

C₂×D₇₃

yes

—

111

#500

—

112

#500

—

113

#500

—

114

#431

—

115

C₂×#107

Dic₇₃

yes

—

116

#423

—

117

C₂×#101

—

146

—

118

C₄×M₁₆

—

119

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

120

#334

—

146

—

121

#332

—

146

—

122

#322

—

146

—

123

#322

—

146

—

124

C₂²×Q₈

—

125

C₂×M₁₆

—

126

C₂×M₁₆

—

146

—

127

C₂²×C₈

C₂×Dic₇₃

yes

—

128

C₄⋊Q₈

—

129

C₄⋊Q₈

—

130

C₄⋊Q₈

—

131

C₄⋊Q₈

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

C₂×Dic₇₃

yes

—

134

C₂×C₄²

C₂²×D₇₃

yes

characteristic

135

#107

—

136

#107

—

137

C₄⋊C₈

—

138

C₄⋊C₈

—

146

—

139

C₄⋊C₈

—

146

—

140

#101

—

292

—

141

C₂²⋊C₈

—

146

—

142

#97

—

146

—

143

C₄×C₈

—

146

—

144

C₄×C₈

—

146

—

145

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

146

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

147

C₄×C₈

C₂²⋊D₇₃

yes

—

148

C₂×Q₈

—

149

C₂×Q₈

—

150

C₂×Q₈

—

151

C₂×Q₈

—

152

C₂²×C₄

—

153

C₂²×C₄

C₂²×Dic₇₃

yes

characteristic

154

C₂²×C₄

D₄×D₇₃

yes

characteristic

155

C₂²×C₄

#11924027

yes

characteristic

156

M₁₆

—

292

—

157

M₁₆

—

146

—

158

C₂×C₈

—

159

C₂×C₈

—

292

—

160

C₂×C₈

—

146

—

161

C₂×C₈

—

146

—

162

C₂×C₈

C₂²⋊Dic₇₃

yes

—

163

C₂×C₈

C₂²⋊Dic₇₃

yes

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₄²

D₄×D₇₃

yes

characteristic

169

C₄²

C₂×C₂²⋊D₇₃

yes

characteristic

170

C₄²

C₂×C₂²⋊D₇₃

yes

characteristic

171

C₄²

#11924027

yes

characteristic

172

C₂³

D₄×Dic₇₃

yes

characteristic

173

Q₈

—

174

Q₈

—

175

Q₈

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

C₂×C₂²⋊Dic₇₃

yes

characteristic

181

C₂×C₄

D₄×Dic₇₃

yes

characteristic

182

C₂×C₄

#14556562

yes

characteristic

183

C₂×C₄

#14556562

yes

characteristic

184

C₂×C₄

#14556578

yes

characteristic

185

C₂×C₄

#14556587

yes

characteristic

186

C₂×C₄

#14556615

yes

—

187

C₈

—

188

C₈

—

292

—

189

C₈

—

292

—

190

C₈

—

584

—

191

C₈

#14556510

yes

complemented

192

C₂²

—

193

C₂²

—

194

C₂²

#17988490

yes

characteristic

195

C₂²

#17988699

yes

—

196

C₄

—

197

C₄

—

198

C₄

—

199

C₄

C₂×#14556510

yes

—

200

C₄

#17988837

yes

—

201

C₄

#17988859

yes

—

202

C₄

#17988975

yes

—

203

C₂

—

204

C₂

#22610776

yes

characteristic

205

C₂

#22610808

yes

characteristic

206

C₂

#22610847

yes

characteristic

	subgroup	G/H
center	C₂²	#17988490
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₂×#500×C₇₃	C₂
socle	C₂²×C₇₃	#390
prime radical	C₂³×C₇₃	C₄×D₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₂×#500	D₇₃
hypocenter	C₇₃	#9808
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

subgroup

G/H

center

C₂²

#17988490

commutator

C₂²×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₂×#500×C₇₃

C₂

socle

C₂²×C₇₃

#390

prime radical

C₂³×C₇₃

C₄×D₄

cosocle

C₂×C₄²×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

N	H ≅ G/N	type
84. C₈×C₇₃	#101	semidirect
109. C₇₃	#9808	semidirect
191. C₈	#14556510	semidirect

H ≅ G/N

type

84. C₈×C₇₃

#101

semidirect

109. C₇₃

#9808

semidirect

191. C₈

#14556510

semidirect

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

subgroup	index
#22610776	2
#22608687	2
#22608603	2
#22608220	2
C₂×#500×C₇₃	2
#9808	73

subgroup

index

C₂×#500×C₇₃

Subgroups — #32985657

Conjugacy classes of subgroups

Complements

Maximal subgroups