Group #32988720 — subgroups

Subgroups — #32988720

Order 18688.

528 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 113 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	#22620879	C₂	1	2	yes	maximal
2	#22610908	C₂	1	1	yes	maximal, characteristic
3	#22608612	C₂	1	2	yes	maximal
4	#22607764	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#2862	C₂	1	1	yes	maximal, characteristic
6	#17989144	C₂²	1	1	yes	characteristic
7	#17988966	—	2	2	no	—
8	#17988937	—	2	2	no	—
9	#17988794	—	2	4	no	—
10	#17988743	—	2	4	no	—
11	#17988640	C₂²	1	2	yes	—
12	#17988611	C₂²	1	1	yes	characteristic
13	#17988347	—	2	2	no	—
14	#17988336	—	2	4	no	—
15	C₂×C₄⋊Q₈×C₇₃	C₂²	1	2	yes	—
16	#391×C₇₃	—	2	4	no	—
17	#382×C₇₃	—	2	2	no	—
18	C₄³×C₇₃	C₂²	1	1	yes	characteristic
19	C₂×#11924018	—	2	8	no	—
20	C₂×#11924012	—	2	1	no	—
21	#14556591	D₄	1	2	yes	—
22	#14556583	—	2	1	no	—
23	#14556570	—	2	2	no	—
24	C₄×Dic₁₄₆	—	2	2	no	—
25	#14556550	—	2	1	no	—
26	C₄⋊#10596831	—	2	2	no	—
27	C₄⋊#10596831	D₄	1	4	yes	—
28	#14556515	—	2	8	no	—
29	C₂²⋊#10596831	—	2	2	no	—
30	C₄×#10596831	—	2	2	no	—
31	C₂²×Q₈×C₇₃	D₄	1	4	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	—	2	2	no	—
38	C₂×C₄²×C₇₃	—	2	2	no	—
39	C₂×C₄²×C₇₃	D₄	1	2	yes	—
40	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
41	#95×C₇₃	—	2	8	no	—
42	C₂×Dic₁₄₆	—	2	2	no	—
43	#11924018	—	8	8	no	—
44	#11924012	—	4	2	no	—
45	C₂×#10596831	—	2	4	no	—
46	C₂×#10596831	—	4	1	no	—
47	C₄⋊Dic₇₃	—	2	2	no	—
48	C₄⋊Dic₇₃	—	2	2	no	—
49	#11924007	—	4	2	no	—
50	C₂²⋊Dic₇₃	—	4	1	no	—
51	C₄×Dic₇₃	—	4	1	no	—
52	C₂×Q₈×C₇₃	—	2	4	no	—
53	C₂×Q₈×C₇₃	—	2	4	no	—
54	C₂×Q₈×C₇₃	—	4	8	no	—
55	C₂²×C₄×C₇₃	—	2	2	no	—
56	C₂²×C₄×C₇₃	—	2	4	no	—
57	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
59	C₄⋊C₄×C₇₃	—	4	2	no	—
60	C₄⋊C₄×C₇₃	—	4	2	no	—
61	C₄⋊C₄×C₇₃	—	4	2	no	—
62	C₄⋊C₄×C₇₃	—	4	2	no	—
63	C₄²×C₇₃	—	2	2	no	—
64	C₄²×C₇₃	—	4	4	no	—
65	C₄²×C₇₃	Q₁₆	1	4	yes	complemented
66	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₇₃	C₂×D₄	1	2	yes	—
68	C₄²×C₇₃	#41	1	1	yes	characteristic
69	C₂³×C₇₃	#120	1	1	yes	characteristic
70	Q₈×C₇₃	—	4	8	no	—
71	Q₈×C₇₃	—	8	4	no	—
72	C₂×C₄×C₇₃	—	2	4	no	—
73	C₂×C₄×C₇₃	—	2	4	no	—
74	C₂×C₄×C₇₃	—	4	4	no	—
75	C₂×C₄×C₇₃	—	4	8	no	—
76	C₂×C₄×C₇₃	—	4	4	no	—
77	C₂×C₄×C₇₃	#120	1	2	yes	—
78	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
79	C₂×C₄×C₇₃	#121	1	2	yes	—
80	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
81	C₂×C₄×C₇₃	C₂×Q₁₆	1	2	yes	—
82	C₂×C₄×C₇₃	#137	1	2	yes	—
83	C₂×Dic₇₃	—	4	2	no	—
84	Dic₁₄₆	—	4	4	no	—
85	#10596831	—	4	4	no	—
86	#10596831	—	8	1	no	—
87	C₂²×C₇₃	—	2	2	no	—
88	C₂²×C₇₃	#392	1	1	yes	characteristic
89	C₂²×C₇₃	#451	1	2	yes	—
90	C₄×C₇₃	—	2	2	no	—
91	C₄×C₇₃	—	4	4	no	—
92	C₄×C₇₃	—	8	4	no	—
93	C₄×C₇₃	#455	1	4	yes	—
94	C₄×C₇₃	#462	1	2	yes	—
95	C₄×C₇₃	#494	1	1	yes	characteristic
96	C₄×C₇₃	#497	1	1	yes	characteristic
97	Dic₇₃	—	8	2	no	—
98	#10297	—	73	1	no	maximal, Sylow
99	C₂×C₇₃	—	2	2	no	—
100	C₂×C₇₃	#2004	1	2	yes	—
101	C₂×C₇₃	#2016	1	1	yes	characteristic
102	#3209	—	73	2	no	—
103	#2862	D₇₃	1	1	yes	characteristic
104	#1994	—	73	1	no	—
105	#1548	—	73	2	no	—
106	#1276	—	73	1	no	—
107	C₇₃	#10297	1	1	yes	characteristic, complemented, Sylow
108	#533	—	73	1	no	—
109	C₂×C₄⋊Q₈	C₂×D₇₃	1	2	yes	—
110	#494	—	146	2	no	—
111	#487	—	146	2	no	—
112	#462	—	146	4	no	—
113	#451	—	146	4	no	—
114	#431	—	73	2	no	—
115	#423	—	73	1	no	—
116	#391	—	2	4	no	—
117	#382	—	2	2	no	—
118	C₄³	C₂×D₇₃	1	1	yes	characteristic
119	#339	—	146	2	no	—
120	#333	—	146	4	no	—
121	C₂²×Q₈	C₂²⋊D₇₃	1	4	yes	—
122	C₂×Q₁₆	—	146	8	no	—
123	C₂×M₁₆	—	146	1	no	—
124	C₄⋊Q₈	—	2	2	no	—
125	C₄⋊Q₈	—	2	2	no	—
126	C₄⋊Q₈	—	2	2	no	—
127	C₄⋊Q₈	—	2	2	no	—
128	C₄⋊Q₈	—	73	2	no	—
129	#125	—	146	1	no	—
130	C₂²⋊Q₈	—	146	2	no	—
131	C₄×Q₈	—	146	2	no	—
132	#117	—	146	1	no	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄²	—	2	2	no	—
136	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
137	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
138	C₄⋊C₈	—	73	4	no	—
139	C₄⋊C₈	—	146	2	no	—
140	#103	—	146	8	no	—
141	C₂²⋊C₈	—	146	2	no	—
142	C₄×C₈	—	146	2	no	—
143	#95	—	2	8	no	—
144	C₂×Q₈	—	2	4	no	—
145	C₂×Q₈	—	2	4	no	—
146	C₂×Q₈	—	4	8	no	—
147	C₂×Q₈	—	146	2	no	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	2	4	no	—
150	C₂²×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
151	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
152	Q₁₆	—	584	8	no	—
153	M₁₆	—	292	2	no	—
154	C₂×C₈	—	292	1	no	—
155	C₂×C₈	—	146	4	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	292	2	no	—
161	C₄⋊C₄	—	146	2	no	—
162	C₄⋊C₄	—	146	2	no	—
163	C₂²⋊C₄	—	292	1	no	—
164	C₄²	—	2	2	no	—
165	C₄²	—	4	4	no	—
166	C₄²	—	292	1	no	—
167	C₄²	#11924018	1	4	yes	complemented
168	C₄²	D₄×D₇₃	1	1	yes	characteristic
169	C₄²	D₄×D₇₃	1	2	yes	—
170	C₄²	#11924027	1	1	yes	characteristic
171	C₂³	#14556560	1	1	yes	characteristic
172	Q₈	—	4	8	no	—
173	Q₈	—	8	4	no	—
174	Q₈	—	292	4	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	4	4	no	—
178	C₂×C₄	—	4	8	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	292	2	no	—
181	C₂×C₄	#14556559	1	2	yes	—
182	C₂×C₄	#14556562	1	1	yes	characteristic
183	C₂×C₄	#14556563	1	2	yes	—
184	C₂×C₄	#14556587	1	1	yes	characteristic
185	C₂×C₄	C₂×#11924018	1	2	yes	—
186	C₂×C₄	#14556622	1	2	yes	—
187	C₈	—	292	4	no	—
188	C₈	—	584	1	no	—
189	C₂²	—	2	2	no	—
190	C₂²	#17988495	1	1	yes	characteristic
191	C₂²	#17988743	1	2	yes	—
192	C₄	—	2	2	no	—
193	C₄	—	4	4	no	—
194	C₄	—	8	4	no	—
195	C₄	—	584	2	no	—
196	C₄	#17988764	1	4	yes	—
197	C₄	#17988794	1	2	yes	—
198	C₄	#17988966	1	1	yes	characteristic
199	C₄	#17988979	1	1	yes	characteristic
200	C₂	—	2	2	no	—
201	C₂	#22610960	1	2	yes	—
202	C₂	#22611031	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22620879

C₂

yes

maximal

#22610908

C₂

yes

maximal, characteristic

#22608612

C₂

yes

maximal

#22607764

C₂

yes

maximal, characteristic

C₇₃×#2862

C₂

yes

maximal, characteristic

#17989144

C₂²

yes

—

—

—

—

C₂²

yes

—

#17988611

C₂²

yes

characteristic

#17988347

—

#17988336

—

C₂×C₄⋊Q₈×C₇₃

C₂²

yes

—

#391×C₇₃

—

#382×C₇₃

—

C₄³×C₇₃

C₂²

yes

—

—

D₄

yes

—

#14556583

—

#14556570

—

C₄×Dic₁₄₆

—

#14556550

—

C₄⋊#10596831

—

C₄⋊#10596831

D₄

yes

—

#14556515

—

C₂²⋊#10596831

—

C₄×#10596831

—

C₂²×Q₈×C₇₃

D₄

yes

—

C₄⋊Q₈×C₇₃

—

C₄⋊Q₈×C₇₃

—

C₄⋊Q₈×C₇₃

—

C₄⋊Q₈×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄²×C₇₃

—

C₂×C₄²×C₇₃

D₄

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

#95×C₇₃

—

C₂×Dic₁₄₆

—

—

—

—

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924007

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

Q₁₆

yes

complemented

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

—

C₄²×C₇₃

yes

C₂³×C₇₃

yes

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#120

yes

—

C₂×C₄×C₇₃

#121

yes

characteristic

C₂×C₄×C₇₃

#121

yes

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₁₆

yes

—

C₂×C₄×C₇₃

#137

yes

—

C₂×Dic₇₃

—

Dic₁₄₆

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

#392

yes

characteristic

C₂²×C₇₃

#451

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#455

yes

—

C₄×C₇₃

#462

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

#10297

—

maximal, Sylow

C₂×C₇₃

—

100

C₂×C₇₃

#2004

yes

—

101

C₂×C₇₃

#2016

yes

characteristic

102

#3209

—

103

#2862

D₇₃

yes

characteristic

104

#1994

—

105

#1548

—

106

#1276

—

107

C₇₃

#10297

yes

characteristic, complemented, Sylow

108

#533

—

109

C₂×C₄⋊Q₈

C₂×D₇₃

yes

—

110

#494

—

146

—

111

#487

—

146

—

112

#462

—

146

—

113

#451

—

146

—

114

#431

—

115

#423

—

116

#391

—

117

#382

—

118

C₄³

C₂×D₇₃

yes

characteristic

119

#339

—

146

—

120

#333

—

146

—

121

C₂²×Q₈

C₂²⋊D₇₃

yes

—

122

C₂×Q₁₆

—

146

—

123

C₂×M₁₆

—

146

—

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

—

129

#125

—

146

—

130

C₂²⋊Q₈

—

146

—

131

C₄×Q₈

—

146

—

132

#117

—

146

—

133

C₂×C₄⋊C₄

—

134

C₂×C₄⋊C₄

—

135

C₂×C₄²

—

136

C₂×C₄²

C₂²⋊D₇₃

yes

—

137

C₂×C₄²

C₂²×D₇₃

yes

characteristic

138

C₄⋊C₈

—

139

C₄⋊C₈

—

146

—

140

#103

—

146

—

141

C₂²⋊C₈

—

146

—

142

C₄×C₈

—

146

—

143

#95

—

144

C₂×Q₈

—

145

C₂×Q₈

—

146

C₂×Q₈

—

147

C₂×Q₈

—

146

—

148

C₂²×C₄

—

149

C₂²×C₄

—

150

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

—

151

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

152

Q₁₆

—

584

—

153

M₁₆

—

292

—

154

C₂×C₈

—

292

—

155

C₂×C₈

—

146

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

292

—

161

C₄⋊C₄

—

146

—

162

C₄⋊C₄

—

146

—

163

C₂²⋊C₄

—

292

—

164

C₄²

—

165

C₄²

—

166

C₄²

—

292

—

167

C₄²

#11924018

yes

complemented

168

C₄²

D₄×D₇₃

yes

characteristic

169

C₄²

D₄×D₇₃

yes

—

170

C₄²

#11924027

yes

characteristic

171

C₂³

#14556560

yes

characteristic

172

Q₈

—

173

Q₈

—

174

Q₈

—

292

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

292

—

181

C₂×C₄

#14556559

yes

—

182

C₂×C₄

#14556562

yes

characteristic

183

C₂×C₄

#14556563

yes

—

184

C₂×C₄

#14556587

yes

characteristic

185

C₂×C₄

C₂×#11924018

yes

—

186

C₂×C₄

#14556622

yes

—

187

C₈

—

292

—

188

C₈

—

584

—

189

C₂²

—

190

C₂²

#17988495

yes

characteristic

191

C₂²

#17988743

yes

—

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

584

—

196

C₄

#17988764

yes

—

197

C₄

#17988794

yes

—

198

C₄

#17988966

yes

characteristic

199

C₄

#17988979

yes

characteristic

200

C₂

—

201

C₂

#22610960

yes

—

202

C₂

#22611031

yes

characteristic

	subgroup	G/H
center	C₂²	#17988495
commutator, cosocle	C₂×C₄²×C₇₃	C₂³
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#2862	C₂
socle	C₂²×C₇₃	#392
prime radical	C₂³×C₇₃	#120
hypercenter	#2862	D₇₃
hypocenter	C₇₃	#10297
LCS	G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2862

subgroup

G/H

center

C₂²

#17988495

commutator, cosocle

C₂×C₄²×C₇₃

C₂³

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#2862

C₂

C₂²×C₇₃

C₂³×C₇₃

D₇₃

C₇₃

G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2862

N	H ≅ G/N	type
65. C₄²×C₇₃	Q₁₆	semidirect
107. C₇₃	#10297	semidirect
167. C₄²	#11924018	semidirect

H ≅ G/N

type

65. C₄²×C₇₃

Q₁₆

semidirect

107. C₇₃

#10297

semidirect

167. C₄²

#11924018

semidirect

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

subgroup	index
#22620879	2
#22610908	2
#22608612	2
#22607764	2
C₇₃×#2862	2
#10297	73

subgroup

index

C₇₃×#2862

Subgroups — #32988720

Conjugacy classes of subgroups

Complements

Maximal subgroups