Group #32990778 — subgroups

Subgroups — #32990778

Order 18688.

310 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 93 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	#22610751	C₂	1	1	yes	maximal, characteristic
2	#22608604	C₂	1	2	yes	maximal
3	#22607791	C₂	1	2	yes	maximal
4	#22607776	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1849	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556591	C₂²	1	1	yes	characteristic
7	#17988640	C₂²	1	2	yes	—
8	C₂×#14556516	—	2	1	no	—
9	C₂×#14556516	—	2	1	no	—
10	#17988453	C₂²	1	1	yes	characteristic
11	#17988334	—	2	4	no	—
12	C₂×#108×C₇₃	—	2	1	no	—
13	C₂×#108×C₇₃	—	2	1	no	—
14	#423×C₇₃	C₂²	1	2	yes	—
15	C₂×#97×C₇₃	C₂²	1	1	yes	characteristic
16	C₂²×Dic₁₄₆	—	2	1	no	—
17	#14556591	—	2	2	no	—
18	#14556591	—	2	1	no	—
19	#14556591	D₄	1	2	yes	—
20	C₂×C₄⋊Dic₇₃	—	2	1	no	—
21	C₂×C₄⋊Dic₇₃	—	2	2	no	—
22	C₄⋊#10596831	—	2	2	no	—
23	#14556516	—	4	2	no	—
24	#14556516	—	4	2	no	—
25	C₂²⋊#10596831	—	2	2	no	—
26	C₄×#10596831	—	2	2	no	—
27	#14556498	—	2	2	no	—
28	C₂×M₁₆×C₇₃	—	2	2	no	—
29	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
32	#108×C₇₃	—	4	2	no	—
33	#108×C₇₃	—	4	2	no	—
34	C₄⋊C₈×C₇₃	—	2	2	no	—
35	C₄⋊C₈×C₇₃	—	2	2	no	—
36	#97×C₇₃	—	2	2	no	—
37	C₂×Dic₁₄₆	—	2	2	no	—
38	C₂×Dic₁₄₆	—	4	1	no	—
39	C₂×Dic₁₄₆	—	4	1	no	—
40	C₂×Dic₁₄₆	—	4	1	no	—
41	C₂×Dic₁₄₆	—	4	1	no	—
42	C₂×Dic₁₄₆	—	4	1	no	—
43	C₂²×Dic₇₃	—	4	1	no	—
44	C₂²×Dic₇₃	—	4	1	no	—
45	C₂×#10596831	—	4	2	no	—
46	C₄⋊Dic₇₃	—	2	4	no	—
47	C₄⋊Dic₇₃	—	4	2	no	—
48	C₄⋊Dic₇₃	—	4	1	no	—
49	C₄⋊Dic₇₃	—	4	1	no	—
50	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₇₃	#41	1	2	yes	—
52	M₁₆×C₇₃	—	4	2	no	—
53	M₁₆×C₇₃	—	4	2	no	—
54	C₂×C₈×C₇₃	—	2	1	no	—
55	C₂×C₈×C₇₃	—	2	1	no	—
56	C₂×C₈×C₇₃	—	2	1	no	—
57	C₂×C₈×C₇₃	—	2	1	no	—
58	C₂×C₈×C₇₃	—	4	2	no	—
59	C₂×C₈×C₇₃	D₈	1	2	yes	—
60	C₂×C₈×C₇₃	SD₁₆	1	2	yes	—
61	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₇₃	#41	1	1	yes	characteristic
63	C₄²×C₇₃	#41	1	2	yes	—
64	C₂³×C₇₃	#124	1	1	yes	characteristic
65	C₂×C₄×C₇₃	—	4	2	no	—
66	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
67	C₂×C₄×C₇₃	#123	1	1	yes	characteristic
68	C₂×C₄×C₇₃	#123	1	2	yes	—
69	C₂×C₄×C₇₃	#126	1	2	yes	—
70	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
73	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
74	C₂×Dic₇₃	—	4	2	no	—
75	C₂×Dic₇₃	—	8	1	no	—
76	C₂×Dic₇₃	—	8	1	no	—
77	C₂×Dic₇₃	—	8	2	no	—
78	C₂×Dic₇₃	—	8	1	no	—
79	Dic₁₄₆	—	4	2	no	—
80	Dic₁₄₆	—	4	2	no	—
81	Dic₁₄₆	—	8	1	no	—
82	Dic₁₄₆	—	8	1	no	—
83	C₈×C₇₃	—	2	2	no	—
84	C₈×C₇₃	—	2	2	no	—
85	C₈×C₇₃	—	8	2	no	—
86	#10596831	—	8	2	no	—
87	C₂²×C₇₃	—	2	2	no	—
88	C₂²×C₇₃	#397	1	1	yes	characteristic
89	C₂²×C₇₃	#486	1	1	yes	characteristic
90	C₂²×C₇₃	#489	1	1	yes	characteristic
91	C₄×C₇₃	—	2	2	no	—
92	C₄×C₇₃	—	2	2	no	—
93	C₄×C₇₃	#459	1	1	yes	characteristic
94	C₄×C₇₃	#461	1	1	yes	characteristic
95	C₄×C₇₃	#480	1	1	yes	characteristic
96	C₄×C₇₃	#481	1	1	yes	characteristic
97	Dic₇₃	—	8	2	no	—
98	Dic₇₃	—	16	1	no	—
99	#10632	—	73	1	no	maximal, Sylow
100	C₂×C₇₃	—	2	1	no	—
101	C₂×C₇₃	—	2	1	no	—
102	C₂×C₇₃	#2071	1	1	yes	characteristic
103	C₂×C₇₃	#2082	1	2	yes	—
104	#1960	—	73	1	no	—
105	#1849	D₇₃	1	1	yes	characteristic
106	#1546	—	73	2	no	—
107	#1286	—	73	2	no	—
108	#1281	—	73	1	no	—
109	C₇₃	#10632	1	1	yes	characteristic, complemented, Sylow
110	C₂×C₄⋊Q₈	—	73	1	no	—
111	#431	—	73	2	no	—
112	C₂×#108	—	2	1	no	—
113	C₂×#108	—	2	1	no	—
114	#423	C₂×D₇₃	1	2	yes	—
115	C₂×#103	—	146	1	no	—
116	C₂×#103	—	146	1	no	—
117	C₂×#97	C₂×D₇₃	1	1	yes	characteristic
118	#382	—	73	1	no	—
119	#332	—	146	4	no	—
120	C₂²×Q₈	—	146	1	no	—
121	C₂×M₁₆	—	2	2	no	—
122	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
123	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
124	C₄⋊Q₈	—	73	2	no	—
125	C₄⋊Q₈	—	146	1	no	—
126	C₄⋊Q₈	—	146	2	no	—
127	C₂×C₄⋊C₄	—	146	1	no	—
128	C₂×C₄⋊C₄	—	146	2	no	—
129	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
130	#108	—	4	2	no	—
131	#108	—	4	2	no	—
132	C₄⋊C₈	—	2	2	no	—
133	C₄⋊C₈	—	2	2	no	—
134	C₄⋊C₈	—	146	2	no	—
135	#103	—	292	2	no	—
136	#103	—	292	2	no	—
137	C₂²⋊C₈	—	146	2	no	—
138	#97	—	2	2	no	—
139	C₄×C₈	—	146	2	no	—
140	#95	—	146	2	no	—
141	C₂×Q₈	—	292	1	no	—
142	C₂×Q₈	—	292	1	no	—
143	C₂×Q₈	—	292	1	no	—
144	C₂×Q₈	—	292	1	no	—
145	C₂×Q₈	—	292	1	no	—
146	C₂×Q₈	—	146	2	no	—
147	C₂²×C₄	—	292	1	no	—
148	C₂²×C₄	—	292	1	no	—
149	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
150	C₂²×C₄	#11924029	1	2	yes	—
151	M₁₆	—	4	2	no	—
152	M₁₆	—	4	2	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	292	2	no	—
159	C₂×C₈	#11924013	1	2	yes	—
160	C₂×C₈	#11924015	1	2	yes	—
161	C₄⋊C₄	—	292	2	no	—
162	C₄⋊C₄	—	292	1	no	—
163	C₄⋊C₄	—	292	1	no	—
164	C₄⋊C₄	—	146	4	no	—
165	C₄²	D₄×D₇₃	1	1	yes	characteristic
166	C₄²	#11924027	1	1	yes	characteristic
167	C₄²	#11924028	1	2	yes	—
168	C₂³	#14556580	1	1	yes	characteristic
169	Q₈	—	292	2	no	—
170	Q₈	—	292	2	no	—
171	Q₈	—	584	1	no	—
172	Q₈	—	584	1	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	292	2	no	—
175	C₂×C₄	—	584	1	no	—
176	C₂×C₄	—	584	1	no	—
177	C₂×C₄	—	584	2	no	—
178	C₂×C₄	—	584	1	no	—
179	C₂×C₄	#14556562	1	1	yes	characteristic
180	C₂×C₄	#14556574	1	1	yes	characteristic
181	C₂×C₄	#14556576	1	2	yes	—
182	C₂×C₄	#14556586	1	2	yes	—
183	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
184	C₂×C₄	C₂×#11924015	1	1	yes	characteristic
185	C₂×C₄	#14556617	1	1	yes	characteristic
186	C₂×C₄	#14556622	1	1	yes	characteristic
187	C₈	—	2	2	no	—
188	C₈	—	2	2	no	—
189	C₈	—	8	2	no	—
190	C₈	—	584	2	no	—
191	C₂²	—	2	2	no	—
192	C₂²	#17988513	1	1	yes	characteristic
193	C₂²	#17988935	1	1	yes	characteristic
194	C₂²	#17988949	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	2	2	no	—
197	C₄	—	584	2	no	—
198	C₄	—	1168	1	no	—
199	C₄	#17988779	1	1	yes	characteristic
200	C₄	#17988789	1	1	yes	characteristic
201	C₄	#17988900	1	1	yes	characteristic
202	C₄	#17988905	1	1	yes	characteristic
203	C₂	—	2	1	no	—
204	C₂	—	2	1	no	—
205	C₂	#22611335	1	1	yes	characteristic
206	C₂	#22611396	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22610751

C₂

yes

maximal, characteristic

#22608604

C₂

yes

maximal

#22607791

C₂

yes

maximal

#22607776

C₂

yes

maximal, characteristic

C₇₃×#1849

C₂

yes

maximal, characteristic

C₂×#14556591

C₂²

yes

characteristic

#17988640

C₂²

yes

—

C₂×#14556516

—

C₂×#14556516

—

#17988453

C₂²

yes

characteristic

#17988334

—

C₂×#108×C₇₃

—

C₂×#108×C₇₃

—

#423×C₇₃

C₂²

yes

—

C₂×#97×C₇₃

C₂²

yes

characteristic

C₂²×Dic₁₄₆

—

#14556591

—

#14556591

—

#14556591

D₄

yes

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

—

—

—

—

—

—

—

C₂×M₁₆×C₇₃

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂×C₄²×C₇₃

C₂³

yes

characteristic

#108×C₇₃

—

#108×C₇₃

—

C₄⋊C₈×C₇₃

—

C₄⋊C₈×C₇₃

—

#97×C₇₃

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

#41

yes

—

M₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

SD₁₆

yes

—

C₄²×C₇₃

C₂×D₄

yes

C₄²×C₇₃

yes

C₄²×C₇₃

yes

—

C₂³×C₇₃

#124

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

—

C₂×C₄×C₇₃

#126

yes

—

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×SD₁₆

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

#10632

—

maximal, Sylow

100

C₂×C₇₃

—

101

C₂×C₇₃

—

102

C₂×C₇₃

#2071

yes

characteristic

103

C₂×C₇₃

#2082

yes

—

104

#1960

—

105

#1849

D₇₃

yes

characteristic

106

#1546

—

107

#1286

—

108

#1281

—

109

C₇₃

#10632

yes

characteristic, complemented, Sylow

110

C₂×C₄⋊Q₈

—

111

#431

—

112

C₂×#108

—

113

C₂×#108

—

114

#423

C₂×D₇₃

yes

—

115

C₂×#103

—

146

—

116

C₂×#103

—

146

—

117

C₂×#97

C₂×D₇₃

yes

characteristic

118

#382

—

119

#332

—

146

—

120

C₂²×Q₈

—

146

—

121

C₂×M₁₆

—

122

C₂²×C₈

C₂²⋊D₇₃

yes

characteristic

123

C₂²×C₈

C₂²⋊D₇₃

yes

characteristic

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

146

—

126

C₄⋊Q₈

—

146

—

127

C₂×C₄⋊C₄

—

146

—

128

C₂×C₄⋊C₄

—

146

—

129

C₂×C₄²

C₂²×D₇₃

yes

characteristic

130

#108

—

131

#108

—

132

C₄⋊C₈

—

133

C₄⋊C₈

—

134

C₄⋊C₈

—

146

—

135

#103

—

292

—

136

#103

—

292

—

137

C₂²⋊C₈

—

146

—

138

#97

—

139

C₄×C₈

—

146

—

140

#95

—

146

—

141

C₂×Q₈

—

292

—

142

C₂×Q₈

—

292

—

143

C₂×Q₈

—

292

—

144

C₂×Q₈

—

292

—

145

C₂×Q₈

—

292

—

146

C₂×Q₈

—

146

—

147

C₂²×C₄

—

292

—

148

C₂²×C₄

—

292

—

149

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

150

C₂²×C₄

#11924029

yes

—

151

M₁₆

—

152

M₁₆

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

292

—

159

C₂×C₈

#11924013

yes

—

160

C₂×C₈

#11924015

yes

—

161

C₄⋊C₄

—

292

—

162

C₄⋊C₄

—

292

—

163

C₄⋊C₄

—

292

—

164

C₄⋊C₄

—

146

—

165

C₄²

D₄×D₇₃

yes

characteristic

166

C₄²

#11924027

yes

characteristic

167

C₄²

#11924028

yes

—

168

C₂³

#14556580

yes

characteristic

169

Q₈

—

292

—

170

Q₈

—

292

—

171

Q₈

—

584

—

172

Q₈

—

584

—

173

C₂×C₄

—

174

C₂×C₄

—

292

—

175

C₂×C₄

—

584

—

176

C₂×C₄

—

584

—

177

C₂×C₄

—

584

—

178

C₂×C₄

—

584

—

179

C₂×C₄

#14556562

yes

characteristic

180

C₂×C₄

#14556574

yes

characteristic

181

C₂×C₄

#14556576

yes

—

182

C₂×C₄

#14556586

yes

—

183

C₂×C₄

C₂×#11924013

yes

characteristic

184

C₂×C₄

C₂×#11924015

yes

characteristic

185

C₂×C₄

#14556617

yes

characteristic

186

C₂×C₄

#14556622

yes

characteristic

187

C₈

—

188

C₈

—

189

C₈

—

190

C₈

—

584

—

191

C₂²

—

192

C₂²

#17988513

yes

characteristic

193

C₂²

#17988935

yes

characteristic

194

C₂²

#17988949

yes

characteristic

195

C₄

—

196

C₄

—

197

C₄

—

584

—

198

C₄

—

1168

—

199

C₄

#17988779

yes

characteristic

200

C₄

#17988789

yes

characteristic

201

C₄

#17988900

yes

characteristic

202

C₄

#17988905

yes

characteristic

203

C₂

—

204

C₂

—

205

C₂

#22611335

yes

characteristic

206

C₂

#22611396

yes

—

	subgroup	G/H
center	C₂²	#17988513
commutator, cosocle	C₂×C₄²×C₇₃	C₂³
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1849	C₂
socle	C₂²×C₇₃	#397
prime radical	C₂³×C₇₃	#124
hypercenter	#1849	D₇₃
hypocenter	C₇₃	#10632
LCS	G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1849

subgroup

G/H

center

C₂²

#17988513

commutator, cosocle

C₂×C₄²×C₇₃

C₂³

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1849

C₂

C₂²×C₇₃

C₂³×C₇₃

D₇₃

C₇₃

G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1849

N	H ≅ G/N	type
109. C₇₃	#10632	semidirect

H ≅ G/N

type

109. C₇₃

#10632

semidirect

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

subgroup	index
#22610751	2
#22608604	2
#22607791	2
#22607776	2
C₇₃×#1849	2
#10632	73

subgroup

index

C₇₃×#1849

Subgroups — #32990778

Conjugacy classes of subgroups

Complements

Maximal subgroups