Group #32985668 — subgroups

Subgroups — #32985668

Order 18688.

402 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 131 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	C₂×#17988992	C₂	1	1	yes	maximal, characteristic
2	#22610920	C₂	1	2	yes	maximal
3	#22608687	C₂	1	1	yes	maximal, characteristic
4	#22608615	C₂	1	2	yes	maximal
5	C₇₃×#1451	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556591	C₂²	1	2	yes	—
7	#17988992	—	2	2	no	—
8	#17988992	—	2	1	no	—
9	#17988992	—	2	1	no	—
10	C₂×#14556525	C₄	1	4	yes	—
11	#17988611	C₂²	1	2	yes	—
12	#17988355	—	2	4	no	—
13	#17988338	—	2	2	no	—
14	#17988334	—	2	4	no	—
15	#431×C₇₃	C₂²	1	1	yes	characteristic
16	C₄×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
18	#322×C₇₃	—	2	1	no	—
19	#322×C₇₃	—	2	1	no	—
20	C₂²×Dic₁₄₆	—	2	2	no	—
21	C₂×#11924012	—	2	2	no	—
22	#14556591	—	2	2	no	—
23	#14556591	—	2	2	no	—
24	#14556591	—	2	2	no	—
25	#14556591	—	2	2	no	—
26	C₂×C₄⋊Dic₇₃	—	2	2	no	—
27	C₂×C₄⋊Dic₇₃	C₂×C₄	1	4	yes	—
28	#14556525	—	2	4	no	—
29	#14556525	—	2	4	no	—
30	C₄⋊#10596831	—	2	2	no	—
31	C₄⋊#10596831	—	2	2	no	—
32	C₂×M₁₆×C₇₃	D₄	1	2	yes	—
33	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₇₃	D₄	1	2	yes	—
36	C₂²⋊C₈×C₇₃	—	2	1	no	—
37	#97×C₇₃	—	2	1	no	—
38	C₄×C₈×C₇₃	—	2	1	no	—
39	C₄×C₈×C₇₃	—	2	1	no	—
40	C₄×C₈×C₇₃	D₄	1	2	yes	—
41	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
42	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
43	C₂×Dic₁₄₆	—	4	2	no	—
44	C₂×Dic₁₄₆	—	4	2	no	—
45	C₂×Dic₁₄₆	—	8	4	no	—
46	C₂²×Dic₇₃	—	2	4	no	—
47	#11924012	—	4	4	no	—
48	C₂×#10596831	—	4	2	no	—
49	C₄⋊Dic₇₃	—	2	4	no	—
50	C₄⋊Dic₇₃	—	2	4	no	—
51	C₄⋊Dic₇₃	—	4	2	no	—
52	C₄⋊Dic₇₃	—	4	2	no	—
53	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
56	M₁₆×C₇₃	—	2	4	no	—
57	C₂×C₈×C₇₃	—	2	4	no	—
58	C₂×C₈×C₇₃	—	2	2	no	—
59	C₂×C₈×C₇₃	—	2	2	no	—
60	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
62	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₇₃	#41	1	1	yes	characteristic
66	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₇₃	—	2	2	no	—
68	C₂×C₄×C₇₃	—	2	2	no	—
69	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
72	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
73	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
74	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
75	C₂×C₄×C₇₃	#137	1	4	yes	—
76	C₂×Dic₇₃	—	4	4	no	—
77	C₂×Dic₇₃	—	8	4	no	—
78	Dic₁₄₆	—	8	4	no	—
79	Dic₁₄₆	—	8	4	no	—
80	C₈×C₇₃	—	2	2	no	—
81	C₈×C₇₃	—	4	2	no	—
82	C₈×C₇₃	—	4	2	no	—
83	C₈×C₇₃	#101	1	4	yes	—
84	#10596831	—	8	2	no	—
85	C₂²×C₇₃	—	2	1	no	—
86	C₂²×C₇₃	—	2	1	no	—
87	C₂²×C₇₃	#390	1	1	yes	characteristic
88	C₂²×C₇₃	#441	1	2	yes	—
89	C₄×C₇₃	—	2	1	no	—
90	C₄×C₇₃	—	2	1	no	—
91	C₄×C₇₃	C₂×#101	1	2	yes	—
92	C₄×C₇₃	#470	1	2	yes	—
93	C₄×C₇₃	#472	1	2	yes	—
94	C₄×C₇₃	#497	1	2	yes	—
95	Dic₇₃	—	8	4	no	—
96	#9810	—	73	1	no	maximal, Sylow
97	C₂×C₇₃	—	2	2	no	—
98	C₂×C₇₃	#1966	1	1	yes	characteristic
99	C₂×C₇₃	#1971	1	1	yes	characteristic
100	C₂×C₇₃	#1978	1	1	yes	characteristic
101	C₂×#500	—	73	1	no	—
102	#1997	—	73	2	no	—
103	#1566	—	73	1	no	—
104	#1549	—	73	2	no	—
105	#1451	D₇₃	1	1	yes	characteristic
106	C₇₃	#9810	1	1	yes	characteristic, complemented, Sylow
107	C₂×C₄⋊Q₈	—	73	2	no	—
108	#500	—	146	2	no	—
109	#500	—	146	1	no	—
110	#500	—	146	1	no	—
111	#431	C₂×D₇₃	1	1	yes	characteristic
112	C₂×#107	—	73	4	no	—
113	#423	—	73	2	no	—
114	C₄×M₁₆	C₂×D₇₃	1	1	yes	characteristic
115	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
116	#343	—	146	4	no	—
117	#334	—	146	2	no	—
118	#332	—	146	4	no	—
119	#322	—	2	1	no	—
120	#322	—	2	1	no	—
121	C₂²×Q₈	—	146	2	no	—
122	C₂×M₁₆	—	146	2	no	—
123	C₂×M₁₆	D₂₉₂	1	2	yes	—
124	C₂²×C₈	C₄×D₇₃	1	2	yes	—
125	C₄⋊Q₈	—	146	2	no	—
126	C₄⋊Q₈	—	146	2	no	—
127	C₄⋊Q₈	—	146	2	no	—
128	C₄⋊Q₈	—	146	2	no	—
129	C₂×C₄⋊C₄	—	73	4	no	—
130	C₂×C₄⋊C₄	—	146	2	no	—
131	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
132	#107	—	146	4	no	—
133	#107	—	146	4	no	—
134	C₄⋊C₈	—	146	2	no	—
135	C₄⋊C₈	—	146	2	no	—
136	C₄⋊C₈	D₂₉₂	1	2	yes	—
137	C₂²⋊C₈	—	2	1	no	—
138	#97	—	2	1	no	—
139	C₄×C₈	—	2	1	no	—
140	C₄×C₈	—	2	1	no	—
141	C₄×C₈	D₂₉₂	1	2	yes	—
142	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
143	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
144	C₂×Q₈	—	292	2	no	—
145	C₂×Q₈	—	292	2	no	—
146	C₂×Q₈	—	584	4	no	—
147	C₂²×C₄	—	146	4	no	—
148	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
149	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
150	C₂²×C₄	#11924028	1	1	yes	characteristic
151	M₁₆	—	2	4	no	—
152	M₁₆	—	292	4	no	—
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	292	2	no	—
157	C₂×C₈	#11924005	1	2	yes	—
158	C₂×C₈	#11924005	1	2	yes	—
159	C₄⋊C₄	—	292	2	no	—
160	C₄⋊C₄	—	292	2	no	—
161	C₄⋊C₄	—	146	4	no	—
162	C₄⋊C₄	—	146	4	no	—
163	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
164	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
165	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
166	C₄²	#11924028	1	1	yes	characteristic
167	C₂³	C₄×D₂₉₂	1	1	yes	characteristic
168	Q₈	—	584	4	no	—
169	Q₈	—	584	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	292	4	no	—
173	C₂×C₄	—	584	4	no	—
174	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
175	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
176	C₂×C₄	#14556565	1	1	yes	characteristic
177	C₂×C₄	#14556565	1	1	yes	characteristic
178	C₂×C₄	#14556579	1	1	yes	characteristic
179	C₂×C₄	#14556588	1	1	yes	characteristic
180	C₂×C₄	#14556624	1	4	yes	—
181	C₈	—	2	2	no	—
182	C₈	—	4	2	no	—
183	C₈	—	4	2	no	—
184	C₈	—	584	2	no	—
185	C₈	#14556509	1	4	yes	—
186	C₂²	—	2	1	no	—
187	C₂²	—	2	1	no	—
188	C₂²	#17988489	1	1	yes	characteristic
189	C₂²	#17988695	1	2	yes	—
190	C₄	—	2	1	no	—
191	C₄	—	2	1	no	—
192	C₄	—	584	4	no	—
193	C₄	C₂×#14556509	1	2	yes	—
194	C₄	#17988845	1	2	yes	—
195	C₄	#17988857	1	2	yes	—
196	C₄	#17988981	1	2	yes	—
197	C₂	—	2	2	no	—
198	C₂	#22610775	1	1	yes	characteristic
199	C₂	#22610802	1	1	yes	characteristic
200	C₂	#22610841	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988992

C₂

yes

maximal, characteristic

#22610920

C₂

yes

maximal

#22608687

C₂

yes

maximal, characteristic

#22608615

C₂

yes

maximal

C₇₃×#1451

C₂

yes

maximal, characteristic

C₂×#14556591

C₂²

yes

—

—

—

—

C₄

yes

—

#17988611

C₂²

yes

—

#17988355

—

#17988338

—

#17988334

—

#431×C₇₃

C₂²

yes

characteristic

C₄×M₁₆×C₇₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#322×C₇₃

—

#322×C₇₃

—

C₂²×Dic₁₄₆

—

—

—

—

—

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

C₂×C₄

yes

—

—

—

—

—

C₂×M₁₆×C₇₃

D₄

yes

—

C₂²×C₈×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄⋊C₈×C₇₃

D₄

yes

—

C₂²⋊C₈×C₇₃

—

#97×C₇₃

—

C₄×C₈×C₇₃

—

C₄×C₈×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

—

#11924012

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

#41

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

—

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

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

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

#101

yes

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#390

yes

characteristic

C₂²×C₇₃

#441

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×#101

yes

—

C₄×C₇₃

#470

yes

—

C₄×C₇₃

#472

yes

—

C₄×C₇₃

#497

yes

—

Dic₇₃

—

#9810

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

100

C₂×C₇₃

#1978

yes

characteristic

101

C₂×#500

—

102

#1997

—

103

#1566

—

104

#1549

—

105

#1451

D₇₃

yes

characteristic

106

C₇₃

#9810

yes

characteristic, complemented, Sylow

107

C₂×C₄⋊Q₈

—

108

#500

—

146

—

109

#500

—

146

—

110

#500

—

146

—

111

#431

C₂×D₇₃

yes

characteristic

112

C₂×#107

—

113

#423

—

114

C₄×M₁₆

C₂×D₇₃

yes

characteristic

115

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

116

#343

—

146

—

117

#334

—

146

—

118

#332

—

146

—

119

#322

—

120

#322

—

121

C₂²×Q₈

—

146

—

122

C₂×M₁₆

—

146

—

123

C₂×M₁₆

D₂₉₂

yes

—

124

C₂²×C₈

C₄×D₇₃

yes

—

125

C₄⋊Q₈

—

146

—

126

C₄⋊Q₈

—

146

—

127

C₄⋊Q₈

—

146

—

128

C₄⋊Q₈

—

146

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

146

—

131

C₂×C₄²

C₂²×D₇₃

yes

characteristic

132

#107

—

146

—

133

#107

—

146

—

134

C₄⋊C₈

—

146

—

135

C₄⋊C₈

—

146

—

136

C₄⋊C₈

D₂₉₂

yes

—

137

C₂²⋊C₈

—

138

#97

—

139

C₄×C₈

—

140

C₄×C₈

—

141

C₄×C₈

D₂₉₂

yes

—

142

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

143

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

144

C₂×Q₈

—

292

—

145

C₂×Q₈

—

292

—

146

C₂×Q₈

—

584

—

147

C₂²×C₄

—

146

—

148

C₂²×C₄

C₂×C₄×D₇₃

yes

characteristic

149

C₂²×C₄

C₂×D₂₉₂

yes

characteristic

150

C₂²×C₄

#11924028

yes

characteristic

151

M₁₆

—

152

M₁₆

—

292

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

292

—

157

C₂×C₈

#11924005

yes

—

158

C₂×C₈

#11924005

yes

—

159

C₄⋊C₄

—

292

—

160

C₄⋊C₄

—

292

—

161

C₄⋊C₄

—

146

—

162

C₄⋊C₄

—

146

—

163

C₄²

C₂×C₂²⋊D₇₃

yes

characteristic

164

C₄²

C₂×D₂₉₂

yes

characteristic

165

C₄²

C₂×D₂₉₂

yes

characteristic

166

C₄²

#11924028

yes

characteristic

167

C₂³

C₄×D₂₉₂

yes

characteristic

168

Q₈

—

584

—

169

Q₈

—

584

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

292

—

173

C₂×C₄

—

584

—

174

C₂×C₄

C₂×#11924005

yes

characteristic

175

C₂×C₄

C₄×D₂₉₂

yes

characteristic

176

C₂×C₄

#14556565

yes

characteristic

177

C₂×C₄

#14556565

yes

characteristic

178

C₂×C₄

#14556579

yes

characteristic

179

C₂×C₄

#14556588

yes

characteristic

180

C₂×C₄

#14556624

yes

—

181

C₈

—

182

C₈

—

183

C₈

—

184

C₈

—

584

—

185

C₈

#14556509

yes

—

186

C₂²

—

187

C₂²

—

188

C₂²

#17988489

yes

characteristic

189

C₂²

#17988695

yes

—

190

C₄

—

191

C₄

—

192

C₄

—

584

—

193

C₄

C₂×#14556509

yes

—

194

C₄

#17988845

yes

—

195

C₄

#17988857

yes

—

196

C₄

#17988981

yes

—

197

C₂

—

198

C₂

#22610775

yes

characteristic

199

C₂

#22610802

yes

characteristic

200

C₂

#22610841

yes

characteristic

	subgroup	G/H
center	C₂²	#17988489
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1451	C₂
socle	C₂²×C₇₃	#390
prime radical	C₂³×C₇₃	C₄×D₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1451	D₇₃
hypocenter	C₇₃	#9810
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1451

subgroup

G/H

center

C₂²

#17988489

commutator

C₂²×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1451

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 ◁ #1451

N	H ≅ G/N	type
106. C₇₃	#9810	semidirect

H ≅ G/N

type

106. C₇₃

#9810

semidirect

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

subgroup	index
C₂×#17988992	2
#22610920	2
#22608687	2
#22608615	2
C₇₃×#1451	2
#9810	73

subgroup

index

C₇₃×#1451

Subgroups — #32985668

Conjugacy classes of subgroups

Complements

Maximal subgroups