Group #32985649 — subgroups

Subgroups — #32985649

Order 18688.

466 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 127 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×#17988967	C₂	1	1	yes	maximal, characteristic
2	#22610919	C₂	1	2	yes	maximal
3	#22608613	C₂	1	2	yes	maximal
4	#22608220	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1566	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₉₂	C₄	1	4	yes	—
7	C₂×#14556591	C₂²	1	2	yes	—
8	#17988967	—	2	1	no	—
9	#17988967	—	2	1	no	—
10	#17988967	—	2	2	no	—
11	#17988640	C₂²	1	1	yes	characteristic
12	C₄×#11924012	C₂²	1	1	yes	characteristic
13	#17988355	—	2	4	no	—
14	#17988335	—	2	4	no	—
15	#17988309	—	2	1	no	—
16	#17988308	—	2	1	no	—
17	#423×C₇₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
19	#334×C₇₃	—	2	2	no	—
20	C₂²×Dic₁₄₆	C₂×C₄	1	4	yes	—
21	C₂×Dic₂₉₂	—	2	4	no	—
22	C₂×Dic₂₉₂	—	2	4	no	—
23	C₂×Dic₂₉₂	—	4	8	no	—
24	C₂×#11924012	D₄	1	2	yes	—
25	#14556591	—	2	2	no	—
26	#14556591	—	2	2	no	—
27	#14556591	—	2	2	no	—
28	#14556591	—	2	2	no	—
29	C₂×C₄⋊Dic₇₃	—	2	2	no	—
30	C₂×C₄⋊Dic₇₃	—	2	2	no	—
31	C₄⋊#10596831	D₄	1	2	yes	—
32	C₂²⋊#10596831	—	2	1	no	—
33	#14556502	—	2	1	no	—
34	C₄×#10596831	—	2	1	no	—
35	C₄×#10596831	—	2	1	no	—
36	C₂×M₁₆×C₇₃	—	2	2	no	—
37	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₇₃	—	2	2	no	—
40	C₄⋊C₈×C₇₃	—	2	2	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	C₂×Dic₁₄₆	—	2	4	no	—
45	C₂×Dic₁₄₆	—	2	4	no	—
46	C₂×Dic₁₄₆	—	4	8	no	—
47	C₂²×Dic₇₃	—	2	4	no	—
48	Dic₂₉₂	—	4	8	no	—
49	Dic₂₉₂	—	4	8	no	—
50	#11924012	—	2	4	no	—
51	C₂×#10596831	—	2	2	no	—
52	C₂×#10596831	—	2	2	no	—
53	C₄⋊Dic₇₃	—	4	2	no	—
54	C₄⋊Dic₇₃	—	4	2	no	—
55	C₄⋊Dic₇₃	—	4	2	no	—
56	C₄⋊Dic₇₃	—	4	2	no	—
57	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
60	M₁₆×C₇₃	—	4	4	no	—
61	C₂×C₈×C₇₃	—	2	4	no	—
62	C₂×C₈×C₇₃	—	4	2	no	—
63	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₇₃	#41	1	1	yes	characteristic
69	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₇₃	—	2	2	no	—
71	C₂×C₄×C₇₃	—	2	2	no	—
72	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
74	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
75	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
76	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
77	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
78	C₂×C₄×C₇₃	#137	1	4	yes	—
79	C₂×Dic₇₃	—	4	4	no	—
80	C₂×Dic₇₃	—	8	4	no	—
81	Dic₁₄₆	—	4	8	no	—
82	Dic₁₄₆	—	8	4	no	—
83	C₈×C₇₃	—	2	2	no	—
84	C₈×C₇₃	—	8	2	no	—
85	C₈×C₇₃	#100	1	4	yes	—
86	#10596831	—	4	2	no	—
87	#10596831	—	4	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	#441	1	2	yes	—
92	C₄×C₇₃	—	2	1	no	—
93	C₄×C₇₃	—	2	1	no	—
94	C₄×C₇₃	C₂×#100	1	2	yes	—
95	C₄×C₇₃	#470	1	2	yes	—
96	C₄×C₇₃	#473	1	2	yes	—
97	C₄×C₇₃	#497	1	2	yes	—
98	Dic₇₃	—	8	4	no	—
99	#9807	—	73	1	no	maximal, Sylow
100	C₂×C₇₃	—	2	2	no	—
101	C₂×C₇₃	#1965	1	1	yes	characteristic
102	C₂×C₇₃	#1971	1	1	yes	characteristic
103	C₂×C₇₃	#1979	1	1	yes	characteristic
104	C₂×#494	—	73	1	no	—
105	#1997	—	73	2	no	—
106	#1566	D₇₃	1	1	yes	characteristic
107	#1548	—	73	2	no	—
108	#1451	—	73	1	no	—
109	C₇₃	#9807	1	1	yes	characteristic, complemented, Sylow
110	C₂²×Q₁₆	—	73	4	no	—
111	C₂×C₄⋊Q₈	—	73	2	no	—
112	#494	—	146	1	no	—
113	#494	—	146	1	no	—
114	#494	—	146	2	no	—
115	#431	—	73	1	no	—
116	#423	C₂×D₇₃	1	2	yes	—
117	C₄×M₁₆	—	73	1	no	—
118	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
119	#343	—	146	4	no	—
120	#334	—	2	2	no	—
121	#333	—	146	4	no	—
122	#322	—	146	1	no	—
123	#322	—	146	1	no	—
124	C₂²×Q₈	—	73	4	no	—
125	C₂×Q₁₆	—	292	8	no	—
126	C₂×Q₁₆	—	146	4	no	—
127	C₂×Q₁₆	—	146	4	no	—
128	C₂×M₁₆	—	2	2	no	—
129	C₂×M₁₆	—	73	2	no	—
130	C₂²×C₈	C₄×D₇₃	1	2	yes	—
131	C₄⋊Q₈	—	146	2	no	—
132	C₄⋊Q₈	—	146	2	no	—
133	C₄⋊Q₈	—	146	2	no	—
134	C₄⋊Q₈	—	146	2	no	—
135	C₂×C₄⋊C₄	—	146	2	no	—
136	C₂×C₄⋊C₄	—	146	2	no	—
137	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
138	C₄⋊C₈	—	2	2	no	—
139	C₄⋊C₈	—	2	2	no	—
140	C₄⋊C₈	—	73	2	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	D₂₉₂	1	1	yes	characteristic
146	C₄×C₈	D₂₉₂	1	1	yes	characteristic
147	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
148	C₂×Q₈	—	292	8	no	—
149	C₂×Q₈	—	146	4	no	—
150	C₂×Q₈	—	146	4	no	—
151	C₂²×C₄	—	146	4	no	—
152	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
153	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
154	C₂²×C₄	#11924029	1	1	yes	characteristic
155	Q₁₆	—	292	8	no	—
156	Q₁₆	—	292	8	no	—
157	M₁₆	—	4	4	no	—
158	M₁₆	—	146	4	no	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	—	4	2	no	—
161	C₂×C₈	—	146	2	no	—
162	C₂×C₈	—	146	2	no	—
163	C₂×C₈	#11924005	1	2	yes	—
164	C₂×C₈	#11924005	1	2	yes	—
165	C₄⋊C₄	—	292	2	no	—
166	C₄⋊C₄	—	292	2	no	—
167	C₄⋊C₄	—	292	2	no	—
168	C₄⋊C₄	—	292	2	no	—
169	C₄²	D₄×D₇₃	1	1	yes	characteristic
170	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
171	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
172	C₄²	#11924029	1	1	yes	characteristic
173	C₂³	#14556553	1	1	yes	characteristic
174	Q₈	—	292	8	no	—
175	Q₈	—	584	4	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	292	4	no	—
179	C₂×C₄	—	584	4	no	—
180	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
181	C₂×C₄	#14556553	1	1	yes	characteristic
182	C₂×C₄	#14556566	1	1	yes	characteristic
183	C₂×C₄	#14556566	1	1	yes	characteristic
184	C₂×C₄	#14556580	1	1	yes	characteristic
185	C₂×C₄	#14556587	1	1	yes	characteristic
186	C₂×C₄	#14556620	1	4	yes	—
187	C₈	—	2	2	no	—
188	C₈	—	8	2	no	—
189	C₈	—	292	2	no	—
190	C₈	—	292	2	no	—
191	C₈	#14556507	1	4	yes	—
192	C₂²	—	2	1	no	—
193	C₂²	—	2	1	no	—
194	C₂²	#17988488	1	1	yes	characteristic
195	C₂²	#17988693	1	2	yes	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	584	4	no	—
199	C₄	C₂×#14556507	1	2	yes	—
200	C₄	#17988846	1	2	yes	—
201	C₄	#17988863	1	2	yes	—
202	C₄	#17988978	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	#22610770	1	1	yes	characteristic
205	C₂	#22610800	1	1	yes	characteristic
206	C₂	#22610844	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988967

C₂

yes

maximal, characteristic

#22610919

C₂

yes

maximal

#22608613

C₂

yes

maximal

#22608220

C₂

yes

maximal, characteristic

C₇₃×#1566

C₂

yes

maximal, characteristic

C₂²×Dic₂₉₂

C₄

yes

—

C₂×#14556591

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄×#11924012

C₂²

yes

—

—

—

—

#423×C₇₃

C₂²

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#334×C₇₃

—

C₂²×Dic₁₄₆

C₂×C₄

yes

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×#11924012

D₄

yes

—

—

—

—

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

—

C₄⋊#10596831

D₄

yes

—

—

—

—

—

C₂×M₁₆×C₇₃

—

C₂²×C₈×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄⋊C₈×C₇₃

—

C₄⋊C₈×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

—

Dic₂₉₂

—

Dic₂₉₂

—

#11924012

—

C₂×#10596831

—

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

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

#100

yes

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#390

yes

characteristic

C₂²×C₇₃

#441

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×#100

yes

—

C₄×C₇₃

#470

yes

—

C₄×C₇₃

#473

yes

—

C₄×C₇₃

#497

yes

—

Dic₇₃

—

#9807

—

maximal, Sylow

100

C₂×C₇₃

—

101

C₂×C₇₃

#1965

yes

characteristic

102

C₂×C₇₃

#1971

yes

characteristic

103

C₂×C₇₃

#1979

yes

characteristic

104

C₂×#494

—

105

#1997

—

106

#1566

D₇₃

yes

characteristic

107

#1548

—

108

#1451

—

109

C₇₃

#9807

yes

characteristic, complemented, Sylow

110

C₂²×Q₁₆

—

111

C₂×C₄⋊Q₈

—

112

#494

—

146

—

113

#494

—

146

—

114

#494

—

146

—

115

#431

—

116

#423

C₂×D₇₃

yes

—

117

C₄×M₁₆

—

118

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

119

#343

—

146

—

120

#334

—

121

#333

—

146

—

122

#322

—

146

—

123

#322

—

146

—

124

C₂²×Q₈

—

125

C₂×Q₁₆

—

292

—

126

C₂×Q₁₆

—

146

—

127

C₂×Q₁₆

—

146

—

128

C₂×M₁₆

—

129

C₂×M₁₆

—

130

C₂²×C₈

C₄×D₇₃

yes

—

131

C₄⋊Q₈

—

146

—

132

C₄⋊Q₈

—

146

—

133

C₄⋊Q₈

—

146

—

134

C₄⋊Q₈

—

146

—

135

C₂×C₄⋊C₄

—

146

—

136

C₂×C₄⋊C₄

—

146

—

137

C₂×C₄²

C₂²×D₇₃

yes

characteristic

138

C₄⋊C₈

—

139

C₄⋊C₈

—

140

C₄⋊C₈

—

141

C₂²⋊C₈

—

146

—

142

#97

—

146

—

143

C₄×C₈

—

146

—

144

C₄×C₈

—

146

—

145

C₄×C₈

D₂₉₂

yes

characteristic

146

C₄×C₈

D₂₉₂

yes

characteristic

147

C₄×C₈

C₂²⋊D₇₃

yes

—

148

C₂×Q₈

—

292

—

149

C₂×Q₈

—

146

—

150

C₂×Q₈

—

146

—

151

C₂²×C₄

—

146

—

152

C₂²×C₄

C₂×C₄×D₇₃

yes

characteristic

153

C₂²×C₄

D₄×D₇₃

yes

characteristic

154

C₂²×C₄

#11924029

yes

characteristic

155

Q₁₆

—

292

—

156

Q₁₆

—

292

—

157

M₁₆

—

158

M₁₆

—

146

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

146

—

162

C₂×C₈

—

146

—

163

C₂×C₈

#11924005

yes

—

164

C₂×C₈

#11924005

yes

—

165

C₄⋊C₄

—

292

—

166

C₄⋊C₄

—

292

—

167

C₄⋊C₄

—

292

—

168

C₄⋊C₄

—

292

—

169

C₄²

D₄×D₇₃

yes

characteristic

170

C₄²

C₂×C₂²⋊D₇₃

yes

characteristic

171

C₄²

C₂×D₂₉₂

yes

characteristic

172

C₄²

#11924029

yes

characteristic

173

C₂³

#14556553

yes

characteristic

174

Q₈

—

292

—

175

Q₈

—

584

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

292

—

179

C₂×C₄

—

584

—

180

C₂×C₄

C₂×#11924005

yes

characteristic

181

C₂×C₄

#14556553

yes

characteristic

182

C₂×C₄

#14556566

yes

characteristic

183

C₂×C₄

#14556566

yes

characteristic

184

C₂×C₄

#14556580

yes

characteristic

185

C₂×C₄

#14556587

yes

characteristic

186

C₂×C₄

#14556620

yes

—

187

C₈

—

188

C₈

—

189

C₈

—

292

—

190

C₈

—

292

—

191

C₈

#14556507

yes

—

192

C₂²

—

193

C₂²

—

194

C₂²

#17988488

yes

characteristic

195

C₂²

#17988693

yes

—

196

C₄

—

197

C₄

—

198

C₄

—

584

—

199

C₄

C₂×#14556507

yes

—

200

C₄

#17988846

yes

—

201

C₄

#17988863

yes

—

202

C₄

#17988978

yes

—

203

C₂

—

204

C₂

#22610770

yes

characteristic

205

C₂

#22610800

yes

characteristic

206

C₂

#22610844

yes

characteristic

	subgroup	G/H
center	C₂²	#17988488
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1566	C₂
socle	C₂²×C₇₃	#390
prime radical	C₂³×C₇₃	C₄×D₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1566	D₇₃
hypocenter	C₇₃	#9807
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1566

subgroup

G/H

center

C₂²

#17988488

commutator

C₂²×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1566

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 ◁ #1566

N	H ≅ G/N	type
109. C₇₃	#9807	semidirect

H ≅ G/N

type

109. C₇₃

#9807

semidirect

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

subgroup	index
C₂×#17988967	2
#22610919	2
#22608613	2
#22608220	2
C₇₃×#1566	2
#9807	73

subgroup

index

C₇₃×#1566

Subgroups — #32985649

Conjugacy classes of subgroups

Complements

Maximal subgroups