Group #32986012 — subgroups

Subgroups — #32986012

Order 18688.

434 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 159 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	#22610048	C₂	1	1	yes	maximal, characteristic
2	#22609677	C₂	1	1	yes	maximal, characteristic
3	#22607772	C₂	1	4	yes	maximal
4	C₄×#125×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
6	C₂×C₄×#10596831	C₂²	1	2	yes	—
7	#17988342	—	2	2	no	—
8	#17988342	—	2	2	no	—
9	C₂×#125×C₇₃	C₄	1	1	yes	characteristic
10	#389×C₇₃	C₄	1	2	yes	—
11	#387×C₇₃	—	2	2	no	—
12	#382×C₇₃	C₄	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₇₃	—	2	2	no	—
14	C₄×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
15	C₄³×C₇₃	C₂²	1	1	yes	characteristic
16	C₂²×#10596831	—	2	2	no	—
17	C₂²×#10596831	—	2	2	no	—
18	C₄⋊#10596831	—	2	4	no	—
19	C₄×#10596831	—	2	2	no	—
20	C₄×#10596831	Q₈	1	4	yes	—
21	#125×C₇₃	—	2	4	no	—
22	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
23	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
24	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
25	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₇₃	—	2	2	no	—
27	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₇₃	D₄	1	2	yes	—
30	C₂×C₄²×C₇₃	D₄	1	2	yes	—
31	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
32	#95×C₇₃	—	2	2	no	—
33	#95×C₇₃	—	2	2	no	—
34	#95×C₇₃	—	2	2	no	—
35	C₂×#10596831	—	2	4	no	—
36	C₂×#10596831	—	4	4	no	—
37	C₂×#10596831	—	4	2	no	—
38	C₂×#10596831	—	4	4	no	—
39	C₂²×C₄×C₇₃	—	2	2	no	—
40	C₂²×C₄×C₇₃	—	2	2	no	—
41	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
46	C₄⋊C₄×C₇₃	—	2	8	no	—
47	C₄⋊C₄×C₇₃	—	2	2	no	—
48	C₄⋊C₄×C₇₃	—	2	4	no	—
49	C₄⋊C₄×C₇₃	—	4	4	no	—
50	C₄⋊C₄×C₇₃	M₁₆	1	4	yes	—
51	C₄²×C₇₃	—	2	2	no	—
52	C₄²×C₇₃	—	2	2	no	—
53	C₄²×C₇₃	—	2	4	no	—
54	C₄²×C₇₃	—	4	2	no	—
55	C₄²×C₇₃	—	4	4	no	—
56	C₄²×C₇₃	C₂×Q₈	1	2	yes	—
57	C₄²×C₇₃	#41	1	2	yes	—
58	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₇₃	—	2	2	no	—
60	C₂×C₄×C₇₃	—	2	2	no	—
61	C₂×C₄×C₇₃	—	2	4	no	—
62	C₂×C₄×C₇₃	—	2	2	no	—
63	C₂×C₄×C₇₃	—	2	2	no	—
64	C₂×C₄×C₇₃	—	2	2	no	—
65	C₂×C₄×C₇₃	—	2	2	no	—
66	C₂×C₄×C₇₃	—	2	2	no	—
67	C₂×C₄×C₇₃	—	4	2	no	—
68	C₂×C₄×C₇₃	—	4	2	no	—
69	C₂×C₄×C₇₃	—	4	2	no	—
70	C₂×C₄×C₇₃	—	4	2	no	—
71	C₂×C₄×C₇₃	—	4	2	no	—
72	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
73	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
74	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
75	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
77	C₂×C₄×C₇₃	#131	1	2	yes	—
78	C₂×C₄×C₇₃	#135	1	4	yes	—
79	#10596831	—	4	4	no	—
80	#10596831	—	8	2	no	—
81	C₂²×C₇₃	#391	1	1	yes	characteristic
82	C₂²×C₇₃	#407	1	1	yes	characteristic
83	C₂²×C₇₃	#408	1	1	yes	characteristic
84	C₂²×C₇₃	#416	1	2	yes	—
85	C₂²×C₇₃	#421	1	2	yes	—
86	C₄×C₇₃	—	2	2	no	—
87	C₄×C₇₃	—	4	2	no	—
88	C₄×C₇₃	—	4	2	no	—
89	C₄×C₇₃	—	4	4	no	—
90	C₄×C₇₃	—	8	2	no	—
91	C₄×C₇₃	#446	1	4	yes	—
92	C₄×C₇₃	#488	1	2	yes	—
93	C₄×C₇₃	#499	1	2	yes	—
94	#9875	—	73	1	no	maximal, Sylow
95	C₂×C₇₃	#1482	1	2	yes	—
96	C₂×C₇₃	#1488	1	2	yes	—
97	C₂×C₇₃	#1985	1	1	yes	characteristic
98	C₂×C₇₃	#1991	1	1	yes	characteristic
99	C₂×C₇₃	#1995	1	1	yes	characteristic
100	C₄×#125	D₇₃	1	1	yes	characteristic
101	#1831	—	73	1	no	—
102	#1744	—	73	1	no	—
103	#1279	—	73	4	no	—
104	C₇₃	#9875	1	1	yes	characteristic, complemented, Sylow
105	C₂×#125	Dic₇₃	1	1	yes	characteristic
106	C₂×C₄⋊C₈	—	73	2	no	—
107	C₂×C₄×C₈	—	73	2	no	—
108	#389	Dic₇₃	1	2	yes	—
109	#387	—	2	2	no	—
110	#382	Dic₇₃	1	1	yes	characteristic
111	C₄×C₄⋊C₄	—	2	2	no	—
112	C₄×C₄⋊C₄	C₂×D₇₃	1	2	yes	—
113	C₄³	C₂×D₇₃	1	1	yes	characteristic
114	#336	—	146	2	no	—
115	#336	—	146	2	no	—
116	C₂²×C₈	—	146	2	no	—
117	C₂²×C₈	—	146	2	no	—
118	#125	—	2	4	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	C₂×Dic₇₃	1	2	yes	—
122	C₂×C₄⋊C₄	C₂×Dic₇₃	1	2	yes	—
123	C₂×C₄²	—	2	2	no	—
124	C₂×C₄²	C₂×Dic₇₃	1	1	yes	characteristic
125	C₂×C₄²	C₂×Dic₇₃	1	1	yes	characteristic
126	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
127	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
128	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
129	C₄⋊C₈	—	146	4	no	—
130	C₄×C₈	—	73	4	no	—
131	C₄×C₈	—	146	2	no	—
132	#95	—	2	2	no	—
133	#95	—	2	2	no	—
134	#95	—	2	2	no	—
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	C₂²⋊Dic₇₃	1	2	yes	—
138	C₂²×C₄	C₂²⋊Dic₇₃	1	2	yes	—
139	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
140	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
141	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
142	C₂×C₈	—	292	4	no	—
143	C₂×C₈	—	292	2	no	—
144	C₂×C₈	—	292	4	no	—
145	C₂×C₈	—	146	4	no	—
146	C₄⋊C₄	—	2	8	no	—
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	4	4	no	—
150	C₄⋊C₄	#11924012	1	4	yes	—
151	C₄²	—	2	2	no	—
152	C₄²	—	2	2	no	—
153	C₄²	—	2	4	no	—
154	C₄²	—	4	2	no	—
155	C₄²	—	4	4	no	—
156	C₄²	Q₈×D₇₃	1	2	yes	—
157	C₄²	#11924029	1	2	yes	—
158	C₂³	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	Q₈×Dic₇₃	1	2	yes	—
173	C₂×C₄	#14556567	1	2	yes	—
174	C₂×C₄	#14556580	1	1	yes	characteristic
175	C₂×C₄	#14556589	1	1	yes	characteristic
176	C₂×C₄	C₂×#11924012	1	2	yes	—
177	C₂×C₄	#14556598	1	2	yes	—
178	C₂×C₄	#14556611	1	4	yes	—
179	C₈	—	292	4	no	—
180	C₈	—	584	2	no	—
181	C₂²	#17988494	1	1	yes	characteristic
182	C₂²	#17988551	1	1	yes	characteristic
183	C₂²	#17988554	1	1	yes	characteristic
184	C₂²	#17988583	1	2	yes	—
185	C₂²	#17988604	1	2	yes	—
186	C₄	—	2	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	4	2	no	—
189	C₄	—	4	4	no	—
190	C₄	—	8	2	no	—
191	C₄	#17988718	1	4	yes	—
192	C₄	#17988944	1	2	yes	—
193	C₄	#17988986	1	2	yes	—
194	C₂	#22608339	1	2	yes	—
195	C₂	#22608365	1	2	yes	—
196	C₂	#22610872	1	1	yes	characteristic
197	C₂	#22610897	1	1	yes	characteristic
198	C₂	#22610913	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22610048

C₂

yes

maximal, characteristic

#22609677

C₂

yes

maximal, characteristic

#22607772

C₂

yes

maximal

C₄×#125×C₇₃

C₂

yes

maximal, characteristic

C₂×C₄⋊#10596831

C₂²

yes

—

C₂×C₄×#10596831

C₂²

yes

—

#17988342

—

#17988342

—

C₂×#125×C₇₃

C₄

yes

characteristic

#389×C₇₃

C₄

yes

—

#387×C₇₃

—

#382×C₇₃

C₄

yes

characteristic

C₄×C₄⋊C₄×C₇₃

—

C₄×C₄⋊C₄×C₇₃

C₂²

yes

—

C₄³×C₇₃

C₂²

yes

—

—

—

—

Q₈

yes

—

#125×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

—

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

—

C₂×C₄²×C₇₃

D₄

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

#95×C₇₃

—

#95×C₇₃

—

#95×C₇₃

—

—

—

—

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

M₁₆

yes

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂×Q₈

yes

—

C₄²×C₇₃

#41

yes

—

C₂³×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄×Q₈

yes

—

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₃

#124

yes

characteristic

C₂×C₄×C₇₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

C₂×M₁₆

yes

—

C₂×C₄×C₇₃

#131

yes

—

C₂×C₄×C₇₃

#135

yes

—

#10596831

—

#10596831

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₂²×C₇₃

#421

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#446

yes

—

C₄×C₇₃

#488

yes

—

C₄×C₇₃

#499

yes

—

#9875

—

maximal, Sylow

C₂×C₇₃

#1482

yes

—

C₂×C₇₃

#1488

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

100

D₇₃

yes

characteristic

101

#1831

—

102

#1744

—

103

#1279

—

104

C₇₃

#9875

yes

characteristic, complemented, Sylow

105

C₂×#125

Dic₇₃

yes

characteristic

106

C₂×C₄⋊C₈

—

107

C₂×C₄×C₈

—

108

#389

Dic₇₃

yes

—

109

#387

—

110

#382

Dic₇₃

yes

characteristic

111

C₄×C₄⋊C₄

—

112

C₄×C₄⋊C₄

C₂×D₇₃

yes

—

113

C₄³

C₂×D₇₃

yes

characteristic

114

#336

—

146

—

115

#336

—

146

—

116

C₂²×C₈

—

146

—

117

C₂²×C₈

—

146

—

118

#125

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

C₂×Dic₇₃

yes

—

122

C₂×C₄⋊C₄

C₂×Dic₇₃

yes

—

123

C₂×C₄²

—

124

C₂×C₄²

C₂×Dic₇₃

yes

characteristic

125

C₂×C₄²

C₂×Dic₇₃

yes

characteristic

126

C₂×C₄²

C₂²⋊D₇₃

yes

—

127

C₂×C₄²

C₂²⋊D₇₃

yes

—

128

C₂×C₄²

C₂²×D₇₃

yes

characteristic

129

C₄⋊C₈

—

146

—

130

C₄×C₈

—

131

C₄×C₈

—

146

—

132

#95

—

133

#95

—

134

#95

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

C₂²⋊Dic₇₃

yes

—

138

C₂²×C₄

C₂²⋊Dic₇₃

yes

—

139

C₂²×C₄

C₂²×Dic₇₃

yes

characteristic

140

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

141

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

142

C₂×C₈

—

292

—

143

C₂×C₈

—

292

—

144

C₂×C₈

—

292

—

145

C₂×C₈

—

146

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

#11924012

yes

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

154

C₄²

—

155

C₄²

—

156

C₄²

Q₈×D₇₃

yes

—

157

C₄²

#11924029

yes

—

158

C₂³

C₂×C₂²⋊Dic₇₃

yes

characteristic

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

Q₈×Dic₇₃

yes

—

173

C₂×C₄

#14556567

yes

—

174

C₂×C₄

#14556580

yes

characteristic

175

C₂×C₄

#14556589

yes

characteristic

176

C₂×C₄

C₂×#11924012

yes

—

177

C₂×C₄

#14556598

yes

—

178

C₂×C₄

#14556611

yes

—

179

C₈

—

292

—

180

C₈

—

584

—

181

C₂²

#17988494

yes

characteristic

182

C₂²

#17988551

yes

characteristic

183

C₂²

#17988554

yes

characteristic

184

C₂²

#17988583

yes

—

185

C₂²

#17988604

yes

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

#17988718

yes

—

192

C₄

#17988944

yes

—

193

C₄

#17988986

yes

—

194

C₂

#22608339

yes

—

195

C₂

#22608365

yes

—

196

C₂

#22610872

yes

characteristic

197

C₂

#22610897

yes

characteristic

198

C₂

#22610913

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₃
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₄×#125×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₄×#125	D₇₃
hypocenter	C₇₃	#9875
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

subgroup

G/H

center

C₂²×C₄

C₂×C₂²⋊D₇₃

commutator

C₂²×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₄×#125×C₇₃

C₂

socle, prime radical

C₂³×C₇₃

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

N	H ≅ G/N	type
104. C₇₃	#9875	semidirect

H ≅ G/N

type

104. C₇₃

#9875

semidirect

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

subgroup	index
#22610048	2
#22609677	2
#22607772	2
C₄×#125×C₇₃	2
#9875	73

subgroup

index

C₄×#125×C₇₃

Subgroups — #32986012

Conjugacy classes of subgroups

Complements

Maximal subgroups