Group #22620887 — subgroups

Subgroups — #22620887

Order 9344.

394 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 125 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	#17989144	C₂	1	1	yes	maximal, characteristic, complemented
2	#17988966	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988959	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988937	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988933	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988806	C₂	1	4	yes	maximal, complemented
7	#17988749	C₂	1	4	yes	maximal, complemented
8	#17988640	C₂	1	1	yes	maximal, characteristic, complemented
9	#532×C₇₃	C₂	1	1	yes	maximal, characteristic
10	C₂×#11924018	C₂²	1	4	yes	—
11	C₂×#11924016	C₂²	1	4	yes	—
12	#14556591	C₂²	1	1	yes	characteristic
13	#14556591	C₂²	1	1	yes	characteristic
14	#14556583	—	2	1	no	—
15	#14556570	—	2	1	no	—
16	#14556570	C₂²	1	2	yes	—
17	C₄×Dic₁₄₆	—	2	1	no	—
18	C₄×Dic₁₄₆	C₂²	1	2	yes	—
19	#14556550	—	2	1	no	—
20	C₄⋊#10596831	C₂²	1	2	yes	—
21	#14556515	C₂²	1	4	yes	—
22	#14556512	C₂²	1	4	yes	—
23	C₂²⋊#10596831	C₂²	1	2	yes	—
24	C₄×#10596831	C₂²	1	1	yes	characteristic
25	C₄×#10596831	C₂²	1	1	yes	characteristic
26	C₂×#41×C₇₃	C₂²	1	2	yes	—
27	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
28	#127×C₇₃	C₂²	1	1	yes	characteristic
29	#124×C₇₃	C₂²	1	2	yes	—
30	#121×C₇₃	—	2	1	no	—
31	#121×C₇₃	—	2	1	no	—
32	C₄×D₄×C₇₃	—	2	1	no	—
33	C₄×D₄×C₇₃	—	2	1	no	—
34	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
35	C₂×Dic₁₄₆	—	2	1	no	—
36	C₂×Dic₁₄₆	C₂³	1	2	yes	—
37	#11924018	—	4	4	no	—
38	#11924016	—	4	4	no	—
39	C₂×#10596831	C₂³	1	2	yes	—
40	C₂×#10596831	C₂³	1	2	yes	—
41	C₄⋊Dic₇₃	—	2	1	no	—
42	C₄⋊Dic₇₃	—	2	1	no	—
43	C₄⋊Dic₇₃	—	2	1	no	—
44	C₄⋊Dic₇₃	C₂³	1	2	yes	—
45	#11924007	—	2	2	no	—
46	#11924007	—	2	2	no	—
47	C₂²⋊Dic₇₃	—	2	2	no	—
48	C₄×Dic₇₃	—	2	2	no	—
49	#41×C₇₃	—	4	4	no	—
50	C₂×Q₈×C₇₃	C₂³	1	2	yes	—
51	C₂×D₄×C₇₃	—	2	2	no	—
52	C₂×D₄×C₇₃	—	2	1	no	—
53	C₂×D₄×C₇₃	—	2	1	no	—
54	C₂×D₄×C₇₃	C₂³	1	2	yes	—
55	C₂²×C₄×C₇₃	—	2	2	no	—
56	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
57	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
58	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
59	C₄⋊C₄×C₇₃	—	2	1	no	—
60	C₄⋊C₄×C₇₃	—	2	1	no	—
61	C₂²⋊C₄×C₇₃	—	2	2	no	—
62	C₂²⋊C₄×C₇₃	—	2	2	no	—
63	C₄²×C₇₃	D₄	1	1	yes	characteristic
64	C₄²×C₇₃	D₄	1	1	yes	characteristic
65	C₄²×C₇₃	C₂³	1	1	yes	characteristic
66	C₄²×C₇₃	C₂³	1	1	yes	characteristic
67	C₂³×C₇₃	—	2	2	no	—
68	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
69	Q₈×C₇₃	—	2	4	no	—
70	D₄×C₇₃	—	2	4	no	—
71	D₄×C₇₃	—	4	4	no	—
72	D₄×C₇₃	—	4	2	no	—
73	D₄×C₇₃	—	4	2	no	—
74	C₂×C₄×C₇₃	—	2	2	no	—
75	C₂×C₄×C₇₃	—	2	2	no	—
76	C₂×C₄×C₇₃	—	2	2	no	—
77	C₂×C₄×C₇₃	—	2	2	no	—
78	C₂×C₄×C₇₃	—	4	4	no	—
79	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
85	C₂×Dic₇₃	—	2	2	no	—
86	C₂×Dic₇₃	—	2	2	no	—
87	Dic₁₄₆	—	2	4	no	—
88	Dic₁₄₆	—	4	2	no	—
89	#10596831	—	2	2	no	—
90	#10596831	—	2	2	no	—
91	C₂²×C₇₃	—	2	1	no	—
92	C₂²×C₇₃	—	2	1	no	—
93	C₂²×C₇₃	—	2	1	no	—
94	C₂²×C₇₃	—	4	2	no	—
95	C₂²×C₇₃	—	4	2	no	—
96	C₂²×C₇₃	—	4	2	no	—
97	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
98	C₄×C₇₃	—	2	1	no	—
99	C₄×C₇₃	—	2	1	no	—
100	C₄×C₇₃	—	2	1	no	—
101	C₄×C₇₃	—	4	2	no	—
102	C₄×C₇₃	#135	1	2	yes	—
103	C₄×C₇₃	#137	1	2	yes	—
104	C₄×C₇₃	H(5,𝔽₂)	1	2	yes	—
105	Dic₇₃	—	4	2	no	—
106	Dic₇₃	—	4	2	no	—
107	C₂×C₇₃	—	4	1	no	—
108	C₂×C₇₃	—	8	2	no	—
109	C₂×C₇₃	#535	1	1	yes	characteristic
110	C₂×C₇₃	C₂×#135	1	1	yes	characteristic
111	C₂×C₇₃	C₂×#137	1	1	yes	characteristic
112	#3210	—	73	1	no	maximal, Sylow
113	C₇₃	#3210	1	1	yes	characteristic, complemented, Sylow
114	#533	—	73	1	no	—
115	#532	D₇₃	1	1	yes	characteristic
116	#494	—	73	1	no	—
117	#492	—	73	1	no	—
118	#487	—	73	1	no	—
119	#486	—	73	1	no	—
120	#464	—	73	4	no	—
121	#452	—	73	4	no	—
122	#431	—	73	1	no	—
123	C₂×#41	C₂×D₇₃	1	2	yes	—
124	C₂×Q₁₆	—	73	4	no	—
125	C₂×SD₁₆	—	73	4	no	—
126	C₄⋊Q₈	—	73	1	no	—
127	C₄⋊Q₈	—	73	1	no	—
128	C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
129	#127	C₂×D₇₃	1	1	yes	characteristic
130	#125	—	146	1	no	—
131	#124	C₂×D₇₃	1	2	yes	—
132	C₂²⋊Q₈	—	73	2	no	—
133	C₂²⋊Q₈	—	146	1	no	—
134	#121	—	2	1	no	—
135	#121	—	2	1	no	—
136	C₄×Q₈	—	73	2	no	—
137	C₄×Q₈	—	146	1	no	—
138	C₄×D₄	—	2	1	no	—
139	C₄×D₄	—	2	1	no	—
140	#117	—	146	1	no	—
141	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
142	C₄⋊C₈	—	73	2	no	—
143	#103	—	73	4	no	—
144	#102	—	73	4	no	—
145	C₂²⋊C₈	—	73	2	no	—
146	C₄×C₈	—	73	1	no	—
147	C₄×C₈	—	73	1	no	—
148	#41	—	4	4	no	—
149	C₂×Q₈	—	73	2	no	—
150	C₂×Q₈	—	146	1	no	—
151	C₂×Q₈	C₂²×D₇₃	1	2	yes	—
152	C₂×D₄	—	2	2	no	—
153	C₂×D₄	—	2	1	no	—
154	C₂×D₄	—	2	1	no	—
155	C₂×D₄	C₂²×D₇₃	1	2	yes	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
158	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
159	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
160	Q₁₆	—	292	4	no	—
161	SD₁₆	—	292	4	no	—
162	C₂×C₈	—	73	2	no	—
163	C₂×C₈	—	73	2	no	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	—	2	1	no	—
166	C₄⋊C₄	—	73	2	no	—
167	C₄⋊C₄	—	146	2	no	—
168	C₄⋊C₄	—	146	2	no	—
169	C₄⋊C₄	—	146	1	no	—
170	C₄⋊C₄	—	146	1	no	—
171	C₄⋊C₄	—	146	1	no	—
172	C₂²⋊C₄	—	2	2	no	—
173	C₂²⋊C₄	—	2	2	no	—
174	C₂²⋊C₄	—	146	2	no	—
175	C₄²	—	146	2	no	—
176	C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
177	C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
178	C₄²	C₂²×D₇₃	1	1	yes	characteristic
179	C₄²	C₂²×D₇₃	1	1	yes	characteristic
180	C₂³	—	2	2	no	—
181	C₂³	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
182	Q₈	—	2	4	no	—
183	Q₈	—	292	2	no	—
184	Q₈	—	146	4	no	—
185	D₄	—	2	4	no	—
186	D₄	—	4	4	no	—
187	D₄	—	4	2	no	—
188	D₄	—	4	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	4	4	no	—
194	C₂×C₄	—	146	2	no	—
195	C₂×C₄	—	146	2	no	—
196	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
197	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
198	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
199	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
200	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
201	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
202	C₈	—	146	2	no	—
203	C₈	—	146	2	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	4	2	no	—
208	C₂²	—	4	2	no	—
209	C₂²	—	4	2	no	—
210	C₂²	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	4	2	no	—
215	C₄	—	292	2	no	—
216	C₄	—	292	2	no	—
217	C₄	#14556611	1	2	yes	—
218	C₄	#14556621	1	2	yes	—
219	C₄	#14556636	1	2	yes	—
220	C₂	—	4	1	no	—
221	C₂	—	8	2	no	—
222	C₂	#17989154	1	1	yes	characteristic
223	C₂	C₂×#14556611	1	1	yes	characteristic
224	C₂	C₂×#14556621	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989144

C₂

yes

maximal, characteristic, complemented

#17988966

C₂

yes

maximal, characteristic, complemented

#17988959

C₂

yes

maximal, characteristic, complemented

#17988937

C₂

yes

maximal, characteristic, complemented

#17988933

C₂

yes

maximal, characteristic, complemented

#17988806

C₂

yes

maximal, complemented

#17988749

C₂

yes

maximal, complemented

#17988640

C₂

yes

maximal, characteristic, complemented

#532×C₇₃

C₂

yes

maximal, characteristic

C₂×#11924018

C₂²

yes

—

C₂×#11924016

C₂²

yes

—

#14556591

C₂²

yes

characteristic

#14556591

C₂²

yes

—

—

C₂²

yes

—

C₄×Dic₁₄₆

—

C₄×Dic₁₄₆

C₂²

yes

—

#14556550

—

C₄⋊#10596831

C₂²

yes

—

#14556515

C₂²

yes

—

#14556512

C₂²

yes

—

C₂²⋊#10596831

C₂²

yes

—

C₄×#10596831

C₂²

yes

characteristic

C₄×#10596831

C₂²

yes

characteristic

C₂×#41×C₇₃

C₂²

yes

—

C₄⋊Q₈×C₇₃

C₂²

yes

characteristic

#127×C₇₃

C₂²

yes

characteristic

#124×C₇₃

C₂²

yes

—

#121×C₇₃

—

#121×C₇₃

—

C₄×D₄×C₇₃

—

C₄×D₄×C₇₃

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

C₂³

yes

—

#11924018

—

#11924016

—

C₂×#10596831

C₂³

yes

—

C₂×#10596831

C₂³

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂³

yes

—

#11924007

—

#11924007

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

#41×C₇₃

—

C₂×Q₈×C₇₃

C₂³

yes

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

C₂³

yes

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

C₂³

yes

characteristic

C₄²×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂⁴

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂²×D₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

100

C₄×C₇₃

—

101

C₄×C₇₃

—

102

C₄×C₇₃

#135

yes

—

103

C₄×C₇₃

#137

yes

—

104

C₄×C₇₃

H(5,𝔽₂)

yes

—

105

Dic₇₃

—

106

Dic₇₃

—

107

C₂×C₇₃

—

108

C₂×C₇₃

—

109

C₂×C₇₃

#535

yes

characteristic

110

C₂×C₇₃

C₂×#135

yes

characteristic

111

C₂×C₇₃

yes

112

—

113

C₇₃

#3210

yes

characteristic, complemented, Sylow

114

#533

—

115

#532

D₇₃

yes

characteristic

116

#494

—

117

#492

—

118

#487

—

119

#486

—

120

#464

—

121

#452

—

122

#431

—

123

C₂×#41

C₂×D₇₃

yes

—

124

C₂×Q₁₆

—

125

C₂×SD₁₆

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

C₂×D₇₃

yes

characteristic

129

#127

C₂×D₇₃

yes

characteristic

130

#125

—

146

—

131

#124

C₂×D₇₃

yes

—

132

C₂²⋊Q₈

—

133

C₂²⋊Q₈

—

146

—

134

#121

—

135

#121

—

136

C₄×Q₈

—

137

C₄×Q₈

—

146

—

138

C₄×D₄

—

139

C₄×D₄

—

140

#117

—

146

—

141

C₂×C₄²

C₂×D₇₃

yes

characteristic

142

C₄⋊C₈

—

143

#103

—

144

#102

—

145

C₂²⋊C₈

—

146

C₄×C₈

—

147

C₄×C₈

—

148

#41

—

149

C₂×Q₈

—

150

C₂×Q₈

—

146

—

151

C₂×Q₈

C₂²×D₇₃

yes

—

152

C₂×D₄

—

153

C₂×D₄

—

154

C₂×D₄

—

155

C₂×D₄

C₂²×D₇₃

yes

—

156

C₂²×C₄

—

157

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

158

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

159

C₂²×C₄

C₂²×D₇₃

yes

characteristic

160

Q₁₆

—

292

—

161

SD₁₆

—

292

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

146

—

168

C₄⋊C₄

—

146

—

169

C₄⋊C₄

—

146

—

170

C₄⋊C₄

—

146

—

171

C₄⋊C₄

—

146

—

172

C₂²⋊C₄

—

173

C₂²⋊C₄

—

174

C₂²⋊C₄

—

146

—

175

C₄²

—

146

—

176

C₄²

C₂²⋊D₇₃

yes

characteristic

177

C₄²

C₂²⋊D₇₃

yes

characteristic

178

C₄²

C₂²×D₇₃

yes

characteristic

179

C₄²

C₂²×D₇₃

yes

characteristic

180

C₂³

—

181

C₂³

C₂×C₂²⋊D₇₃

yes

characteristic

182

Q₈

—

183

Q₈

—

292

—

184

Q₈

—

146

—

185

D₄

—

186

D₄

—

187

D₄

—

188

D₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

146

—

195

C₂×C₄

—

146

—

196

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

197

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

198

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

199

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

200

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

201

C₂×C₄

C₂³×D₇₃

yes

characteristic

202

C₈

—

146

—

203

C₈

—

146

—

204

C₂²

—

205

C₂²

—

206

C₂²

—

207

C₂²

—

208

C₂²

—

209

C₂²

—

210

C₂²

C₂²×C₂²⋊D₇₃

yes

characteristic

211

C₄

—

212

C₄

—

213

C₄

—

214

C₄

—

215

C₄

—

292

—

216

C₄

—

292

—

217

C₄

#14556611

yes

—

218

C₄

#14556621

yes

—

219

C₄

#14556636

yes

—

220

C₂

—

221

C₂

—

222

C₂

#17989154

yes

characteristic

223

C₂

C₂×#14556611

yes

characteristic

224

C₂

C₂×#14556621

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₃
Fitting	#532×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₄
hypercenter	#532	D₇₃
hypocenter	C₇₃	#3210
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₇₃

commutator

C₂×C₄×C₇₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₃

Fitting

#532×C₇₃

C₂

socle

C₂²×C₇₃

C₂²×D₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. #17989144	C₂	semidirect
2. #17988966	C₂	semidirect
3. #17988959	C₂	semidirect
4. #17988937	C₂	semidirect
5. #17988933	C₂	semidirect
6. #17988806	C₂	semidirect
7. #17988749	C₂	semidirect
8. #17988640	C₂	semidirect
113. C₇₃	#3210	semidirect

H ≅ G/N

type

1. #17989144

C₂

semidirect

2. #17988966

C₂

semidirect

3. #17988959

C₂

semidirect

4. #17988937

C₂

semidirect

5. #17988933

C₂

semidirect

6. #17988806

C₂

semidirect

7. #17988749

C₂

semidirect

8. #17988640

C₂

semidirect

113. C₇₃

#3210

semidirect

Subgroups — #22620887

Conjugacy classes of subgroups

Complements