Group #22621377 — subgroups

Subgroups — #22621377

Order 9344.

346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 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	#17989199	C₂	1	1	yes	maximal, characteristic, complemented
2	#17988891	C₂	1	2	yes	maximal, complemented
3	#17988887	C₂	1	2	yes	maximal, complemented
4	#17988831	C₂	1	2	yes	maximal, complemented
5	#17988814	C₂	1	2	yes	maximal, complemented
6	#17988792	C₂	1	2	yes	maximal, complemented
7	#17988784	C₂	1	2	yes	maximal, complemented
8	#17988614	C₂	1	1	yes	maximal, characteristic, complemented
9	#540×C₇₃	C₂	1	1	yes	maximal, characteristic
10	C₂×#11924016	C₂²	1	2	yes	—
11	C₂×#11924016	C₂²	1	2	yes	—
12	C₂×#11924012	C₂²	1	1	yes	characteristic
13	C₂²×#10596831	C₂²	1	1	yes	characteristic
14	#14556591	C₂²	1	1	yes	characteristic
15	#14556584	—	2	2	no	—
16	#14556583	C₂²	1	1	yes	characteristic
17	#14556581	—	2	2	no	—
18	#14556570	C₂²	1	2	yes	—
19	C₄×Dic₁₄₆	C₂²	1	2	yes	—
20	#14556524	C₂²	1	2	yes	—
21	#14556522	C₂²	1	2	yes	—
22	C₄⋊#10596831	C₂²	1	2	yes	—
23	C₄⋊#10596831	C₂²	1	2	yes	—
24	#14556514	C₂²	1	2	yes	—
25	#14556514	C₂²	1	2	yes	—
26	#14556512	C₂²	1	2	yes	—
27	#14556512	C₂²	1	2	yes	—
28	#127×C₇₃	C₂²	1	1	yes	characteristic
29	#125×C₇₃	C₂²	1	1	yes	characteristic
30	#123×C₇₃	—	2	2	no	—
31	#121×C₇₃	—	2	2	no	—
32	#121×C₇₃	C₂²	1	2	yes	—
33	C₄×D₄×C₇₃	C₂²	1	2	yes	—
34	#117×C₇₃	C₂²	1	1	yes	characteristic
35	C₂×Dic₁₄₆	C₂³	1	2	yes	—
36	#11924016	—	4	2	no	—
37	#11924016	—	4	2	no	—
38	#11924012	—	4	1	no	—
39	C₂×#10596831	—	4	1	no	—
40	C₂×#10596831	C₂³	1	2	yes	—
41	C₂×#10596831	C₂³	1	2	yes	—
42	C₄⋊Dic₇₃	—	2	2	no	—
43	C₄⋊Dic₇₃	C₂³	1	2	yes	—
44	#11924007	—	2	2	no	—
45	#11924007	—	2	2	no	—
46	#11924007	—	2	2	no	—
47	C₂²⋊Dic₇₃	—	2	2	no	—
48	C₄×Dic₇₃	—	2	2	no	—
49	C₂×D₄×C₇₃	—	2	2	no	—
50	C₂×D₄×C₇₃	—	2	2	no	—
51	C₂×D₄×C₇₃	C₂³	1	2	yes	—
52	C₂²×C₄×C₇₃	—	2	2	no	—
53	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₇₃	—	2	2	no	—
55	C₄⋊C₄×C₇₃	D₄	1	2	yes	—
56	C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
57	C₂²⋊C₄×C₇₃	—	2	2	no	—
58	C₂²⋊C₄×C₇₃	—	2	2	no	—
59	C₂²⋊C₄×C₇₃	D₄	1	2	yes	—
60	C₄²×C₇₃	C₂³	1	1	yes	characteristic
61	C₄²×C₇₃	C₂³	1	1	yes	characteristic
62	C₂³×C₇₃	—	2	2	no	—
63	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
64	D₄×C₇₃	—	2	2	no	—
65	D₄×C₇₃	—	2	2	no	—
66	D₄×C₇₃	—	8	2	no	—
67	D₄×C₇₃	—	8	2	no	—
68	C₂×C₄×C₇₃	—	2	1	no	—
69	C₂×C₄×C₇₃	—	2	1	no	—
70	C₂×C₄×C₇₃	—	2	2	no	—
71	C₂×C₄×C₇₃	—	8	2	no	—
72	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
73	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
74	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
76	C₂×Dic₇₃	—	2	2	no	—
77	C₂×Dic₇₃	—	2	2	no	—
78	Dic₁₄₆	—	2	2	no	—
79	Dic₁₄₆	—	2	2	no	—
80	#10596831	—	2	2	no	—
81	#10596831	—	2	2	no	—
82	C₂²×C₇₃	—	2	1	no	—
83	C₂²×C₇₃	—	2	2	no	—
84	C₂²×C₇₃	—	4	2	no	—
85	C₂²×C₇₃	—	4	2	no	—
86	C₂²×C₇₃	—	4	2	no	—
87	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
88	C₄×C₇₃	—	2	1	no	—
89	C₄×C₇₃	—	2	2	no	—
90	C₄×C₇₃	—	2	2	no	—
91	C₄×C₇₃	—	4	2	no	—
92	C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
93	C₄×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
94	Dic₇₃	—	4	2	no	—
95	Dic₇₃	—	4	2	no	—
96	C₂×C₇₃	—	4	1	no	—
97	C₂×C₇₃	—	8	2	no	—
98	C₂×C₇₃	#537	1	1	yes	characteristic
99	C₂×C₇₃	#577	1	2	yes	—
100	#3260	—	73	1	no	maximal, Sylow
101	C₇₃	#3260	1	1	yes	characteristic, complemented, Sylow
102	#541	—	73	1	no	—
103	#540	D₇₃	1	1	yes	characteristic
104	#479	—	73	2	no	—
105	#478	—	73	2	no	—
106	#468	—	73	2	no	—
107	#465	—	73	2	no	—
108	#461	—	73	2	no	—
109	#460	—	73	2	no	—
110	#424	—	73	1	no	—
111	C₂×SD₁₆	—	73	2	no	—
112	C₂×SD₁₆	—	73	2	no	—
113	C₂×M₁₆	—	73	1	no	—
114	C₂²×C₈	—	73	1	no	—
115	C₄⋊Q₈	—	73	1	no	—
116	#127	C₂×D₇₃	1	1	yes	characteristic
117	#126	—	146	2	no	—
118	#125	—	73	1	no	—
119	#125	—	146	2	no	—
120	#125	C₂×D₇₃	1	1	yes	characteristic
121	#123	—	2	2	no	—
122	C₂²⋊Q₈	—	73	2	no	—
123	#121	—	2	2	no	—
124	#121	C₂×D₇₃	1	2	yes	—
125	C₄×Q₈	—	73	2	no	—
126	C₄×D₄	C₂×D₇₃	1	2	yes	—
127	#117	C₂×D₇₃	1	1	yes	characteristic
128	#107	—	73	2	no	—
129	#106	—	73	2	no	—
130	C₄⋊C₈	—	73	2	no	—
131	C₄⋊C₈	—	73	2	no	—
132	#103	—	73	2	no	—
133	#103	—	73	2	no	—
134	#102	—	73	2	no	—
135	#102	—	73	2	no	—
136	C₂×Q₈	—	73	2	no	—
137	C₂×D₄	—	2	2	no	—
138	C₂×D₄	—	2	2	no	—
139	C₂×D₄	C₂²×D₇₃	1	2	yes	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
142	SD₁₆	—	292	2	no	—
143	SD₁₆	—	292	2	no	—
144	M₁₆	—	292	1	no	—
145	C₂×C₈	—	292	1	no	—
146	C₂×C₈	—	73	2	no	—
147	C₂×C₈	—	73	2	no	—
148	C₄⋊C₄	—	2	2	no	—
149	C₄⋊C₄	—	73	2	no	—
150	C₄⋊C₄	—	146	2	no	—
151	C₄⋊C₄	—	146	2	no	—
152	C₄⋊C₄	—	146	2	no	—
153	C₄⋊C₄	—	146	2	no	—
154	C₄⋊C₄	C₂²⋊D₇₃	1	2	yes	—
155	C₄⋊C₄	C₂²×D₇₃	1	2	yes	—
156	C₂²⋊C₄	—	2	2	no	—
157	C₂²⋊C₄	—	2	2	no	—
158	C₂²⋊C₄	—	146	2	no	—
159	C₂²⋊C₄	C₂²⋊D₇₃	1	2	yes	—
160	C₄²	—	146	2	no	—
161	C₄²	C₂²×D₇₃	1	1	yes	characteristic
162	C₄²	C₂²×D₇₃	1	1	yes	characteristic
163	C₂³	—	2	2	no	—
164	C₂³	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
165	Q₈	—	146	2	no	—
166	Q₈	—	146	2	no	—
167	D₄	—	2	2	no	—
168	D₄	—	2	2	no	—
169	D₄	—	8	2	no	—
170	D₄	—	8	2	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	8	2	no	—
175	C₂×C₄	—	146	2	no	—
176	C₂×C₄	—	146	2	no	—
177	C₂×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
178	C₂×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
179	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
180	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
181	C₈	—	146	2	no	—
182	C₈	—	146	2	no	—
183	C₂²	—	2	2	no	—
184	C₂²	—	2	1	no	—
185	C₂²	—	4	2	no	—
186	C₂²	—	4	2	no	—
187	C₂²	—	4	2	no	—
188	C₂²	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
189	C₄	—	2	2	no	—
190	C₄	—	2	2	no	—
191	C₄	—	2	1	no	—
192	C₄	—	4	2	no	—
193	C₄	—	292	2	no	—
194	C₄	—	292	2	no	—
195	C₄	#14556637	1	1	yes	characteristic
196	C₄	#14556638	1	1	yes	characteristic
197	C₂	—	4	1	no	—
198	C₂	—	8	2	no	—
199	C₂	#17989165	1	1	yes	characteristic
200	C₂	#17989398	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17989199

C₂

yes

maximal, characteristic, complemented

#17988891

C₂

yes

maximal, complemented

#17988887

C₂

yes

maximal, complemented

#17988831

C₂

yes

maximal, complemented

#17988814

C₂

yes

maximal, complemented

#17988792

C₂

yes

maximal, complemented

#17988784

C₂

yes

maximal, complemented

#17988614

C₂

yes

maximal, characteristic, complemented

#540×C₇₃

C₂

yes

maximal, characteristic

C₂×#11924016

C₂²

yes

—

C₂×#11924016

C₂²

yes

—

C₂×#11924012

C₂²

yes

characteristic

C₂²×#10596831

C₂²

yes

characteristic

#14556591

C₂²

yes

characteristic

#14556584

—

#14556583

C₂²

yes

characteristic

#14556581

—

#14556570

C₂²

yes

—

C₄×Dic₁₄₆

C₂²

yes

—

#14556524

C₂²

yes

—

#14556522

C₂²

yes

—

C₄⋊#10596831

C₂²

yes

—

C₄⋊#10596831

C₂²

yes

—

#14556514

C₂²

yes

—

#14556514

C₂²

yes

—

#14556512

C₂²

yes

—

#14556512

C₂²

yes

—

#127×C₇₃

C₂²

yes

characteristic

#125×C₇₃

C₂²

yes

characteristic

#123×C₇₃

—

#121×C₇₃

—

#121×C₇₃

C₂²

yes

—

C₄×D₄×C₇₃

C₂²

yes

—

#117×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

C₂³

yes

—

—

—

—

—

C₂³

yes

—

C₂×#10596831

C₂³

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂³

yes

—

#11924007

—

#11924007

—

#11924007

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

C₂³

yes

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

D₄

yes

—

C₄⋊C₄×C₇₃

C₂³

yes

—

C₂²⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

D₄

yes

—

C₄²×C₇₃

C₂³

yes

characteristic

C₄²×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

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×D₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

H(5,𝔽₂)

yes

characteristic

C₄×C₇₃

2₋¹⁺⁴

yes

characteristic

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#537

yes

characteristic

C₂×C₇₃

#577

yes

—

100

#3260

—

maximal, Sylow

101

C₇₃

#3260

yes

characteristic, complemented, Sylow

102

#541

—

103

#540

D₇₃

yes

characteristic

104

#479

—

105

#478

—

106

#468

—

107

#465

—

108

#461

—

109

#460

—

110

#424

—

111

C₂×SD₁₆

—

112

C₂×SD₁₆

—

113

C₂×M₁₆

—

114

C₂²×C₈

—

115

C₄⋊Q₈

—

116

#127

C₂×D₇₃

yes

characteristic

117

#126

—

146

—

118

#125

—

119

#125

—

146

—

120

#125

C₂×D₇₃

yes

characteristic

121

#123

—

122

C₂²⋊Q₈

—

123

#121

—

124

#121

C₂×D₇₃

yes

—

125

C₄×Q₈

—

126

C₄×D₄

C₂×D₇₃

yes

—

127

#117

C₂×D₇₃

yes

characteristic

128

#107

—

129

#106

—

130

C₄⋊C₈

—

131

C₄⋊C₈

—

132

#103

—

133

#103

—

134

#102

—

135

#102

—

136

C₂×Q₈

—

137

C₂×D₄

—

138

C₂×D₄

—

139

C₂×D₄

C₂²×D₇₃

yes

—

140

C₂²×C₄

—

141

C₂²×C₄

C₂²×D₇₃

yes

characteristic

142

SD₁₆

—

292

—

143

SD₁₆

—

292

—

144

M₁₆

—

292

—

145

C₂×C₈

—

292

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

146

—

151

C₄⋊C₄

—

146

—

152

C₄⋊C₄

—

146

—

153

C₄⋊C₄

—

146

—

154

C₄⋊C₄

C₂²⋊D₇₃

yes

—

155

C₄⋊C₄

C₂²×D₇₃

yes

—

156

C₂²⋊C₄

—

157

C₂²⋊C₄

—

158

C₂²⋊C₄

—

146

—

159

C₂²⋊C₄

C₂²⋊D₇₃

yes

—

160

C₄²

—

146

—

161

C₄²

C₂²×D₇₃

yes

characteristic

162

C₄²

C₂²×D₇₃

yes

characteristic

163

C₂³

—

164

C₂³

C₂×C₂²⋊D₇₃

yes

characteristic

165

Q₈

—

146

—

166

Q₈

—

146

—

167

D₄

—

168

D₄

—

169

D₄

—

170

D₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

146

—

176

C₂×C₄

—

146

—

177

C₂×C₄

C₂×C₂²⋊D₇₃

yes

—

178

C₂×C₄

C₂×C₂²⋊D₇₃

yes

—

179

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

180

C₂×C₄

C₂³×D₇₃

yes

characteristic

181

C₈

—

146

—

182

C₈

—

146

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

C₂²×C₂²⋊D₇₃

yes

characteristic

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

—

292

—

194

C₄

—

292

—

195

C₄

#14556637

yes

characteristic

196

C₄

#14556638

yes

characteristic

197

C₂

—

198

C₂

—

199

C₂

#17989165

yes

characteristic

200

C₂

#17989398

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₃
Fitting	#540×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₄
hypercenter	#540	D₇₃
hypocenter	C₇₃	#3260
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #540

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₇₃

commutator

C₂×C₄×C₇₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₃

Fitting

#540×C₇₃

C₂

socle

C₂²×C₇₃

C₂²×D₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ #117 ◁ #540

N	H ≅ G/N	type
1. #17989199	C₂	semidirect
2. #17988891	C₂	semidirect
3. #17988887	C₂	semidirect
4. #17988831	C₂	semidirect
5. #17988814	C₂	semidirect
6. #17988792	C₂	semidirect
7. #17988784	C₂	semidirect
8. #17988614	C₂	semidirect
101. C₇₃	#3260	semidirect

H ≅ G/N

type

1. #17989199

C₂

semidirect

2. #17988891

C₂

semidirect

3. #17988887

C₂

semidirect

4. #17988831

C₂

semidirect

5. #17988814

C₂

semidirect

6. #17988792

C₂

semidirect

7. #17988784

C₂

semidirect

8. #17988614

C₂

semidirect

101. C₇₃

#3260

semidirect

Subgroups — #22621377

Conjugacy classes of subgroups

Complements