Group #22611193 — subgroups

Subgroups — #22611193

Order 9344.

276 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₂²×#11924016	C₂	1	1	yes	maximal, characteristic
2	#17988614	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988595	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14556509	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14556508	C₂	1	1	yes	maximal, characteristic
6	#17988351	C₂	1	1	yes	maximal, characteristic, complemented
7	#418×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₄₆	C₂²	1	1	yes	characteristic
9	C₂×#11924016	—	2	2	no	—
10	C₂×#11924016	—	2	2	no	—
11	C₂×#11924016	—	2	2	no	—
12	C₂×#11924012	C₂²	1	1	yes	characteristic
13	C₂×#11924012	C₂²	1	1	yes	characteristic
14	C₂²×#10596831	C₂²	1	1	yes	characteristic
15	C₄⋊#10596831	—	2	2	no	—
16	#14556514	—	2	2	no	—
17	#14556509	—	2	2	no	—
18	#14556508	—	2	2	no	—
19	C₂²×D₄×C₇₃	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
21	#117×C₇₃	C₂²	1	1	yes	characteristic
22	#102×C₇₃	—	2	2	no	—
23	C₂×Dic₁₄₆	—	2	1	no	—
24	C₂×Dic₁₄₆	—	2	1	no	—
25	C₂×Dic₁₄₆	—	2	1	no	—
26	C₂×Dic₁₄₆	—	2	1	no	—
27	C₂×Dic₁₄₆	—	2	1	no	—
28	C₂×Dic₁₄₆	D₄	1	2	yes	—
29	C₂²×Dic₇₃	—	2	1	no	—
30	#11924016	—	2	8	no	—
31	#11924012	—	2	1	no	—
32	#11924012	—	2	1	no	—
33	#11924012	—	2	1	no	—
34	#11924012	—	2	1	no	—
35	C₂×#10596831	—	2	1	no	—
36	C₂×#10596831	—	2	1	no	—
37	C₂×#10596831	—	2	1	no	—
38	C₂×#10596831	—	2	1	no	—
39	C₂×#10596831	D₄	1	2	yes	—
40	C₂⁴×C₇₃	—	2	1	no	—
41	C₂×D₄×C₇₃	—	2	1	no	—
42	C₂×D₄×C₇₃	—	2	1	no	—
43	C₂×D₄×C₇₃	—	2	1	no	—
44	C₂×D₄×C₇₃	—	2	1	no	—
45	C₂×D₄×C₇₃	—	2	1	no	—
46	C₂×D₄×C₇₃	D₄	1	2	yes	—
47	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
48	M₁₆×C₇₃	—	2	1	no	—
49	M₁₆×C₇₃	—	2	1	no	—
50	C₂×C₈×C₇₃	—	2	1	no	—
51	C₄⋊C₄×C₇₃	D₄	1	2	yes	—
52	C₂²⋊C₄×C₇₃	—	2	1	no	—
53	C₄²×C₇₃	—	2	1	no	—
54	C₂³×C₇₃	—	2	2	no	—
55	C₂³×C₇₃	—	4	1	no	—
56	C₂³×C₇₃	—	4	1	no	—
57	C₂³×C₇₃	—	8	2	no	—
58	C₂³×C₇₃	#41	1	1	yes	characteristic
59	D₄×C₇₃	—	2	2	no	—
60	D₄×C₇₃	—	2	2	no	—
61	D₄×C₇₃	—	4	1	no	—
62	D₄×C₇₃	—	4	1	no	—
63	C₂×C₄×C₇₃	—	2	2	no	—
64	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
69	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
70	C₂×Dic₇₃	—	2	2	no	—
71	C₂×Dic₇₃	—	4	1	no	—
72	C₂×Dic₇₃	—	4	1	no	—
73	Dic₁₄₆	—	2	2	no	—
74	Dic₁₄₆	—	2	2	no	—
75	Dic₁₄₆	—	4	1	no	—
76	Dic₁₄₆	—	4	1	no	—
77	C₈×C₇₃	—	4	1	no	—
78	#10596831	—	2	2	no	—
79	#10596831	—	4	1	no	—
80	#10596831	—	4	1	no	—
81	C₂²×C₇₃	—	2	2	no	—
82	C₂²×C₇₃	—	4	2	no	—
83	C₂²×C₇₃	—	8	2	no	—
84	C₂²×C₇₃	—	8	2	no	—
85	C₂²×C₇₃	—	8	2	no	—
86	C₂²×C₇₃	#121	1	1	yes	characteristic
87	C₂²×C₇₃	#123	1	1	yes	characteristic
88	C₂²×C₇₃	#123	1	1	yes	characteristic
89	C₄×C₇₃	—	4	2	no	—
90	C₄×C₇₃	#120	1	1	yes	characteristic
91	C₄×C₇₃	#121	1	1	yes	characteristic
92	C₄×C₇₃	#121	1	1	yes	characteristic
93	C₄×C₇₃	#124	1	1	yes	characteristic
94	Dic₇₃	—	4	2	no	—
95	C₂×C₇₃	—	2	1	no	—
96	C₂×C₇₃	—	2	1	no	—
97	C₂×C₇₃	—	8	2	no	—
98	C₂×C₇₃	#394	1	1	yes	characteristic
99	C₂×C₇₃	#471	1	2	yes	—
100	#2046	—	73	1	no	maximal, Sylow
101	C₇₃	#2046	1	1	yes	characteristic, complemented, Sylow
102	C₂²×SD₁₆	—	73	1	no	—
103	#424	—	73	1	no	—
104	#419	—	73	1	no	—
105	#418	D₇₃	1	1	yes	characteristic
106	C₂×#101	—	73	1	no	—
107	C₂×#100	—	73	1	no	—
108	#341	—	73	1	no	—
109	C₂²×Q₈	—	73	1	no	—
110	C₂²×D₄	C₂×D₇₃	1	1	yes	characteristic
111	C₂×SD₁₆	—	146	2	no	—
112	C₂×SD₁₆	—	146	2	no	—
113	C₂×SD₁₆	—	146	2	no	—
114	C₂×M₁₆	—	73	1	no	—
115	C₂×M₁₆	—	73	1	no	—
116	C₂×M₁₆	C₂×D₇₃	1	1	yes	characteristic
117	C₂²×C₈	—	73	1	no	—
118	#117	C₂×D₇₃	1	1	yes	characteristic
119	C₄⋊C₈	—	146	2	no	—
120	#103	—	146	2	no	—
121	#102	—	2	2	no	—
122	#101	—	146	2	no	—
123	#100	—	146	2	no	—
124	C₂⁴	—	2	1	no	—
125	C₂×Q₈	—	73	2	no	—
126	C₂×Q₈	—	146	1	no	—
127	C₂×Q₈	—	146	1	no	—
128	C₂×Q₈	—	146	1	no	—
129	C₂×Q₈	—	146	1	no	—
130	C₂×Q₈	—	146	1	no	—
131	C₂×D₄	—	2	1	no	—
132	C₂×D₄	—	2	1	no	—
133	C₂×D₄	—	2	1	no	—
134	C₂×D₄	—	2	1	no	—
135	C₂×D₄	—	2	1	no	—
136	C₂×D₄	D₂₉₂	1	2	yes	—
137	C₂²×C₄	—	146	1	no	—
138	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
139	SD₁₆	—	146	8	no	—
140	M₁₆	—	2	1	no	—
141	M₁₆	—	2	1	no	—
142	M₁₆	—	146	1	no	—
143	M₁₆	—	146	1	no	—
144	M₁₆	—	146	1	no	—
145	M₁₆	—	146	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	73	2	no	—
148	C₂×C₈	—	146	1	no	—
149	C₂×C₈	—	146	1	no	—
150	C₂×C₈	—	146	1	no	—
151	C₂×C₈	—	146	1	no	—
152	C₄⋊C₄	C₂²⋊D₇₃	1	2	yes	—
153	C₂²⋊C₄	—	2	1	no	—
154	C₄²	—	2	1	no	—
155	C₂³	—	2	2	no	—
156	C₂³	—	4	1	no	—
157	C₂³	—	4	1	no	—
158	C₂³	—	8	2	no	—
159	C₂³	#11924027	1	1	yes	characteristic
160	Q₈	—	292	1	no	—
161	Q₈	—	292	1	no	—
162	Q₈	—	146	2	no	—
163	Q₈	—	146	2	no	—
164	D₄	—	2	2	no	—
165	D₄	—	2	2	no	—
166	D₄	—	4	1	no	—
167	D₄	—	4	1	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	292	1	no	—
170	C₂×C₄	—	292	1	no	—
171	C₂×C₄	—	146	2	no	—
172	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
173	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
174	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
175	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
176	C₂×C₄	#11924027	1	1	yes	characteristic
177	C₂×C₄	#11924028	1	1	yes	characteristic
178	C₈	—	4	1	no	—
179	C₈	—	292	1	no	—
180	C₈	—	292	1	no	—
181	C₈	—	146	2	no	—
182	C₂²	—	2	2	no	—
183	C₂²	—	4	2	no	—
184	C₂²	—	8	2	no	—
185	C₂²	—	8	2	no	—
186	C₂²	—	8	2	no	—
187	C₂²	#14556563	1	1	yes	characteristic
188	C₂²	#14556573	1	1	yes	characteristic
189	C₂²	#14556575	1	1	yes	characteristic
190	C₄	—	4	2	no	—
191	C₄	—	292	2	no	—
192	C₄	#14556561	1	1	yes	characteristic
193	C₄	#14556562	1	1	yes	characteristic
194	C₄	#14556565	1	1	yes	characteristic
195	C₄	#14556577	1	1	yes	characteristic
196	C₂	—	2	1	no	—
197	C₂	—	2	1	no	—
198	C₂	—	8	2	no	—
199	C₂	#17988503	1	1	yes	characteristic
200	C₂	#17988848	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11924016

C₂

yes

maximal, characteristic

#17988614

C₂

yes

maximal, characteristic, complemented

#17988595

C₂

yes

maximal, characteristic, complemented

C₂×#14556509

C₂

yes

maximal, characteristic, complemented

C₂×#14556508

C₂

yes

maximal, characteristic

#17988351

C₂

yes

maximal, characteristic, complemented

#418×C₇₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₆

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₂×#11924012

C₂²

yes

characteristic

C₂²×#10596831

C₂²

yes

—

—

—

—

C₂²×D₄×C₇₃

C₂²

yes

characteristic

C₂×M₁₆×C₇₃

C₂²

yes

characteristic

#117×C₇₃

C₂²

yes

characteristic

#102×C₇₃

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

D₄

yes

—

C₂²×Dic₇₃

—

—

—

—

—

—

—

—

—

—

D₄

yes

—

C₂⁴×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

D₄

yes

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

D₄

yes

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

#41

yes

characteristic

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×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

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

#10596831

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#394

yes

characteristic

C₂×C₇₃

#471

yes

—

100

#2046

—

maximal, Sylow

101

C₇₃

#2046

yes

characteristic, complemented, Sylow

102

C₂²×SD₁₆

—

103

#424

—

104

#419

—

105

#418

D₇₃

yes

characteristic

106

C₂×#101

—

107

C₂×#100

—

108

#341

—

109

C₂²×Q₈

—

110

C₂²×D₄

C₂×D₇₃

yes

characteristic

111

C₂×SD₁₆

—

146

—

112

C₂×SD₁₆

—

146

—

113

C₂×SD₁₆

—

146

—

114

C₂×M₁₆

—

115

C₂×M₁₆

—

116

C₂×M₁₆

C₂×D₇₃

yes

characteristic

117

C₂²×C₈

—

118

#117

C₂×D₇₃

yes

characteristic

119

C₄⋊C₈

—

146

—

120

#103

—

146

—

121

#102

—

122

#101

—

146

—

123

#100

—

146

—

124

C₂⁴

—

125

C₂×Q₈

—

126

C₂×Q₈

—

146

—

127

C₂×Q₈

—

146

—

128

C₂×Q₈

—

146

—

129

C₂×Q₈

—

146

—

130

C₂×Q₈

—

146

—

131

C₂×D₄

—

132

C₂×D₄

—

133

C₂×D₄

—

134

C₂×D₄

—

135

C₂×D₄

—

136

C₂×D₄

D₂₉₂

yes

—

137

C₂²×C₄

—

146

—

138

C₂²×C₄

C₂²×D₇₃

yes

characteristic

139

SD₁₆

—

146

—

140

M₁₆

—

141

M₁₆

—

142

M₁₆

—

146

—

143

M₁₆

—

146

—

144

M₁₆

—

146

—

145

M₁₆

—

146

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

146

—

149

C₂×C₈

—

146

—

150

C₂×C₈

—

146

—

151

C₂×C₈

—

146

—

152

C₄⋊C₄

C₂²⋊D₇₃

yes

—

153

C₂²⋊C₄

—

154

C₄²

—

155

C₂³

—

156

C₂³

—

157

C₂³

—

158

C₂³

—

159

C₂³

#11924027

yes

characteristic

160

Q₈

—

292

—

161

Q₈

—

292

—

162

Q₈

—

146

—

163

Q₈

—

146

—

164

D₄

—

165

D₄

—

166

D₄

—

167

D₄

—

168

C₂×C₄

—

169

C₂×C₄

—

292

—

170

C₂×C₄

—

292

—

171

C₂×C₄

—

146

—

172

C₂×C₄

D₄×D₇₃

yes

characteristic

173

C₂×C₄

D₄×D₇₃

yes

characteristic

174

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

175

C₂×C₄

C₂×D₂₉₂

yes

characteristic

176

C₂×C₄

#11924027

yes

characteristic

177

C₂×C₄

#11924028

yes

characteristic

178

C₈

—

179

C₈

—

292

—

180

C₈

—

292

—

181

C₈

—

146

—

182

C₂²

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

#14556563

yes

characteristic

188

C₂²

#14556573

yes

characteristic

189

C₂²

#14556575

yes

characteristic

190

C₄

—

191

C₄

—

292

—

192

C₄

#14556561

yes

characteristic

193

C₄

#14556562

yes

characteristic

194

C₄

#14556565

yes

characteristic

195

C₄

#14556577

yes

characteristic

196

C₂

—

197

C₂

—

198

C₂

—

199

C₂

#17988503

yes

characteristic

200

C₂

#17988848

yes

—

	subgroup	G/H
center	C₂²	#14556563
commutator	C₂²×C₄×C₇₃	C₂³
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	#418×C₇₃	C₂
socle	C₂²×C₇₃	#121
hypercenter	#418	D₇₃
hypocenter	C₇₃	#2046
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #418

subgroup

G/H

center

C₂²

#14556563

commutator

C₂²×C₄×C₇₃

C₂³

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

#418×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #418

N	H ≅ G/N	type
2. #17988614	C₂	semidirect
3. #17988595	C₂	semidirect
4. C₂×#14556509	C₂	semidirect
6. #17988351	C₂	semidirect
101. C₇₃	#2046	semidirect

H ≅ G/N

type

2. #17988614

C₂

semidirect

3. #17988595

C₂

semidirect

4. C₂×#14556509

C₂

semidirect

6. #17988351

C₂

semidirect

101. C₇₃

#2046

semidirect

Subgroups — #22611193

Conjugacy classes of subgroups

Complements