Group #22609055 — subgroups

Subgroups — #22609055

Order 9344.

256 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 55 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	#17989257	C₂	1	1	yes	maximal, characteristic
2	#17988879	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988874	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988331	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988319	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988316	C₂	1	1	yes	maximal, characteristic, complemented
7	#492×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11924027	—	2	1	no	—
9	#14556587	—	2	1	no	—
10	#14556563	—	2	2	no	—
11	#14556562	—	2	1	no	—
12	C₄×Dic₁₄₆	C₂²	1	1	yes	characteristic
13	C₄×D₂₉₂	C₂²	1	1	yes	characteristic
14	D₄×Dic₇₃	—	2	1	no	—
15	#14556524	—	2	1	no	—
16	#14556522	—	2	1	no	—
17	C₄⋊#10596831	C₂²	1	1	yes	characteristic
18	C₄⋊#10596831	C₂²	1	1	yes	characteristic
19	#14556514	—	2	1	no	—
20	#14556511	—	2	1	no	—
21	C₂×SD₁₆×C₇₃	—	2	2	no	—
22	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
23	#127×C₇₃	C₂²	1	1	yes	characteristic
24	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
25	#11924027	—	4	2	no	—
26	#11924027	—	4	2	no	—
27	C₂×Dic₁₄₆	D₄	1	1	yes	characteristic
28	C₂×D₂₉₂	D₄	1	1	yes	characteristic
29	C₂×C₂²⋊D₇₃	—	2	2	no	—
30	C₂×C₂²⋊D₇₃	—	2	2	no	—
31	C₂²×Dic₇₃	—	2	2	no	—
32	C₂×C₄×D₇₃	—	2	1	no	—
33	C₂×#10596831	—	2	1	no	—
34	C₂×#10596831	—	2	1	no	—
35	C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
36	C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
37	#11924007	—	2	1	no	—
38	C₂²⋊Dic₇₃	—	2	2	no	—
39	#11924005	—	2	1	no	—
40	C₄×Dic₇₃	—	2	1	no	—
41	C₂×Q₈×C₇₃	—	2	1	no	—
42	C₂×D₄×C₇₃	—	2	1	no	—
43	C₂×D₄×C₇₃	—	2	1	no	—
44	C₂×D₄×C₇₃	D₄	1	2	yes	—
45	SD₁₆×C₇₃	—	4	4	no	—
46	C₂×C₈×C₇₃	—	2	1	no	—
47	C₄⋊C₄×C₇₃	—	2	1	no	—
48	C₄⋊C₄×C₇₃	—	2	1	no	—
49	C₄²×C₇₃	C₂³	1	1	yes	characteristic
50	C₂³×C₇₃	—	2	2	no	—
51	C₂²×D₇₃	—	2	1	no	—
52	Q₈×C₇₃	—	4	2	no	—
53	D₄×C₇₃	—	4	2	no	—
54	D₄×C₇₃	—	4	1	no	—
55	D₄×C₇₃	—	4	1	no	—
56	D₄×C₇₃	—	4	1	no	—
57	D₄×C₇₃	—	4	1	no	—
58	C₂×C₄×C₇₃	—	4	1	no	—
59	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
62	C₂²⋊D₇₃	—	4	4	no	—
63	C₂²⋊D₇₃	—	8	2	no	—
64	C₂×Dic₇₃	—	2	1	no	—
65	C₂×Dic₇₃	—	2	1	no	—
66	C₂×Dic₇₃	—	2	1	no	—
67	C₂×Dic₇₃	—	4	4	no	—
68	D₂₉₂	—	2	1	no	—
69	D₂₉₂	—	2	1	no	—
70	C₄×D₇₃	—	4	1	no	—
71	C₄×D₇₃	—	4	1	no	—
72	Dic₁₄₆	—	2	1	no	—
73	Dic₁₄₆	SD₁₆	1	2	yes	complemented
74	C₈×C₇₃	—	2	2	no	—
75	#10596831	—	4	1	no	—
76	#10596831	—	4	1	no	—
77	C₂²×C₇₃	—	4	2	no	—
78	C₂²×C₇₃	—	4	2	no	—
79	C₂²×C₇₃	—	4	2	no	—
80	C₂²×C₇₃	#120	1	1	yes	characteristic
81	C₂×D₇₃	—	4	1	no	—
82	C₂×D₇₃	—	4	1	no	—
83	C₂×D₇₃	—	4	1	no	—
84	C₄×C₇₃	—	2	1	no	—
85	C₄×C₇₃	—	8	1	no	—
86	C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
87	C₄×C₇₃	#135	1	1	yes	characteristic
88	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
89	C₄×C₇₃	#137	1	1	yes	characteristic
90	Dic₇₃	—	2	2	no	—
91	Dic₇₃	—	4	1	no	—
92	Dic₇₃	—	4	1	no	—
93	C₂×C₇₃	—	8	2	no	—
94	C₂×C₇₃	#450	1	1	yes	characteristic
95	C₂×C₇₃	#452	1	1	yes	characteristic
96	C₂×C₇₃	#453	1	1	yes	characteristic
97	D₇₃	—	8	1	no	—
98	#1631	—	73	1	no	maximal, Sylow
99	C₇₃	#1631	1	1	yes	characteristic, complemented, Sylow
100	#550	—	73	1	no	—
101	#492	D₇₃	1	1	yes	characteristic, complemented
102	#477	—	73	1	no	—
103	#476	—	73	1	no	—
104	#331	—	73	1	no	—
105	Q₈⋊C₈	—	73	1	no	—
106	D₄⋊C₈	—	73	1	no	—
107	C₂×#41	—	146	1	no	—
108	C₂×SD₁₆	—	2	2	no	—
109	C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
110	#127	—	146	1	no	—
111	#127	C₂×D₇₃	1	1	yes	characteristic
112	#121	—	146	1	no	—
113	#121	—	146	2	no	—
114	C₄×Q₈	—	73	1	no	—
115	C₄×D₄	—	73	1	no	—
116	C₄×D₄	—	146	1	no	—
117	#107	—	146	1	no	—
118	#106	—	146	1	no	—
119	C₄⋊C₈	—	73	1	no	—
120	C₄⋊C₈	—	73	1	no	—
121	#103	—	146	1	no	—
122	#102	—	146	1	no	—
123	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
124	#41	—	292	2	no	—
125	#41	—	292	2	no	—
126	C₂×Q₈	—	2	1	no	—
127	C₂×Q₈	—	73	1	no	—
128	C₂×D₄	—	2	1	no	—
129	C₂×D₄	—	2	1	no	—
130	C₂×D₄	—	73	1	no	—
131	C₂×D₄	—	146	2	no	—
132	C₂×D₄	—	146	2	no	—
133	C₂×D₄	C₂²⋊D₇₃	1	2	yes	—
134	C₂²×C₄	—	146	2	no	—
135	C₂²×C₄	—	146	1	no	—
136	SD₁₆	—	4	4	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	146	1	no	—
139	C₂×C₈	—	146	1	no	—
140	C₄⋊C₄	—	2	1	no	—
141	C₄⋊C₄	—	2	1	no	—
142	C₄⋊C₄	—	73	1	no	—
143	C₄⋊C₄	—	73	1	no	—
144	C₄⋊C₄	—	146	1	no	—
145	C₂²⋊C₄	—	146	2	no	—
146	C₂²⋊C₄	—	146	1	no	—
147	C₄²	—	146	1	no	—
148	C₄²	C₂²×D₇₃	1	1	yes	characteristic
149	C₂³	—	2	2	no	—
150	C₂³	—	146	1	no	—
151	Q₈	—	4	2	no	—
152	Q₈	—	73	2	no	—
153	Q₈	—	146	1	no	—
154	D₄	—	4	2	no	—
155	D₄	—	4	1	no	—
156	D₄	—	4	1	no	—
157	D₄	—	4	1	no	—
158	D₄	—	4	1	no	—
159	D₄	—	292	4	no	—
160	D₄	—	584	2	no	—
161	D₄	—	146	1	no	—
162	D₄	—	146	1	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	292	4	no	—
165	C₂×C₄	—	292	1	no	—
166	C₂×C₄	—	292	1	no	—
167	C₂×C₄	—	146	1	no	—
168	C₂×C₄	—	146	1	no	—
169	C₂×C₄	—	146	1	no	—
170	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
171	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
172	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
173	C₈	—	2	2	no	—
174	C₈	—	292	1	no	—
175	C₈	—	292	1	no	—
176	C₂²	—	4	2	no	—
177	C₂²	—	4	2	no	—
178	C₂²	—	4	2	no	—
179	C₂²	—	292	1	no	—
180	C₂²	—	292	1	no	—
181	C₂²	—	292	1	no	—
182	C₂²	#14556559	1	1	yes	characteristic
183	C₄	—	2	1	no	—
184	C₄	—	8	1	no	—
185	C₄	—	292	1	no	—
186	C₄	—	292	1	no	—
187	C₄	—	146	2	no	—
188	C₄	SD₁₆×D₇₃	1	1	yes	characteristic
189	C₄	#14556612	1	1	yes	characteristic
190	C₄	#14556618	1	1	yes	characteristic
191	C₄	#14556620	1	1	yes	characteristic
192	C₂	—	8	2	no	—
193	C₂	—	584	1	no	—
194	C₂	#17988736	1	1	yes	characteristic
195	C₂	#17988746	1	1	yes	characteristic
196	C₂	#17988754	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989257

C₂

yes

maximal, characteristic

#17988879

C₂

yes

maximal, characteristic, complemented

#17988874

C₂

yes

maximal, characteristic, complemented

#17988331

C₂

yes

maximal, characteristic, complemented

#17988319

C₂

yes

maximal, characteristic, complemented

#17988316

C₂

yes

maximal, characteristic, complemented

#492×C₇₃

C₂

yes

maximal, characteristic, complemented

—

—

—

—

C₄×Dic₁₄₆

C₂²

yes

characteristic

C₄×D₂₉₂

C₂²

yes

characteristic

D₄×Dic₇₃

—

#14556524

—

#14556522

—

C₄⋊#10596831

C₂²

yes

characteristic

C₄⋊#10596831

C₂²

yes

characteristic

#14556514

—

#14556511

—

C₂×SD₁₆×C₇₃

—

C₄⋊Q₈×C₇₃

C₂²

yes

characteristic

#127×C₇₃

C₂²

yes

characteristic

C₄×C₈×C₇₃

C₂²

yes

characteristic

#11924027

—

#11924027

—

C₂×Dic₁₄₆

D₄

yes

characteristic

C₂×D₂₉₂

D₄

yes

characteristic

C₂×C₂²⋊D₇₃

—

C₂×C₂²⋊D₇₃

—

C₂²×Dic₇₃

—

C₂×C₄×D₇₃

—

C₂×#10596831

—

C₂×#10596831

—

C₄⋊Dic₇₃

D₄

yes

characteristic

C₄⋊Dic₇₃

D₄

yes

characteristic

#11924007

—

C₂²⋊Dic₇₃

—

#11924005

—

C₄×Dic₇₃

—

C₂×Q₈×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

D₄

yes

—

SD₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

—

C₂²×D₇₃

—

Q₈×C₇₃

—

D₄×C₇₃

—

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

C₂²⋊D₇₃

—

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

Dic₁₄₆

SD₁₆

yes

complemented

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#120

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×SD₁₆

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#1631

—

maximal, Sylow

C₇₃

#1631

yes

characteristic, complemented, Sylow

100

#550

—

101

#492

D₇₃

yes

characteristic, complemented

102

#477

—

103

#476

—

104

#331

—

105

Q₈⋊C₈

—

106

D₄⋊C₈

—

107

C₂×#41

—

146

—

108

C₂×SD₁₆

—

109

C₄⋊Q₈

C₂×D₇₃

yes

characteristic

110

#127

—

146

—

111

#127

C₂×D₇₃

yes

characteristic

112

#121

—

146

—

113

#121

—

146

—

114

C₄×Q₈

—

115

C₄×D₄

—

116

C₄×D₄

—

146

—

117

#107

—

146

—

118

#106

—

146

—

119

C₄⋊C₈

—

120

C₄⋊C₈

—

121

#103

—

146

—

122

#102

—

146

—

123

C₄×C₈

C₂×D₇₃

yes

characteristic

124

#41

—

292

—

125

#41

—

292

—

126

C₂×Q₈

—

127

C₂×Q₈

—

128

C₂×D₄

—

129

C₂×D₄

—

130

C₂×D₄

—

131

C₂×D₄

—

146

—

132

C₂×D₄

—

146

—

133

C₂×D₄

C₂²⋊D₇₃

yes

—

134

C₂²×C₄

—

146

—

135

C₂²×C₄

—

146

—

136

SD₁₆

—

137

C₂×C₈

—

138

C₂×C₈

—

146

—

139

C₂×C₈

—

146

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

146

—

145

C₂²⋊C₄

—

146

—

146

C₂²⋊C₄

—

146

—

147

C₄²

—

146

—

148

C₄²

C₂²×D₇₃

yes

characteristic

149

C₂³

—

150

C₂³

—

146

—

151

Q₈

—

152

Q₈

—

153

Q₈

—

146

—

154

D₄

—

155

D₄

—

156

D₄

—

157

D₄

—

158

D₄

—

159

D₄

—

292

—

160

D₄

—

584

—

161

D₄

—

146

—

162

D₄

—

146

—

163

C₂×C₄

—

164

C₂×C₄

—

292

—

165

C₂×C₄

—

292

—

166

C₂×C₄

—

292

—

167

C₂×C₄

—

146

—

168

C₂×C₄

—

146

—

169

C₂×C₄

—

146

—

170

C₂×C₄

D₄×D₇₃

yes

characteristic

171

C₂×C₄

D₄×D₇₃

yes

characteristic

172

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

173

C₈

—

174

C₈

—

292

—

175

C₈

—

292

—

176

C₂²

—

177

C₂²

—

178

C₂²

—

179

C₂²

—

292

—

180

C₂²

—

292

—

181

C₂²

—

292

—

182

C₂²

#14556559

yes

characteristic

183

C₄

—

184

C₄

—

185

C₄

—

292

—

186

C₄

—

292

—

187

C₄

—

146

—

188

C₄

SD₁₆×D₇₃

yes

characteristic

189

C₄

#14556612

yes

characteristic

190

C₄

#14556618

yes

characteristic

191

C₄

#14556620

yes

characteristic

192

C₂

—

193

C₂

—

584

—

194

C₂

#17988736

yes

characteristic

195

C₂

#17988746

yes

characteristic

196

C₂

#17988754

yes

characteristic

	subgroup	G/H
center	C₂²	#14556559
commutator	C₄²×C₇₃	C₂³
Frattini	C₄²	C₂²×D₇₃
Fitting	#492×C₇₃	C₂
socle	C₂²×C₇₃	#120
hypercenter	#492	D₇₃
hypocenter	C₇₃	#1631
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#14556559

commutator

C₄²×C₇₃

C₂³

Frattini

C₄²

C₂²×D₇₃

Fitting

#492×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
2. #17988879	C₂	semidirect
3. #17988874	C₂	semidirect
4. #17988331	C₂	semidirect
5. #17988319	C₂	semidirect
6. #17988316	C₂	semidirect
7. #492×C₇₃	C₂	semidirect
73. Dic₁₄₆	SD₁₆	semidirect
99. C₇₃	#1631	semidirect
101. #492	D₇₃	semidirect

H ≅ G/N

type

2. #17988879

C₂

semidirect

3. #17988874

C₂

semidirect

4. #17988331

C₂

semidirect

5. #17988319

C₂

semidirect

6. #17988316

C₂

semidirect

7. #492×C₇₃

C₂

semidirect

73. Dic₁₄₆

SD₁₆

semidirect

99. C₇₃

#1631

semidirect

101. #492

D₇₃

semidirect

Subgroups — #22609055

Conjugacy classes of subgroups

Complements