Group #22611067 — subgroups

Subgroups — #22611067

Order 9344.

326 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₂×#14556591	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14556512	C₂	1	2	yes	maximal
3	#17988349	C₂	1	2	yes	maximal, complemented
4	#17988342	C₂	1	1	yes	maximal, characteristic, complemented
5	#390×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₄₆	—	2	1	no	—
7	C₂²×#10596831	C₂²	1	2	yes	—
8	#14556591	—	2	4	no	—
9	C₂×C₄⋊Dic₇₃	—	2	1	no	—
10	C₂×C₄⋊Dic₇₃	C₂²	1	2	yes	—
11	#14556512	—	2	4	no	—
12	C₂²×D₄×C₇₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊C₄×C₇₃	—	2	1	no	—
15	C₂×C₂²⋊C₄×C₇₃	—	2	1	no	—
16	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₄₆	—	2	4	no	—
18	C₂×Dic₁₄₆	—	4	2	no	—
19	C₂×Dic₁₄₆	—	4	2	no	—
20	C₂²×Dic₇₃	—	2	2	no	—
21	C₂×#10596831	—	2	2	no	—
22	C₂×#10596831	—	2	2	no	—
23	C₂×#10596831	—	2	2	no	—
24	C₄⋊Dic₇₃	—	2	4	no	—
25	C₄⋊Dic₇₃	—	2	2	no	—
26	C₄⋊Dic₇₃	D₄	1	4	yes	—
27	C₂⁴×C₇₃	—	2	1	no	—
28	C₂×D₄×C₇₃	—	2	2	no	—
29	C₂×D₄×C₇₃	—	2	2	no	—
30	C₂×D₄×C₇₃	—	2	1	no	—
31	C₂×D₄×C₇₃	Q₈	1	2	yes	complemented
32	C₂²×C₄×C₇₃	—	2	1	no	—
33	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₇₃	—	2	2	no	—
37	C₂²⋊C₄×C₇₃	—	4	2	no	—
38	C₂²⋊C₄×C₇₃	—	4	2	no	—
39	C₄²×C₇₃	—	2	2	no	—
40	C₂³×C₇₃	—	2	2	no	—
41	C₂³×C₇₃	—	4	1	no	—
42	C₂³×C₇₃	—	4	1	no	—
43	C₂³×C₇₃	—	4	1	no	—
44	C₂³×C₇₃	—	4	1	no	—
45	C₂³×C₇₃	—	4	1	no	—
46	C₂³×C₇₃	—	4	1	no	—
47	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
48	D₄×C₇₃	—	2	4	no	—
49	D₄×C₇₃	—	4	2	no	—
50	C₂×C₄×C₇₃	—	2	1	no	—
51	C₂×C₄×C₇₃	—	2	1	no	—
52	C₂×C₄×C₇₃	—	2	1	no	—
53	C₂×C₄×C₇₃	—	2	1	no	—
54	C₂×C₄×C₇₃	—	4	1	no	—
55	C₂×C₄×C₇₃	—	4	1	no	—
56	C₂×C₄×C₇₃	—	4	1	no	—
57	C₂×C₄×C₇₃	D₈	1	2	yes	—
58	C₂×C₄×C₇₃	SD₁₆	1	2	yes	—
59	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₄×C₇₃	#41	1	2	yes	—
62	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
63	C₂×Dic₇₃	—	2	4	no	—
64	C₂×Dic₇₃	—	4	2	no	—
65	C₂×Dic₇₃	—	4	2	no	—
66	Dic₁₄₆	—	4	8	no	—
67	#10596831	—	4	2	no	—
68	C₂²×C₇₃	—	4	2	no	—
69	C₂²×C₇₃	—	4	2	no	—
70	C₂²×C₇₃	—	4	2	no	—
71	C₂²×C₇₃	—	8	1	no	—
72	C₂²×C₇₃	—	8	1	no	—
73	C₂²×C₇₃	—	8	1	no	—
74	C₂²×C₇₃	—	8	1	no	—
75	C₂²×C₇₃	#120	1	1	yes	characteristic
76	C₂²×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
77	C₂²×C₇₃	#124	1	1	yes	characteristic
78	C₂²×C₇₃	C₂×D₈	1	1	yes	characteristic
79	C₂²×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
80	C₂²×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₂²×C₇₃	#137	1	1	yes	characteristic
82	C₄×C₇₃	—	2	2	no	—
83	C₄×C₇₃	—	2	2	no	—
84	C₄×C₇₃	—	8	1	no	—
85	C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
86	C₄×C₇₃	#124	1	2	yes	—
87	Dic₇₃	—	4	4	no	—
88	C₂×C₇₃	—	8	2	no	—
89	C₂×C₇₃	#393	1	1	yes	characteristic
90	C₂×C₇₃	#447	1	1	yes	characteristic
91	C₂×C₇₃	#448	1	1	yes	characteristic
92	C₂×C₇₃	#474	1	1	yes	characteristic
93	C₂×C₇₃	#476	1	1	yes	characteristic
94	C₂×C₇₃	#486	1	1	yes	characteristic
95	C₂×C₇₃	#489	1	1	yes	characteristic
96	#2023	—	73	1	no	maximal, Sylow
97	C₇₃	#2023	1	1	yes	characteristic, complemented, Sylow
98	C₂×C₄⋊Q₈	—	73	1	no	—
99	C₂×#102	—	73	2	no	—
100	#390	D₇₃	1	1	yes	characteristic
101	#340	—	73	2	no	—
102	#336	—	73	1	no	—
103	C₂²×Q₈	—	146	1	no	—
104	C₂²×D₄	C₂×D₇₃	1	1	yes	characteristic
105	C₂²×C₈	—	73	2	no	—
106	C₄⋊Q₈	—	146	4	no	—
107	C₂×C₄⋊C₄	—	73	2	no	—
108	C₂×C₄⋊C₄	—	146	1	no	—
109	C₂×C₄⋊C₄	C₂×D₇₃	1	1	yes	characteristic
110	C₂×C₂²⋊C₄	—	2	1	no	—
111	C₂×C₂²⋊C₄	—	2	1	no	—
112	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
113	#102	—	146	4	no	—
114	C₂⁴	—	2	1	no	—
115	C₂×Q₈	—	292	2	no	—
116	C₂×Q₈	—	292	2	no	—
117	C₂×Q₈	—	146	4	no	—
118	C₂×D₄	—	2	2	no	—
119	C₂×D₄	—	2	2	no	—
120	C₂×D₄	—	2	1	no	—
121	C₂×D₄	Dic₁₄₆	1	2	yes	complemented
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	—	146	2	no	—
124	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
125	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
126	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
127	C₂×C₈	—	146	2	no	—
128	C₂×C₈	—	146	2	no	—
129	C₂×C₈	—	146	2	no	—
130	C₄⋊C₄	—	2	2	no	—
131	C₄⋊C₄	—	73	4	no	—
132	C₄⋊C₄	—	146	4	no	—
133	C₄⋊C₄	—	146	2	no	—
134	C₂²⋊C₄	—	4	2	no	—
135	C₂²⋊C₄	—	4	2	no	—
136	C₄²	—	2	2	no	—
137	C₂³	—	2	2	no	—
138	C₂³	—	4	1	no	—
139	C₂³	—	4	1	no	—
140	C₂³	—	4	1	no	—
141	C₂³	—	4	1	no	—
142	C₂³	—	4	1	no	—
143	C₂³	—	4	1	no	—
144	C₂³	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
145	Q₈	—	292	8	no	—
146	D₄	—	2	4	no	—
147	D₄	—	4	2	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	292	2	no	—
156	C₂×C₄	—	292	2	no	—
157	C₂×C₄	—	146	4	no	—
158	C₂×C₄	#11924013	1	2	yes	—
159	C₂×C₄	#11924016	1	2	yes	—
160	C₂×C₄	D₄×D₇₃	1	2	yes	—
161	C₂×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
162	C₂×C₄	#11924027	1	2	yes	—
163	C₂×C₄	#11924028	1	1	yes	characteristic
164	C₈	—	292	2	no	—
165	C₂²	—	4	2	no	—
166	C₂²	—	4	2	no	—
167	C₂²	—	4	2	no	—
168	C₂²	—	8	1	no	—
169	C₂²	—	8	1	no	—
170	C₂²	—	8	1	no	—
171	C₂²	—	8	1	no	—
172	C₂²	#14556559	1	1	yes	characteristic
173	C₂²	#14556570	1	1	yes	characteristic
174	C₂²	#14556578	1	1	yes	characteristic
175	C₂²	C₂×#11924013	1	1	yes	characteristic
176	C₂²	C₂×#11924016	1	1	yes	characteristic
177	C₂²	#14556616	1	1	yes	characteristic
178	C₂²	#14556621	1	1	yes	characteristic
179	C₄	—	2	2	no	—
180	C₄	—	2	2	no	—
181	C₄	—	8	1	no	—
182	C₄	—	292	4	no	—
183	C₄	#14556568	1	2	yes	—
184	C₄	#14556577	1	2	yes	—
185	C₂	—	8	2	no	—
186	C₂	#17988499	1	1	yes	characteristic
187	C₂	#17988720	1	1	yes	characteristic
188	C₂	#17988725	1	1	yes	characteristic
189	C₂	#17988867	1	1	yes	characteristic
190	C₂	#17988877	1	1	yes	characteristic
191	C₂	#17988933	1	1	yes	characteristic
192	C₂	#17988946	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556591

C₂

yes

maximal, characteristic, complemented

C₂×#14556512

C₂

yes

maximal

#17988349

C₂

yes

maximal, complemented

#17988342

C₂

yes

maximal, characteristic, complemented

#390×C₇₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₆

—

C₂²×#10596831

C₂²

yes

—

#14556591

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

C₂²

yes

—

#14556512

—

C₂²×D₄×C₇₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₃

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₇₃

—

C₂×C₂²⋊C₄×C₇₃

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

—

C₂×#10596831

—

C₂×#10596831

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

D₄

yes

—

C₂⁴×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

Q₈

yes

complemented

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

D₄×C₇₃

—

D₄×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

D₈

yes

—

C₂×C₄×C₇₃

SD₁₆

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₃

#41

yes

—

C₂×C₄×C₇₃

#41

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#120

yes

characteristic

C₂²×C₇₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₇₃

#124

yes

characteristic

C₂²×C₇₃

C₂×D₈

yes

characteristic

C₂²×C₇₃

C₂×SD₁₆

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂²⋊Q₈

yes

—

C₄×C₇₃

#124

yes

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×#102

—

100

#390

D₇₃

yes

characteristic

101

#340

—

102

#336

—

103

C₂²×Q₈

—

146

—

104

C₂²×D₄

C₂×D₇₃

yes

characteristic

105

C₂²×C₈

—

106

C₄⋊Q₈

—

146

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

146

—

109

C₂×C₄⋊C₄

C₂×D₇₃

yes

characteristic

110

C₂×C₂²⋊C₄

—

111

C₂×C₂²⋊C₄

—

112

C₂×C₄²

C₂×D₇₃

yes

characteristic

113

#102

—

146

—

114

C₂⁴

—

115

C₂×Q₈

—

292

—

116

C₂×Q₈

—

292

—

117

C₂×Q₈

—

146

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂×D₄

—

121

C₂×D₄

Dic₁₄₆

yes

complemented

122

C₂²×C₄

—

123

C₂²×C₄

—

146

—

124

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

125

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

126

C₂²×C₄

C₂²×D₇₃

yes

characteristic

127

C₂×C₈

—

146

—

128

C₂×C₈

—

146

—

129

C₂×C₈

—

146

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

146

—

133

C₄⋊C₄

—

146

—

134

C₂²⋊C₄

—

135

C₂²⋊C₄

—

136

C₄²

—

137

C₂³

—

138

C₂³

—

139

C₂³

—

140

C₂³

—

141

C₂³

—

142

C₂³

—

143

C₂³

—

144

C₂³

C₂×C₂²⋊D₇₃

yes

characteristic

145

Q₈

—

292

—

146

D₄

—

147

D₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

292

—

156

C₂×C₄

—

292

—

157

C₂×C₄

—

146

—

158

C₂×C₄

#11924013

yes

—

159

C₂×C₄

#11924016

yes

—

160

C₂×C₄

D₄×D₇₃

yes

—

161

C₂×C₄

C₂×Dic₁₄₆

yes

characteristic

162

C₂×C₄

#11924027

yes

—

163

C₂×C₄

#11924028

yes

characteristic

164

C₈

—

292

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

#14556559

yes

characteristic

173

C₂²

#14556570

yes

characteristic

174

C₂²

#14556578

yes

characteristic

175

C₂²

C₂×#11924013

yes

characteristic

176

C₂²

C₂×#11924016

yes

characteristic

177

C₂²

#14556616

yes

characteristic

178

C₂²

#14556621

yes

characteristic

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

292

—

183

C₄

#14556568

yes

—

184

C₄

#14556577

yes

—

185

C₂

—

186

C₂

#17988499

yes

characteristic

187

C₂

#17988720

yes

characteristic

188

C₂

#17988725

yes

characteristic

189

C₂

#17988867

yes

characteristic

190

C₂

#17988877

yes

characteristic

191

C₂

#17988933

yes

characteristic

192

C₂

#17988946

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₃
commutator	C₂²×C₄×C₇₃	C₂³
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	#390×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₄
hypercenter	#390	D₇₃
hypocenter	C₇₃	#2023
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₃

commutator

C₂²×C₄×C₇₃

C₂³

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

#390×C₇₃

C₂

socle

C₂³×C₇₃

C₂×D₄

D₇₃

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

N	H ≅ G/N	type
1. C₂×#14556591	C₂	semidirect
3. #17988349	C₂	semidirect
4. #17988342	C₂	semidirect
31. C₂×D₄×C₇₃	Q₈	semidirect
97. C₇₃	#2023	semidirect
121. C₂×D₄	Dic₁₄₆	semidirect

H ≅ G/N

type

1. C₂×#14556591

C₂

semidirect

3. #17988349

C₂

semidirect

4. #17988342

C₂

semidirect

31. C₂×D₄×C₇₃

Q₈

semidirect

97. C₇₃

#2023

semidirect

121. C₂×D₄

Dic₁₄₆

semidirect

Subgroups — #22611067

Conjugacy classes of subgroups

Complements