Group #22610522 — subgroups

Subgroups — #22610522

Order 9344.

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

normal subgroups only

This group has 230 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×#14556525	C₂	1	1	yes	maximal, characteristic
2	#17988483	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×Dic₇₃	C₂	1	1	yes	maximal, characteristic
4	#17988349	C₂	1	1	yes	maximal, characteristic
5	#17988349	C₂	1	1	yes	maximal, characteristic
6	#17988341	C₂	1	1	yes	maximal, characteristic
7	#340×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10596831	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
11	C₂×#11924007	—	2	1	no	—
12	C₂×C₄×Dic₇₃	—	2	1	no	—
13	C₂×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
14	#14556525	C₄	1	4	yes	complemented
15	#14556498	—	2	1	no	—
16	#14556498	—	2	1	no	—
17	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
20	C₂²×Dic₇₃	—	2	1	no	—
21	C₂²×Dic₇₃	—	2	1	no	—
22	C₂²×Dic₇₃	D₄	1	1	yes	characteristic
23	C₂²×Dic₇₃	D₄	1	1	yes	characteristic
24	C₂×#10596831	—	2	1	no	—
25	C₂×#10596831	—	2	1	no	—
26	C₂×#10596831	—	2	1	no	—
27	C₄⋊Dic₇₃	—	2	1	no	—
28	C₄⋊Dic₇₃	—	2	1	no	—
29	C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
30	C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
31	#11924007	—	4	2	no	—
32	C₄×Dic₇₃	—	2	1	no	—
33	C₄×Dic₇₃	—	2	1	no	—
34	C₄×Dic₇₃	—	4	2	no	—
35	C₂²×C₄×C₇₃	—	2	1	no	—
36	C₂²×C₄×C₇₃	—	2	1	no	—
37	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
38	C₂×C₈×C₇₃	—	2	1	no	—
39	C₂×C₈×C₇₃	—	2	1	no	—
40	C₂×C₈×C₇₃	C₂×C₄	1	2	yes	—
41	C₄⋊C₄×C₇₃	—	2	1	no	—
42	C₄⋊C₄×C₇₃	—	2	1	no	—
43	C₄⋊C₄×C₇₃	—	2	1	no	—
44	C₄⋊C₄×C₇₃	Q₈	1	2	yes	—
45	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₇₃	—	2	2	no	—
47	C₂×C₄×C₇₃	—	4	1	no	—
48	C₂×C₄×C₇₃	—	4	1	no	—
49	C₂×C₄×C₇₃	—	4	1	no	—
50	C₂×C₄×C₇₃	—	4	1	no	—
51	C₂×C₄×C₇₃	—	4	1	no	—
52	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
55	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
56	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
57	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
58	C₂×Dic₇₃	—	2	1	no	—
59	C₂×Dic₇₃	—	2	1	no	—
60	C₂×Dic₇₃	—	2	2	no	—
61	C₂×Dic₇₃	—	2	2	no	—
62	C₂×Dic₇₃	—	2	1	no	—
63	C₂×Dic₇₃	—	2	1	no	—
64	C₂×Dic₇₃	—	2	1	no	—
65	C₂×Dic₇₃	—	4	1	no	—
66	C₂×Dic₇₃	—	4	1	no	—
67	C₂×Dic₇₃	—	4	1	no	—
68	C₂×Dic₇₃	—	4	1	no	—
69	C₂×Dic₇₃	Q₁₆	1	2	yes	—
70	C₈×C₇₃	—	2	2	no	—
71	#10596831	—	4	1	no	—
72	C₂²×C₇₃	C₄×D₄	1	1	yes	characteristic
73	C₂²×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂²×C₇₃	#123	1	1	yes	characteristic
75	C₂²×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
76	C₂²×C₇₃	#135	1	1	yes	characteristic
77	C₂²×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂²×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₄×C₇₃	—	4	2	no	—
80	C₄×C₇₃	—	8	1	no	—
81	C₄×C₇₃	#117	1	1	yes	characteristic
82	C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
83	C₄×C₇₃	#125	1	1	yes	characteristic
84	C₄×C₇₃	#126	1	1	yes	characteristic
85	Dic₇₃	—	2	2	no	—
86	Dic₇₃	—	4	2	no	—
87	Dic₇₃	—	4	2	no	—
88	Dic₇₃	—	4	1	no	—
89	C₂×C₇₃	#387	1	1	yes	characteristic
90	C₂×C₇₃	C₄×Q₁₆	1	1	yes	characteristic
91	C₂×C₇₃	#442	1	1	yes	characteristic
92	C₂×C₇₃	#477	1	1	yes	characteristic
93	C₂×C₇₃	#478	1	1	yes	characteristic
94	C₂×C₇₃	#482	1	1	yes	characteristic
95	C₂×C₇₃	#484	1	1	yes	characteristic
96	#1921	—	73	1	no	maximal, Sylow
97	C₇₃	#1921	1	1	yes	characteristic, complemented, Sylow
98	C₂×#107	—	73	1	no	—
99	#389	—	73	1	no	—
100	C₄×C₄⋊C₄	—	73	1	no	—
101	#340	—	73	1	no	—
102	#340	—	73	1	no	—
103	#340	D₇₃	1	1	yes	characteristic
104	#336	—	73	1	no	—
105	C₂²×C₈	—	73	1	no	—
106	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
107	C₂×C₄⋊C₄	—	73	1	no	—
108	C₂×C₄⋊C₄	—	73	1	no	—
109	C₂×C₄⋊C₄	—	146	1	no	—
110	C₂×C₄⋊C₄	C₂×D₇₃	1	1	yes	characteristic
111	C₂×C₄⋊C₄	C₂×D₇₃	1	1	yes	characteristic
112	C₂×C₄²	—	73	1	no	—
113	C₂×C₄²	—	146	1	no	—
114	#107	—	73	4	no	—
115	#95	—	146	1	no	—
116	#95	—	146	1	no	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	73	1	no	—
120	C₂²×C₄	—	73	1	no	—
121	C₂²×C₄	—	146	1	no	—
122	C₂²×C₄	—	146	1	no	—
123	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	146	1	no	—
127	C₂×C₈	—	146	1	no	—
128	C₂×C₈	—	146	1	no	—
129	C₂×C₈	C₄×D₇₃	1	2	yes	—
130	C₄⋊C₄	—	2	1	no	—
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	—	292	2	no	—
134	C₄⋊C₄	—	73	2	no	—
135	C₄⋊C₄	—	73	2	no	—
136	C₄⋊C₄	—	146	1	no	—
137	C₄⋊C₄	—	146	1	no	—
138	C₄⋊C₄	Dic₁₄₆	1	2	yes	—
139	C₄²	—	292	2	no	—
140	C₄²	—	146	1	no	—
141	C₄²	—	146	1	no	—
142	C₂³	D₄×D₇₃	1	1	yes	characteristic
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	292	1	no	—
150	C₂×C₄	—	292	1	no	—
151	C₂×C₄	—	292	1	no	—
152	C₂×C₄	—	292	1	no	—
153	C₂×C₄	—	73	2	no	—
154	C₂×C₄	—	146	2	no	—
155	C₂×C₄	—	146	2	no	—
156	C₂×C₄	—	146	1	no	—
157	C₂×C₄	—	146	1	no	—
158	C₂×C₄	—	146	1	no	—
159	C₂×C₄	—	146	1	no	—
160	C₂×C₄	—	146	1	no	—
161	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
162	C₂×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
163	C₂×C₄	#11924027	1	1	yes	characteristic
164	C₂×C₄	#11924027	1	1	yes	characteristic
165	C₂×C₄	#11924028	1	1	yes	characteristic
166	C₂×C₄	#11924029	1	1	yes	characteristic
167	C₈	—	2	2	no	—
168	C₈	—	292	1	no	—
169	C₂²	#14556553	1	1	yes	characteristic
170	C₂²	#14556568	1	1	yes	characteristic
171	C₂²	#14556573	1	1	yes	characteristic
172	C₂²	Q₁₆×D₇₃	1	1	yes	characteristic
173	C₂²	#14556610	1	1	yes	characteristic
174	C₂²	#14556618	1	1	yes	characteristic
175	C₂²	#14556618	1	1	yes	characteristic
176	C₄	—	4	2	no	—
177	C₄	—	8	1	no	—
178	C₄	—	292	1	no	—
179	C₄	—	292	2	no	—
180	C₄	—	292	2	no	—
181	C₄	—	146	2	no	—
182	C₄	#14556547	1	1	yes	characteristic
183	C₄	C₄×Dic₁₄₆	1	1	yes	characteristic
184	C₄	#14556582	1	1	yes	characteristic
185	C₄	#14556586	1	1	yes	characteristic
186	C₂	#17988470	1	1	yes	characteristic
187	C₂	#17988681	1	1	yes	characteristic
188	C₂	#17988698	1	1	yes	characteristic
189	C₂	#17988880	1	1	yes	characteristic
190	C₂	#17988885	1	1	yes	characteristic
191	C₂	#17988909	1	1	yes	characteristic
192	C₂	#17988921	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556525

C₂

yes

maximal, characteristic

#17988483

C₂

yes

maximal, characteristic

C₄⋊C₄×Dic₇₃

C₂

yes

maximal, characteristic

#17988349

C₂

yes

maximal, characteristic

#17988349

C₂

yes

maximal, characteristic

#17988341

C₂

yes

maximal, characteristic

#340×C₇₃

C₂

yes

maximal, characteristic

C₂²×#10596831

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₃

C₂²

yes

characteristic

C₂×#11924007

—

C₂×C₄×Dic₇₃

—

C₂×C₄×Dic₇₃

C₂²

yes

characteristic

#14556525

C₄

yes

complemented

#14556498

—

#14556498

—

C₂²×C₈×C₇₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₃

C₂²

yes

characteristic

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

D₄

yes

characteristic

C₂²×Dic₇₃

D₄

yes

—

—

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂×C₄

yes

—

C₄⋊Dic₇₃

C₂×C₄

yes

—

#11924007

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×C₄

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

Q₈

yes

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₈

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

Q₁₆

yes

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

C₄×D₄

yes

characteristic

C₂²×C₇₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₇₃

#123

yes

characteristic

C₂²×C₇₃

C₂×Q₁₆

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#117

yes

characteristic

C₄×C₇₃

C₄×Q₈

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

#387

yes

characteristic

C₂×C₇₃

C₄×Q₁₆

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×#107

—

#389

—

100

C₄×C₄⋊C₄

—

101

#340

—

102

#340

—

103

#340

D₇₃

yes

characteristic

104

#336

—

105

C₂²×C₈

—

106

C₂²×C₈

C₂×D₇₃

yes

characteristic

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

146

—

110

C₂×C₄⋊C₄

C₂×D₇₃

yes

characteristic

111

C₂×C₄⋊C₄

C₂×D₇₃

yes

characteristic

112

C₂×C₄²

—

113

C₂×C₄²

—

146

—

114

#107

—

115

#95

—

146

—

116

#95

—

146

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

146

—

122

C₂²×C₄

—

146

—

123

C₂²×C₄

C₂²×D₇₃

yes

characteristic

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

146

—

127

C₂×C₈

—

146

—

128

C₂×C₈

—

146

—

129

C₂×C₈

C₄×D₇₃

yes

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

292

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

146

—

137

C₄⋊C₄

—

146

—

138

C₄⋊C₄

Dic₁₄₆

yes

—

139

C₄²

—

292

—

140

C₄²

—

146

—

141

C₄²

—

146

—

142

C₂³

D₄×D₇₃

yes

characteristic

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

292

—

150

C₂×C₄

—

292

—

151

C₂×C₄

—

292

—

152

C₂×C₄

—

292

—

153

C₂×C₄

—

154

C₂×C₄

—

146

—

155

C₂×C₄

—

146

—

156

C₂×C₄

—

146

—

157

C₂×C₄

—

146

—

158

C₂×C₄

—

146

—

159

C₂×C₄

—

146

—

160

C₂×C₄

—

146

—

161

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

162

C₂×C₄

C₂×Dic₁₄₆

yes

characteristic

163

C₂×C₄

#11924027

yes

characteristic

164

C₂×C₄

#11924027

yes

characteristic

165

C₂×C₄

#11924028

yes

characteristic

166

C₂×C₄

#11924029

yes

characteristic

167

C₈

—

168

C₈

—

292

—

169

C₂²

#14556553

yes

characteristic

170

C₂²

#14556568

yes

characteristic

171

C₂²

#14556573

yes

characteristic

172

C₂²

Q₁₆×D₇₃

yes

characteristic

173

C₂²

#14556610

yes

characteristic

174

C₂²

#14556618

yes

characteristic

175

C₂²

#14556618

yes

characteristic

176

C₄

—

177

C₄

—

178

C₄

—

292

—

179

C₄

—

292

—

180

C₄

—

292

—

181

C₄

—

146

—

182

C₄

#14556547

yes

characteristic

183

C₄

C₄×Dic₁₄₆

yes

characteristic

184

C₄

#14556582

yes

characteristic

185

C₄

#14556586

yes

characteristic

186

C₂

#17988470

yes

characteristic

187

C₂

#17988681

yes

characteristic

188

C₂

#17988698

yes

characteristic

189

C₂

#17988880

yes

characteristic

190

C₂

#17988885

yes

characteristic

191

C₂

#17988909

yes

characteristic

192

C₂

#17988921

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	#340×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₄
hypercenter	#340	D₇₃
hypocenter	C₇₃	#1921
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₇₃

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

#340×C₇₃

C₂

socle

C₂³×C₇₃

C₂×D₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
14. #14556525	C₄	semidirect
97. C₇₃	#1921	semidirect

H ≅ G/N

type

14. #14556525

C₄

semidirect

97. C₇₃

#1921

semidirect

Subgroups — #22610522

Conjugacy classes of subgroups

Complements