Group #22611413 — subgroups

Subgroups — #22611413

Order 9344.

214 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 83 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×#14556524	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556522	C₂	1	1	yes	maximal, characteristic
3	#17988485	C₂	1	1	yes	maximal, characteristic
4	#17988349	C₂	1	1	yes	maximal, characteristic
5	#17988349	C₂	1	1	yes	maximal, characteristic
6	#17988342	C₂	1	1	yes	maximal, characteristic
7	#382×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10596831	C₂²	1	1	yes	characteristic
9	C₂²×#10596831	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
12	C₂×#11924007	—	2	1	no	—
13	#14556524	—	2	2	no	—
14	#14556522	—	2	2	no	—
15	#14556498	—	2	1	no	—
16	C₂×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
19	#95×C₇₃	—	2	1	no	—
20	#95×C₇₃	—	2	1	no	—
21	C₂²×Dic₇₃	—	2	1	no	—
22	C₂²×Dic₇₃	—	2	1	no	—
23	C₂×#10596831	—	2	1	no	—
24	C₂×#10596831	—	2	1	no	—
25	C₂×#10596831	—	2	1	no	—
26	C₂×#10596831	—	2	1	no	—
27	C₂×#10596831	Q₈	1	2	yes	—
28	C₂×#10596831	Q₈	1	2	yes	—
29	C₄⋊Dic₇₃	—	2	1	no	—
30	C₄⋊Dic₇₃	—	2	1	no	—
31	C₄⋊Dic₇₃	—	2	1	no	—
32	C₄⋊Dic₇₃	D₄	1	2	yes	—
33	#11924007	—	4	2	no	—
34	C₂²×C₄×C₇₃	—	2	1	no	—
35	C₂²×C₄×C₇₃	—	2	1	no	—
36	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₇₃	—	2	1	no	—
40	C₄⋊C₄×C₇₃	—	2	1	no	—
41	C₄⋊C₄×C₇₃	—	2	1	no	—
42	C₄⋊C₄×C₇₃	—	2	1	no	—
43	C₄²×C₇₃	—	2	1	no	—
44	C₄²×C₇₃	—	2	1	no	—
45	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₇₃	—	2	1	no	—
47	C₂×C₄×C₇₃	—	2	1	no	—
48	C₂×C₄×C₇₃	—	2	1	no	—
49	C₂×C₄×C₇₃	—	2	1	no	—
50	C₂×C₄×C₇₃	—	2	1	no	—
51	C₂×C₄×C₇₃	—	2	1	no	—
52	C₂×C₄×C₇₃	—	4	1	no	—
53	C₂×C₄×C₇₃	—	4	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	—	4	1	no	—
58	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
62	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
63	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
64	C₂×Dic₇₃	—	2	2	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	—	4	1	no	—
70	#10596831	—	2	2	no	—
71	#10596831	—	2	2	no	—
72	C₂²×C₇₃	#121	1	1	yes	characteristic
73	C₂²×C₇₃	#123	1	1	yes	characteristic
74	C₂²×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
75	C₂²×C₇₃	#135	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	#137	1	1	yes	characteristic
79	C₄×C₇₃	—	4	1	no	—
80	C₄×C₇₃	—	4	1	no	—
81	C₄×C₇₃	—	8	1	no	—
82	C₄×C₇₃	—	8	1	no	—
83	C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
84	C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
85	C₄×C₇₃	#125	1	1	yes	characteristic
86	C₄×C₇₃	#125	1	1	yes	characteristic
87	Dic₇₃	—	4	2	no	—
88	Dic₇₃	—	8	1	no	—
89	C₂×C₇₃	#398	1	1	yes	characteristic
90	C₂×C₇₃	#463	1	1	yes	characteristic
91	C₂×C₇₃	#464	1	1	yes	characteristic
92	C₂×C₇₃	#482	1	1	yes	characteristic
93	C₂×C₇₃	#485	1	1	yes	characteristic
94	C₂×C₇₃	#499	1	1	yes	characteristic
95	C₂×C₇₃	#501	1	1	yes	characteristic
96	#2085	—	73	1	no	maximal, Sylow
97	C₇₃	#2085	1	1	yes	characteristic, complemented, Sylow
98	C₂×#107	—	73	1	no	—
99	C₂×#106	—	73	1	no	—
100	#389	—	73	1	no	—
101	#382	D₇₃	1	1	yes	characteristic
102	#340	—	73	1	no	—
103	#340	—	73	1	no	—
104	#336	—	73	1	no	—
105	C₂²×C₈	—	73	1	no	—
106	C₂²×C₈	—	73	1	no	—
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	C₂×D₇₃	1	1	yes	characteristic
113	#107	—	146	2	no	—
114	#106	—	146	2	no	—
115	#95	—	2	1	no	—
116	#95	—	2	1	no	—
117	#95	—	146	1	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	—	146	1	no	—
121	C₂²×C₄	—	146	1	no	—
122	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
123	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
124	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
125	C₂×C₈	—	73	2	no	—
126	C₂×C₈	—	73	2	no	—
127	C₂×C₈	—	146	1	no	—
128	C₂×C₈	—	146	1	no	—
129	C₂×C₈	—	146	1	no	—
130	C₂×C₈	—	146	1	no	—
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	—	2	1	no	—
134	C₄⋊C₄	—	2	1	no	—
135	C₄⋊C₄	—	292	2	no	—
136	C₄⋊C₄	—	73	2	no	—
137	C₄⋊C₄	—	146	1	no	—
138	C₄⋊C₄	—	146	1	no	—
139	C₄⋊C₄	—	146	1	no	—
140	C₄²	—	2	1	no	—
141	C₄²	—	2	1	no	—
142	C₂³	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	292	1	no	—
156	C₂×C₄	—	292	1	no	—
157	C₂×C₄	—	292	1	no	—
158	C₂×C₄	—	292	1	no	—
159	C₂×C₄	—	292	1	no	—
160	C₂×C₄	—	146	2	no	—
161	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
162	C₂×C₄	Q₈×D₇₃	1	1	yes	characteristic
163	C₂×C₄	Q₈×D₇₃	1	1	yes	characteristic
164	C₂×C₄	#11924027	1	1	yes	characteristic
165	C₂×C₄	#11924028	1	1	yes	characteristic
166	C₂×C₄	#11924028	1	1	yes	characteristic
167	C₈	—	146	2	no	—
168	C₈	—	146	2	no	—
169	C₂²	#14556562	1	1	yes	characteristic
170	C₂²	#14556574	1	1	yes	characteristic
171	C₂²	#14556589	1	1	yes	characteristic
172	C₂²	#14556611	1	1	yes	characteristic
173	C₂²	#14556611	1	1	yes	characteristic
174	C₂²	#14556616	1	1	yes	characteristic
175	C₂²	#14556621	1	1	yes	characteristic
176	C₄	—	4	1	no	—
177	C₄	—	4	1	no	—
178	C₄	—	8	1	no	—
179	C₄	—	8	1	no	—
180	C₄	—	292	2	no	—
181	C₄	—	584	1	no	—
182	C₄	#14556569	1	1	yes	characteristic
183	C₄	#14556569	1	1	yes	characteristic
184	C₄	#14556581	1	1	yes	characteristic
185	C₄	#14556581	1	1	yes	characteristic
186	C₂	#17988518	1	1	yes	characteristic
187	C₂	#17988799	1	1	yes	characteristic
188	C₂	#17988806	1	1	yes	characteristic
189	C₂	#17988911	1	1	yes	characteristic
190	C₂	#17988928	1	1	yes	characteristic
191	C₂	#17988986	1	1	yes	characteristic
192	C₂	#17988997	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556524

C₂

yes

maximal, characteristic

C₂×#14556522

C₂

yes

maximal, characteristic

#17988485

C₂

yes

maximal, characteristic

#17988349

C₂

yes

maximal, characteristic

#17988349

C₂

yes

maximal, characteristic

#17988342

C₂

yes

maximal, characteristic

#382×C₇₃

C₂

yes

maximal, characteristic

C₂²×#10596831

C₂²

yes

characteristic

C₂²×#10596831

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₃

C₂²

yes

—

—

—

—

C₂×C₄⋊C₄×C₇₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₃

C₂²

yes

characteristic

C₂×C₄²×C₇₃

C₂²

yes

characteristic

#95×C₇₃

—

#95×C₇₃

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

—

—

—

—

Q₈

yes

—

C₂×#10596831

Q₈

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

D₄

yes

—

#11924007

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

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

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₈

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

—

#10596831

—

#10596831

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

C₄⋊Q₈

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂²⋊Q₈

yes

characteristic

C₄×C₇₃

C₂²⋊Q₈

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

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×#107

—

C₂×#106

—

100

#389

—

101

#382

D₇₃

yes

characteristic

102

#340

—

103

#340

—

104

#336

—

105

C₂²×C₈

—

106

C₂²×C₈

—

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

C₂×D₇₃

yes

characteristic

113

#107

—

146

—

114

#106

—

146

—

115

#95

—

116

#95

—

117

#95

—

146

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

146

—

121

C₂²×C₄

—

146

—

122

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

123

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

124

C₂²×C₄

C₂²×D₇₃

yes

characteristic

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

146

—

128

C₂×C₈

—

146

—

129

C₂×C₈

—

146

—

130

C₂×C₈

—

146

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

292

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

146

—

138

C₄⋊C₄

—

146

—

139

C₄⋊C₄

—

146

—

140

C₄²

—

141

C₄²

—

142

C₂³

C₂×C₂²⋊D₇₃

yes

characteristic

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

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

—

292

—

158

C₂×C₄

—

292

—

159

C₂×C₄

—

292

—

160

C₂×C₄

—

146

—

161

C₂×C₄

D₄×D₇₃

yes

characteristic

162

C₂×C₄

Q₈×D₇₃

yes

characteristic

163

C₂×C₄

Q₈×D₇₃

yes

characteristic

164

C₂×C₄

#11924027

yes

characteristic

165

C₂×C₄

#11924028

yes

characteristic

166

C₂×C₄

#11924028

yes

characteristic

167

C₈

—

146

—

168

C₈

—

146

—

169

C₂²

#14556562

yes

characteristic

170

C₂²

#14556574

yes

characteristic

171

C₂²

#14556589

yes

characteristic

172

C₂²

#14556611

yes

characteristic

173

C₂²

#14556611

yes

characteristic

174

C₂²

#14556616

yes

characteristic

175

C₂²

#14556621

yes

characteristic

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

292

—

181

C₄

—

584

—

182

C₄

#14556569

yes

characteristic

183

C₄

#14556569

yes

characteristic

184

C₄

#14556581

yes

characteristic

185

C₄

#14556581

yes

characteristic

186

C₂

#17988518

yes

characteristic

187

C₂

#17988799

yes

characteristic

188

C₂

#17988806

yes

characteristic

189

C₂

#17988911

yes

characteristic

190

C₂

#17988928

yes

characteristic

191

C₂

#17988986

yes

characteristic

192

C₂

#17988997

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₃
commutator	C₂²×C₄×C₇₃	C₂³
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	#382×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₄
hypercenter	#382	D₇₃
hypocenter	C₇₃	#2085
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₃

commutator

C₂²×C₄×C₇₃

C₂³

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

#382×C₇₃

C₂

socle

C₂³×C₇₃

C₂×D₄

D₇₃

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
97. C₇₃	#2085	semidirect

H ≅ G/N

type

97. C₇₃

#2085

semidirect

Subgroups — #22611413

Conjugacy classes of subgroups

Complements