Group #22611586 — subgroups

Subgroups — #22611586

Order 9344.

234 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 65 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	#17989158	C₂	1	1	yes	maximal, characteristic
2	#17988561	C₂	1	1	yes	maximal, characteristic
3	#17988378	C₂	1	4	yes	maximal, complemented
4	#413×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₄₆	C₄	1	1	yes	characteristic, complemented
6	#14556589	—	2	1	no	—
7	#14556578	—	2	1	no	—
8	#14556572	C₄	1	2	yes	complemented
9	#14556572	C₂²	1	2	yes	—
10	#14556570	—	2	1	no	—
11	#14556570	—	2	1	no	—
12	#14556550	—	2	1	no	—
13	#14556550	C₄	1	1	yes	characteristic, complemented
14	#14556506	—	2	1	no	—
15	#14556506	C₂²	1	2	yes	—
16	C₂×#41×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×M₁₆×C₇₃	—	2	1	no	—
18	#101×C₇₃	—	2	1	no	—
19	#100×C₇₃	C₂²	1	2	yes	—
20	C₂×Dic₁₄₆	—	2	1	no	—
21	C₂×Dic₁₄₆	—	2	1	no	—
22	C₂×Dic₁₄₆	—	4	1	no	—
23	C₂×Dic₁₄₆	—	4	1	no	—
24	C₂²×Dic₇₃	C₂×C₄	1	2	yes	—
25	C₄⋊Dic₇₃	—	2	1	no	—
26	C₄⋊Dic₇₃	—	4	1	no	—
27	#11924007	—	2	2	no	—
28	#11924007	—	2	2	no	—
29	C₂²⋊Dic₇₃	—	2	2	no	—
30	C₂²⋊Dic₇₃	—	2	2	no	—
31	C₂²⋊Dic₇₃	—	2	2	no	—
32	C₂²⋊Dic₇₃	C₂×C₄	1	2	yes	—
33	C₄×Dic₇₃	—	2	1	no	—
34	C₄×Dic₇₃	—	4	1	no	—
35	#41×C₇₃	—	4	1	no	—
36	#41×C₇₃	—	4	1	no	—
37	C₂×Q₈×C₇₃	D₄	1	1	yes	characteristic
38	C₂×D₄×C₇₃	C₂×C₄	1	1	yes	characteristic
39	C₂×D₄×C₇₃	D₄	1	1	yes	characteristic
40	C₂×D₄×C₇₃	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
43	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
44	M₁₆×C₇₃	—	2	2	no	—
45	M₁₆×C₇₃	—	4	1	no	—
46	C₂×C₈×C₇₃	—	4	1	no	—
47	C₂³×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂³×C₇₃	C₂²×C₄	1	1	yes	characteristic
49	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
50	Q₈×C₇₃	—	4	1	no	—
51	D₄×C₇₃	—	4	1	no	—
52	D₄×C₇₃	—	4	1	no	—
53	D₄×C₇₃	—	4	1	no	—
54	C₂×C₄×C₇₃	—	2	1	no	—
55	C₂×C₄×C₇₃	—	2	1	no	—
56	C₂×C₄×C₇₃	—	2	1	no	—
57	C₂×C₄×C₇₃	—	2	1	no	—
58	C₂×C₄×C₇₃	—	2	1	no	—
59	C₂×C₄×C₇₃	—	2	1	no	—
60	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
64	C₂×Dic₇₃	—	2	2	no	—
65	C₂×Dic₇₃	—	2	2	no	—
66	C₂×Dic₇₃	—	4	2	no	—
67	C₂×Dic₇₃	—	4	2	no	—
68	Dic₁₄₆	—	4	2	no	—
69	Dic₁₄₆	—	4	1	no	—
70	Dic₁₄₆	—	4	1	no	—
71	C₈×C₇₃	—	4	2	no	—
72	C₂²×C₇₃	—	2	1	no	—
73	C₂²×C₇₃	—	2	1	no	—
74	C₂²×C₇₃	—	2	1	no	—
75	C₂²×C₇₃	—	4	1	no	—
76	C₂²×C₇₃	—	4	1	no	—
77	C₂²×C₇₃	—	4	1	no	—
78	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₄×C₇₃	—	2	1	no	—
80	C₄×C₇₃	—	2	1	no	—
81	C₄×C₇₃	—	2	1	no	—
82	C₄×C₇₃	#99	1	1	yes	characteristic
83	C₄×C₇₃	#99	1	1	yes	characteristic
84	Dic₇₃	—	4	2	no	—
85	Dic₇₃	—	4	2	no	—
86	Dic₇₃	—	8	2	no	—
87	C₂×C₇₃	—	2	1	no	—
88	C₂×C₇₃	—	4	1	no	—
89	C₂×C₇₃	—	4	1	no	—
90	C₂×C₇₃	—	4	1	no	—
91	C₂×C₇₃	C₂×#99	1	1	yes	characteristic
92	#2123	—	73	1	no	maximal, Sylow
93	C₇₃	#2123	1	1	yes	characteristic, complemented, Sylow
94	#536	—	73	1	no	—
95	#413	D₇₃	1	1	yes	characteristic
96	#410	—	73	1	no	—
97	#352	—	73	4	no	—
98	C₂×#41	C₂×D₇₃	1	1	yes	characteristic
99	C₂²×Q₈	—	73	1	no	—
100	C₂×M₁₆	—	2	1	no	—
101	C₄⋊Q₈	—	146	1	no	—
102	#124	—	146	1	no	—
103	#123	—	73	2	no	—
104	#123	—	73	2	no	—
105	C₂²⋊Q₈	—	146	1	no	—
106	C₂²⋊Q₈	—	146	1	no	—
107	#117	—	73	1	no	—
108	#117	—	146	1	no	—
109	#101	—	2	1	no	—
110	#100	C₂×D₇₃	1	2	yes	—
111	#99	—	73	2	no	—
112	#99	—	146	1	no	—
113	#41	—	4	1	no	—
114	#41	—	4	1	no	—
115	C₂×Q₈	—	292	1	no	—
116	C₂×Q₈	—	292	1	no	—
117	C₂×Q₈	—	146	1	no	—
118	C₂×Q₈	—	146	1	no	—
119	C₂×Q₈	C₂²⋊D₇₃	1	1	yes	characteristic
120	C₂×D₄	C₄×D₇₃	1	1	yes	characteristic
121	C₂×D₄	D₂₉₂	1	1	yes	characteristic
122	C₂×D₄	C₂²×D₇₃	1	1	yes	characteristic
123	C₂²×C₄	—	73	2	no	—
124	C₂²×C₄	C₄×D₇₃	1	1	yes	characteristic
125	C₂²×C₄	D₂₉₂	1	1	yes	characteristic
126	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
127	M₁₆	—	2	2	no	—
128	M₁₆	—	4	1	no	—
129	C₂×C₈	—	4	1	no	—
130	C₄⋊C₄	—	292	1	no	—
131	C₄⋊C₄	—	146	2	no	—
132	C₄⋊C₄	—	146	1	no	—
133	C₄⋊C₄	—	146	2	no	—
134	C₂²⋊C₄	—	73	2	no	—
135	C₂²⋊C₄	—	146	2	no	—
136	C₂²⋊C₄	—	146	2	no	—
137	C₂²⋊C₄	—	146	2	no	—
138	C₄²	—	292	1	no	—
139	C₄²	—	146	1	no	—
140	C₂³	#11924005	1	1	yes	characteristic
141	C₂³	C₂×C₄×D₇₃	1	1	yes	characteristic
142	C₂³	C₂×D₂₉₂	1	1	yes	characteristic
143	Q₈	—	4	1	no	—
144	Q₈	—	292	2	no	—
145	Q₈	—	292	1	no	—
146	Q₈	—	292	1	no	—
147	D₄	—	4	1	no	—
148	D₄	—	4	1	no	—
149	D₄	—	4	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	292	2	no	—
157	C₂×C₄	—	292	2	no	—
158	C₂×C₄	—	146	2	no	—
159	C₂×C₄	—	146	2	no	—
160	C₂×C₄	#11924005	1	1	yes	characteristic
161	C₂×C₄	#11924005	1	1	yes	characteristic
162	C₂×C₄	#11924005	1	1	yes	characteristic
163	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
164	C₈	—	4	2	no	—
165	C₂²	—	2	1	no	—
166	C₂²	—	2	1	no	—
167	C₂²	—	2	1	no	—
168	C₂²	—	4	1	no	—
169	C₂²	—	4	1	no	—
170	C₂²	—	4	1	no	—
171	C₂²	C₂×#11924005	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	292	2	no	—
176	C₄	—	292	2	no	—
177	C₄	—	584	2	no	—
178	C₄	#14556505	1	1	yes	characteristic
179	C₄	#14556505	1	1	yes	characteristic
180	C₂	—	2	1	no	—
181	C₂	—	4	1	no	—
182	C₂	—	4	1	no	—
183	C₂	—	4	1	no	—
184	C₂	C₂×#14556505	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989158

C₂

yes

maximal, characteristic

#17988561

C₂

yes

maximal, characteristic

#17988378

C₂

yes

maximal, complemented

#413×C₇₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₆

C₄

yes

characteristic, complemented

#14556589

—

#14556578

—

#14556572

C₄

yes

complemented

#14556572

C₂²

yes

—

—

—

—

C₄

yes

characteristic, complemented

#14556506

—

#14556506

C₂²

yes

—

C₂×#41×C₇₃

C₂²

yes

characteristic

C₂×M₁₆×C₇₃

—

#101×C₇₃

—

#100×C₇₃

C₂²

yes

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

C₂×C₄

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924007

—

#11924007

—

C₂²⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

C₂²⋊Dic₇₃

C₂×C₄

yes

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

#41×C₇₃

—

#41×C₇₃

—

C₂×Q₈×C₇₃

D₄

yes

characteristic

C₂×D₄×C₇₃

C₂×C₄

yes

characteristic

C₂×D₄×C₇₃

D₄

yes

characteristic

C₂×D₄×C₇₃

C₂³

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂³×C₇₃

C₂²⋊C₄

yes

characteristic

C₂³×C₇₃

C₂²×C₄

yes

characteristic

C₂³×C₇₃

C₂×D₄

yes

characteristic

Q₈×C₇₃

—

D₄×C₇₃

—

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

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

C₂×#99

yes

—

C₇₃

yes

characteristic, complemented, Sylow

#536

—

#413

D₇₃

yes

characteristic

#410

—

#352

—

C₂×#41

C₂×D₇₃

yes

characteristic

C₂²×Q₈

—

100

C₂×M₁₆

—

101

C₄⋊Q₈

—

146

—

102

#124

—

146

—

103

#123

—

104

#123

—

105

C₂²⋊Q₈

—

146

—

106

C₂²⋊Q₈

—

146

—

107

#117

—

108

#117

—

146

—

109

#101

—

110

#100

C₂×D₇₃

yes

—

111

#99

—

112

#99

—

146

—

113

#41

—

114

#41

—

115

C₂×Q₈

—

292

—

116

C₂×Q₈

—

292

—

117

C₂×Q₈

—

146

—

118

C₂×Q₈

—

146

—

119

C₂×Q₈

C₂²⋊D₇₃

yes

characteristic

120

C₂×D₄

C₄×D₇₃

yes

characteristic

121

C₂×D₄

D₂₉₂

yes

characteristic

122

C₂×D₄

C₂²×D₇₃

yes

characteristic

123

C₂²×C₄

—

124

C₂²×C₄

C₄×D₇₃

yes

characteristic

125

C₂²×C₄

D₂₉₂

yes

characteristic

126

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

127

M₁₆

—

128

M₁₆

—

129

C₂×C₈

—

130

C₄⋊C₄

—

292

—

131

C₄⋊C₄

—

146

—

132

C₄⋊C₄

—

146

—

133

C₄⋊C₄

—

146

—

134

C₂²⋊C₄

—

135

C₂²⋊C₄

—

146

—

136

C₂²⋊C₄

—

146

—

137

C₂²⋊C₄

—

146

—

138

C₄²

—

292

—

139

C₄²

—

146

—

140

C₂³

#11924005

yes

characteristic

141

C₂³

C₂×C₄×D₇₃

yes

characteristic

142

C₂³

C₂×D₂₉₂

yes

characteristic

143

Q₈

—

144

Q₈

—

292

—

145

Q₈

—

292

—

146

Q₈

—

292

—

147

D₄

—

148

D₄

—

149

D₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

292

—

157

C₂×C₄

—

292

—

158

C₂×C₄

—

146

—

159

C₂×C₄

—

146

—

160

C₂×C₄

#11924005

yes

characteristic

161

C₂×C₄

#11924005

yes

characteristic

162

C₂×C₄

#11924005

yes

characteristic

163

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

164

C₈

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

C₂×#11924005

yes

characteristic

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

292

—

176

C₄

—

292

—

177

C₄

—

584

—

178

C₄

#14556505

yes

characteristic

179

C₄

#14556505

yes

characteristic

180

C₂

—

181

C₂

—

182

C₂

—

183

C₂

—

184

C₂

C₂×#14556505

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#14556505
commutator	C₂³×C₇₃	C₂²×C₄
Frattini	C₂×D₄	C₂²×D₇₃
Fitting	#413×C₇₃	C₂
socle	C₂×C₇₃	C₂×#99
hypercenter	#413	D₇₃
hypocenter	C₇₃	#2123
LCS	G ▷ C₂³×C₇₃ ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #413

subgroup

G/H

center

C₂

C₂×#14556505

commutator

C₂³×C₇₃

C₂²×C₄

Frattini

C₂×D₄

C₂²×D₇₃

Fitting

#413×C₇₃

C₂

socle

C₂×C₇₃

C₂×#99

D₇₃

C₇₃

G ▷ C₂³×C₇₃ ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #413

N	H ≅ G/N	type
3. #17988378	C₂	semidirect
5. C₂²×Dic₁₄₆	C₄	semidirect
8. #14556572	C₄	semidirect
13. #14556550	C₄	semidirect
93. C₇₃	#2123	semidirect

H ≅ G/N

type

3. #17988378

C₂

semidirect

5. C₂²×Dic₁₄₆

C₄

semidirect

8. #14556572

C₄

semidirect

13. #14556550

C₄

semidirect

93. C₇₃

#2123

semidirect

Subgroups — #22611586

Conjugacy classes of subgroups

Complements