Group #22619419 — subgroups

Subgroups — #22619419

Order 9344.

480 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14556624	C₂	1	2	yes	maximal
2	C₂×#14556591	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#11924028	C₂	1	1	yes	maximal, characteristic
4	#17988798	C₂	1	4	yes	maximal, complemented
5	#17988793	C₂	1	4	yes	maximal, complemented
6	#17988597	C₂	1	2	yes	maximal, complemented
7	#423×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11924028	C₂²	1	1	yes	characteristic
9	C₂²×Dic₁₄₆	C₂²	1	2	yes	—
10	#14556624	—	2	8	no	—
11	C₂×Dic₂₉₂	C₂²	1	4	yes	—
12	C₂×#11924017	C₂²	1	4	yes	—
13	#14556591	—	2	1	no	—
14	#14556591	—	2	1	no	—
15	#14556591	C₂²	1	4	yes	—
16	C₄×Dic₁₄₆	C₂²	1	2	yes	—
17	C₄×D₂₉₂	C₂²	1	2	yes	—
18	C₄×C₂²⋊D₇₃	—	2	2	no	—
19	#14556550	C₂²	1	1	yes	characteristic
20	#14556548	—	2	1	no	—
21	C₂×C₄⋊Dic₇₃	—	2	2	no	—
22	C₄²×D₇₃	—	2	1	no	—
23	#14556516	C₂²	1	8	yes	—
24	C₂×M₁₆×C₇₃	C₂²	1	2	yes	—
25	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
26	C₄⋊C₈×C₇₃	C₂²	1	4	yes	—
27	#11924028	—	2	2	no	—
28	#11924028	D₄	1	4	yes	—
29	C₂×Dic₁₄₆	—	2	4	no	—
30	C₂×Dic₁₄₆	—	2	4	no	—
31	C₂×Dic₁₄₆	—	2	2	no	—
32	C₂×Dic₁₄₆	C₂³	1	4	yes	—
33	C₂×Dic₁₄₆	C₂³	1	1	yes	characteristic
34	C₂×D₂₉₂	C₂³	1	1	yes	characteristic
35	C₂×C₂²⋊D₇₃	—	2	1	no	—
36	C₂²×Dic₇₃	—	2	2	no	—
37	C₂×C₄×D₇₃	—	2	1	no	—
38	C₂×C₄×D₇₃	—	2	2	no	—
39	Dic₂₉₂	—	2	8	no	—
40	#11924017	—	2	8	no	—
41	C₄⋊Dic₇₃	—	2	4	no	—
42	C₄⋊Dic₇₃	—	2	4	no	—
43	C₄⋊Dic₇₃	C₂³	1	2	yes	—
44	#11924007	—	2	2	no	—
45	C₂²⋊Dic₇₃	—	2	1	no	—
46	#11924005	—	2	2	no	—
47	C₄×Dic₇₃	—	2	1	no	—
48	C₄×Dic₇₃	—	2	2	no	—
49	C₂²×C₄×C₇₃	D₄	1	2	yes	—
50	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
51	M₁₆×C₇₃	—	2	4	no	—
52	C₂×C₈×C₇₃	C₂³	1	4	yes	—
53	C₄²×C₇₃	D₄	1	1	yes	characteristic
54	C₄²×C₇₃	D₄	1	1	yes	characteristic
55	C₄²×C₇₃	C₂³	1	2	yes	—
56	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×D₇₃	—	2	1	no	—
58	C₂×C₄×C₇₃	—	2	4	no	—
59	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₇₃	#41	1	2	yes	—
64	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
65	C₂²⋊D₇₃	—	2	4	no	—
66	C₂×Dic₇₃	—	2	1	no	—
67	C₂×Dic₇₃	—	2	2	no	—
68	C₂×Dic₇₃	—	2	4	no	—
69	C₂×Dic₇₃	—	4	4	no	—
70	D₂₉₂	—	2	1	no	—
71	D₂₉₂	C₂×D₄	1	2	yes	—
72	C₄×D₇₃	—	2	4	no	—
73	C₄×D₇₃	—	4	4	no	—
74	Dic₁₄₆	—	2	1	no	—
75	Dic₁₄₆	—	2	4	no	—
76	Dic₁₄₆	—	2	4	no	—
77	Dic₁₄₆	—	4	4	no	—
78	Dic₁₄₆	C₂×D₄	1	2	yes	—
79	C₈×C₇₃	—	2	4	no	—
80	C₂²×C₇₃	—	2	2	no	—
81	C₂²×C₇₃	#121	1	2	yes	—
82	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
83	C₂×D₇₃	—	2	2	no	—
84	C₂×D₇₃	—	4	2	no	—
85	C₄×C₇₃	—	2	2	no	—
86	C₄×C₇₃	#121	1	2	yes	—
87	C₄×C₇₃	#137	1	4	yes	complemented
88	C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
89	C₄×C₇₃	C₂×#41	1	1	yes	characteristic
90	Dic₇₃	—	2	2	no	—
91	Dic₇₃	—	4	4	no	—
92	Dic₇₃	—	4	2	no	—
93	C₂×C₇₃	—	2	2	no	—
94	C₂×C₇₃	C₂×#121	1	1	yes	characteristic
95	C₂×C₇₃	C₂×#137	1	2	yes	—
96	D₇₃	—	4	2	no	—
97	#3039	—	73	1	no	maximal, Sylow
98	C₇₃	#3039	1	1	yes	characteristic, complemented, Sylow
99	C₂×#137	—	73	2	no	—
100	C₂×C₄⋊Q₈	—	73	1	no	—
101	C₄×#41	—	73	1	no	—
102	#462	—	73	4	no	—
103	#461	—	73	4	no	—
104	#423	D₇₃	1	1	yes	characteristic, complemented
105	#419	—	73	2	no	—
106	C₂×#41	—	73	1	no	—
107	C₂²×Q₈	—	73	2	no	—
108	#137	—	146	8	no	—
109	C₂×Q₁₆	—	73	4	no	—
110	C₂×SD₁₆	—	73	4	no	—
111	C₂×M₁₆	C₂×D₇₃	1	2	yes	—
112	C₄⋊Q₈	—	73	4	no	—
113	C₄⋊Q₈	—	146	1	no	—
114	C₄⋊Q₈	—	146	1	no	—
115	C₄×Q₈	—	73	2	no	—
116	C₄×D₄	—	73	2	no	—
117	C₄×D₄	—	146	2	no	—
118	#117	—	73	1	no	—
119	#117	—	146	1	no	—
120	C₂×C₄⋊C₄	—	146	2	no	—
121	C₂×C₄²	—	146	1	no	—
122	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
123	C₄⋊C₈	C₂×D₇₃	1	4	yes	—
124	#103	—	73	8	no	—
125	#41	—	73	4	no	—
126	#41	—	146	2	no	—
127	C₂×Q₈	—	73	1	no	—
128	C₂×Q₈	—	73	4	no	—
129	C₂×Q₈	—	146	4	no	—
130	C₂×Q₈	—	146	4	no	—
131	C₂×Q₈	—	146	2	no	—
132	C₂×D₄	—	73	1	no	—
133	C₂×D₄	—	146	1	no	—
134	C₂²×C₄	—	146	2	no	—
135	C₂²×C₄	—	146	2	no	—
136	C₂²×C₄	—	146	1	no	—
137	C₂²×C₄	D₂₉₂	1	2	yes	—
138	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
139	Q₁₆	—	146	8	no	—
140	SD₁₆	—	146	8	no	—
141	M₁₆	—	2	4	no	—
142	C₂×C₈	C₂²×D₇₃	1	4	yes	—
143	C₄⋊C₄	—	73	2	no	—
144	C₄⋊C₄	—	146	2	no	—
145	C₄⋊C₄	—	146	4	no	—
146	C₄⋊C₄	—	146	4	no	—
147	C₂²⋊C₄	—	146	1	no	—
148	C₂²⋊C₄	—	146	2	no	—
149	C₄²	—	146	1	no	—
150	C₄²	—	146	2	no	—
151	C₄²	D₂₉₂	1	1	yes	characteristic
152	C₄²	D₂₉₂	1	1	yes	characteristic
153	C₄²	C₂²×D₇₃	1	2	yes	—
154	C₂³	—	146	1	no	—
155	C₂³	C₂×D₂₉₂	1	1	yes	characteristic
156	Q₈	—	292	4	no	—
157	Q₈	—	73	2	no	—
158	Q₈	—	146	4	no	—
159	Q₈	—	146	4	no	—
160	Q₈	—	146	1	no	—
161	D₄	—	73	2	no	—
162	D₄	—	146	4	no	—
163	D₄	—	146	1	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	292	4	no	—
166	C₂×C₄	—	292	4	no	—
167	C₂×C₄	—	146	2	no	—
168	C₂×C₄	—	146	4	no	—
169	C₂×C₄	—	146	1	no	—
170	C₂×C₄	—	146	4	no	—
171	C₂×C₄	D₄×D₇₃	1	2	yes	—
172	C₂×C₄	C₂×D₂₉₂	1	2	yes	—
173	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
174	C₂×C₄	C₂×D₂₉₂	1	2	yes	—
175	C₂×C₄	#11924029	1	2	yes	—
176	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
177	C₈	—	2	4	no	—
178	C₂²	—	2	2	no	—
179	C₂²	—	292	2	no	—
180	C₂²	—	146	2	no	—
181	C₂²	#14556566	1	2	yes	—
182	C₂²	C₂²×D₂₉₂	1	1	yes	characteristic
183	C₄	—	2	2	no	—
184	C₄	—	292	2	no	—
185	C₄	—	292	4	no	—
186	C₄	—	146	2	no	—
187	C₄	#14556566	1	2	yes	—
188	C₄	#14556624	1	4	yes	complemented
189	C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
190	C₄	C₂×#11924029	1	1	yes	characteristic
191	C₂	—	2	2	no	—
192	C₂	—	292	2	no	—
193	C₂	C₂×#14556566	1	1	yes	characteristic
194	C₂	C₂×#14556624	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556624

C₂

yes

maximal

C₂×#14556591

C₂

yes

maximal, characteristic, complemented

C₄×#11924028

C₂

yes

maximal, characteristic

#17988798

C₂

yes

maximal, complemented

#17988793

C₂

yes

maximal, complemented

#17988597

C₂

yes

maximal, complemented

#423×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#11924028

C₂²

yes

characteristic

C₂²×Dic₁₄₆

C₂²

yes

—

#14556624

—

C₂×Dic₂₉₂

C₂²

yes

—

C₂×#11924017

C₂²

yes

—

#14556591

—

#14556591

—

#14556591

C₂²

yes

—

C₄×Dic₁₄₆

C₂²

yes

—

C₄×D₂₉₂

C₂²

yes

—

C₄×C₂²⋊D₇₃

—

#14556550

C₂²

yes

characteristic

#14556548

—

C₂×C₄⋊Dic₇₃

—

C₄²×D₇₃

—

#14556516

C₂²

yes

—

C₂×M₁₆×C₇₃

C₂²

yes

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

C₂²

yes

—

#11924028

—

#11924028

D₄

yes

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

C₂³

yes

—

C₂×Dic₁₄₆

C₂³

yes

characteristic

C₂×D₂₉₂

C₂³

yes

characteristic

C₂×C₂²⋊D₇₃

—

C₂²×Dic₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

Dic₂₉₂

—

#11924017

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂³

yes

—

#11924007

—

C₂²⋊Dic₇₃

—

#11924005

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

D₄

yes

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

C₂³

yes

—

C₄²×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

C₂³

yes

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

C₂²×D₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

#41

yes

—

C₂×C₄×C₇₃

C₂⁴

yes

characteristic

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

C₂×D₄

yes

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

C₂×D₄

yes

—

C₈×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#121

yes

—

C₂²×C₇₃

C₂²×D₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#121

yes

—

C₄×C₇₃

#137

yes

complemented

C₄×C₇₃

C₂²×D₄

yes

characteristic

C₄×C₇₃

C₂×#41

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

C₂×#121

yes

characteristic

C₂×C₇₃

C₂×#137

yes

—

D₇₃

—

#3039

—

maximal, Sylow

C₇₃

#3039

yes

characteristic, complemented, Sylow

C₂×#137

—

100

C₂×C₄⋊Q₈

—

101

C₄×#41

—

102

#462

—

103

#461

—

104

#423

D₇₃

yes

characteristic, complemented

105

#419

—

106

C₂×#41

—

107

C₂²×Q₈

—

108

#137

—

146

—

109

C₂×Q₁₆

—

110

C₂×SD₁₆

—

111

C₂×M₁₆

C₂×D₇₃

yes

—

112

C₄⋊Q₈

—

113

C₄⋊Q₈

—

146

—

114

C₄⋊Q₈

—

146

—

115

C₄×Q₈

—

116

C₄×D₄

—

117

C₄×D₄

—

146

—

118

#117

—

119

#117

—

146

—

120

C₂×C₄⋊C₄

—

146

—

121

C₂×C₄²

—

146

—

122

C₂×C₄²

C₂×D₇₃

yes

characteristic

123

C₄⋊C₈

C₂×D₇₃

yes

—

124

#103

—

125

#41

—

126

#41

—

146

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂×Q₈

—

146

—

130

C₂×Q₈

—

146

—

131

C₂×Q₈

—

146

—

132

C₂×D₄

—

133

C₂×D₄

—

146

—

134

C₂²×C₄

—

146

—

135

C₂²×C₄

—

146

—

136

C₂²×C₄

—

146

—

137

C₂²×C₄

D₂₉₂

yes

—

138

C₂²×C₄

C₂²×D₇₃

yes

characteristic

139

Q₁₆

—

146

—

140

SD₁₆

—

146

—

141

M₁₆

—

142

C₂×C₈

C₂²×D₇₃

yes

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

146

—

145

C₄⋊C₄

—

146

—

146

C₄⋊C₄

—

146

—

147

C₂²⋊C₄

—

146

—

148

C₂²⋊C₄

—

146

—

149

C₄²

—

146

—

150

C₄²

—

146

—

151

C₄²

D₂₉₂

yes

characteristic

152

C₄²

D₂₉₂

yes

characteristic

153

C₄²

C₂²×D₇₃

yes

—

154

C₂³

—

146

—

155

C₂³

C₂×D₂₉₂

yes

characteristic

156

Q₈

—

292

—

157

Q₈

—

158

Q₈

—

146

—

159

Q₈

—

146

—

160

Q₈

—

146

—

161

D₄

—

162

D₄

—

146

—

163

D₄

—

146

—

164

C₂×C₄

—

165

C₂×C₄

—

292

—

166

C₂×C₄

—

292

—

167

C₂×C₄

—

146

—

168

C₂×C₄

—

146

—

169

C₂×C₄

—

146

—

170

C₂×C₄

—

146

—

171

C₂×C₄

D₄×D₇₃

yes

—

172

C₂×C₄

C₂×D₂₉₂

yes

—

173

C₂×C₄

C₂×D₂₉₂

yes

characteristic

174

C₂×C₄

C₂×D₂₉₂

yes

—

175

C₂×C₄

#11924029

yes

—

176

C₂×C₄

C₂³×D₇₃

yes

characteristic

177

C₈

—

178

C₂²

—

179

C₂²

—

292

—

180

C₂²

—

146

—

181

C₂²

#14556566

yes

—

182

C₂²

C₂²×D₂₉₂

yes

characteristic

183

C₄

—

184

C₄

—

292

—

185

C₄

—

292

—

186

C₄

—

146

—

187

C₄

#14556566

yes

—

188

C₄

#14556624

yes

complemented

189

C₄

C₂×D₄×D₇₃

yes

characteristic

190

C₄

C₂×#11924029

yes

characteristic

191

C₂

—

192

C₂

—

292

—

193

C₂

C₂×#14556566

yes

characteristic

194

C₂

C₂×#14556624

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₃
Fitting	#423×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₄
hypercenter	#423	D₇₃
hypocenter	C₇₃	#3039
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂²×D₂₉₂

commutator

C₂×C₄×C₇₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₃

Fitting

#423×C₇₃

C₂

socle

C₂²×C₇₃

C₂²×D₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
2. C₂×#14556591	C₂	semidirect
4. #17988798	C₂	semidirect
5. #17988793	C₂	semidirect
6. #17988597	C₂	semidirect
7. #423×C₇₃	C₂	semidirect
87. C₄×C₇₃	#137	semidirect
98. C₇₃	#3039	semidirect
104. #423	D₇₃	semidirect
188. C₄	#14556624	semidirect

H ≅ G/N

type

2. C₂×#14556591

C₂

semidirect

4. #17988798

C₂

semidirect

5. #17988793

C₂

semidirect

6. #17988597

C₂

semidirect

7. #423×C₇₃

C₂

semidirect

87. C₄×C₇₃

#137

semidirect

98. C₇₃

#3039

semidirect

104. #423

D₇₃

semidirect

188. C₄

#14556624

semidirect

Subgroups — #22619419

Conjugacy classes of subgroups

Complements