Group #21456273 — subgroups

Subgroups — #21456273

Order 7808.

552 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 145 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×#14519806	C₂	1	4	yes	maximal, complemented
2	#16840310	C₂	1	1	yes	maximal, characteristic
3	C₄⋊Q₈×D₆₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#16840153	C₂	1	4	yes	maximal, complemented
5	#16840142	C₂	1	1	yes	maximal, characteristic, complemented
6	#16840141	C₂	1	1	yes	maximal, characteristic, complemented
7	#16840134	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#11758189	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#492	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11758224	C₂²	1	2	yes	—
11	C₂×Q₈×D₆₁	C₂²	1	2	yes	—
12	#14519806	—	2	16	no	—
13	C₂×Dic₂₄₄	C₂²	1	4	yes	—
14	C₂×#11758202	C₂²	1	4	yes	—
15	C₂×#11758200	C₂²	1	4	yes	—
16	C₂×#11758189	C₂²	1	2	yes	—
17	#14519777	C₂²	1	1	yes	characteristic
18	#14519777	C₂²	1	1	yes	characteristic
19	#14519776	—	2	2	no	—
20	#14519775	C₂²	1	2	yes	—
21	#14519764	C₂²	1	2	yes	—
22	#14519748	—	2	2	no	—
23	D₄×Dic₆₁	—	2	2	no	—
24	C₄⋊C₄×D₆₁	—	2	2	no	—
25	C₄²×D₆₁	C₂²	1	1	yes	characteristic
26	#14519687	C₂²	1	2	yes	—
27	C₄×#92418	C₂²	1	1	yes	characteristic
28	C₂×SD₁₆×C₆₁	C₂²	1	4	yes	—
29	C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
30	#127×C₆₁	C₂²	1	1	yes	characteristic
31	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
32	C₂×Q₈×C₆₁	C₂³	1	2	yes	—
33	C₂×D₄×C₆₁	—	2	2	no	—
34	C₂×D₄×C₆₁	C₂³	1	2	yes	—
35	C₂×C₂²⋊D₆₁	—	2	2	no	—
36	C₂²×Dic₆₁	—	2	2	no	—
37	Q₈×D₆₁	—	2	8	no	—
38	#11758224	—	2	8	no	—
39	C₂×C₄×D₆₁	—	2	2	no	—
40	C₂×C₄×D₆₁	D₄	1	2	yes	—
41	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
42	C₂×Dic₁₂₂	—	2	2	no	—
43	C₂×Dic₁₂₂	C₂³	1	2	yes	—
44	C₂×Dic₁₂₂	C₂³	1	2	yes	—
45	SD₁₆×C₆₁	—	2	8	no	—
46	C₂×C₈×C₆₁	C₂³	1	2	yes	—
47	C₄⋊C₄×C₆₁	—	2	2	no	—
48	C₄²×C₆₁	C₂³	1	1	yes	characteristic
49	C₂²⋊Dic₆₁	—	2	4	no	—
50	#11758202	—	2	8	no	—
51	#11758200	—	2	8	no	—
52	C₄⋊Dic₆₁	—	2	2	no	—
53	C₄⋊Dic₆₁	—	2	2	no	—
54	#11758196	—	2	4	no	—
55	C₄×Dic₆₁	D₄	1	2	yes	—
56	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
57	C₂×#92418	C₂³	1	2	yes	—
58	Dic₂₄₄	—	2	8	no	—
59	#11758189	D₄	1	8	yes	complemented
60	C₂³×C₆₁	—	2	2	no	—
61	C₂²×D₆₁	C₂×D₄	1	1	yes	characteristic
62	Q₈×C₆₁	—	2	4	no	—
63	D₄×C₆₁	—	2	4	no	—
64	D₄×C₆₁	—	4	4	no	—
65	C₂×C₄×C₆₁	—	2	2	no	—
66	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
68	C₂²⋊D₆₁	—	4	4	no	—
69	C₂×Dic₆₁	—	2	2	no	—
70	C₂×Dic₆₁	—	2	2	no	—
71	C₂×Dic₆₁	—	4	4	no	—
72	C₂×Dic₆₁	C₂×D₄	1	2	yes	—
73	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
74	C₄×D₆₁	—	2	4	no	—
75	C₄×D₆₁	—	4	4	no	—
76	C₄×D₆₁	C₂×D₄	1	4	yes	—
77	Dic₁₂₂	—	2	4	no	—
78	Dic₁₂₂	—	2	4	no	—
79	Dic₁₂₂	—	4	4	no	—
80	C₈×C₆₁	C₂×D₄	1	4	yes	—
81	#92418	C₂×D₄	1	4	yes	—
82	C₂²×C₆₁	—	4	2	no	—
83	C₂²×C₆₁	—	4	4	no	—
84	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
85	C₂×D₆₁	—	2	2	no	—
86	C₂×D₆₁	#127	1	2	yes	—
87	C₄×C₆₁	—	4	2	no	—
88	C₄×C₆₁	#137	1	4	yes	complemented
89	C₄×C₆₁	C₂²×D₄	1	2	yes	—
90	Dic₆₁	—	2	2	no	—
91	Dic₆₁	—	4	2	no	—
92	Dic₆₁	—	4	2	no	—
93	Dic₆₁	#127	1	2	yes	—
94	#3152	—	61	1	no	maximal, Sylow
95	C₂×C₆₁	—	8	2	no	—
96	C₂×C₆₁	C₂×#127	1	1	yes	characteristic
97	C₂×C₆₁	C₂×#137	1	2	yes	—
98	D₆₁	—	2	2	no	—
99	C₂×#137	—	61	4	no	—
100	#532	—	61	1	no	—
101	C₂×C₄⋊Q₈	—	61	1	no	—
102	#497	—	61	4	no	—
103	#494	—	61	1	no	—
104	#494	—	61	1	no	—
105	#492	—	61	1	no	—
106	#492	D₆₁	1	1	yes	characteristic, complemented
107	C₄×M₁₆	—	61	1	no	—
108	C₆₁	#3152	1	1	yes	characteristic, complemented, Sylow
109	C₂×#41	—	61	2	no	—
110	C₂²×Q₈	—	61	2	no	—
111	#137	—	122	16	no	—
112	C₂×Q₁₆	—	61	4	no	—
113	C₂×Q₁₆	—	61	4	no	—
114	C₂×SD₁₆	—	61	4	no	—
115	C₂×SD₁₆	C₂×D₆₁	1	4	yes	—
116	C₂×M₁₆	—	61	2	no	—
117	C₄⋊Q₈	—	61	2	no	—
118	C₄⋊Q₈	—	61	1	no	—
119	C₄⋊Q₈	—	61	1	no	—
120	C₄⋊Q₈	—	122	2	no	—
121	C₄⋊Q₈	C₂×D₆₁	1	1	yes	characteristic
122	#127	C₂×D₆₁	1	1	yes	characteristic
123	#124	—	61	2	no	—
124	#121	—	122	2	no	—
125	C₄×D₄	—	122	2	no	—
126	C₂×C₄⋊C₄	—	122	2	no	—
127	C₂×C₄²	—	61	1	no	—
128	#97	—	61	2	no	—
129	C₄×C₈	—	61	1	no	—
130	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
131	#41	—	122	8	no	—
132	C₂×Q₈	—	61	2	no	—
133	C₂×Q₈	—	61	2	no	—
134	C₂×Q₈	—	122	2	no	—
135	C₂×Q₈	—	122	8	no	—
136	C₂×Q₈	C₂²×D₆₁	1	2	yes	—
137	C₂×D₄	—	2	2	no	—
138	C₂×D₄	—	122	2	no	—
139	C₂×D₄	C₂²×D₆₁	1	2	yes	—
140	C₂²×C₄	—	61	1	no	—
141	C₂²×C₄	—	61	2	no	—
142	C₂²×C₄	—	122	2	no	—
143	C₂²×C₄	—	122	2	no	—
144	Q₁₆	—	122	8	no	—
145	Q₁₆	—	122	8	no	—
146	SD₁₆	—	2	8	no	—
147	SD₁₆	—	122	8	no	—
148	M₁₆	—	61	8	no	—
149	C₂×C₈	—	61	2	no	—
150	C₂×C₈	C₂²×D₆₁	1	2	yes	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	122	4	no	—
153	C₄⋊C₄	—	122	2	no	—
154	C₄⋊C₄	—	122	2	no	—
155	C₂²⋊C₄	—	122	4	no	—
156	C₄²	—	61	2	no	—
157	C₄²	—	61	1	no	—
158	C₄²	C₂²×D₆₁	1	1	yes	characteristic
159	C₂³	—	2	2	no	—
160	C₂³	—	61	1	no	—
161	Q₈	—	2	4	no	—
162	Q₈	—	122	4	no	—
163	Q₈	—	122	4	no	—
164	Q₈	—	244	4	no	—
165	D₄	—	2	4	no	—
166	D₄	—	4	4	no	—
167	D₄	—	244	4	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	61	1	no	—
170	C₂×C₄	—	61	4	no	—
171	C₂×C₄	—	61	2	no	—
172	C₂×C₄	—	122	4	no	—
173	C₂×C₄	—	122	2	no	—
174	C₂×C₄	—	122	2	no	—
175	C₂×C₄	—	244	4	no	—
176	C₂×C₄	—	244	4	no	—
177	C₂×C₄	D₄×D₆₁	1	2	yes	—
178	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
179	C₈	—	61	4	no	—
180	C₈	D₄×D₆₁	1	4	yes	—
181	C₂²	—	4	4	no	—
182	C₂²	—	4	2	no	—
183	C₂²	—	61	2	no	—
184	C₂²	—	122	2	no	—
185	C₂²	C₂×D₄×D₆₁	1	1	yes	characteristic
186	C₄	—	4	2	no	—
187	C₄	—	61	2	no	—
188	C₄	—	122	2	no	—
189	C₄	—	244	2	no	—
190	C₄	—	244	2	no	—
191	C₄	#14519806	1	4	yes	complemented
192	C₄	C₂×D₄×D₆₁	1	2	yes	—
193	C₂	—	8	2	no	—
194	C₂	—	122	2	no	—
195	C₂	#127×D₆₁	1	1	yes	characteristic
196	C₂	C₂×#14519806	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519806

C₂

yes

maximal, complemented

#16840310

C₂

yes

maximal, characteristic

C₄⋊Q₈×D₆₁

C₂

yes

maximal, characteristic, complemented

#16840153

C₂

yes

maximal, complemented

#16840142

C₂

yes

maximal, characteristic, complemented

#16840141

C₂

yes

maximal, characteristic, complemented

#16840134

C₂

yes

maximal, characteristic, complemented

C₄×#11758189

C₂

yes

maximal, characteristic, complemented

C₆₁×#492

C₂

yes

maximal, characteristic, complemented

C₂×#11758224

C₂²

yes

—

C₂×Q₈×D₆₁

C₂²

yes

—

#14519806

—

C₂×Dic₂₄₄

C₂²

yes

—

C₂×#11758202

C₂²

yes

—

C₂×#11758200

C₂²

yes

—

C₂×#11758189

C₂²

yes

—

#14519777

C₂²

yes

characteristic

#14519777

C₂²

yes

characteristic

#14519776

—

#14519775

C₂²

yes

—

#14519764

C₂²

yes

—

#14519748

—

D₄×Dic₆₁

—

C₄⋊C₄×D₆₁

—

C₄²×D₆₁

C₂²

yes

characteristic

#14519687

C₂²

yes

—

C₄×#92418

C₂²

yes

characteristic

C₂×SD₁₆×C₆₁

C₂²

yes

—

C₄⋊Q₈×C₆₁

C₂²

yes

characteristic

#127×C₆₁

C₂²

yes

characteristic

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

C₂³

yes

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

C₂³

yes

—

C₂×C₂²⋊D₆₁

—

C₂²×Dic₆₁

—

Q₈×D₆₁

—

#11758224

—

C₂×C₄×D₆₁

—

C₂×C₄×D₆₁

D₄

yes

—

C₂×C₄×D₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

C₂³

yes

—

C₂×Dic₁₂₂

C₂³

yes

—

SD₁₆×C₆₁

—

C₂×C₈×C₆₁

C₂³

yes

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂³

yes

characteristic

C₂²⋊Dic₆₁

—

#11758202

—

#11758200

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

#11758196

—

C₄×Dic₆₁

D₄

yes

—

C₄×Dic₆₁

C₂³

yes

characteristic

C₂×#92418

C₂³

yes

—

Dic₂₄₄

—

#11758189

D₄

yes

complemented

C₂³×C₆₁

—

C₂²×D₆₁

C₂×D₄

yes

characteristic

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂²⋊D₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂×D₄

yes

—

C₂×Dic₆₁

C₂×D₄

yes

characteristic

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

C₂×D₄

yes

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

C₂×D₄

yes

—

#92418

C₂×D₄

yes

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

#127

yes

—

C₄×C₆₁

—

C₄×C₆₁

#137

yes

complemented

C₄×C₆₁

C₂²×D₄

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

#127

yes

—

#3152

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

C₂×#127

yes

characteristic

C₂×C₆₁

C₂×#137

yes

—

D₆₁

—

C₂×#137

—

100

#532

—

101

C₂×C₄⋊Q₈

—

102

#497

—

103

#494

—

104

#494

—

105

#492

—

106

#492

D₆₁

yes

characteristic, complemented

107

C₄×M₁₆

—

108

C₆₁

#3152

yes

characteristic, complemented, Sylow

109

C₂×#41

—

110

C₂²×Q₈

—

111

#137

—

122

—

112

C₂×Q₁₆

—

113

C₂×Q₁₆

—

114

C₂×SD₁₆

—

115

C₂×SD₁₆

C₂×D₆₁

yes

—

116

C₂×M₁₆

—

117

C₄⋊Q₈

—

118

C₄⋊Q₈

—

119

C₄⋊Q₈

—

120

C₄⋊Q₈

—

122

—

121

C₄⋊Q₈

C₂×D₆₁

yes

characteristic

122

#127

C₂×D₆₁

yes

characteristic

123

#124

—

124

#121

—

122

—

125

C₄×D₄

—

122

—

126

C₂×C₄⋊C₄

—

122

—

127

C₂×C₄²

—

128

#97

—

129

C₄×C₈

—

130

C₄×C₈

C₂×D₆₁

yes

characteristic

131

#41

—

122

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

122

—

135

C₂×Q₈

—

122

—

136

C₂×Q₈

C₂²×D₆₁

yes

—

137

C₂×D₄

—

138

C₂×D₄

—

122

—

139

C₂×D₄

C₂²×D₆₁

yes

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂²×C₄

—

122

—

143

C₂²×C₄

—

122

—

144

Q₁₆

—

122

—

145

Q₁₆

—

122

—

146

SD₁₆

—

147

SD₁₆

—

122

—

148

M₁₆

—

149

C₂×C₈

—

150

C₂×C₈

C₂²×D₆₁

yes

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

122

—

153

C₄⋊C₄

—

122

—

154

C₄⋊C₄

—

122

—

155

C₂²⋊C₄

—

122

—

156

C₄²

—

157

C₄²

—

158

C₄²

C₂²×D₆₁

yes

characteristic

159

C₂³

—

160

C₂³

—

161

Q₈

—

162

Q₈

—

122

—

163

Q₈

—

122

—

164

Q₈

—

244

—

165

D₄

—

166

D₄

—

167

D₄

—

244

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

122

—

173

C₂×C₄

—

122

—

174

C₂×C₄

—

122

—

175

C₂×C₄

—

244

—

176

C₂×C₄

—

244

—

177

C₂×C₄

D₄×D₆₁

yes

—

178

C₂×C₄

C₂³×D₆₁

yes

characteristic

179

C₈

—

180

C₈

D₄×D₆₁

yes

—

181

C₂²

—

182

C₂²

—

183

C₂²

—

184

C₂²

—

122

—

185

C₂²

C₂×D₄×D₆₁

yes

characteristic

186

C₄

—

187

C₄

—

188

C₄

—

122

—

189

C₄

—

244

—

190

C₄

—

244

—

191

C₄

#14519806

yes

complemented

192

C₄

C₂×D₄×D₆₁

yes

—

193

C₂

—

194

C₂

—

122

—

195

C₂

#127×D₆₁

yes

characteristic

196

C₂

C₂×#14519806

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₆₁×#492	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	#492	D₆₁
hypocenter	C₆₁	#3152
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

C₂×D₄×D₆₁

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₆₁×#492

C₂

socle

C₂²×C₆₁

C₂²×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. C₂×#14519806	C₂	semidirect
3. C₄⋊Q₈×D₆₁	C₂	semidirect
4. #16840153	C₂	semidirect
5. #16840142	C₂	semidirect
6. #16840141	C₂	semidirect
7. #16840134	C₂	semidirect
8. C₄×#11758189	C₂	semidirect
9. C₆₁×#492	C₂	semidirect
59. #11758189	D₄	semidirect
88. C₄×C₆₁	#137	semidirect
106. #492	D₆₁	semidirect
108. C₆₁	#3152	semidirect
191. C₄	#14519806	semidirect

H ≅ G/N

type

1. C₂×#14519806

C₂

semidirect

3. C₄⋊Q₈×D₆₁

C₂

semidirect

4. #16840153

C₂

semidirect

5. #16840142

C₂

semidirect

6. #16840141

C₂

semidirect

7. #16840134

C₂

semidirect

8. C₄×#11758189

C₂

semidirect

9. C₆₁×#492

C₂

semidirect

59. #11758189

D₄

semidirect

88. C₄×C₆₁

#137

semidirect

106. #492

D₆₁

semidirect

108. C₆₁

#3152

semidirect

191. C₄

#14519806

semidirect

Subgroups — #21456273

Conjugacy classes of subgroups

Complements