Group #21457281 — subgroups

Subgroups — #21457281

Order 7808.

346 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 109 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	#16840385	C₂	1	2	yes	maximal, complemented
2	#16840055	C₂	1	2	yes	maximal, complemented
3	#16840041	C₂	1	2	yes	maximal, complemented
4	#16840010	C₂	1	2	yes	maximal, complemented
5	#16839998	C₂	1	1	yes	maximal, characteristic
6	#16839986	C₂	1	2	yes	maximal, complemented
7	#16839980	C₂	1	2	yes	maximal, complemented
8	#16839788	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#467	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11758201	C₂²	1	2	yes	—
11	C₂×D₄₈₈	C₂²	1	1	yes	characteristic
12	C₂×#11758194	C₂²	1	1	yes	characteristic
13	#14519776	C₂²	1	2	yes	—
14	#14519772	—	2	2	no	—
15	#14519766	C₂²	1	2	yes	—
16	#14519763	—	2	2	no	—
17	#14519759	—	2	2	no	—
18	#14519757	—	2	2	no	—
19	#14519751	C₂²	1	2	yes	—
20	Q₈×Dic₆₁	C₂²	1	2	yes	—
21	#14519739	C₂²	1	2	yes	—
22	#14519736	C₂²	1	1	yes	characteristic
23	#14519711	C₂²	1	1	yes	characteristic
24	#14519710	C₂²	1	2	yes	—
25	#14519706	C₂²	1	2	yes	—
26	#14519706	C₂²	1	2	yes	—
27	#14519701	C₂²	1	2	yes	—
28	#14519699	C₂²	1	2	yes	—
29	#14519697	C₂²	1	2	yes	—
30	C₂×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
31	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
32	C₂²⋊Q₈×C₆₁	C₂²	1	2	yes	—
33	#107×C₆₁	C₂²	1	1	yes	characteristic
34	#103×C₆₁	C₂²	1	2	yes	—
35	C₂×Q₈×C₆₁	C₂³	1	2	yes	—
36	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
37	C₂×C₂²⋊D₆₁	—	2	2	no	—
38	C₂×D₂₄₄	C₂³	1	2	yes	—
39	C₂×C₄×D₆₁	—	2	2	no	—
40	C₂×Dic₁₂₂	—	2	2	no	—
41	Q₁₆×C₆₁	—	4	1	no	—
42	C₂×C₈×C₆₁	—	4	1	no	—
43	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₆₁	—	2	2	no	—
46	C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
47	C₂²⋊C₄×C₆₁	—	2	2	no	—
48	C₂²⋊Dic₆₁	D₄	1	2	yes	—
49	#11758201	—	4	2	no	—
50	#11758198	—	2	2	no	—
51	#11758198	—	2	2	no	—
52	#11758198	—	2	2	no	—
53	#11758198	—	2	2	no	—
54	C₄⋊Dic₆₁	—	2	2	no	—
55	C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
56	C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
57	C₄⋊Dic₆₁	C₂³	1	2	yes	—
58	#11758196	—	2	2	no	—
59	C₄×Dic₆₁	—	2	2	no	—
60	C₄×Dic₆₁	C₂³	1	2	yes	—
61	#11758194	—	4	1	no	—
62	C₂×#92418	C₂³	1	2	yes	—
63	D₄₈₈	—	4	1	no	—
64	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
65	C₂²×D₆₁	—	2	2	no	—
66	Q₈×C₆₁	—	2	2	no	—
67	Q₈×C₆₁	—	2	2	no	—
68	C₂×C₄×C₆₁	—	2	1	no	—
69	C₂×C₄×C₆₁	—	2	1	no	—
70	C₂×C₄×C₆₁	—	2	2	no	—
71	C₂×C₄×C₆₁	—	2	2	no	—
72	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
74	C₂²⋊D₆₁	—	8	2	no	—
75	C₂×Dic₆₁	—	2	2	no	—
76	C₂×Dic₆₁	C₂×D₄	1	2	yes	—
77	C₂×Dic₆₁	C₂×D₄	1	2	yes	—
78	D₂₄₄	—	2	2	no	—
79	D₂₄₄	—	2	2	no	—
80	C₄×D₆₁	—	8	2	no	—
81	Dic₁₂₂	—	8	2	no	—
82	C₈×C₆₁	—	2	1	no	—
83	C₈×C₆₁	—	2	1	no	—
84	#92418	—	2	2	no	—
85	C₂²×C₆₁	—	2	1	no	—
86	C₂²×C₆₁	—	2	1	no	—
87	C₂²×C₆₁	—	2	1	no	—
88	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
89	C₂×D₆₁	—	4	2	no	—
90	C₂×D₆₁	—	4	2	no	—
91	C₂×D₆₁	—	4	2	no	—
92	C₄×C₆₁	—	2	1	no	—
93	C₄×C₆₁	—	4	2	no	—
94	C₄×C₆₁	—	4	2	no	—
95	C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
96	C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
97	Dic₆₁	—	2	2	no	—
98	Dic₆₁	—	2	2	no	—
99	Dic₆₁	—	4	2	no	—
100	#3262	—	61	1	no	maximal, Sylow
101	C₂×C₆₁	—	4	1	no	—
102	C₂×C₆₁	#537	1	1	yes	characteristic
103	C₂×C₆₁	#576	1	1	yes	characteristic
104	C₂×C₆₁	#578	1	1	yes	characteristic
105	D₆₁	—	8	2	no	—
106	#542	—	61	2	no	—
107	#477	—	61	2	no	—
108	#474	—	61	2	no	—
109	#468	—	61	2	no	—
110	#467	D₆₁	1	1	yes	characteristic, complemented
111	#466	—	61	1	no	—
112	#464	—	61	2	no	—
113	#463	—	61	2	no	—
114	#424	—	61	1	no	—
115	C₆₁	#3262	1	1	yes	characteristic, complemented, Sylow
116	C₂×Q₁₆	C₂×D₆₁	1	1	yes	characteristic
117	C₂×SD₁₆	—	61	2	no	—
118	C₂×D₈	—	61	1	no	—
119	C₂×M₁₆	—	61	1	no	—
120	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
121	C₄⋊Q₈	—	61	2	no	—
122	#126	—	122	2	no	—
123	#124	—	61	2	no	—
124	#124	—	122	2	no	—
125	#123	—	122	2	no	—
126	C₂²⋊Q₈	—	122	2	no	—
127	C₂²⋊Q₈	C₂×D₆₁	1	2	yes	—
128	#121	—	61	2	no	—
129	C₄×Q₈	—	61	2	no	—
130	C₄×D₄	—	61	2	no	—
131	#117	—	61	1	no	—
132	#107	—	61	1	no	—
133	#107	—	61	2	no	—
134	#107	C₂×D₆₁	1	1	yes	characteristic
135	C₄⋊C₈	—	61	2	no	—
136	C₄⋊C₈	—	61	2	no	—
137	#103	—	61	2	no	—
138	#103	C₂×D₆₁	1	2	yes	—
139	#102	—	61	2	no	—
140	#102	—	61	2	no	—
141	C₂×Q₈	—	122	2	no	—
142	C₂×Q₈	C₂²×D₆₁	1	2	yes	—
143	C₂×D₄	—	61	2	no	—
144	C₂×D₄	—	122	2	no	—
145	C₂²×C₄	—	122	2	no	—
146	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
147	Q₁₆	—	4	1	no	—
148	SD₁₆	—	244	2	no	—
149	D₈	—	244	1	no	—
150	M₁₆	—	244	1	no	—
151	C₂×C₈	—	4	1	no	—
152	C₂×C₈	—	61	2	no	—
153	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
154	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	61	1	no	—
157	C₄⋊C₄	—	61	1	no	—
158	C₄⋊C₄	—	61	2	no	—
159	C₄⋊C₄	—	122	2	no	—
160	C₄⋊C₄	—	122	2	no	—
161	C₄⋊C₄	C₂²×D₆₁	1	2	yes	—
162	C₂²⋊C₄	—	2	2	no	—
163	C₂²⋊C₄	—	61	2	no	—
164	C₂²⋊C₄	—	122	2	no	—
165	C₂²⋊C₄	—	122	2	no	—
166	C₂²⋊C₄	—	122	2	no	—
167	C₂²⋊C₄	—	122	2	no	—
168	C₄²	—	61	2	no	—
169	C₄²	—	122	2	no	—
170	C₂³	—	122	2	no	—
171	C₂³	D₄×D₆₁	1	1	yes	characteristic
172	Q₈	—	2	2	no	—
173	Q₈	—	2	2	no	—
174	Q₈	—	488	2	no	—
175	D₄	—	122	2	no	—
176	D₄	—	122	2	no	—
177	D₄	—	488	2	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	61	2	no	—
183	C₂×C₄	—	61	2	no	—
184	C₂×C₄	—	122	2	no	—
185	C₂×C₄	—	488	2	no	—
186	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
187	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
188	C₈	—	2	1	no	—
189	C₈	—	2	1	no	—
190	C₈	—	122	2	no	—
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	244	2	no	—
195	C₂²	—	244	2	no	—
196	C₂²	—	244	2	no	—
197	C₂²	C₂×D₄×D₆₁	1	1	yes	characteristic
198	C₄	—	2	1	no	—
199	C₄	—	4	2	no	—
200	C₄	—	4	2	no	—
201	C₄	—	122	2	no	—
202	C₄	—	122	2	no	—
203	C₄	—	244	2	no	—
204	C₄	#14519822	1	1	yes	characteristic
205	C₄	#14519824	1	1	yes	characteristic
206	C₂	—	4	1	no	—
207	C₂	—	488	2	no	—
208	C₂	#16840339	1	1	yes	characteristic
209	C₂	#16840564	1	1	yes	characteristic
210	C₂	#16840575	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840385

C₂

yes

maximal, complemented

#16840055

C₂

yes

maximal, complemented

#16840041

C₂

yes

maximal, complemented

#16840010

C₂

yes

maximal, complemented

#16839998

C₂

yes

maximal, characteristic

#16839986

C₂

yes

maximal, complemented

#16839980

C₂

yes

maximal, complemented

#16839788

C₂

yes

maximal, characteristic, complemented

C₆₁×#467

C₂

yes

maximal, characteristic, complemented

C₂×#11758201

C₂²

yes

—

C₂×D₄₈₈

C₂²

yes

characteristic

C₂×#11758194

C₂²

yes

characteristic

#14519776

C₂²

yes

—

#14519772

—

#14519766

C₂²

yes

—

—

—

—

C₂²

yes

—

Q₈×Dic₆₁

C₂²

yes

—

#14519739

C₂²

yes

—

#14519736

C₂²

yes

characteristic

#14519711

C₂²

yes

characteristic

#14519710

C₂²

yes

—

#14519706

C₂²

yes

—

#14519706

C₂²

yes

—

#14519701

C₂²

yes

—

#14519699

C₂²

yes

—

#14519697

C₂²

yes

—

C₂×Q₁₆×C₆₁

C₂²

yes

characteristic

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂²⋊Q₈×C₆₁

C₂²

yes

—

#107×C₆₁

C₂²

yes

characteristic

#103×C₆₁

C₂²

yes

—

C₂×Q₈×C₆₁

C₂³

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂×C₂²⋊D₆₁

—

C₂×D₂₄₄

C₂³

yes

—

C₂×C₄×D₆₁

—

C₂×Dic₁₂₂

—

Q₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂³

yes

—

C₂²⋊C₄×C₆₁

—

C₂²⋊Dic₆₁

D₄

yes

—

—

—

—

—

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

D₄

yes

characteristic

C₄⋊Dic₆₁

D₄

yes

characteristic

C₄⋊Dic₆₁

C₂³

yes

—

#11758196

—

C₄×Dic₆₁

—

C₄×Dic₆₁

C₂³

yes

—

#11758194

—

C₂×#92418

C₂³

yes

—

D₄₈₈

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

C₂²×D₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂²⋊D₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂×D₄

yes

—

C₂×Dic₆₁

C₂×D₄

yes

—

D₂₄₄

—

D₂₄₄

—

C₄×D₆₁

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

H(5,𝔽₂)

yes

characteristic

C₄×C₆₁

2₋¹⁺⁴

yes

characteristic

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

100

#3262

—

maximal, Sylow

101

C₂×C₆₁

—

102

C₂×C₆₁

#537

yes

characteristic

103

C₂×C₆₁

#576

yes

characteristic

104

C₂×C₆₁

#578

yes

characteristic

105

D₆₁

—

106

#542

—

107

#477

—

108

#474

—

109

#468

—

110

#467

D₆₁

yes

characteristic, complemented

111

#466

—

112

#464

—

113

#463

—

114

#424

—

115

C₆₁

#3262

yes

characteristic, complemented, Sylow

116

C₂×Q₁₆

C₂×D₆₁

yes

characteristic

117

C₂×SD₁₆

—

118

C₂×D₈

—

119

C₂×M₁₆

—

120

C₂²×C₈

C₂×D₆₁

yes

characteristic

121

C₄⋊Q₈

—

122

#126

—

122

—

123

#124

—

124

#124

—

122

—

125

#123

—

122

—

126

C₂²⋊Q₈

—

122

—

127

C₂²⋊Q₈

C₂×D₆₁

yes

—

128

#121

—

129

C₄×Q₈

—

130

C₄×D₄

—

131

#117

—

132

#107

—

133

#107

—

134

#107

C₂×D₆₁

yes

characteristic

135

C₄⋊C₈

—

136

C₄⋊C₈

—

137

#103

—

138

#103

C₂×D₆₁

yes

—

139

#102

—

140

#102

—

141

C₂×Q₈

—

122

—

142

C₂×Q₈

C₂²×D₆₁

yes

—

143

C₂×D₄

—

144

C₂×D₄

—

122

—

145

C₂²×C₄

—

122

—

146

C₂²×C₄

C₂²×D₆₁

yes

characteristic

147

Q₁₆

—

148

SD₁₆

—

244

—

149

D₈

—

244

—

150

M₁₆

—

244

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

C₂²×D₆₁

yes

characteristic

154

C₂×C₈

C₂²×D₆₁

yes

characteristic

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

122

—

160

C₄⋊C₄

—

122

—

161

C₄⋊C₄

C₂²×D₆₁

yes

—

162

C₂²⋊C₄

—

163

C₂²⋊C₄

—

164

C₂²⋊C₄

—

122

—

165

C₂²⋊C₄

—

122

—

166

C₂²⋊C₄

—

122

—

167

C₂²⋊C₄

—

122

—

168

C₄²

—

169

C₄²

—

122

—

170

C₂³

—

122

—

171

C₂³

D₄×D₆₁

yes

characteristic

172

Q₈

—

173

Q₈

—

174

Q₈

—

488

—

175

D₄

—

122

—

176

D₄

—

122

—

177

D₄

—

488

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

122

—

185

C₂×C₄

—

488

—

186

C₂×C₄

D₄×D₆₁

yes

characteristic

187

C₂×C₄

C₂³×D₆₁

yes

characteristic

188

C₈

—

189

C₈

—

190

C₈

—

122

—

191

C₂²

—

192

C₂²

—

193

C₂²

—

194

C₂²

—

244

—

195

C₂²

—

244

—

196

C₂²

—

244

—

197

C₂²

C₂×D₄×D₆₁

yes

characteristic

198

C₄

—

199

C₄

—

200

C₄

—

201

C₄

—

122

—

202

C₄

—

122

—

203

C₄

—

244

—

204

C₄

#14519822

yes

characteristic

205

C₄

#14519824

yes

characteristic

206

C₂

—

207

C₂

—

488

—

208

C₂

#16840339

yes

characteristic

209

C₂

#16840564

yes

characteristic

210

C₂

#16840575

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₆₁×#467	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	#467	D₆₁
hypocenter	C₆₁	#3262
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #467

subgroup

G/H

center

C₂²

C₂×D₄×D₆₁

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₆₁×#467

C₂

socle

C₂²×C₆₁

C₂²×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #467

N	H ≅ G/N	type
1. #16840385	C₂	semidirect
2. #16840055	C₂	semidirect
3. #16840041	C₂	semidirect
4. #16840010	C₂	semidirect
6. #16839986	C₂	semidirect
7. #16839980	C₂	semidirect
8. #16839788	C₂	semidirect
9. C₆₁×#467	C₂	semidirect
110. #467	D₆₁	semidirect
115. C₆₁	#3262	semidirect

H ≅ G/N

type

1. #16840385

C₂

semidirect

2. #16840055

C₂

semidirect

3. #16840041

C₂

semidirect

4. #16840010

C₂

semidirect

6. #16839986

C₂

semidirect

7. #16839980

C₂

semidirect

8. #16839788

C₂

semidirect

9. C₆₁×#467

C₂

semidirect

110. #467

D₆₁

semidirect

115. C₆₁

#3262

semidirect

Subgroups — #21457281

Conjugacy classes of subgroups

Complements