Group #21458901 — subgroups

Subgroups — #21458901

Order 7808.

356 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 113 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	#16840481	C₂	1	2	yes	maximal, complemented
2	#16840153	C₂	1	2	yes	maximal, complemented
3	#16839981	C₂	1	2	yes	maximal, complemented
4	#16839975	C₂	1	2	yes	maximal
5	#16839970	C₂	1	2	yes	maximal, complemented
6	#16839892	C₂	1	1	yes	maximal, characteristic, complemented
7	#16839863	C₂	1	2	yes	maximal, complemented
8	C₄×Dic₂₄₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#495	C₂	1	1	yes	maximal, characteristic
10	C₂×Dic₂₄₄	C₂²	1	1	yes	characteristic
11	C₂×#11758202	C₂²	1	2	yes	—
12	C₂×#11758200	C₂²	1	2	yes	—
13	#14519775	C₂²	1	2	yes	—
14	#14519770	—	2	2	no	—
15	#14519770	—	2	2	no	—
16	#14519764	C₂²	1	2	yes	—
17	#14519754	—	2	2	no	—
18	Q₈×Dic₆₁	C₂²	1	2	yes	—
19	C₄×Dic₁₂₂	C₂²	1	2	yes	—
20	C₄×Dic₁₂₂	C₂²	1	1	yes	characteristic
21	D₄×Dic₆₁	C₂²	1	2	yes	—
22	#14519733	—	2	2	no	—
23	#14519711	C₂²	1	1	yes	characteristic
24	C₄⋊#92418	C₂²	1	1	yes	characteristic
25	#14519706	C₂²	1	2	yes	—
26	#14519702	C₂²	1	2	yes	—
27	#14519701	C₂²	1	2	yes	—
28	#14519698	C₂²	1	2	yes	—
29	#14519687	C₂²	1	2	yes	—
30	C₂×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
31	C₂×SD₁₆×C₆₁	C₂²	1	2	yes	—
32	C₂×D₈×C₆₁	C₂²	1	1	yes	characteristic
33	#124×C₆₁	C₂²	1	2	yes	—
34	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
35	C₂×Q₈×C₆₁	C₂³	1	2	yes	—
36	C₂×D₄×C₆₁	C₂³	1	2	yes	—
37	C₂²×Dic₆₁	—	2	2	no	—
38	C₂×Dic₁₂₂	D₄	1	1	yes	characteristic
39	C₂×Dic₁₂₂	C₂³	1	2	yes	—
40	Q₁₆×C₆₁	—	4	1	no	—
41	SD₁₆×C₆₁	—	2	4	no	—
42	D₈×C₆₁	—	4	1	no	—
43	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
45	C₂²⋊C₄×C₆₁	—	2	2	no	—
46	C₂²⋊C₄×C₆₁	—	2	2	no	—
47	C₄²×C₆₁	C₂³	1	1	yes	characteristic
48	C₂²⋊Dic₆₁	—	2	2	no	—
49	C₂²⋊Dic₆₁	—	2	2	no	—
50	C₂²⋊Dic₆₁	—	2	2	no	—
51	#11758202	—	4	2	no	—
52	#11758200	—	4	2	no	—
53	C₄⋊Dic₆₁	—	2	2	no	—
54	C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
55	C₄⋊Dic₆₁	C₂³	1	2	yes	—
56	#11758196	—	2	2	no	—
57	#11758196	—	2	2	no	—
58	#11758196	—	2	2	no	—
59	#11758196	D₄	1	2	yes	—
60	C₄×Dic₆₁	—	2	2	no	—
61	C₄×Dic₆₁	—	2	2	no	—
62	C₄×Dic₆₁	C₂³	1	2	yes	—
63	C₂×#92418	C₂³	1	2	yes	—
64	Dic₂₄₄	—	2	2	no	—
65	C₂³×C₆₁	—	2	2	no	—
66	Q₈×C₆₁	—	2	2	no	—
67	Q₈×C₆₁	—	2	2	no	—
68	D₄×C₆₁	—	2	2	no	—
69	D₄×C₆₁	—	2	2	no	—
70	C₂×C₄×C₆₁	—	2	2	no	—
71	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
74	C₂×Dic₆₁	—	2	2	no	—
75	C₂×Dic₆₁	—	2	2	no	—
76	C₂×Dic₆₁	—	4	4	no	—
77	C₂×Dic₆₁	C₂×D₄	1	2	yes	—
78	C₂×Dic₆₁	C₂×D₄	1	2	yes	—
79	Dic₁₂₂	—	2	2	no	—
80	Dic₁₂₂	—	2	2	no	—
81	Dic₁₂₂	—	4	1	no	—
82	C₈×C₆₁	—	2	1	no	—
83	C₈×C₆₁	#41	1	2	yes	—
84	#92418	—	2	2	no	—
85	C₂²×C₆₁	—	4	2	no	—
86	C₂²×C₆₁	—	4	2	no	—
87	C₂²×C₆₁	—	4	2	no	—
88	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
89	C₄×C₆₁	—	2	1	no	—
90	C₄×C₆₁	—	2	1	no	—
91	C₄×C₆₁	—	4	2	no	—
92	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
93	C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
94	Dic₆₁	—	2	2	no	—
95	Dic₆₁	—	2	2	no	—
96	Dic₆₁	—	4	2	no	—
97	Dic₆₁	—	4	2	no	—
98	#3402	—	61	1	no	maximal, Sylow
99	C₂×C₆₁	—	8	2	no	—
100	C₂×C₆₁	#550	1	1	yes	characteristic
101	C₂×C₆₁	#575	1	1	yes	characteristic
102	C₂×C₆₁	#578	1	1	yes	characteristic
103	#556	—	61	2	no	—
104	#497	—	61	2	no	—
105	#495	D₆₁	1	1	yes	characteristic
106	#464	—	61	2	no	—
107	#463	—	61	2	no	—
108	#462	—	61	2	no	—
109	#446	—	61	1	no	—
110	#440	—	61	2	no	—
111	C₄×Q₁₆	—	61	1	no	—
112	C₆₁	#3402	1	1	yes	characteristic, complemented, Sylow
113	C₂×Q₁₆	—	61	2	no	—
114	C₂×Q₁₆	—	61	1	no	—
115	C₂×Q₁₆	C₂×D₆₁	1	1	yes	characteristic
116	C₂×SD₁₆	—	61	2	no	—
117	C₂×SD₁₆	C₂×D₆₁	1	2	yes	—
118	C₂×D₈	C₂×D₆₁	1	1	yes	characteristic
119	C₄⋊Q₈	—	61	2	no	—
120	#126	—	122	2	no	—
121	#126	—	122	2	no	—
122	#124	—	61	2	no	—
123	#124	C₂×D₆₁	1	2	yes	—
124	C₂²⋊Q₈	—	122	2	no	—
125	C₄×Q₈	—	61	1	no	—
126	C₄×Q₈	—	61	2	no	—
127	C₄×Q₈	—	61	2	no	—
128	C₄×D₄	—	61	2	no	—
129	#117	—	122	2	no	—
130	#107	—	61	1	no	—
131	C₄⋊C₈	—	61	2	no	—
132	C₄⋊C₈	—	61	1	no	—
133	#103	—	61	2	no	—
134	#103	—	61	2	no	—
135	#102	—	61	2	no	—
136	#97	—	61	2	no	—
137	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
138	C₂×Q₈	—	61	1	no	—
139	C₂×Q₈	—	61	2	no	—
140	C₂×Q₈	C₂²×D₆₁	1	2	yes	—
141	C₂×D₄	C₂²×D₆₁	1	2	yes	—
142	C₂²×C₄	—	122	2	no	—
143	Q₁₆	—	4	1	no	—
144	Q₁₆	—	122	2	no	—
145	Q₁₆	—	244	2	no	—
146	SD₁₆	—	2	4	no	—
147	SD₁₆	—	244	2	no	—
148	D₈	—	4	1	no	—
149	C₂×C₈	—	61	2	no	—
150	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
151	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
152	C₄⋊C₄	—	61	2	no	—
153	C₄⋊C₄	—	61	1	no	—
154	C₄⋊C₄	—	61	2	no	—
155	C₄⋊C₄	—	122	2	no	—
156	C₄⋊C₄	—	122	2	no	—
157	C₄⋊C₄	—	122	2	no	—
158	C₄⋊C₄	—	122	2	no	—
159	C₂²⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	—	2	2	no	—
161	C₂²⋊C₄	—	122	2	no	—
162	C₂²⋊C₄	—	122	2	no	—
163	C₂²⋊C₄	—	122	2	no	—
164	C₄²	—	61	2	no	—
165	C₄²	—	122	2	no	—
166	C₄²	—	122	2	no	—
167	C₄²	C₂²×D₆₁	1	1	yes	characteristic
168	C₂³	—	2	2	no	—
169	Q₈	—	2	2	no	—
170	Q₈	—	2	2	no	—
171	Q₈	—	122	2	no	—
172	Q₈	—	122	2	no	—
173	Q₈	—	244	1	no	—
174	D₄	—	2	2	no	—
175	D₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	61	2	no	—
178	C₂×C₄	—	61	2	no	—
179	C₂×C₄	—	122	2	no	—
180	C₂×C₄	—	122	2	no	—
181	C₂×C₄	—	244	4	no	—
182	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
183	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
184	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
185	C₈	—	2	1	no	—
186	C₈	—	122	2	no	—
187	C₈	#11758224	1	2	yes	—
188	C₂²	—	4	2	no	—
189	C₂²	—	4	2	no	—
190	C₂²	—	4	2	no	—
191	C₂²	C₂×D₄×D₆₁	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	—	4	2	no	—
195	C₄	—	122	2	no	—
196	C₄	—	122	2	no	—
197	C₄	—	244	2	no	—
198	C₄	—	244	2	no	—
199	C₄	C₂×#11758224	1	1	yes	characteristic
200	C₄	#14519822	1	1	yes	characteristic
201	C₂	—	8	2	no	—
202	C₂	#16840432	1	1	yes	characteristic
203	C₂	#16840559	1	1	yes	characteristic
204	C₂	#16840576	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840481

C₂

yes

maximal, complemented

#16840153

C₂

yes

maximal, complemented

#16839981

C₂

yes

maximal, complemented

#16839975

C₂

yes

maximal

#16839970

C₂

yes

maximal, complemented

#16839892

C₂

yes

maximal, characteristic, complemented

#16839863

C₂

yes

maximal, complemented

C₄×Dic₂₄₄

C₂

yes

maximal, characteristic, complemented

C₆₁×#495

C₂

yes

maximal, characteristic

C₂×Dic₂₄₄

C₂²

yes

characteristic

C₂×#11758202

C₂²

yes

—

C₂×#11758200

C₂²

yes

—

#14519775

C₂²

yes

—

#14519770

—

#14519770

—

#14519764

C₂²

yes

—

#14519754

—

Q₈×Dic₆₁

C₂²

yes

—

C₄×Dic₁₂₂

C₂²

yes

—

C₄×Dic₁₂₂

C₂²

yes

characteristic

D₄×Dic₆₁

C₂²

yes

—

#14519733

—

#14519711

C₂²

yes

characteristic

C₄⋊#92418

C₂²

yes

characteristic

#14519706

C₂²

yes

—

#14519702

C₂²

yes

—

#14519701

C₂²

yes

—

#14519698

C₂²

yes

—

#14519687

C₂²

yes

—

C₂×Q₁₆×C₆₁

C₂²

yes

characteristic

C₂×SD₁₆×C₆₁

C₂²

yes

—

C₂×D₈×C₆₁

C₂²

yes

characteristic

#124×C₆₁

C₂²

yes

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

C₂³

yes

—

C₂×D₄×C₆₁

C₂³

yes

—

C₂²×Dic₆₁

—

C₂×Dic₁₂₂

D₄

yes

characteristic

C₂×Dic₁₂₂

C₂³

yes

—

Q₁₆×C₆₁

—

SD₁₆×C₆₁

—

D₈×C₆₁

—

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₂²⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

C₂³

yes

characteristic

C₂²⋊Dic₆₁

—

C₂²⋊Dic₆₁

—

C₂²⋊Dic₆₁

—

#11758202

—

#11758200

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

D₄

yes

characteristic

C₄⋊Dic₆₁

C₂³

yes

—

—

—

—

D₄

yes

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

C₂³

yes

—

C₂×#92418

C₂³

yes

—

Dic₂₄₄

—

C₂³×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂×D₄

yes

—

C₂×Dic₆₁

C₂×D₄

yes

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

#41

yes

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×#41

yes

characteristic

C₄×C₆₁

H(5,𝔽₂)

yes

characteristic

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

#3402

—

maximal, Sylow

C₂×C₆₁

—

100

C₂×C₆₁

#550

yes

characteristic

101

C₂×C₆₁

#575

yes

characteristic

102

C₂×C₆₁

#578

yes

characteristic

103

#556

—

104

#497

—

105

#495

D₆₁

yes

characteristic

106

#464

—

107

#463

—

108

#462

—

109

#446

—

110

#440

—

111

C₄×Q₁₆

—

112

C₆₁

#3402

yes

characteristic, complemented, Sylow

113

C₂×Q₁₆

—

114

C₂×Q₁₆

—

115

C₂×Q₁₆

C₂×D₆₁

yes

characteristic

116

C₂×SD₁₆

—

117

C₂×SD₁₆

C₂×D₆₁

yes

—

118

C₂×D₈

C₂×D₆₁

yes

characteristic

119

C₄⋊Q₈

—

120

#126

—

122

—

121

#126

—

122

—

122

#124

—

123

#124

C₂×D₆₁

yes

—

124

C₂²⋊Q₈

—

122

—

125

C₄×Q₈

—

126

C₄×Q₈

—

127

C₄×Q₈

—

128

C₄×D₄

—

129

#117

—

122

—

130

#107

—

131

C₄⋊C₈

—

132

C₄⋊C₈

—

133

#103

—

134

#103

—

135

#102

—

136

#97

—

137

C₄×C₈

C₂×D₆₁

yes

characteristic

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

C₂²×D₆₁

yes

—

141

C₂×D₄

C₂²×D₆₁

yes

—

142

C₂²×C₄

—

122

—

143

Q₁₆

—

144

Q₁₆

—

122

—

145

Q₁₆

—

244

—

146

SD₁₆

—

147

SD₁₆

—

244

—

148

D₈

—

149

C₂×C₈

—

150

C₂×C₈

C₂²×D₆₁

yes

characteristic

151

C₂×C₈

C₂²×D₆₁

yes

characteristic

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

122

—

156

C₄⋊C₄

—

122

—

157

C₄⋊C₄

—

122

—

158

C₄⋊C₄

—

122

—

159

C₂²⋊C₄

—

160

C₂²⋊C₄

—

161

C₂²⋊C₄

—

122

—

162

C₂²⋊C₄

—

122

—

163

C₂²⋊C₄

—

122

—

164

C₄²

—

165

C₄²

—

122

—

166

C₄²

—

122

—

167

C₄²

C₂²×D₆₁

yes

characteristic

168

C₂³

—

169

Q₈

—

170

Q₈

—

171

Q₈

—

122

—

172

Q₈

—

122

—

173

Q₈

—

244

—

174

D₄

—

175

D₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

122

—

180

C₂×C₄

—

122

—

181

C₂×C₄

—

244

—

182

C₂×C₄

D₄×D₆₁

yes

characteristic

183

C₂×C₄

D₄×D₆₁

yes

characteristic

184

C₂×C₄

C₂³×D₆₁

yes

characteristic

185

C₈

—

186

C₈

—

122

—

187

C₈

#11758224

yes

—

188

C₂²

—

189

C₂²

—

190

C₂²

—

191

C₂²

C₂×D₄×D₆₁

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

122

—

196

C₄

—

122

—

197

C₄

—

244

—

198

C₄

—

244

—

199

C₄

C₂×#11758224

yes

characteristic

200

C₄

#14519822

yes

characteristic

201

C₂

—

202

C₂

#16840432

yes

characteristic

203

C₂

#16840559

yes

characteristic

204

C₂

#16840576

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₆₁×#495	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	#495	D₆₁
hypocenter	C₆₁	#3402
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #495

subgroup

G/H

center

C₂²

C₂×D₄×D₆₁

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₆₁×#495

C₂

socle

C₂²×C₆₁

C₂²×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #495

N	H ≅ G/N	type
1. #16840481	C₂	semidirect
2. #16840153	C₂	semidirect
3. #16839981	C₂	semidirect
5. #16839970	C₂	semidirect
6. #16839892	C₂	semidirect
7. #16839863	C₂	semidirect
8. C₄×Dic₂₄₄	C₂	semidirect
112. C₆₁	#3402	semidirect

H ≅ G/N

type

1. #16840481

C₂

semidirect

2. #16840153

C₂

semidirect

3. #16839981

C₂

semidirect

5. #16839970

C₂

semidirect

6. #16839892

C₂

semidirect

7. #16839863

C₂

semidirect

8. C₄×Dic₂₄₄

C₂

semidirect

112. C₆₁

#3402

semidirect

Subgroups — #21458901

Conjugacy classes of subgroups

Complements