Group #21458900 — subgroups

Subgroups — #21458900

Order 7808.

356 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 121 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	#16840485	C₂	1	2	yes	maximal, complemented
2	#16840156	C₂	1	2	yes	maximal, complemented
3	#16840146	C₂	1	1	yes	maximal, characteristic
4	#16839985	C₂	1	2	yes	maximal, complemented
5	#16839978	C₂	1	2	yes	maximal
6	#16839973	C₂	1	2	yes	maximal, complemented
7	#16839864	C₂	1	2	yes	maximal, complemented
8	C₄×Dic₂₄₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#446	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₂₄₄	C₂²	1	1	yes	characteristic
11	C₂×Dic₂₄₄	C₂²	1	1	yes	characteristic
12	C₂×Dic₂₄₄	C₂²	1	2	yes	—
13	C₂×#11758190	C₂²	1	2	yes	—
14	C₂×#11758190	C₂²	1	2	yes	—
15	C₂×D₄₈₈	C₂²	1	1	yes	characteristic
16	#14519777	C₂²	1	2	yes	—
17	#14519772	—	2	2	no	—
18	#14519772	—	2	2	no	—
19	#14519765	C₂²	1	2	yes	—
20	#14519765	C₂²	1	2	yes	—
21	#14519753	—	2	2	no	—
22	C₄×Dic₁₂₂	C₂²	1	2	yes	—
23	C₄×Dic₁₂₂	C₂²	1	2	yes	—
24	C₄×D₂₄₄	C₂²	1	2	yes	—
25	#14519735	—	2	2	no	—
26	#14519711	C₂²	1	1	yes	characteristic
27	#14519702	C₂²	1	2	yes	—
28	#14519702	C₂²	1	2	yes	—
29	#14519699	C₂²	1	2	yes	—
30	C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
31	C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
32	C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
33	#97×C₆₁	C₂²	1	2	yes	—
34	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
35	C₂×Q₈×C₆₁	D₄	1	1	yes	characteristic
36	C₂×D₂₄₄	C₂³	1	2	yes	—
37	C₂×C₄×D₆₁	—	2	2	no	—
38	C₂×Dic₁₂₂	C₂³	1	2	yes	—
39	C₂×Dic₁₂₂	C₂³	1	2	yes	—
40	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
41	C₂×C₈×C₆₁	C₂³	1	2	yes	—
42	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
43	C₄⋊C₄×C₆₁	D₄	1	2	yes	—
44	C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
45	C₄²×C₆₁	C₂³	1	1	yes	characteristic
46	C₄²×C₆₁	C₂³	1	2	yes	—
47	#11758198	—	2	2	no	—
48	#11758198	—	2	2	no	—
49	#11758198	—	2	2	no	—
50	#11758198	—	2	2	no	—
51	#11758198	—	2	2	no	—
52	C₄⋊Dic₆₁	—	2	2	no	—
53	C₄⋊Dic₆₁	—	2	2	no	—
54	C₄⋊Dic₆₁	C₂³	1	2	yes	—
55	#11758196	—	2	2	no	—
56	#11758196	—	2	2	no	—
57	C₄×Dic₆₁	—	2	2	no	—
58	C₄×Dic₆₁	—	2	2	no	—
59	Dic₂₄₄	—	2	2	no	—
60	Dic₂₄₄	—	4	2	no	—
61	Dic₂₄₄	—	4	1	no	—
62	D₄₈₈	—	4	1	no	—
63	#11758190	—	2	4	no	—
64	#11758190	—	4	2	no	—
65	C₂²×D₆₁	—	2	2	no	—
66	Q₈×C₆₁	—	4	1	no	—
67	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
71	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
72	C₂×Dic₆₁	—	2	2	no	—
73	C₂×Dic₆₁	—	2	2	no	—
74	C₂×Dic₆₁	—	2	2	no	—
75	D₂₄₄	—	2	2	no	—
76	D₂₄₄	—	2	2	no	—
77	C₄×D₆₁	—	4	4	no	—
78	Dic₁₂₂	—	2	2	no	—
79	Dic₁₂₂	—	2	2	no	—
80	Dic₁₂₂	—	2	2	no	—
81	Dic₁₂₂	—	2	2	no	—
82	C₈×C₆₁	—	2	2	no	—
83	C₈×C₆₁	—	2	1	no	—
84	C₈×C₆₁	#41	1	2	yes	—
85	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
86	C₂×D₆₁	—	4	2	no	—
87	C₂×D₆₁	—	4	2	no	—
88	C₂×D₆₁	—	4	2	no	—
89	C₄×C₆₁	—	2	2	no	—
90	C₄×C₆₁	—	2	1	no	—
91	C₄×C₆₁	—	2	1	no	—
92	C₄×C₆₁	—	2	2	no	—
93	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
94	C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
95	Dic₆₁	—	4	2	no	—
96	Dic₆₁	—	4	2	no	—
97	Dic₆₁	—	4	2	no	—
98	#3402	—	61	1	no	maximal, Sylow
99	C₂×C₆₁	#550	1	1	yes	characteristic
100	C₂×C₆₁	#575	1	1	yes	characteristic
101	C₂×C₆₁	#578	1	1	yes	characteristic
102	D₆₁	—	8	2	no	—
103	#556	—	61	2	no	—
104	#497	—	61	2	no	—
105	#495	—	61	1	no	—
106	#464	—	61	2	no	—
107	#463	—	61	2	no	—
108	#462	—	61	2	no	—
109	#446	D₆₁	1	1	yes	characteristic, complemented
110	#440	—	61	2	no	—
111	C₄×Q₁₆	—	61	1	no	—
112	C₆₁	#3402	1	1	yes	characteristic, complemented, Sylow
113	C₂×Q₁₆	—	61	1	no	—
114	C₂×Q₁₆	—	61	1	no	—
115	C₂×Q₁₆	—	61	2	no	—
116	C₂×SD₁₆	—	61	2	no	—
117	C₂×SD₁₆	—	61	2	no	—
118	C₂×D₈	—	61	1	no	—
119	C₄⋊Q₈	—	61	2	no	—
120	#126	—	122	2	no	—
121	#126	—	122	2	no	—
122	#124	—	61	2	no	—
123	#124	—	61	2	no	—
124	C₂²⋊Q₈	—	122	2	no	—
125	C₄×Q₈	—	61	2	no	—
126	C₄×Q₈	—	61	2	no	—
127	C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
128	C₄×D₄	—	61	2	no	—
129	#117	—	122	2	no	—
130	#107	—	61	1	no	—
131	C₄⋊C₈	C₂×D₆₁	1	2	yes	—
132	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
133	#103	—	61	2	no	—
134	#103	—	61	2	no	—
135	#102	—	61	2	no	—
136	#97	C₂×D₆₁	1	2	yes	—
137	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
138	C₂×Q₈	—	61	2	no	—
139	C₂×Q₈	—	61	2	no	—
140	C₂×Q₈	D₂₄₄	1	1	yes	characteristic
141	C₂×D₄	—	61	2	no	—
142	C₂²×C₄	—	122	2	no	—
143	Q₁₆	—	122	2	no	—
144	Q₁₆	—	244	1	no	—
145	Q₁₆	—	244	2	no	—
146	SD₁₆	—	122	4	no	—
147	SD₁₆	—	244	2	no	—
148	D₈	—	244	1	no	—
149	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
150	C₂×C₈	C₂²×D₆₁	1	2	yes	—
151	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
152	C₄⋊C₄	—	61	2	no	—
153	C₄⋊C₄	—	122	2	no	—
154	C₄⋊C₄	—	122	2	no	—
155	C₄⋊C₄	—	122	2	no	—
156	C₄⋊C₄	—	122	2	no	—
157	C₄⋊C₄	D₂₄₄	1	2	yes	—
158	C₄⋊C₄	D₂₄₄	1	1	yes	characteristic
159	C₂²⋊C₄	—	122	2	no	—
160	C₂²⋊C₄	—	122	2	no	—
161	C₂²⋊C₄	—	122	2	no	—
162	C₂²⋊C₄	—	122	2	no	—
163	C₂²⋊C₄	—	122	2	no	—
164	C₄²	—	122	2	no	—
165	C₄²	—	122	2	no	—
166	C₄²	C₂²×D₆₁	1	1	yes	characteristic
167	C₄²	C₂²×D₆₁	1	2	yes	—
168	C₂³	—	122	2	no	—
169	Q₈	—	4	1	no	—
170	Q₈	—	122	2	no	—
171	Q₈	—	122	2	no	—
172	Q₈	—	122	2	no	—
173	Q₈	—	122	2	no	—
174	D₄	—	122	2	no	—
175	D₄	—	122	2	no	—
176	C₂×C₄	—	122	2	no	—
177	C₂×C₄	—	122	2	no	—
178	C₂×C₄	—	122	2	no	—
179	C₂×C₄	—	244	4	no	—
180	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
181	C₂×C₄	C₂×D₂₄₄	1	2	yes	—
182	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
183	C₂×C₄	C₂×D₂₄₄	1	2	yes	—
184	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
185	C₈	—	2	1	no	—
186	C₈	—	2	2	no	—
187	C₈	#11758226	1	2	yes	—
188	C₂²	—	244	2	no	—
189	C₂²	—	244	2	no	—
190	C₂²	—	244	2	no	—
191	C₂²	C₂²×D₂₄₄	1	1	yes	characteristic
192	C₄	—	2	2	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	1	no	—
195	C₄	—	2	2	no	—
196	C₄	—	244	2	no	—
197	C₄	—	244	2	no	—
198	C₄	—	244	2	no	—
199	C₄	C₂×#11758226	1	1	yes	characteristic
200	C₄	#14519823	1	1	yes	characteristic
201	C₂	—	488	2	no	—
202	C₂	#16840434	1	1	yes	characteristic
203	C₂	#16840562	1	1	yes	characteristic
204	C₂	#16840579	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840485

C₂

yes

maximal, complemented

#16840156

C₂

yes

maximal, complemented

#16840146

C₂

yes

maximal, characteristic

#16839985

C₂

yes

maximal, complemented

#16839978

C₂

yes

maximal

#16839973

C₂

yes

maximal, complemented

#16839864

C₂

yes

maximal, complemented

C₄×Dic₂₄₄

C₂

yes

maximal, characteristic, complemented

C₆₁×#446

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₄₄

C₂²

yes

characteristic

C₂×Dic₂₄₄

C₂²

yes

characteristic

C₂×Dic₂₄₄

C₂²

yes

—

C₂×#11758190

C₂²

yes

—

C₂×#11758190

C₂²

yes

—

C₂×D₄₈₈

C₂²

yes

characteristic

#14519777

C₂²

yes

—

#14519772

—

#14519772

—

#14519765

C₂²

yes

—

#14519765

C₂²

yes

—

#14519753

—

C₄×Dic₁₂₂

C₂²

yes

—

C₄×Dic₁₂₂

C₂²

yes

—

C₄×D₂₄₄

C₂²

yes

—

#14519735

—

#14519711

C₂²

yes

characteristic

#14519702

C₂²

yes

—

#14519702

C₂²

yes

—

#14519699

C₂²

yes

—

C₄×Q₈×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

C₂²

yes

—

C₄⋊C₈×C₆₁

C₂²

yes

characteristic

#97×C₆₁

C₂²

yes

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

D₄

yes

characteristic

C₂×D₂₄₄

C₂³

yes

—

C₂×C₄×D₆₁

—

C₂×Dic₁₂₂

C₂³

yes

—

C₂×Dic₁₂₂

C₂³

yes

—

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₈×C₆₁

C₂³

yes

—

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₄⋊C₄×C₆₁

D₄

yes

—

C₄⋊C₄×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

C₂³

yes

characteristic

C₄²×C₆₁

C₂³

yes

—

—

—

—

—

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂³

yes

—

#11758196

—

#11758196

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

Dic₂₄₄

—

Dic₂₄₄

—

Dic₂₄₄

—

D₄₈₈

—

#11758190

—

#11758190

—

C₂²×D₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

—

C₄×D₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

#41

yes

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×#41

yes

characteristic

C₄×C₆₁

H(5,𝔽₂)

yes

characteristic

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

100

C₂×C₆₁

#575

yes

characteristic

101

C₂×C₆₁

#578

yes

characteristic

102

D₆₁

—

103

#556

—

104

#497

—

105

#495

—

106

#464

—

107

#463

—

108

#462

—

109

#446

D₆₁

yes

characteristic, complemented

110

#440

—

111

C₄×Q₁₆

—

112

C₆₁

#3402

yes

characteristic, complemented, Sylow

113

C₂×Q₁₆

—

114

C₂×Q₁₆

—

115

C₂×Q₁₆

—

116

C₂×SD₁₆

—

117

C₂×SD₁₆

—

118

C₂×D₈

—

119

C₄⋊Q₈

—

120

#126

—

122

—

121

#126

—

122

—

122

#124

—

123

#124

—

124

C₂²⋊Q₈

—

122

—

125

C₄×Q₈

—

126

C₄×Q₈

—

127

C₄×Q₈

C₂×D₆₁

yes

characteristic

128

C₄×D₄

—

129

#117

—

122

—

130

#107

—

131

C₄⋊C₈

C₂×D₆₁

yes

—

132

C₄⋊C₈

C₂×D₆₁

yes

characteristic

133

#103

—

134

#103

—

135

#102

—

136

#97

C₂×D₆₁

yes

—

137

C₄×C₈

C₂×D₆₁

yes

characteristic

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

D₂₄₄

yes

characteristic

141

C₂×D₄

—

142

C₂²×C₄

—

122

—

143

Q₁₆

—

122

—

144

Q₁₆

—

244

—

145

Q₁₆

—

244

—

146

SD₁₆

—

122

—

147

SD₁₆

—

244

—

148

D₈

—

244

—

149

C₂×C₈

C₂²×D₆₁

yes

characteristic

150

C₂×C₈

C₂²×D₆₁

yes

—

151

C₂×C₈

C₂²×D₆₁

yes

characteristic

152

C₄⋊C₄

—

153

C₄⋊C₄

—

122

—

154

C₄⋊C₄

—

122

—

155

C₄⋊C₄

—

122

—

156

C₄⋊C₄

—

122

—

157

C₄⋊C₄

D₂₄₄

yes

—

158

C₄⋊C₄

D₂₄₄

yes

characteristic

159

C₂²⋊C₄

—

122

—

160

C₂²⋊C₄

—

122

—

161

C₂²⋊C₄

—

122

—

162

C₂²⋊C₄

—

122

—

163

C₂²⋊C₄

—

122

—

164

C₄²

—

122

—

165

C₄²

—

122

—

166

C₄²

C₂²×D₆₁

yes

characteristic

167

C₄²

C₂²×D₆₁

yes

—

168

C₂³

—

122

—

169

Q₈

—

170

Q₈

—

122

—

171

Q₈

—

122

—

172

Q₈

—

122

—

173

Q₈

—

122

—

174

D₄

—

122

—

175

D₄

—

122

—

176

C₂×C₄

—

122

—

177

C₂×C₄

—

122

—

178

C₂×C₄

—

122

—

179

C₂×C₄

—

244

—

180

C₂×C₄

C₂×D₂₄₄

yes

characteristic

181

C₂×C₄

C₂×D₂₄₄

yes

—

182

C₂×C₄

C₂×D₂₄₄

yes

characteristic

183

C₂×C₄

C₂×D₂₄₄

yes

—

184

C₂×C₄

C₂³×D₆₁

yes

characteristic

185

C₈

—

186

C₈

—

187

C₈

#11758226

yes

—

188

C₂²

—

244

—

189

C₂²

—

244

—

190

C₂²

—

244

—

191

C₂²

C₂²×D₂₄₄

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

244

—

197

C₄

—

244

—

198

C₄

—

244

—

199

C₄

C₂×#11758226

yes

characteristic

200

C₄

#14519823

yes

characteristic

201

C₂

—

488

—

202

C₂

#16840434

yes

characteristic

203

C₂

#16840562

yes

characteristic

204

C₂

#16840579

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₆₁×#446	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	#446	D₆₁
hypocenter	C₆₁	#3402
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

subgroup

G/H

center

C₂²

C₂²×D₂₄₄

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₆₁×#446

C₂

socle

C₂²×C₆₁

C₂²×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

N	H ≅ G/N	type
1. #16840485	C₂	semidirect
2. #16840156	C₂	semidirect
4. #16839985	C₂	semidirect
6. #16839973	C₂	semidirect
7. #16839864	C₂	semidirect
8. C₄×Dic₂₄₄	C₂	semidirect
9. C₆₁×#446	C₂	semidirect
109. #446	D₆₁	semidirect
112. C₆₁	#3402	semidirect

H ≅ G/N

type

1. #16840485

C₂

semidirect

2. #16840156

C₂

semidirect

4. #16839985

C₂

semidirect

6. #16839973

C₂

semidirect

7. #16839864

C₂

semidirect

8. C₄×Dic₂₄₄

C₂

semidirect

9. C₆₁×#446

C₂

semidirect

109. #446

D₆₁

semidirect

112. C₆₁

#3402

semidirect

Subgroups — #21458900

Conjugacy classes of subgroups

Complements