Group #21446006 — subgroups

Subgroups — #21446006

Order 7808.

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

normal subgroups only

This group has 360 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	C₂²×C₄⋊Dic₆₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#14519710	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊#92418	C₂	1	1	yes	maximal, characteristic
4	#16839524	C₂	1	2	yes	maximal, complemented
5	C₆₁×#380	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₆₁	—	2	1	no	—
7	C₂²×#92418	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₆₁	—	2	1	no	—
9	C₂×C₄⋊Dic₆₁	—	2	4	no	—
10	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
11	#14519710	—	2	4	no	—
12	C₂²⋊#92418	C₄	1	4	yes	complemented
13	C₂³×C₄×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
17	#95×C₆₁	—	2	1	no	—
18	C₂⁴×C₆₁	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₆₁	—	2	2	no	—
20	C₂²×C₄×C₆₁	—	2	2	no	—
21	C₂²×C₄×C₆₁	—	2	1	no	—
22	C₂²×C₄×C₆₁	—	2	1	no	—
23	C₂²×C₄×C₆₁	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₆₁	Q₈	1	2	yes	—
26	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
27	C₂²×Dic₆₁	—	2	4	no	—
28	C₂²×Dic₆₁	—	2	2	no	—
29	C₂²×Dic₆₁	—	4	2	no	—
30	C₂²×Dic₆₁	—	4	2	no	—
31	C₄⋊C₄×C₆₁	—	2	2	no	—
32	C₄⋊C₄×C₆₁	—	2	2	no	—
33	C₂²⋊C₄×C₆₁	—	4	2	no	—
34	C₄⋊Dic₆₁	—	2	4	no	—
35	C₄⋊Dic₆₁	—	2	2	no	—
36	C₄⋊Dic₆₁	D₄	1	4	yes	—
37	C₂×#92418	—	2	2	no	—
38	C₂×#92418	—	2	2	no	—
39	C₂×#92418	C₂×C₄	1	4	yes	complemented
40	C₂³×C₆₁	—	2	1	no	—
41	C₂³×C₆₁	—	2	1	no	—
42	C₂³×C₆₁	—	2	1	no	—
43	C₂³×C₆₁	—	2	1	no	—
44	C₂³×C₆₁	C₄⋊C₄	1	2	yes	—
45	C₂³×C₆₁	D₈	1	2	yes	—
46	C₂³×C₆₁	Q₁₆	1	2	yes	—
47	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₆₁	—	2	1	no	—
49	C₂×C₄×C₆₁	—	2	1	no	—
50	C₂×C₄×C₆₁	—	2	4	no	—
51	C₂×C₄×C₆₁	—	2	4	no	—
52	C₂×C₄×C₆₁	—	4	1	no	—
53	C₂×C₄×C₆₁	—	4	1	no	—
54	C₂×C₄×C₆₁	—	4	1	no	—
55	C₂×C₄×C₆₁	—	4	1	no	—
56	C₂×C₄×C₆₁	—	4	1	no	—
57	C₂×C₄×C₆₁	—	4	1	no	—
58	C₂×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
59	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₄×C₆₁	#41	1	2	yes	—
63	C₂×Dic₆₁	—	2	4	no	—
64	C₂×Dic₆₁	—	4	8	no	—
65	C₂×Dic₆₁	—	4	2	no	—
66	C₂×Dic₆₁	—	4	2	no	—
67	#92418	—	2	4	no	—
68	C₂²×C₆₁	—	2	2	no	—
69	C₂²×C₆₁	—	2	2	no	—
70	C₂²×C₆₁	—	2	2	no	—
71	C₂²×C₆₁	—	2	2	no	—
72	C₂²×C₆₁	—	2	2	no	—
73	C₂²×C₆₁	—	2	2	no	—
74	C₂²×C₆₁	#107	1	4	yes	complemented
75	C₂²×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₆₁	#120	1	1	yes	characteristic
77	C₂²×C₆₁	#123	1	1	yes	characteristic
78	C₂²×C₆₁	C₂×D₈	1	1	yes	characteristic
79	C₂²×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
80	C₂²×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₂²×C₆₁	#137	1	1	yes	characteristic
82	C₄×C₆₁	—	2	2	no	—
83	C₄×C₆₁	—	8	1	no	—
84	C₄×C₆₁	—	8	1	no	—
85	C₄×C₆₁	C₄×D₄	1	2	yes	—
86	C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
87	Dic₆₁	—	4	4	no	—
88	#1852	—	61	1	no	maximal, Sylow
89	C₂×C₆₁	—	2	4	no	—
90	C₂×C₆₁	#385	1	1	yes	characteristic
91	C₂×C₆₁	C₂×#107	1	1	yes	characteristic
92	C₂×C₆₁	#428	1	1	yes	characteristic
93	C₂×C₆₁	#447	1	1	yes	characteristic
94	C₂×C₆₁	#451	1	1	yes	characteristic
95	C₂×C₆₁	#480	1	1	yes	characteristic
96	C₂×C₆₁	#484	1	1	yes	characteristic
97	C₂²×C₄⋊C₄	—	61	1	no	—
98	C₂×#107	—	61	2	no	—
99	C₂×C₂²⋊C₈	—	61	1	no	—
100	#380	D₆₁	1	1	yes	characteristic
101	#340	—	61	2	no	—
102	C₆₁	#1852	1	1	yes	characteristic, complemented, Sylow
103	C₂³×C₄	—	122	1	no	—
104	C₂³×C₄	C₂×D₆₁	1	1	yes	characteristic
105	C₂²×C₈	—	61	2	no	—
106	C₂×C₄⋊C₄	—	61	2	no	—
107	C₂×C₄⋊C₄	—	122	4	no	—
108	C₂×C₄⋊C₄	—	122	1	no	—
109	C₂×C₄⋊C₄	C₂×D₆₁	1	1	yes	characteristic
110	C₂×C₄⋊C₄	C₂×D₆₁	1	1	yes	characteristic
111	C₂×C₂²⋊C₄	—	2	1	no	—
112	#107	—	122	4	no	—
113	C₂²⋊C₈	—	61	4	no	—
114	#95	—	2	1	no	—
115	C₂⁴	C₂²⋊D₆₁	1	1	yes	characteristic
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	—	2	2	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	—	122	4	no	—
121	C₂²×C₄	—	122	2	no	—
122	C₂²×C₄	—	244	2	no	—
123	C₂²×C₄	—	244	2	no	—
124	C₂²×C₄	Dic₁₂₂	1	2	yes	—
125	C₂²×C₄	C₄×D₆₁	1	2	yes	—
126	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
128	C₂×C₈	—	61	4	no	—
129	C₂×C₈	—	122	2	no	—
130	C₂×C₈	—	122	2	no	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	2	2	no	—
133	C₄⋊C₄	—	61	4	no	—
134	C₄⋊C₄	—	122	4	no	—
135	C₄⋊C₄	—	122	2	no	—
136	C₂²⋊C₄	—	4	2	no	—
137	C₂³	—	2	1	no	—
138	C₂³	—	2	1	no	—
139	C₂³	—	2	1	no	—
140	C₂³	—	2	1	no	—
141	C₂³	#11758196	1	2	yes	—
142	C₂³	#11758199	1	2	yes	—
143	C₂³	#11758202	1	2	yes	—
144	C₂³	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	122	4	no	—
156	C₂×C₄	—	244	8	no	—
157	C₂×C₄	—	244	2	no	—
158	C₂×C₄	—	244	2	no	—
159	C₂×C₄	#11758196	1	2	yes	—
160	C₂×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
161	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
162	C₂×C₄	D₄×D₆₁	1	2	yes	—
163	C₂×C₄	#11758224	1	2	yes	—
164	C₈	—	122	4	no	—
165	C₂²	—	2	2	no	—
166	C₂²	—	2	2	no	—
167	C₂²	—	2	2	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	2	no	—
170	C₂²	—	2	2	no	—
171	C₂²	#14519710	1	4	yes	complemented
172	C₂²	C₂×#11758196	1	1	yes	characteristic
173	C₂²	#14519745	1	1	yes	characteristic
174	C₂²	#14519758	1	1	yes	characteristic
175	C₂²	C₂×#11758199	1	1	yes	characteristic
176	C₂²	C₂×#11758202	1	1	yes	characteristic
177	C₂²	#14519802	1	1	yes	characteristic
178	C₂²	#14519807	1	1	yes	characteristic
179	C₄	—	2	2	no	—
180	C₄	—	8	1	no	—
181	C₄	—	8	1	no	—
182	C₄	—	244	4	no	—
183	C₄	#14519737	1	2	yes	—
184	C₄	#14519754	1	2	yes	—
185	C₂	—	2	4	no	—
186	C₂	#16839634	1	1	yes	characteristic
187	C₂	C₂×#14519710	1	1	yes	characteristic
188	C₂	#16839805	1	1	yes	characteristic
189	C₂	#16839895	1	1	yes	characteristic
190	C₂	#16839917	1	1	yes	characteristic
191	C₂	#16840073	1	1	yes	characteristic
192	C₂	#16840095	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊Dic₆₁

C₂

yes

maximal, characteristic

C₂×#14519710

C₂

yes

maximal, complemented

C₂×C₂²⋊#92418

C₂

yes

maximal, characteristic

#16839524

C₂

yes

maximal, complemented

C₆₁×#380

C₂

yes

maximal, characteristic

C₂³×Dic₆₁

—

C₂²×#92418

C₂²

yes

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

#14519710

—

C₂²⋊#92418

C₄

yes

complemented

C₂³×C₄×C₆₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₁

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₆₁

—

#95×C₆₁

—

C₂⁴×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

Q₈

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

D₄

yes

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

C₂×C₄

yes

complemented

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

C₄⋊C₄

yes

—

C₂³×C₆₁

D₈

yes

—

C₂³×C₆₁

Q₁₆

yes

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄⋊C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₁

#41

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#107

yes

complemented

C₂²×C₆₁

C₂×C₄⋊C₄

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

C₂×D₈

yes

characteristic

C₂²×C₆₁

C₂×Q₁₆

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₄×D₄

yes

—

C₄×C₆₁

C₂²⋊Q₈

yes

—

Dic₆₁

—

#1852

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂²×C₄⋊C₄

—

C₂×#107

—

C₂×C₂²⋊C₈

—

100

#380

D₆₁

yes

characteristic

101

#340

—

102

C₆₁

#1852

yes

characteristic, complemented, Sylow

103

C₂³×C₄

—

122

—

104

C₂³×C₄

C₂×D₆₁

yes

characteristic

105

C₂²×C₈

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

122

—

108

C₂×C₄⋊C₄

—

122

—

109

C₂×C₄⋊C₄

C₂×D₆₁

yes

characteristic

110

C₂×C₄⋊C₄

C₂×D₆₁

yes

characteristic

111

C₂×C₂²⋊C₄

—

112

#107

—

122

—

113

C₂²⋊C₈

—

114

#95

—

115

C₂⁴

C₂²⋊D₆₁

yes

characteristic

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

122

—

121

C₂²×C₄

—

122

—

122

C₂²×C₄

—

244

—

123

C₂²×C₄

—

244

—

124

C₂²×C₄

Dic₁₂₂

yes

—

125

C₂²×C₄

C₄×D₆₁

yes

—

126

C₂²×C₄

C₂²⋊D₆₁

yes

characteristic

127

C₂²×C₄

C₂²×D₆₁

yes

characteristic

128

C₂×C₈

—

129

C₂×C₈

—

122

—

130

C₂×C₈

—

122

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

122

—

135

C₄⋊C₄

—

122

—

136

C₂²⋊C₄

—

137

C₂³

—

138

C₂³

—

139

C₂³

—

140

C₂³

—

141

C₂³

#11758196

yes

—

142

C₂³

#11758199

yes

—

143

C₂³

#11758202

yes

—

144

C₂³

C₂×C₂²⋊D₆₁

yes

characteristic

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

122

—

156

C₂×C₄

—

244

—

157

C₂×C₄

—

244

—

158

C₂×C₄

—

244

—

159

C₂×C₄

#11758196

yes

—

160

C₂×C₄

C₂×Dic₁₂₂

yes

characteristic

161

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

162

C₂×C₄

D₄×D₆₁

yes

—

163

C₂×C₄

#11758224

yes

—

164

C₈

—

122

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

#14519710

yes

complemented

172

C₂²

C₂×#11758196

yes

characteristic

173

C₂²

#14519745

yes

characteristic

174

C₂²

#14519758

yes

characteristic

175

C₂²

C₂×#11758199

yes

characteristic

176

C₂²

C₂×#11758202

yes

characteristic

177

C₂²

#14519802

yes

characteristic

178

C₂²

#14519807

yes

characteristic

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

244

—

183

C₄

#14519737

yes

—

184

C₄

#14519754

yes

—

185

C₂

—

186

C₂

#16839634

yes

characteristic

187

C₂

C₂×#14519710

yes

characteristic

188

C₂

#16839805

yes

characteristic

189

C₂

#16839895

yes

characteristic

190

C₂

#16839917

yes

characteristic

191

C₂

#16840073

yes

characteristic

192

C₂

#16840095

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₆₁×#380	C₂
socle	C₂³×C₆₁	C₂×D₄
hypercenter	#380	D₆₁
hypocenter	C₆₁	#1852
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #380

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₆₁

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₆₁×#380

C₂

socle

C₂³×C₆₁

C₂×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #380

N	H ≅ G/N	type
2. C₂×#14519710	C₂	semidirect
4. #16839524	C₂	semidirect
12. C₂²⋊#92418	C₄	semidirect
39. C₂×#92418	C₂×C₄	semidirect
74. C₂²×C₆₁	#107	semidirect
102. C₆₁	#1852	semidirect
171. C₂²	#14519710	semidirect

H ≅ G/N

type

2. C₂×#14519710

C₂

semidirect

4. #16839524

C₂

semidirect

12. C₂²⋊#92418

C₄

semidirect

39. C₂×#92418

C₂×C₄

semidirect

74. C₂²×C₆₁

#107

semidirect

102. C₆₁

#1852

semidirect

171. C₂²

#14519710

semidirect

Subgroups — #21446006

Conjugacy classes of subgroups

Complements