Group #21446413 — subgroups

Subgroups — #21446413

Order 7808.

230 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 99 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	#16839661	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₄×Dic₆₁	C₂	1	1	yes	maximal, characteristic
3	#16839525	C₂	1	1	yes	maximal, characteristic
4	#16839524	C₂	1	1	yes	maximal, characteristic
5	#16839524	C₂	1	1	yes	maximal, characteristic
6	#16839516	C₂	1	1	yes	maximal, characteristic
7	C₂×#107×C₆₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×#92418	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₆₁	—	2	1	no	—
10	C₂×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₆₁	—	2	1	no	—
13	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
14	#14519684	—	2	1	no	—
15	#14519684	—	2	1	no	—
16	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
19	#107×C₆₁	C₄	1	4	yes	complemented
20	C₂²×C₄×C₆₁	—	2	1	no	—
21	C₂²×C₄×C₆₁	—	2	1	no	—
22	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
23	C₂²×Dic₆₁	—	2	1	no	—
24	C₂²×Dic₆₁	—	2	1	no	—
25	C₂²×Dic₆₁	D₄	1	1	yes	characteristic
26	C₂²×Dic₆₁	D₄	1	1	yes	characteristic
27	C₂×C₈×C₆₁	—	2	1	no	—
28	C₂×C₈×C₆₁	—	2	1	no	—
29	C₂×C₈×C₆₁	C₂×C₄	1	2	yes	—
30	C₄⋊C₄×C₆₁	—	2	1	no	—
31	C₄⋊C₄×C₆₁	—	2	1	no	—
32	C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
33	C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
34	C₄⋊Dic₆₁	—	2	1	no	—
35	C₄⋊Dic₆₁	—	2	1	no	—
36	C₄⋊Dic₆₁	—	2	1	no	—
37	C₄⋊Dic₆₁	—	4	2	no	—
38	C₄⋊Dic₆₁	Q₈	1	2	yes	—
39	C₄×Dic₆₁	—	2	1	no	—
40	C₄×Dic₆₁	—	2	1	no	—
41	C₄×Dic₆₁	—	4	2	no	—
42	C₂×#92418	—	2	1	no	—
43	C₂×#92418	—	2	1	no	—
44	C₂×#92418	—	2	1	no	—
45	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₆₁	—	2	2	no	—
47	C₂×C₄×C₆₁	—	2	2	no	—
48	C₂×C₄×C₆₁	—	4	1	no	—
49	C₂×C₄×C₆₁	—	4	1	no	—
50	C₂×C₄×C₆₁	—	4	1	no	—
51	C₂×C₄×C₆₁	—	4	1	no	—
52	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
55	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
56	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
57	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
58	C₂×Dic₆₁	—	2	1	no	—
59	C₂×Dic₆₁	—	2	1	no	—
60	C₂×Dic₆₁	—	2	2	no	—
61	C₂×Dic₆₁	—	2	1	no	—
62	C₂×Dic₆₁	—	2	1	no	—
63	C₂×Dic₆₁	—	2	1	no	—
64	C₂×Dic₆₁	—	4	1	no	—
65	C₂×Dic₆₁	—	4	1	no	—
66	C₂×Dic₆₁	—	4	1	no	—
67	C₂×Dic₆₁	—	4	1	no	—
68	C₂×Dic₆₁	—	4	1	no	—
69	C₂×Dic₆₁	Q₁₆	1	2	yes	—
70	C₈×C₆₁	—	2	2	no	—
71	#92418	—	4	1	no	—
72	C₂²×C₆₁	C₄×D₄	1	1	yes	characteristic
73	C₂²×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂²×C₆₁	#123	1	1	yes	characteristic
75	C₂²×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
76	C₂²×C₆₁	#135	1	1	yes	characteristic
77	C₂²×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂²×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₄×C₆₁	—	4	2	no	—
80	C₄×C₆₁	—	4	2	no	—
81	C₄×C₆₁	#117	1	1	yes	characteristic
82	C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
83	C₄×C₆₁	#125	1	1	yes	characteristic
84	C₄×C₆₁	#126	1	1	yes	characteristic
85	Dic₆₁	—	2	2	no	—
86	Dic₆₁	—	4	2	no	—
87	Dic₆₁	—	4	1	no	—
88	Dic₆₁	—	8	1	no	—
89	#1921	—	61	1	no	maximal, Sylow
90	C₂×C₆₁	#387	1	1	yes	characteristic
91	C₂×C₆₁	C₄×Q₁₆	1	1	yes	characteristic
92	C₂×C₆₁	#442	1	1	yes	characteristic
93	C₂×C₆₁	#477	1	1	yes	characteristic
94	C₂×C₆₁	#478	1	1	yes	characteristic
95	C₂×C₆₁	#482	1	1	yes	characteristic
96	C₂×C₆₁	#484	1	1	yes	characteristic
97	C₂×#107	D₆₁	1	1	yes	characteristic
98	#389	—	61	1	no	—
99	C₄×C₄⋊C₄	—	61	1	no	—
100	#340	—	61	1	no	—
101	#340	—	61	1	no	—
102	#340	—	61	1	no	—
103	#336	—	61	1	no	—
104	C₆₁	#1921	1	1	yes	characteristic, complemented, Sylow
105	C₂²×C₈	—	61	1	no	—
106	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
107	C₂×C₄⋊C₄	—	61	1	no	—
108	C₂×C₄⋊C₄	—	61	1	no	—
109	C₂×C₄⋊C₄	—	122	1	no	—
110	C₂×C₄⋊C₄	C₂×D₆₁	1	1	yes	characteristic
111	C₂×C₄⋊C₄	C₂×D₆₁	1	1	yes	characteristic
112	C₂×C₄²	—	61	1	no	—
113	C₂×C₄²	—	122	1	no	—
114	#107	Dic₆₁	1	4	yes	complemented
115	#95	—	122	1	no	—
116	#95	—	122	1	no	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	61	1	no	—
120	C₂²×C₄	—	61	1	no	—
121	C₂²×C₄	—	122	1	no	—
122	C₂²×C₄	—	122	1	no	—
123	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	122	1	no	—
127	C₂×C₈	—	122	1	no	—
128	C₂×C₈	—	122	1	no	—
129	C₂×C₈	C₂×Dic₆₁	1	2	yes	—
130	C₄⋊C₄	—	2	1	no	—
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	61	2	no	—
133	C₄⋊C₄	—	122	1	no	—
134	C₄⋊C₄	—	122	1	no	—
135	C₄⋊C₄	—	122	1	no	—
136	C₄⋊C₄	—	244	2	no	—
137	C₄⋊C₄	C₂×Dic₆₁	1	2	yes	—
138	C₄⋊C₄	C₂×Dic₆₁	1	2	yes	—
139	C₄²	—	122	1	no	—
140	C₄²	—	122	1	no	—
141	C₄²	—	244	2	no	—
142	C₂³	D₄×D₆₁	1	1	yes	characteristic
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	61	2	no	—
150	C₂×C₄	—	122	1	no	—
151	C₂×C₄	—	122	1	no	—
152	C₂×C₄	—	122	2	no	—
153	C₂×C₄	—	122	1	no	—
154	C₂×C₄	—	122	1	no	—
155	C₂×C₄	—	122	1	no	—
156	C₂×C₄	—	244	1	no	—
157	C₂×C₄	—	244	1	no	—
158	C₂×C₄	—	244	1	no	—
159	C₂×C₄	—	244	1	no	—
160	C₂×C₄	—	244	1	no	—
161	C₂×C₄	#11758222	1	1	yes	characteristic
162	C₂×C₄	#11758222	1	1	yes	characteristic
163	C₂×C₄	#11758224	1	1	yes	characteristic
164	C₂×C₄	Q₈×D₆₁	1	1	yes	characteristic
165	C₂×C₄	#11758226	1	1	yes	characteristic
166	C₂×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
167	C₈	—	2	2	no	—
168	C₈	—	244	1	no	—
169	C₂²	D₄×Dic₆₁	1	1	yes	characteristic
170	C₂²	#14519755	1	1	yes	characteristic
171	C₂²	#14519762	1	1	yes	characteristic
172	C₂²	Q₁₆×D₆₁	1	1	yes	characteristic
173	C₂²	#14519800	1	1	yes	characteristic
174	C₂²	#14519801	1	1	yes	characteristic
175	C₂²	#14519801	1	1	yes	characteristic
176	C₄	—	4	2	no	—
177	C₄	—	4	2	no	—
178	C₄	—	122	2	no	—
179	C₄	—	488	1	no	—
180	C₄	—	244	2	no	—
181	C₄	—	244	1	no	—
182	C₄	#14519736	1	1	yes	characteristic
183	C₄	Q₈×Dic₆₁	1	1	yes	characteristic
184	C₄	#14519767	1	1	yes	characteristic
185	C₄	#14519772	1	1	yes	characteristic
186	C₂	#16839648	1	1	yes	characteristic
187	C₂	Q₁₆×Dic₆₁	1	1	yes	characteristic
188	C₂	#16839874	1	1	yes	characteristic
189	C₂	#16840056	1	1	yes	characteristic
190	C₂	#16840065	1	1	yes	characteristic
191	C₂	#16840089	1	1	yes	characteristic
192	C₂	#16840099	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16839661

C₂

yes

maximal, characteristic

C₄⋊C₄×Dic₆₁

C₂

yes

maximal, characteristic

#16839525

C₂

yes

maximal, characteristic

#16839524

C₂

yes

maximal, characteristic

#16839524

C₂

yes

maximal, characteristic

#16839516

C₂

yes

maximal, characteristic

C₂×#107×C₆₁

C₂

yes

maximal, characteristic

C₂²×#92418

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₁

C₂²

yes

characteristic

C₂×C₄×Dic₆₁

—

C₂×C₄×Dic₆₁

C₂²

yes

characteristic

#14519684

—

#14519684

—

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₁

C₂²

yes

characteristic

#107×C₆₁

C₄

yes

complemented

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₂²×Dic₆₁

D₄

yes

characteristic

C₂²×Dic₆₁

D₄

yes

characteristic

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂×C₄

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

Q₈

yes

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

—

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

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

Q₁₆

yes

—

C₈×C₆₁

—

#92418

—

C₂²×C₆₁

C₄×D₄

yes

characteristic

C₂²×C₆₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₆₁

#123

yes

characteristic

C₂²×C₆₁

C₂×Q₁₆

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#117

yes

characteristic

C₄×C₆₁

C₄×Q₈

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

C₄×Q₁₆

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

yes

characteristic

#389

—

C₄×C₄⋊C₄

—

100

#340

—

101

#340

—

102

#340

—

103

#336

—

104

C₆₁

#1921

yes

characteristic, complemented, Sylow

105

C₂²×C₈

—

106

C₂²×C₈

C₂×D₆₁

yes

characteristic

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

122

—

110

C₂×C₄⋊C₄

C₂×D₆₁

yes

characteristic

111

C₂×C₄⋊C₄

C₂×D₆₁

yes

characteristic

112

C₂×C₄²

—

113

C₂×C₄²

—

122

—

114

#107

Dic₆₁

yes

complemented

115

#95

—

122

—

116

#95

—

122

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

—

122

C₂²×C₄

—

122

—

123

C₂²×C₄

C₂²×D₆₁

yes

characteristic

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

122

—

127

C₂×C₈

—

122

—

128

C₂×C₈

—

122

—

129

C₂×C₈

C₂×Dic₆₁

yes

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

122

—

134

C₄⋊C₄

—

122

—

135

C₄⋊C₄

—

122

—

136

C₄⋊C₄

—

244

—

137

C₄⋊C₄

C₂×Dic₆₁

yes

—

138

C₄⋊C₄

C₂×Dic₆₁

yes

—

139

C₄²

—

122

—

140

C₄²

—

122

—

141

C₄²

—

244

—

142

C₂³

D₄×D₆₁

yes

characteristic

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

122

—

151

C₂×C₄

—

122

—

152

C₂×C₄

—

122

—

153

C₂×C₄

—

122

—

154

C₂×C₄

—

122

—

155

C₂×C₄

—

122

—

156

C₂×C₄

—

244

—

157

C₂×C₄

—

244

—

158

C₂×C₄

—

244

—

159

C₂×C₄

—

244

—

160

C₂×C₄

—

244

—

161

C₂×C₄

#11758222

yes

characteristic

162

C₂×C₄

#11758222

yes

characteristic

163

C₂×C₄

#11758224

yes

characteristic

164

C₂×C₄

Q₈×D₆₁

yes

characteristic

165

C₂×C₄

#11758226

yes

characteristic

166

C₂×C₄

C₂²×Dic₆₁

yes

characteristic

167

C₈

—

168

C₈

—

244

—

169

C₂²

D₄×Dic₆₁

yes

characteristic

170

C₂²

#14519755

yes

characteristic

171

C₂²

#14519762

yes

characteristic

172

C₂²

Q₁₆×D₆₁

yes

characteristic

173

C₂²

#14519800

yes

characteristic

174

C₂²

#14519801

yes

characteristic

175

C₂²

#14519801

yes

characteristic

176

C₄

—

177

C₄

—

178

C₄

—

122

—

179

C₄

—

488

—

180

C₄

—

244

—

181

C₄

—

244

—

182

C₄

#14519736

yes

characteristic

183

C₄

Q₈×Dic₆₁

yes

characteristic

184

C₄

#14519767

yes

characteristic

185

C₄

#14519772

yes

characteristic

186

C₂

#16839648

yes

characteristic

187

C₂

Q₁₆×Dic₆₁

yes

characteristic

188

C₂

#16839874

yes

characteristic

189

C₂

#16840056

yes

characteristic

190

C₂

#16840065

yes

characteristic

191

C₂

#16840089

yes

characteristic

192

C₂

#16840099

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₂×#107×C₆₁	C₂
socle	C₂³×C₆₁	C₂×D₄
hypercenter	C₂×#107	D₆₁
hypocenter	C₆₁	#1921
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₆₁

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₂×#107×C₆₁

C₂

socle

C₂³×C₆₁

C₂×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
19. #107×C₆₁	C₄	semidirect
104. C₆₁	#1921	semidirect
114. #107	Dic₆₁	semidirect

H ≅ G/N

type

19. #107×C₆₁

C₄

semidirect

104. C₆₁

#1921

semidirect

114. #107

Dic₆₁

semidirect

Subgroups — #21446413

Conjugacy classes of subgroups

Complements