Group #21447359 — subgroups

Subgroups — #21447359

Order 7808.

254 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 87 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	C₂×#14519708	C₂	1	1	yes	maximal, characteristic
2	C₂×#14519702	C₂	1	1	yes	maximal, characteristic
3	C₂×#14519700	C₂	1	1	yes	maximal, characteristic
4	#16839667	C₂	1	1	yes	maximal, characteristic
5	#16839631	C₂	1	1	yes	maximal, characteristic
6	#16839516	C₂	1	1	yes	maximal, characteristic
7	C₂×#106×C₆₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₂₂	C₂²	1	1	yes	characteristic
9	C₂²×#92418	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
11	C₂×#11758196	—	2	2	no	—
12	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
13	#14519708	—	2	2	no	—
14	#14519702	—	2	2	no	—
15	#14519700	—	2	2	no	—
16	#14519684	—	2	2	no	—
17	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
20	#106×C₆₁	—	2	2	no	—
21	C₂²×C₄×C₆₁	—	2	1	no	—
22	C₂²×C₄×C₆₁	—	2	1	no	—
23	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
24	C₂²×Dic₆₁	—	2	1	no	—
25	C₂²×Dic₆₁	—	2	1	no	—
26	C₂²×Dic₆₁	D₄	1	2	yes	—
27	C₂×Dic₁₂₂	—	2	1	no	—
28	C₂×Dic₁₂₂	—	2	1	no	—
29	C₂×Dic₁₂₂	—	2	1	no	—
30	C₂×Dic₁₂₂	—	2	1	no	—
31	C₂×Dic₁₂₂	—	2	1	no	—
32	C₂×Dic₁₂₂	Q₈	1	2	yes	—
33	C₂×C₈×C₆₁	—	2	1	no	—
34	C₂×C₈×C₆₁	—	2	1	no	—
35	C₂×C₈×C₆₁	D₄	1	2	yes	—
36	C₄⋊C₄×C₆₁	—	2	1	no	—
37	C₄⋊C₄×C₆₁	—	2	1	no	—
38	C₄⋊C₄×C₆₁	—	2	1	no	—
39	C₄⋊C₄×C₆₁	—	2	1	no	—
40	C₄⋊Dic₆₁	—	2	1	no	—
41	C₄⋊Dic₆₁	—	2	1	no	—
42	#11758196	—	4	4	no	—
43	C₄×Dic₆₁	—	2	1	no	—
44	C₄×Dic₆₁	—	2	1	no	—
45	C₂×#92418	—	2	1	no	—
46	C₂×#92418	—	2	1	no	—
47	C₂×#92418	D₄	1	2	yes	—
48	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₆₁	—	4	1	no	—
50	C₂×C₄×C₆₁	—	4	1	no	—
51	C₂×C₄×C₆₁	—	4	2	no	—
52	C₂×C₄×C₆₁	—	4	2	no	—
53	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
57	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
58	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
59	C₂×Dic₆₁	—	2	2	no	—
60	C₂×Dic₆₁	—	2	2	no	—
61	C₂×Dic₆₁	—	2	2	no	—
62	C₂×Dic₆₁	—	4	1	no	—
63	C₂×Dic₆₁	—	4	1	no	—
64	C₂×Dic₆₁	—	4	1	no	—
65	C₂×Dic₆₁	—	4	2	no	—
66	C₂×Dic₆₁	SD₁₆	1	4	yes	—
67	Dic₁₂₂	—	2	2	no	—
68	Dic₁₂₂	—	2	2	no	—
69	Dic₁₂₂	—	4	1	no	—
70	Dic₁₂₂	—	4	1	no	—
71	C₈×C₆₁	—	2	2	no	—
72	#92418	—	2	2	no	—
73	C₂²×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂²×C₆₁	#123	1	1	yes	characteristic
75	C₂²×C₆₁	#127	1	1	yes	characteristic
76	C₂²×C₆₁	C₂×SD₁₆	1	2	yes	—
77	C₂²×C₆₁	#137	1	2	yes	—
78	C₄×C₆₁	—	8	1	no	—
79	C₄×C₆₁	—	8	1	no	—
80	C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
81	C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
82	C₄×C₆₁	#123	1	1	yes	characteristic
83	C₄×C₆₁	#123	1	1	yes	characteristic
84	Dic₆₁	—	2	4	no	—
85	Dic₆₁	—	4	2	no	—
86	Dic₆₁	—	8	1	no	—
87	#2096	—	61	1	no	maximal, Sylow
88	C₂×C₆₁	#399	1	1	yes	characteristic
89	C₂×C₆₁	#475	1	2	yes	—
90	C₂×C₆₁	#483	1	2	yes	—
91	C₂×C₆₁	#492	1	1	yes	characteristic
92	C₂×C₆₁	#497	1	1	yes	characteristic
93	C₂×#106	—	61	1	no	—
94	C₂×#106	D₆₁	1	1	yes	characteristic
95	C₂×#103	—	61	1	no	—
96	C₂×#103	—	61	1	no	—
97	#391	—	61	1	no	—
98	#384	—	61	1	no	—
99	#336	—	61	1	no	—
100	C₆₁	#2096	1	1	yes	characteristic, complemented, Sylow
101	C₂²×Q₈	—	61	1	no	—
102	C₂²×C₈	—	61	1	no	—
103	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
104	C₂×C₄⋊C₄	—	61	1	no	—
105	C₂×C₄⋊C₄	—	122	2	no	—
106	C₂×C₄⋊C₄	C₂×D₆₁	1	1	yes	characteristic
107	C₂×C₄⋊C₄	C₂×D₆₁	1	1	yes	characteristic
108	C₂×C₄²	—	61	1	no	—
109	#106	—	2	2	no	—
110	#106	—	122	2	no	—
111	#103	—	122	2	no	—
112	#103	—	122	2	no	—
113	#95	—	122	2	no	—
114	C₂×Q₈	—	61	2	no	—
115	C₂×Q₈	—	122	1	no	—
116	C₂×Q₈	—	122	1	no	—
117	C₂×Q₈	—	122	1	no	—
118	C₂×Q₈	—	122	1	no	—
119	C₂×Q₈	—	122	1	no	—
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	—	2	1	no	—
122	C₂²×C₄	—	61	2	no	—
123	C₂²×C₄	—	122	1	no	—
124	C₂²×C₄	—	122	1	no	—
125	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	61	2	no	—
129	C₂×C₈	—	122	1	no	—
130	C₂×C₈	—	122	1	no	—
131	C₂×C₈	C₂²⋊D₆₁	1	2	yes	—
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	—	2	1	no	—
134	C₄⋊C₄	—	2	1	no	—
135	C₄⋊C₄	—	2	1	no	—
136	C₄⋊C₄	—	122	1	no	—
137	C₄⋊C₄	—	122	1	no	—
138	C₄⋊C₄	—	244	4	no	—
139	C₄²	—	122	1	no	—
140	C₄²	—	122	1	no	—
141	C₂³	D₄×D₆₁	1	1	yes	characteristic
142	Q₈	—	122	2	no	—
143	Q₈	—	122	2	no	—
144	Q₈	—	244	1	no	—
145	Q₈	—	244	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	61	4	no	—
151	C₂×C₄	—	122	2	no	—
152	C₂×C₄	—	122	2	no	—
153	C₂×C₄	—	122	2	no	—
154	C₂×C₄	—	244	1	no	—
155	C₂×C₄	—	244	2	no	—
156	C₂×C₄	—	244	1	no	—
157	C₂×C₄	—	244	1	no	—
158	C₂×C₄	#11758222	1	1	yes	characteristic
159	C₂×C₄	#11758222	1	1	yes	characteristic
160	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
161	C₂×C₄	Q₈×D₆₁	1	1	yes	characteristic
162	C₂×C₄	#11758226	1	1	yes	characteristic
163	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
164	C₈	—	2	2	no	—
165	C₈	—	122	2	no	—
166	C₂²	#14519755	1	1	yes	characteristic
167	C₂²	#14519762	1	1	yes	characteristic
168	C₂²	#14519773	1	1	yes	characteristic
169	C₂²	SD₁₆×D₆₁	1	2	yes	—
170	C₂²	#14519806	1	2	yes	—
171	C₄	—	8	1	no	—
172	C₄	—	8	1	no	—
173	C₄	—	122	4	no	—
174	C₄	—	488	1	no	—
175	C₄	—	244	2	no	—
176	C₄	#14519753	1	1	yes	characteristic
177	C₄	#14519755	1	1	yes	characteristic
178	C₄	#14519760	1	1	yes	characteristic
179	C₄	#14519762	1	1	yes	characteristic
180	C₂	#16839697	1	1	yes	characteristic
181	C₂	#16840046	1	2	yes	—
182	C₂	#16840094	1	2	yes	—
183	C₂	#16840133	1	1	yes	characteristic
184	C₂	#16840153	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519708

C₂

yes

maximal, characteristic

C₂×#14519702

C₂

yes

maximal, characteristic

C₂×#14519700

C₂

yes

maximal, characteristic

#16839667

C₂

yes

maximal, characteristic

#16839631

C₂

yes

maximal, characteristic

#16839516

C₂

yes

maximal, characteristic

C₂×#106×C₆₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₂₂

C₂²

yes

characteristic

C₂²×#92418

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₁

C₂²

yes

characteristic

C₂×#11758196

—

C₂×C₄×Dic₆₁

C₂²

yes

—

—

—

—

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₁

C₂²

yes

characteristic

#106×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₂²×Dic₆₁

D₄

yes

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

Q₈

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

D₄

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

#11758196

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

D₄

yes

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

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

SD₁₆

yes

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

#92418

—

C₂²×C₆₁

C₂²⋊Q₈

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

C₂×SD₁₆

yes

—

C₂²×C₆₁

#137

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂²⋊Q₈

yes

characteristic

C₄×C₆₁

C₂²⋊Q₈

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×C₆₁

#483

yes

—

C₂×C₆₁

yes

C₂×C₆₁

yes

—

D₆₁

yes

—

—

—

—

—

100

C₆₁

#2096

yes

characteristic, complemented, Sylow

101

C₂²×Q₈

—

102

C₂²×C₈

—

103

C₂²×C₈

C₂×D₆₁

yes

characteristic

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

—

122

—

106

C₂×C₄⋊C₄

C₂×D₆₁

yes

characteristic

107

C₂×C₄⋊C₄

C₂×D₆₁

yes

characteristic

108

C₂×C₄²

—

109

#106

—

110

#106

—

122

—

111

#103

—

122

—

112

#103

—

122

—

113

#95

—

122

—

114

C₂×Q₈

—

115

C₂×Q₈

—

122

—

116

C₂×Q₈

—

122

—

117

C₂×Q₈

—

122

—

118

C₂×Q₈

—

122

—

119

C₂×Q₈

—

122

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

122

—

124

C₂²×C₄

—

122

—

125

C₂²×C₄

C₂²×D₆₁

yes

characteristic

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

122

—

130

C₂×C₈

—

122

—

131

C₂×C₈

C₂²⋊D₆₁

yes

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

122

—

137

C₄⋊C₄

—

122

—

138

C₄⋊C₄

—

244

—

139

C₄²

—

122

—

140

C₄²

—

122

—

141

C₂³

D₄×D₆₁

yes

characteristic

142

Q₈

—

122

—

143

Q₈

—

122

—

144

Q₈

—

244

—

145

Q₈

—

244

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

122

—

152

C₂×C₄

—

122

—

153

C₂×C₄

—

122

—

154

C₂×C₄

—

244

—

155

C₂×C₄

—

244

—

156

C₂×C₄

—

244

—

157

C₂×C₄

—

244

—

158

C₂×C₄

#11758222

yes

characteristic

159

C₂×C₄

#11758222

yes

characteristic

160

C₂×C₄

D₄×D₆₁

yes

characteristic

161

C₂×C₄

Q₈×D₆₁

yes

characteristic

162

C₂×C₄

#11758226

yes

characteristic

163

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

164

C₈

—

165

C₈

—

122

—

166

C₂²

#14519755

yes

characteristic

167

C₂²

#14519762

yes

characteristic

168

C₂²

#14519773

yes

characteristic

169

C₂²

SD₁₆×D₆₁

yes

—

170

C₂²

#14519806

yes

—

171

C₄

—

172

C₄

—

173

C₄

—

122

—

174

C₄

—

488

—

175

C₄

—

244

—

176

C₄

#14519753

yes

characteristic

177

C₄

#14519755

yes

characteristic

178

C₄

#14519760

yes

characteristic

179

C₄

#14519762

yes

characteristic

180

C₂

#16839697

yes

characteristic

181

C₂

#16840046

yes

—

182

C₂

#16840094

yes

—

183

C₂

#16840133

yes

characteristic

184

C₂

#16840153

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₆₁
commutator	C₂²×C₄×C₆₁	C₂³
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₂×#106×C₆₁	C₂
socle	C₂³×C₆₁	C₂×D₄
hypercenter	C₂×#106	D₆₁
hypocenter	C₆₁	#2096
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₆₁

commutator

C₂²×C₄×C₆₁

C₂³

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₂×#106×C₆₁

C₂

socle

C₂³×C₆₁

C₂×D₄

D₆₁

C₆₁

G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
100. C₆₁	#2096	semidirect

H ≅ G/N

type

100. C₆₁

#2096

semidirect

Subgroups — #21447359

Conjugacy classes of subgroups

Complements