Group #29585129 — subgroups

Subgroups — #29585129

Order 15616.

442 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 165 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	#21446780	C₂	1	2	yes	maximal
2	#21446621	C₂	1	1	yes	maximal, characteristic
3	#21445976	C₂	1	2	yes	maximal
4	#21445976	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1273	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519777	C₂²	1	1	yes	characteristic
7	C₂×#14519709	—	2	2	no	—
8	C₂×#14519709	—	2	4	no	—
9	C₂×#14519702	—	2	2	no	—
10	#16839632	C₂²	1	2	yes	—
11	#16839632	C₂²	1	1	yes	characteristic
12	#16839525	—	2	4	no	—
13	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₂₂	—	2	1	no	—
16	#14519777	—	2	2	no	—
17	#14519777	—	2	1	no	—
18	#14519777	Q₈	1	2	yes	—
19	C₂×C₄⋊Dic₆₁	—	2	2	no	—
20	C₂×C₄⋊Dic₆₁	—	2	1	no	—
21	C₂×C₄⋊Dic₆₁	—	2	2	no	—
22	C₂×C₄⋊Dic₆₁	—	2	2	no	—
23	C₂×C₄⋊Dic₆₁	—	2	2	no	—
24	C₂×C₄⋊Dic₆₁	—	2	2	no	—
25	#14519709	—	4	4	no	—
26	#14519709	—	4	8	no	—
27	#14519702	—	4	4	no	—
28	C₂²×C₈×C₆₁	D₄	1	2	yes	—
29	C₂²×C₈×C₆₁	D₄	1	4	yes	—
30	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₆₁	—	2	2	no	—
32	C₄×C₈×C₆₁	—	2	2	no	—
33	C₄×C₈×C₆₁	—	2	2	no	—
34	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
36	C₂²×Dic₆₁	—	4	1	no	—
37	C₂²×Dic₆₁	—	4	1	no	—
38	C₂²×Dic₆₁	—	4	2	no	—
39	C₂×Dic₁₂₂	—	2	2	no	—
40	C₂×Dic₁₂₂	—	4	2	no	—
41	C₂×Dic₁₂₂	—	4	2	no	—
42	C₂×Dic₁₂₂	—	4	1	no	—
43	C₂×C₈×C₆₁	—	2	2	no	—
44	C₂×C₈×C₆₁	—	2	4	no	—
45	C₂×C₈×C₆₁	—	2	2	no	—
46	C₂×C₈×C₆₁	—	2	4	no	—
47	C₂×C₈×C₆₁	SD₁₆	1	4	yes	—
48	C₂×C₈×C₆₁	SD₁₆	1	8	yes	—
49	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₆₁	#41	1	2	yes	—
51	C₄²×C₆₁	#41	1	1	yes	characteristic
52	C₄⋊Dic₆₁	—	2	4	no	—
53	C₄⋊Dic₆₁	—	4	2	no	—
54	C₄⋊Dic₆₁	—	4	2	no	—
55	C₄⋊Dic₆₁	—	4	4	no	—
56	C₄⋊Dic₆₁	—	4	4	no	—
57	C₄⋊Dic₆₁	—	4	4	no	—
58	C₄⋊Dic₆₁	—	4	4	no	—
59	C₂³×C₆₁	#127	1	1	yes	characteristic
60	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
61	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₆₁	#123	1	1	yes	characteristic
63	C₂×C₄×C₆₁	#123	1	2	yes	—
64	C₂×C₄×C₆₁	C₂×SD₁₆	1	4	yes	—
65	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
66	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
67	C₂×C₄×C₆₁	#137	1	4	yes	—
68	C₂×Dic₆₁	—	4	2	no	—
69	C₂×Dic₆₁	—	8	1	no	—
70	C₂×Dic₆₁	—	8	1	no	—
71	C₂×Dic₆₁	—	8	1	no	—
72	C₂×Dic₆₁	—	8	2	no	—
73	C₂×Dic₆₁	—	8	2	no	—
74	C₂×Dic₆₁	—	8	2	no	—
75	C₂×Dic₆₁	—	8	2	no	—
76	Dic₁₂₂	—	4	4	no	—
77	Dic₁₂₂	—	8	2	no	—
78	C₈×C₆₁	—	2	4	no	—
79	C₈×C₆₁	—	2	8	no	—
80	#10685	—	61	1	no	maximal, Sylow
81	C₂²×C₆₁	#399	1	1	yes	characteristic
82	C₂²×C₆₁	#492	1	1	yes	characteristic
83	C₂²×C₆₁	#492	1	2	yes	—
84	C₂²×C₆₁	#496	1	1	yes	characteristic
85	C₂²×C₆₁	#497	1	2	yes	—
86	C₄×C₆₁	#475	1	4	yes	—
87	C₄×C₆₁	#476	1	2	yes	—
88	C₄×C₆₁	#481	1	2	yes	—
89	C₄×C₆₁	#483	1	4	yes	—
90	Dic₆₁	—	8	2	no	—
91	Dic₆₁	—	16	1	no	—
92	Dic₆₁	—	16	2	no	—
93	#1990	—	61	2	no	—
94	#1956	—	61	1	no	—
95	#1844	—	61	2	no	—
96	#1844	—	61	1	no	—
97	#1273	D₆₁	1	1	yes	characteristic
98	C₂×C₆₁	#1705	1	1	yes	characteristic
99	C₂×C₆₁	#1714	1	2	yes	—
100	C₂×C₆₁	#1716	1	1	yes	characteristic
101	C₂×C₆₁	#2094	1	1	yes	characteristic
102	C₂×C₆₁	#2096	1	2	yes	—
103	C₂×C₄⋊Q₈	—	61	1	no	—
104	C₂×#106	—	122	2	no	—
105	C₂×#106	—	122	4	no	—
106	C₂×#103	—	122	2	no	—
107	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
108	C₂×C₄×C₈	C₂×D₆₁	1	2	yes	—
109	#384	—	61	1	no	—
110	#384	—	61	2	no	—
111	#340	—	122	4	no	—
112	C₆₁	#10685	1	1	yes	characteristic, complemented, Sylow
113	C₂²×Q₈	—	122	1	no	—
114	C₂²×C₈	D₂₄₄	1	4	yes	—
115	C₂²×C₈	D₂₄₄	1	2	yes	—
116	C₄⋊Q₈	—	61	2	no	—
117	C₄⋊Q₈	—	122	2	no	—
118	C₄⋊Q₈	—	122	1	no	—
119	C₂×C₄⋊C₄	—	122	2	no	—
120	C₂×C₄⋊C₄	—	122	2	no	—
121	C₂×C₄⋊C₄	—	122	1	no	—
122	C₂×C₄⋊C₄	—	122	2	no	—
123	C₂×C₄⋊C₄	—	122	2	no	—
124	C₂×C₄⋊C₄	—	122	2	no	—
125	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
126	#106	—	244	4	no	—
127	#106	—	244	8	no	—
128	#103	—	244	4	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	—	2	2	no	—
132	C₂×Q₈	—	122	2	no	—
133	C₂×Q₈	—	244	2	no	—
134	C₂×Q₈	—	244	2	no	—
135	C₂×Q₈	—	244	1	no	—
136	C₂²×C₄	—	244	1	no	—
137	C₂²×C₄	—	244	2	no	—
138	C₂²×C₄	—	244	1	no	—
139	C₂²×C₄	C₂×D₂₄₄	1	2	yes	—
140	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	#11758190	1	4	yes	—
146	C₂×C₈	#11758190	1	8	yes	—
147	C₄⋊C₄	—	122	4	no	—
148	C₄⋊C₄	—	244	4	no	—
149	C₄⋊C₄	—	244	2	no	—
150	C₄⋊C₄	—	244	4	no	—
151	C₄⋊C₄	—	244	4	no	—
152	C₄⋊C₄	—	244	4	no	—
153	C₄⋊C₄	—	244	2	no	—
154	C₄²	#11758224	1	2	yes	—
155	C₄²	#11758224	1	1	yes	characteristic
156	C₄²	Q₈×D₆₁	1	1	yes	characteristic
157	C₂³	#14519774	1	1	yes	characteristic
158	Q₈	—	488	2	no	—
159	Q₈	—	244	4	no	—
160	C₂×C₄	—	488	2	no	—
161	C₂×C₄	—	488	1	no	—
162	C₂×C₄	—	488	2	no	—
163	C₂×C₄	—	488	2	no	—
164	C₂×C₄	—	488	2	no	—
165	C₂×C₄	—	488	1	no	—
166	C₂×C₄	—	488	1	no	—
167	C₂×C₄	—	244	2	no	—
168	C₂×C₄	#14519757	1	2	yes	—
169	C₂×C₄	#14519757	1	1	yes	characteristic
170	C₂×C₄	#14519761	1	1	yes	characteristic
171	C₂×C₄	#14519761	1	2	yes	—
172	C₂×C₄	C₂×#11758190	1	4	yes	—
173	C₂×C₄	C₂×#11758190	1	2	yes	—
174	C₂×C₄	#14519805	1	2	yes	—
175	C₂×C₄	#14519810	1	4	yes	—
176	C₈	—	2	4	no	—
177	C₈	—	2	8	no	—
178	C₂²	#16839699	1	1	yes	characteristic
179	C₂²	#16840135	1	2	yes	—
180	C₂²	#16840135	1	1	yes	characteristic
181	C₂²	#16840151	1	1	yes	characteristic
182	C₂²	#16840156	1	2	yes	—
183	C₄	—	976	1	no	—
184	C₄	—	976	2	no	—
185	C₄	—	488	2	no	—
186	C₄	#16840048	1	4	yes	—
187	C₄	#16840053	1	2	yes	—
188	C₄	#16840081	1	2	yes	—
189	C₄	#16840093	1	4	yes	—
190	C₂	#21445406	1	1	yes	characteristic
191	C₂	#21445441	1	2	yes	—
192	C₂	#21445452	1	1	yes	characteristic
193	C₂	#21447351	1	1	yes	characteristic
194	C₂	#21447361	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21446780

C₂

yes

maximal

#21446621

C₂

yes

maximal, characteristic

#21445976

C₂

yes

maximal

#21445976

C₂

yes

maximal, characteristic

C₆₁×#1273

C₂

yes

maximal, characteristic

C₂×#14519777

C₂²

yes

—

—

—

C₂²

yes

—

#16839632

C₂²

yes

characteristic

#16839525

—

C₂×C₄×C₈×C₆₁

C₂²

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂²×Dic₁₂₂

—

#14519777

—

#14519777

—

#14519777

Q₈

yes

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

#14519709

—

#14519709

—

#14519702

—

C₂²×C₈×C₆₁

D₄

yes

—

C₂²×C₈×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄×C₈×C₆₁

—

C₄×C₈×C₆₁

—

C₄×C₈×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×D₄

yes

—

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

SD₁₆

yes

—

C₂×C₈×C₆₁

SD₁₆

yes

—

C₄²×C₆₁

C₂×Q₈

yes

characteristic

C₄²×C₆₁

#41

yes

—

C₄²×C₆₁

#41

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₂³×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₁

C₂²⋊Q₈

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

—

C₂×C₄×C₆₁

C₂×SD₁₆

yes

—

C₂×C₄×C₆₁

C₂×SD₁₆

yes

—

C₂×C₄×C₆₁

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₆₁

#137

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

#496

yes

characteristic

C₂²×C₆₁

#497

yes

—

C₄×C₆₁

#475

yes

—

C₄×C₆₁

#476

yes

—

C₄×C₆₁

#481

yes

—

C₄×C₆₁

#483

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

—

—

—

—

D₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

100

C₂×C₆₁

#1716

yes

characteristic

101

C₂×C₆₁

#2094

yes

characteristic

102

C₂×C₆₁

#2096

yes

—

103

C₂×C₄⋊Q₈

—

104

C₂×#106

—

122

—

105

C₂×#106

—

122

—

106

C₂×#103

—

122

—

107

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

108

C₂×C₄×C₈

C₂×D₆₁

yes

—

109

#384

—

110

#384

—

111

#340

—

122

—

112

C₆₁

#10685

yes

characteristic, complemented, Sylow

113

C₂²×Q₈

—

122

—

114

C₂²×C₈

D₂₄₄

yes

—

115

C₂²×C₈

D₂₄₄

yes

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

—

122

—

118

C₄⋊Q₈

—

122

—

119

C₂×C₄⋊C₄

—

122

—

120

C₂×C₄⋊C₄

—

122

—

121

C₂×C₄⋊C₄

—

122

—

122

C₂×C₄⋊C₄

—

122

—

123

C₂×C₄⋊C₄

—

122

—

124

C₂×C₄⋊C₄

—

122

—

125

C₂×C₄²

C₂²×D₆₁

yes

characteristic

126

#106

—

244

—

127

#106

—

244

—

128

#103

—

244

—

129

C₄×C₈

—

130

C₄×C₈

—

131

C₄×C₈

—

132

C₂×Q₈

—

122

—

133

C₂×Q₈

—

244

—

134

C₂×Q₈

—

244

—

135

C₂×Q₈

—

244

—

136

C₂²×C₄

—

244

—

137

C₂²×C₄

—

244

—

138

C₂²×C₄

—

244

—

139

C₂²×C₄

C₂×D₂₄₄

yes

—

140

C₂²×C₄

C₂×D₂₄₄

yes

characteristic

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

#11758190

yes

—

146

C₂×C₈

#11758190

yes

—

147

C₄⋊C₄

—

122

—

148

C₄⋊C₄

—

244

—

149

C₄⋊C₄

—

244

—

150

C₄⋊C₄

—

244

—

151

C₄⋊C₄

—

244

—

152

C₄⋊C₄

—

244

—

153

C₄⋊C₄

—

244

—

154

C₄²

#11758224

yes

—

155

C₄²

#11758224

yes

characteristic

156

C₄²

Q₈×D₆₁

yes

characteristic

157

C₂³

#14519774

yes

characteristic

158

Q₈

—

488

—

159

Q₈

—

244

—

160

C₂×C₄

—

488

—

161

C₂×C₄

—

488

—

162

C₂×C₄

—

488

—

163

C₂×C₄

—

488

—

164

C₂×C₄

—

488

—

165

C₂×C₄

—

488

—

166

C₂×C₄

—

488

—

167

C₂×C₄

—

244

—

168

C₂×C₄

#14519757

yes

—

169

C₂×C₄

#14519757

yes

characteristic

170

C₂×C₄

#14519761

yes

characteristic

171

C₂×C₄

#14519761

yes

—

172

C₂×C₄

C₂×#11758190

yes

—

173

C₂×C₄

C₂×#11758190

yes

—

174

C₂×C₄

#14519805

yes

—

175

C₂×C₄

#14519810

yes

—

176

C₈

—

177

C₈

—

178

C₂²

#16839699

yes

characteristic

179

C₂²

#16840135

yes

—

180

C₂²

#16840135

yes

characteristic

181

C₂²

#16840151

yes

characteristic

182

C₂²

#16840156

yes

—

183

C₄

—

976

—

184

C₄

—

976

—

185

C₄

—

488

—

186

C₄

#16840048

yes

—

187

C₄

#16840053

yes

—

188

C₄

#16840081

yes

—

189

C₄

#16840093

yes

—

190

C₂

#21445406

yes

characteristic

191

C₂

#21445441

yes

—

192

C₂

#21445452

yes

characteristic

193

C₂

#21447351

yes

characteristic

194

C₂

#21447361

yes

—

	subgroup	G/H
center	C₂³	#14519774
commutator, cosocle	C₂×C₄²×C₆₁	C₂³
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1273	C₂
socle, prime radical	C₂³×C₆₁	#127
hypercenter	#1273	D₆₁
hypocenter	C₆₁	#10685
LCS	G ▷ C₂×C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center

C₂³

#14519774

commutator, cosocle

C₂×C₄²×C₆₁

C₂³

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#1273

C₂

C₂³×C₆₁

D₆₁

C₆₁

G ▷ C₂×C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
112. C₆₁	#10685	semidirect

H ≅ G/N

type

112. C₆₁

#10685

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#21446780	2
#21446621	2
#21445976	2
C₆₁×#1273	2
#10685	61

subgroup

index

C₆₁×#1273

Subgroups — #29585129

Conjugacy classes of subgroups

Complements

Maximal subgroups