Group #29599550 — subgroups

Subgroups — #29599550

Order 15616.

418 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 95 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	#21447577	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16839575	C₂	1	2	yes	maximal, complemented
3	#21446730	C₂	1	2	yes	maximal, complemented
4	#21443913	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519717	C₂²	1	1	yes	characteristic
7	C₂×#14519713	C₂²	1	2	yes	—
8	#16839767	—	2	2	no	—
9	C₂×#14519694	—	2	2	no	—
10	#16839717	C₂²	1	1	yes	characteristic
11	#16839580	—	2	2	no	—
12	#16839575	—	2	4	no	—
13	C₂²×D₈×C₆₁	C₂²	1	2	yes	—
14	C₂×D₁₆×C₆₁	—	2	4	no	—
15	C₂×D₁₆×C₆₁	—	2	2	no	—
16	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×#11758194	—	2	2	no	—
18	C₂×#11758194	D₄	1	1	yes	characteristic
19	#14519736	D₄	1	1	yes	characteristic
20	#14519717	—	2	2	no	—
21	C₂×#11758185	—	2	1	no	—
22	#14519713	—	2	2	no	—
23	#14519713	—	2	2	no	—
24	#14519698	—	4	4	no	—
25	#14519694	—	4	4	no	—
26	#14519687	—	2	1	no	—
27	C₈×Dic₆₁	—	2	1	no	—
28	C₂²×D₄×C₆₁	—	2	2	no	—
29	C₂×D₈×C₆₁	—	2	2	no	—
30	C₂×D₈×C₆₁	—	2	2	no	—
31	C₂×D₈×C₆₁	—	2	2	no	—
32	C₂×D₈×C₆₁	—	2	2	no	—
33	C₂×D₈×C₆₁	—	2	2	no	—
34	C₂×D₈×C₆₁	D₄	1	4	yes	—
35	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
36	D₁₆×C₆₁	—	2	8	no	—
37	C₂×C₁₆×C₆₁	—	2	2	no	—
38	C₂×C₁₆×C₆₁	D₄	1	2	yes	—
39	C₂⁴×C₆₁	—	4	2	no	—
40	C₂×D₄×C₆₁	—	2	4	no	—
41	C₂×D₄×C₆₁	—	4	2	no	—
42	C₂×D₄×C₆₁	—	4	2	no	—
43	C₂×D₄×C₆₁	—	4	2	no	—
44	C₂×D₄×C₆₁	—	4	2	no	—
45	C₂×D₄×C₆₁	—	4	2	no	—
46	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
47	D₈×C₆₁	—	2	4	no	—
48	D₈×C₆₁	—	2	4	no	—
49	D₈×C₆₁	—	4	2	no	—
50	D₈×C₆₁	—	4	2	no	—
51	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
52	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
54	C₂×C₈×C₆₁	#41	1	2	yes	—
55	C₂²⋊Dic₆₁	—	2	1	no	—
56	C₄⋊Dic₆₁	D₈	1	2	yes	—
57	C₄×Dic₆₁	—	2	1	no	—
58	#11758194	—	2	2	no	—
59	#11758194	—	4	2	no	—
60	#11758194	—	4	2	no	—
61	C₂×#92418	—	2	1	no	—
62	C₂×#92418	—	4	2	no	—
63	C₁₆×C₆₁	—	2	2	no	—
64	#11758185	—	4	1	no	—
65	C₂³×C₆₁	—	4	4	no	—
66	C₂³×C₆₁	—	8	2	no	—
67	C₂³×C₆₁	—	8	2	no	—
68	C₂³×C₆₁	—	16	4	no	—
69	C₂³×C₆₁	#135	1	1	yes	characteristic
70	D₄×C₆₁	—	4	4	no	—
71	D₄×C₆₁	—	4	4	no	—
72	D₄×C₆₁	—	8	2	no	—
73	D₄×C₆₁	—	8	2	no	—
74	C₂×C₄×C₆₁	#121	1	2	yes	—
75	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
76	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
78	C₂×Dic₆₁	—	2	2	no	—
79	C₈×C₆₁	#120	1	1	yes	characteristic
80	C₈×C₆₁	#123	1	2	yes	—
81	C₈×C₆₁	#124	1	1	yes	characteristic
82	#92418	—	4	1	no	—
83	#92418	—	8	2	no	—
84	#13439	—	61	1	no	maximal, Sylow
85	C₂²×C₆₁	—	2	2	no	—
86	C₂²×C₆₁	—	8	4	no	—
87	C₂²×C₆₁	—	16	4	no	—
88	C₂²×C₆₁	—	16	4	no	—
89	C₂²×C₆₁	—	16	4	no	—
90	C₂²×C₆₁	#463	1	2	yes	—
91	C₂²×C₆₁	#466	1	1	yes	characteristic
92	C₄×C₆₁	#394	1	1	yes	characteristic
93	C₄×C₆₁	#459	1	2	yes	—
94	C₄×C₆₁	#469	1	1	yes	characteristic
95	Dic₆₁	—	4	2	no	—
96	C₂²×D₁₆	D₆₁	1	1	yes	characteristic
97	#2151	—	61	1	no	—
98	C₂×#361	—	61	2	no	—
99	#1978	—	61	2	no	—
100	#1381	—	61	1	no	—
101	C₂×C₆₁	—	2	2	no	—
102	C₂×C₆₁	—	16	4	no	—
103	C₂×C₆₁	#2054	1	1	yes	characteristic
104	C₂×C₆₁	#2221	1	2	yes	—
105	C₂²×D₈	C₂×D₆₁	1	2	yes	—
106	C₂×D₁₆	—	2	4	no	—
107	C₂×D₁₆	—	2	2	no	—
108	C₂×M₃₂	—	61	1	no	—
109	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
110	C₂×#108	—	61	2	no	—
111	#418	—	122	2	no	—
112	C₂×#100	—	122	2	no	—
113	#405	—	61	1	no	—
114	C₄⋊C₁₆	—	122	2	no	—
115	#361	—	122	4	no	—
116	C₆₁	#13439	1	1	yes	characteristic, complemented, Sylow
117	C₂²×D₄	—	2	2	no	—
118	C₂×D₈	—	2	2	no	—
119	C₂×D₈	—	2	2	no	—
120	C₂×D₈	—	2	2	no	—
121	C₂×D₈	—	2	2	no	—
122	C₂×D₈	—	2	2	no	—
123	C₂×D₈	C₂²⋊D₆₁	1	4	yes	—
124	C₂×M₁₆	—	61	1	no	—
125	C₂×M₁₆	—	122	2	no	—
126	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
127	#117	—	61	1	no	—
128	D₁₆	—	2	8	no	—
129	M₃₂	—	122	2	no	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	122	1	no	—
132	C₂×C₁₆	C₂²⋊D₆₁	1	2	yes	—
133	#108	—	122	2	no	—
134	#108	—	122	2	no	—
135	#102	—	244	4	no	—
136	#100	—	244	4	no	—
137	#97	—	122	1	no	—
138	C₄×C₈	—	122	1	no	—
139	C₂⁴	—	4	2	no	—
140	C₂×D₄	—	2	4	no	—
141	C₂×D₄	—	4	2	no	—
142	C₂×D₄	—	4	2	no	—
143	C₂×D₄	—	4	2	no	—
144	C₂×D₄	—	4	2	no	—
145	C₂×D₄	—	4	2	no	—
146	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
147	D₈	—	2	4	no	—
148	D₈	—	2	4	no	—
149	D₈	—	4	2	no	—
150	D₈	—	4	2	no	—
151	M₁₆	—	122	2	no	—
152	M₁₆	—	244	2	no	—
153	M₁₆	—	244	2	no	—
154	C₂×C₈	—	122	1	no	—
155	C₂×C₈	—	244	2	no	—
156	C₂×C₈	#11758224	1	1	yes	characteristic
157	C₂×C₈	#11758224	1	2	yes	—
158	C₂×C₈	C₂×C₂²⋊D₆₁	1	2	yes	—
159	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
160	C₄⋊C₄	—	61	2	no	—
161	C₂²⋊C₄	—	122	1	no	—
162	C₄²	—	122	1	no	—
163	C₁₆	—	2	2	no	—
164	C₁₆	—	244	1	no	—
165	C₂³	—	4	4	no	—
166	C₂³	—	8	2	no	—
167	C₂³	—	8	2	no	—
168	C₂³	—	16	4	no	—
169	C₂³	#14519796	1	1	yes	characteristic
170	D₄	—	4	4	no	—
171	D₄	—	4	4	no	—
172	D₄	—	8	2	no	—
173	D₄	—	8	2	no	—
174	C₂×C₄	—	122	2	no	—
175	C₂×C₄	#14519748	1	1	yes	characteristic
176	C₂×C₄	#14519749	1	2	yes	—
177	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
178	C₂×C₄	#14519801	1	2	yes	—
179	C₈	—	488	2	no	—
180	C₈	—	244	1	no	—
181	C₈	#14519746	1	1	yes	characteristic
182	C₈	#14519758	1	2	yes	—
183	C₈	#14519764	1	1	yes	characteristic
184	C₂²	—	2	2	no	—
185	C₂²	—	8	4	no	—
186	C₂²	—	16	4	no	—
187	C₂²	—	16	4	no	—
188	C₂²	—	16	4	no	—
189	C₂²	#16839975	1	2	yes	—
190	C₂²	#16839995	1	1	yes	characteristic
191	C₄	—	244	2	no	—
192	C₄	#16839676	1	1	yes	characteristic
193	C₄	#16839955	1	2	yes	—
194	C₄	#16840012	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	—	16	4	no	—
197	C₂	#21447127	1	1	yes	characteristic
198	C₂	#21447936	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21447577

C₂

yes

maximal, characteristic, complemented

C₂×#16839575

C₂

yes

maximal, complemented

#21446730

C₂

yes

maximal, complemented

#21443913

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₆×C₆₁

C₂

yes

maximal, characteristic

C₂×#14519717

C₂²

yes

characteristic

C₂×#14519713

C₂²

yes

—

#16839767

—

C₂×#14519694

—

#16839717

C₂²

yes

characteristic

#16839580

—

#16839575

—

C₂²×D₈×C₆₁

C₂²

yes

—

C₂×D₁₆×C₆₁

—

C₂×D₁₆×C₆₁

—

C₂²×C₁₆×C₆₁

C₂²

yes

characteristic

C₂×#11758194

—

C₂×#11758194

D₄

yes

characteristic

#14519736

D₄

yes

—

—

—

—

—

—

—

C₈×Dic₆₁

—

C₂²×D₄×C₆₁

—

C₂×D₈×C₆₁

—

C₂×D₈×C₆₁

—

C₂×D₈×C₆₁

—

C₂×D₈×C₆₁

—

C₂×D₈×C₆₁

—

C₂×D₈×C₆₁

D₄

yes

—

C₂²×C₈×C₆₁

C₂³

yes

characteristic

D₁₆×C₆₁

—

C₂×C₁₆×C₆₁

—

C₂×C₁₆×C₆₁

D₄

yes

—

C₂⁴×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

D₈×C₆₁

—

D₈×C₆₁

—

D₈×C₆₁

—

D₈×C₆₁

—

C₂×C₈×C₆₁

C₂×D₄

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

C₂×C₈×C₆₁

yes

C₂×C₈×C₆₁

yes

—

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

D₈

yes

—

C₄×Dic₆₁

—

—

—

—

—

—

C₁₆×C₆₁

—

#11758185

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

#135

yes

characteristic

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

#121

yes

—

C₂×C₄×C₆₁

#121

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×C₄×C₆₁

AGL(1,ℤ/8)

yes

—

C₂×Dic₆₁

—

C₈×C₆₁

#120

yes

characteristic

C₈×C₆₁

#123

yes

—

C₈×C₆₁

yes

—

—

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#463

yes

—

C₂²×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

C₄×C₆₁

#469

yes

characteristic

Dic₆₁

—

C₂²×D₁₆

D₆₁

yes

—

—

—

100

#1381

—

101

C₂×C₆₁

—

102

C₂×C₆₁

—

103

C₂×C₆₁

#2054

yes

characteristic

104

C₂×C₆₁

#2221

yes

—

105

C₂²×D₈

C₂×D₆₁

yes

—

106

C₂×D₁₆

—

107

C₂×D₁₆

—

108

C₂×M₃₂

—

109

C₂²×C₁₆

C₂×D₆₁

yes

characteristic

110

C₂×#108

—

111

#418

—

122

—

112

C₂×#100

—

122

—

113

#405

—

114

C₄⋊C₁₆

—

122

—

115

#361

—

122

—

116

C₆₁

#13439

yes

characteristic, complemented, Sylow

117

C₂²×D₄

—

118

C₂×D₈

—

119

C₂×D₈

—

120

C₂×D₈

—

121

C₂×D₈

—

122

C₂×D₈

—

123

C₂×D₈

C₂²⋊D₆₁

yes

—

124

C₂×M₁₆

—

125

C₂×M₁₆

—

122

—

126

C₂²×C₈

C₂²×D₆₁

yes

characteristic

127

#117

—

128

D₁₆

—

129

M₃₂

—

122

—

130

C₂×C₁₆

—

131

C₂×C₁₆

—

122

—

132

C₂×C₁₆

C₂²⋊D₆₁

yes

—

133

#108

—

122

—

134

#108

—

122

—

135

#102

—

244

—

136

#100

—

244

—

137

#97

—

122

—

138

C₄×C₈

—

122

—

139

C₂⁴

—

140

C₂×D₄

—

141

C₂×D₄

—

142

C₂×D₄

—

143

C₂×D₄

—

144

C₂×D₄

—

145

C₂×D₄

—

146

C₂²×C₄

D₄×D₆₁

yes

characteristic

147

D₈

—

148

D₈

—

149

D₈

—

150

D₈

—

151

M₁₆

—

122

—

152

M₁₆

—

244

—

153

M₁₆

—

244

—

154

C₂×C₈

—

122

—

155

C₂×C₈

—

244

—

156

C₂×C₈

#11758224

yes

characteristic

157

C₂×C₈

#11758224

yes

—

158

C₂×C₈

C₂×C₂²⋊D₆₁

yes

—

159

C₂×C₈

C₂×C₂²⋊D₆₁

yes

characteristic

160

C₄⋊C₄

—

161

C₂²⋊C₄

—

122

—

162

C₄²

—

122

—

163

C₁₆

—

164

C₁₆

—

244

—

165

C₂³

—

166

C₂³

—

167

C₂³

—

168

C₂³

—

169

C₂³

#14519796

yes

characteristic

170

D₄

—

171

D₄

—

172

D₄

—

173

D₄

—

174

C₂×C₄

—

122

—

175

C₂×C₄

#14519748

yes

characteristic

176

C₂×C₄

#14519749

yes

—

177

C₂×C₄

D₈×D₆₁

yes

characteristic

178

C₂×C₄

#14519801

yes

—

179

C₈

—

488

—

180

C₈

—

244

—

181

C₈

#14519746

yes

characteristic

182

C₈

#14519758

yes

—

183

C₈

#14519764

yes

characteristic

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

#16839975

yes

—

190

C₂²

#16839995

yes

characteristic

191

C₄

—

244

—

192

C₄

#16839676

yes

characteristic

193

C₄

#16839955

yes

—

194

C₄

#16840012

yes

characteristic

195

C₂

—

196

C₂

—

197

C₂

#21447127

yes

characteristic

198

C₂

#21447936

yes

—

	subgroup	G/H
center	C₂²	#16839995
commutator, cosocle	C₂²×C₈×C₆₁	C₂³
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting, prime radical	C₂²×D₁₆×C₆₁	C₂
socle	C₂²×C₆₁	#466
hypercenter	C₂²×D₁₆	D₆₁
hypocenter	C₆₁	#13439
LCS	G ▷ C₂²×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₁₆

subgroup

G/H

center

C₂²

#16839995

commutator, cosocle

C₂²×C₈×C₆₁

C₂³

Frattini

C₂²×C₈

C₂²×D₆₁

Fitting, prime radical

C₂²×D₁₆×C₆₁

C₂

socle

C₂²×C₆₁

#466

hypercenter

C₂²×D₁₆

D₆₁

hypocenter

C₆₁

#13439

LCS

G ▷ C₂²×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₁₆

N	H ≅ G/N	type
1. #21447577	C₂	semidirect
2. C₂×#16839575	C₂	semidirect
3. #21446730	C₂	semidirect
4. #21443913	C₂	semidirect
116. C₆₁	#13439	semidirect

H ≅ G/N

type

1. #21447577

C₂

semidirect

2. C₂×#16839575

C₂

semidirect

3. #21446730

C₂

semidirect

4. #21443913

C₂

semidirect

116. C₆₁

#13439

semidirect

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

subgroup	index
#21447577	2
C₂×#16839575	2
#21446730	2
#21443913	2
C₂²×D₁₆×C₆₁	2
#13439	61

subgroup

index

C₂²×D₁₆×C₆₁

Subgroups — #29599550

Conjugacy classes of subgroups

Complements

Maximal subgroups