Group #29596351 — subgroups

Subgroups — #29596351

Order 15616.

366 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 127 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	#21447433	C₂	1	1	yes	maximal, characteristic
2	#21445865	C₂	1	1	yes	maximal, characteristic
3	#21443933	C₂	1	2	yes	maximal, complemented
4	#21443916	C₂	1	2	yes	maximal, complemented
5	C₆₁×#2916	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519717	C₂²	1	1	yes	characteristic
7	C₂²×#11758185	C₂²	1	1	yes	characteristic
8	#16839800	C₂²	1	2	yes	—
9	#16839736	—	2	1	no	—
10	C₂²⋊#11758185	—	2	2	no	—
11	#16839519	—	2	1	no	—
12	C₆₁×#568	—	2	1	no	—
13	C₂×#131×C₆₁	C₂²	1	1	yes	characteristic
14	C₆₁×#430	—	2	1	no	—
15	C₆₁×#430	C₄	1	4	yes	complemented
16	C₂×#108×C₆₁	—	2	2	no	—
17	C₂×#108×C₆₁	C₂²	1	2	yes	—
18	#14519736	—	2	2	no	—
19	#14519717	—	2	2	no	—
20	C₂×#11758185	—	2	1	no	—
21	C₂×#11758185	—	2	1	no	—
22	C₂×#11758185	—	2	2	no	—
23	#14519711	—	2	2	no	—
24	#14519709	—	2	2	no	—
25	#14519692	—	4	2	no	—
26	C₂×#41×C₆₁	D₄	1	1	yes	characteristic
27	#131×C₆₁	—	2	2	no	—
28	#131×C₆₁	—	2	2	no	—
29	#131×C₆₁	—	4	4	no	—
30	C₂×M₁₆×C₆₁	—	2	4	no	—
31	C₂×M₁₆×C₆₁	—	2	2	no	—
32	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
33	C₂×M₁₆×C₆₁	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₆₁	D₄	1	2	yes	—
35	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
36	#108×C₆₁	—	2	4	no	—
37	#108×C₆₁	—	2	2	no	—
38	#108×C₆₁	C₂×C₄	1	4	yes	—
39	#41×C₆₁	—	2	2	no	—
40	#41×C₆₁	—	2	2	no	—
41	C₂×Q₈×C₆₁	Q₁₆	1	1	yes	characteristic
42	C₂×D₄×C₆₁	SD₁₆	1	2	yes	—
43	C₂×D₄×C₆₁	Q₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
46	M₁₆×C₆₁	—	2	2	no	—
47	M₁₆×C₆₁	—	2	4	no	—
48	M₁₆×C₆₁	—	2	4	no	—
49	M₁₆×C₆₁	—	4	4	no	—
50	M₁₆×C₆₁	—	4	2	no	—
51	C₂×C₈×C₆₁	—	2	2	no	—
52	C₂×C₈×C₆₁	—	2	2	no	—
53	C₂×C₈×C₆₁	—	4	4	no	—
54	C₂×C₈×C₆₁	—	4	2	no	—
55	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₆₁	#41	1	2	yes	—
59	C₂×C₈×C₆₁	#41	1	2	yes	—
60	C₂²⋊Dic₆₁	—	4	2	no	—
61	C₄⋊Dic₆₁	—	4	2	no	—
62	C₄×Dic₆₁	—	4	2	no	—
63	#11758185	—	4	1	no	—
64	#11758185	—	4	1	no	—
65	C₂³×C₆₁	#103	1	2	yes	—
66	C₂³×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
67	Q₈×C₆₁	—	2	2	no	—
68	D₄×C₆₁	—	2	4	no	—
69	D₄×C₆₁	—	2	2	no	—
70	C₂×C₄×C₆₁	—	2	2	no	—
71	C₂×C₄×C₆₁	—	2	2	no	—
72	C₂×C₄×C₆₁	#103	1	2	yes	—
73	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₆₁	#123	1	2	yes	—
75	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
77	C₂×Dic₆₁	—	8	2	no	—
78	C₈×C₆₁	—	2	2	no	—
79	C₈×C₆₁	—	4	4	no	—
80	C₈×C₆₁	#117	1	2	yes	—
81	C₈×C₆₁	#123	1	1	yes	characteristic
82	C₈×C₆₁	#123	1	1	yes	characteristic
83	#12877	—	61	1	no	maximal, Sylow
84	C₂²×C₆₁	—	2	2	no	—
85	C₂²×C₆₁	—	4	1	no	—
86	C₂²×C₆₁	—	4	2	no	—
87	C₂²×C₆₁	C₂×#103	1	1	yes	characteristic
88	C₂²×C₆₁	#481	1	2	yes	—
89	C₄×C₆₁	—	2	2	no	—
90	C₄×C₆₁	#381	1	1	yes	characteristic
91	C₄×C₆₁	#418	1	1	yes	characteristic
92	C₄×C₆₁	#484	1	2	yes	—
93	Dic₆₁	—	16	2	no	—
94	#2916	D₆₁	1	1	yes	characteristic
95	#2113	—	61	1	no	—
96	#1812	—	61	1	no	—
97	#1389	—	61	2	no	—
98	#1382	—	61	2	no	—
99	C₂×C₆₁	—	2	2	no	—
100	C₂×C₆₁	—	2	1	no	—
101	C₂×C₆₁	—	4	2	no	—
102	C₂×C₆₁	#1824	1	1	yes	characteristic
103	#568	—	2	1	no	—
104	C₂×#131	C₂×D₆₁	1	1	yes	characteristic
105	C₂×M₃₂	—	61	1	no	—
106	C₂²×C₁₆	—	61	1	no	—
107	#430	—	2	1	no	—
108	#430	Dic₆₁	1	4	yes	complemented
109	C₂×#108	—	2	2	no	—
110	C₂×#108	C₂×D₆₁	1	2	yes	—
111	#427	—	61	2	no	—
112	#410	—	122	1	no	—
113	C₂²⋊C₁₆	—	122	2	no	—
114	H(3,ℤ/4)	—	122	1	no	—
115	C₆₁	#12877	1	1	yes	characteristic, complemented, Sylow
116	C₂×#41	C₂²⋊D₆₁	1	1	yes	characteristic
117	#131	—	2	2	no	—
118	#131	—	2	2	no	—
119	#131	—	4	4	no	—
120	C₂×M₁₆	—	2	4	no	—
121	C₂×M₁₆	—	2	2	no	—
122	C₂×M₁₆	C₂×Dic₆₁	1	2	yes	—
123	C₂×M₁₆	C₂²⋊D₆₁	1	1	yes	characteristic
124	C₂²×C₈	C₂²⋊D₆₁	1	2	yes	—
125	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
126	#117	—	122	2	no	—
127	M₃₂	—	122	2	no	—
128	C₂×C₁₆	—	122	2	no	—
129	C₂×C₁₆	—	122	1	no	—
130	C₂×C₁₆	—	122	1	no	—
131	#108	—	2	4	no	—
132	#108	—	2	2	no	—
133	#108	C₂×Dic₆₁	1	4	yes	—
134	#107	—	122	2	no	—
135	#106	—	122	2	no	—
136	#99	—	244	2	no	—
137	#41	—	2	2	no	—
138	#41	—	2	2	no	—
139	C₂×Q₈	#11758202	1	1	yes	characteristic
140	C₂×D₄	#11758201	1	2	yes	—
141	C₂×D₄	#11758202	1	1	yes	characteristic
142	C₂²×C₄	C₂²⋊Dic₆₁	1	2	yes	—
143	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
144	M₁₆	—	2	2	no	—
145	M₁₆	—	2	4	no	—
146	M₁₆	—	2	4	no	—
147	M₁₆	—	4	4	no	—
148	M₁₆	—	4	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	4	4	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
154	C₂×C₈	#11758222	1	2	yes	—
155	C₂×C₈	#11758222	1	2	yes	—
156	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
157	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
158	C₄⋊C₄	—	244	2	no	—
159	C₂²⋊C₄	—	244	2	no	—
160	C₄²	—	244	2	no	—
161	C₁₆	—	244	1	no	—
162	C₁₆	—	244	1	no	—
163	C₂³	#14519701	1	2	yes	—
164	C₂³	C₂×#11758201	1	1	yes	characteristic
165	Q₈	—	2	2	no	—
166	D₄	—	2	4	no	—
167	D₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	488	2	no	—
171	C₂×C₄	#14519701	1	2	yes	—
172	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
173	C₂×C₄	#14519760	1	2	yes	—
174	C₂×C₄	C₂×#11758202	1	1	yes	characteristic
175	C₂×C₄	#14519803	1	2	yes	—
176	C₈	—	2	2	no	—
177	C₈	—	4	4	no	—
178	C₈	#14519736	1	2	yes	—
179	C₈	#14519760	1	1	yes	characteristic
180	C₈	#14519760	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	—	4	1	no	—
183	C₂²	—	4	2	no	—
184	C₂²	C₂×#14519701	1	1	yes	characteristic
185	C₂²	#16840080	1	2	yes	—
186	C₄	—	2	2	no	—
187	C₄	—	976	2	no	—
188	C₄	#16839625	1	1	yes	characteristic
189	C₄	#16839767	1	1	yes	characteristic
190	C₄	#16840097	1	2	yes	—
191	C₂	—	2	2	no	—
192	C₂	—	2	1	no	—
193	C₂	—	4	2	no	—
194	C₂	#21445909	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21447433

C₂

yes

maximal, characteristic

#21445865

C₂

yes

maximal, characteristic

#21443933

C₂

yes

maximal, complemented

#21443916

C₂

yes

maximal, complemented

C₆₁×#2916

C₂

yes

maximal, characteristic

C₂×#14519717

C₂²

yes

characteristic

C₂²×#11758185

C₂²

yes

characteristic

#16839800

C₂²

yes

—

#16839736

—

C₂²⋊#11758185

—

#16839519

—

C₆₁×#568

—

C₂×#131×C₆₁

C₂²

yes

characteristic

C₆₁×#430

—

C₆₁×#430

C₄

yes

complemented

C₂×#108×C₆₁

—

C₂×#108×C₆₁

C₂²

yes

—

—

—

—

—

—

—

—

—

C₂×#41×C₆₁

D₄

yes

characteristic

#131×C₆₁

—

#131×C₆₁

—

#131×C₆₁

—

C₂×M₁₆×C₆₁

—

C₂×M₁₆×C₆₁

—

C₂×M₁₆×C₆₁

C₂×C₄

yes

—

C₂×M₁₆×C₆₁

D₄

yes

characteristic

C₂²×C₈×C₆₁

D₄

yes

—

C₂²×C₈×C₆₁

C₂³

yes

characteristic

#108×C₆₁

—

#108×C₆₁

—

#108×C₆₁

C₂×C₄

yes

—

#41×C₆₁

—

#41×C₆₁

—

C₂×Q₈×C₆₁

Q₁₆

yes

characteristic

C₂×D₄×C₆₁

SD₁₆

yes

—

C₂×D₄×C₆₁

Q₁₆

yes

characteristic

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

M₁₆×C₆₁

—

M₁₆×C₆₁

—

M₁₆×C₆₁

—

M₁₆×C₆₁

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂²⋊C₄

yes

—

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

#41

yes

—

C₂×C₈×C₆₁

#41

yes

—

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄×Dic₆₁

—

#11758185

—

#11758185

—

C₂³×C₆₁

#103

yes

—

C₂³×C₆₁

C₂×SD₁₆

yes

characteristic

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#103

yes

—

C₂×C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

#123

yes

—

C₂×C₄×C₆₁

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₆₁

AGL(1,ℤ/8)

yes

—

C₂×Dic₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

#117

yes

—

C₈×C₆₁

yes

C₈×C₆₁

yes

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×#103

yes

characteristic

C₂²×C₆₁

#481

yes

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

Dic₆₁

—

#2916

D₆₁

yes

—

—

—

—

C₂×C₆₁

—

100

C₂×C₆₁

—

101

C₂×C₆₁

—

102

C₂×C₆₁

#1824

yes

characteristic

103

#568

—

104

C₂×#131

C₂×D₆₁

yes

characteristic

105

C₂×M₃₂

—

106

C₂²×C₁₆

—

107

#430

—

108

#430

Dic₆₁

yes

complemented

109

C₂×#108

—

110

C₂×#108

C₂×D₆₁

yes

—

111

#427

—

112

#410

—

122

—

113

C₂²⋊C₁₆

—

122

—

114

H(3,ℤ/4)

—

122

—

115

C₆₁

#12877

yes

characteristic, complemented, Sylow

116

C₂×#41

C₂²⋊D₆₁

yes

characteristic

117

#131

—

118

#131

—

119

#131

—

120

C₂×M₁₆

—

121

C₂×M₁₆

—

122

C₂×M₁₆

C₂×Dic₆₁

yes

—

123

C₂×M₁₆

C₂²⋊D₆₁

yes

characteristic

124

C₂²×C₈

C₂²⋊D₆₁

yes

—

125

C₂²×C₈

C₂²×D₆₁

yes

characteristic

126

#117

—

122

—

127

M₃₂

—

122

—

128

C₂×C₁₆

—

122

—

129

C₂×C₁₆

—

122

—

130

C₂×C₁₆

—

122

—

131

#108

—

132

#108

—

133

#108

C₂×Dic₆₁

yes

—

134

#107

—

122

—

135

#106

—

122

—

136

#99

—

244

—

137

#41

—

138

#41

—

139

C₂×Q₈

#11758202

yes

characteristic

140

C₂×D₄

#11758201

yes

—

141

C₂×D₄

#11758202

yes

characteristic

142

C₂²×C₄

C₂²⋊Dic₆₁

yes

—

143

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

144

M₁₆

—

145

M₁₆

—

146

M₁₆

—

147

M₁₆

—

148

M₁₆

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

C₂²⋊Dic₆₁

yes

—

154

C₂×C₈

#11758222

yes

—

155

C₂×C₈

#11758222

yes

—

156

C₂×C₈

C₂²×Dic₆₁

yes

characteristic

157

C₂×C₈

C₂×C₂²⋊D₆₁

yes

characteristic

158

C₄⋊C₄

—

244

—

159

C₂²⋊C₄

—

244

—

160

C₄²

—

244

—

161

C₁₆

—

244

—

162

C₁₆

—

244

—

163

C₂³

#14519701

yes

—

164

C₂³

C₂×#11758201

yes

characteristic

165

Q₈

—

166

D₄

—

167

D₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

488

—

171

C₂×C₄

#14519701

yes

—

172

C₂×C₄

C₂×C₂²⋊Dic₆₁

yes

characteristic

173

C₂×C₄

#14519760

yes

—

174

C₂×C₄

C₂×#11758202

yes

characteristic

175

C₂×C₄

#14519803

yes

—

176

C₈

—

177

C₈

—

178

C₈

#14519736

yes

—

179

C₈

#14519760

yes

characteristic

180

C₈

#14519760

yes

characteristic

181

C₂²

—

182

C₂²

—

183

C₂²

—

184

C₂²

C₂×#14519701

yes

characteristic

185

C₂²

#16840080

yes

—

186

C₄

—

187

C₄

—

976

—

188

C₄

#16839625

yes

characteristic

189

C₄

#16839767

yes

characteristic

190

C₄

#16840097

yes

—

191

C₂

—

192

C₂

—

193

C₂

—

194

C₂

#21445909

yes

characteristic

	subgroup	G/H
center	C₄	#16839767
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#2916	C₂
socle	C₂×C₆₁	#1824
prime radical	C₂×#41×C₆₁	D₄
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#2916	D₆₁
hypocenter	C₆₁	#12877
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #2916

subgroup

G/H

center

C₄

#16839767

commutator

C₂×C₈×C₆₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₁

Fitting

C₆₁×#2916

C₂

socle

C₂×C₆₁

#1824

prime radical

C₂×#41×C₆₁

D₄

cosocle

C₂²×C₈×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #2916

N	H ≅ G/N	type
3. #21443933	C₂	semidirect
4. #21443916	C₂	semidirect
15. C₆₁×#430	C₄	semidirect
108. #430	Dic₆₁	semidirect
115. C₆₁	#12877	semidirect

H ≅ G/N

type

3. #21443933

C₂

semidirect

4. #21443916

C₂

semidirect

15. C₆₁×#430

C₄

semidirect

108. #430

Dic₆₁

semidirect

115. C₆₁

#12877

semidirect

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

subgroup	index
#21447433	2
#21445865	2
#21443933	2
#21443916	2
C₆₁×#2916	2
#12877	61

subgroup

index

C₆₁×#2916

Subgroups — #29596351

Conjugacy classes of subgroups

Complements

Maximal subgroups