Group #29578640 — subgroups

Subgroups — #29578640

Order 15616.

294 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 131 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	#21444571	C₂	1	2	yes	maximal
2	C₂×#16839515	C₂	1	1	yes	maximal, characteristic
3	#21444111	C₂	1	1	yes	maximal, characteristic
4	#21443670	C₂	1	2	yes	maximal
5	C₄×#108×C₆₁	C₂	1	1	yes	maximal, characteristic
6	#16839815	C₂²	1	1	yes	characteristic
7	#16839815	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊#92418	C₂²	1	2	yes	—
9	#16839515	—	2	2	no	—
10	#16839515	—	2	2	no	—
11	#16839513	—	2	2	no	—
12	#16839484	—	2	2	no	—
13	C₂×#108×C₆₁	C₄	1	2	yes	—
14	C₄×M₁₆×C₆₁	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×#341	C₄	1	1	yes	characteristic
17	C₆₁×#341	C₄	1	1	yes	characteristic
18	C₂²×#92418	—	2	2	no	—
19	C₄⋊#92418	—	2	1	no	—
20	C₄⋊#92418	—	2	2	no	—
21	C₄⋊#92418	—	2	2	no	—
22	C₄⋊#92418	D₄	1	1	yes	characteristic
23	C₄⋊#92418	D₄	1	1	yes	characteristic
24	C₂²⋊#92418	—	2	1	no	—
25	C₂²⋊#92418	Q₈	1	2	yes	—
26	C₄×#92418	—	2	1	no	—
27	C₄×#92418	—	2	1	no	—
28	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
29	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
33	#108×C₆₁	—	2	4	no	—
34	#97×C₆₁	—	2	2	no	—
35	C₄×C₈×C₆₁	—	2	2	no	—
36	C₄×C₈×C₆₁	D₄	1	2	yes	—
37	C₄×C₈×C₆₁	Q₈	1	2	yes	—
38	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
41	M₁₆×C₆₁	—	2	2	no	—
42	M₁₆×C₆₁	—	2	2	no	—
43	M₁₆×C₆₁	—	2	2	no	—
44	M₁₆×C₆₁	M₁₆	1	4	yes	—
45	C₂×C₈×C₆₁	—	2	2	no	—
46	C₂×C₈×C₆₁	—	2	2	no	—
47	C₂×C₈×C₆₁	—	2	2	no	—
48	C₂×C₈×C₆₁	—	2	2	no	—
49	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
50	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
51	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
52	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
54	C₄²×C₆₁	#41	1	1	yes	characteristic
55	C₂×#92418	—	2	1	no	—
56	C₂×#92418	—	2	1	no	—
57	C₂×#92418	—	2	2	no	—
58	C₂×#92418	—	4	2	no	—
59	C₂×#92418	—	8	2	no	—
60	C₂³×C₆₁	C₄×Q₈	1	1	yes	characteristic
61	C₂×C₄×C₆₁	—	2	1	no	—
62	C₂×C₄×C₆₁	—	2	1	no	—
63	C₂×C₄×C₆₁	—	4	1	no	—
64	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
67	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₆₁	#125	1	1	yes	characteristic
69	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
71	C₂×C₄×C₆₁	#131	1	2	yes	—
72	C₈×C₆₁	—	2	2	no	—
73	C₈×C₆₁	—	2	2	no	—
74	C₈×C₆₁	—	2	4	no	—
75	C₈×C₆₁	—	4	2	no	—
76	#92418	—	4	2	no	—
77	#92418	—	4	1	no	—
78	#92418	—	4	1	no	—
79	#92418	—	8	2	no	—
80	#9610	—	61	1	no	maximal, Sylow
81	C₂²×C₆₁	—	2	1	no	—
82	C₂²×C₆₁	—	2	1	no	—
83	C₂²×C₆₁	#389	1	1	yes	characteristic
84	C₂²×C₆₁	#446	1	2	yes	—
85	C₄×C₆₁	—	2	1	no	—
86	C₄×C₆₁	—	2	1	no	—
87	C₄×C₆₁	—	2	1	no	—
88	C₄×C₆₁	#423	1	1	yes	characteristic
89	C₄×C₆₁	#424	1	1	yes	characteristic
90	C₄×C₆₁	#435	1	2	yes	—
91	C₄×C₆₁	#472	1	1	yes	characteristic
92	C₄×C₆₁	#473	1	1	yes	characteristic
93	C₄×#108	D₆₁	1	1	yes	characteristic
94	#1565	—	61	2	no	—
95	C₂×#335	—	61	1	no	—
96	#1452	—	61	1	no	—
97	#1283	—	61	2	no	—
98	C₂×C₆₁	—	2	2	no	—
99	C₂×C₆₁	#1940	1	1	yes	characteristic
100	C₂×C₆₁	#1952	1	1	yes	characteristic
101	C₂×C₆₁	#1953	1	1	yes	characteristic
102	#431	—	61	1	no	—
103	#431	—	61	1	no	—
104	C₂×#108	Dic₆₁	1	2	yes	—
105	C₂×C₄⋊C₈	—	61	2	no	—
106	C₄×M₁₆	C₂×D₆₁	1	2	yes	—
107	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
108	#341	Dic₆₁	1	1	yes	characteristic
109	#341	Dic₆₁	1	1	yes	characteristic
110	#335	—	122	2	no	—
111	#335	—	122	2	no	—
112	#334	—	122	2	no	—
113	#322	—	122	2	no	—
114	C₆₁	#9610	1	1	yes	characteristic, complemented, Sylow
115	C₂×M₁₆	C₂×Dic₆₁	1	2	yes	—
116	C₂×M₁₆	C₂×Dic₆₁	1	2	yes	—
117	C₂²×C₈	—	122	2	no	—
118	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
119	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
120	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
121	#108	—	2	4	no	—
122	C₄⋊C₈	—	61	1	no	—
123	C₄⋊C₈	—	61	1	no	—
124	C₄⋊C₈	—	122	2	no	—
125	C₄⋊C₈	—	122	2	no	—
126	C₄⋊C₈	—	122	1	no	—
127	C₂²⋊C₈	—	61	2	no	—
128	C₂²⋊C₈	—	122	1	no	—
129	#97	—	2	2	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	—	122	1	no	—
132	C₄×C₈	—	122	1	no	—
133	C₄×C₈	Dic₁₂₂	1	2	yes	—
134	C₄×C₈	D₂₄₄	1	2	yes	—
135	C₂²×C₄	Q₈×D₆₁	1	1	yes	characteristic
136	C₂²×C₄	#11758226	1	1	yes	characteristic
137	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
138	M₁₆	—	2	2	no	—
139	M₁₆	—	2	2	no	—
140	M₁₆	—	2	2	no	—
141	M₁₆	#11758194	1	4	yes	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	122	1	no	—
147	C₂×C₈	—	122	1	no	—
148	C₂×C₈	—	122	2	no	—
149	C₂×C₈	—	488	2	no	—
150	C₂×C₈	—	244	2	no	—
151	C₂×C₈	C₄⋊Dic₆₁	1	2	yes	—
152	C₂×C₈	C₄⋊Dic₆₁	1	2	yes	—
153	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
154	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
155	C₄²	D₄×D₆₁	1	1	yes	characteristic
156	C₄²	#11758224	1	1	yes	characteristic
157	C₂³	Q₈×Dic₆₁	1	1	yes	characteristic
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
162	C₂×C₄	D₄×Dic₆₁	1	1	yes	characteristic
163	C₂×C₄	#14519752	1	1	yes	characteristic
164	C₂×C₄	#14519757	1	1	yes	characteristic
165	C₂×C₄	#14519768	1	1	yes	characteristic
166	C₂×C₄	#14519776	1	1	yes	characteristic
167	C₂×C₄	C₂×#11758194	1	2	yes	—
168	C₂×C₄	#14519784	1	2	yes	—
169	C₈	—	2	2	no	—
170	C₈	—	2	2	no	—
171	C₈	—	2	4	no	—
172	C₈	—	4	2	no	—
173	C₈	—	488	2	no	—
174	C₈	—	244	2	no	—
175	C₈	—	244	1	no	—
176	C₈	—	244	1	no	—
177	C₂²	—	2	1	no	—
178	C₂²	—	2	1	no	—
179	C₂²	#16839661	1	1	yes	characteristic
180	C₂²	#16839893	1	2	yes	—
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	#16839786	1	1	yes	characteristic
185	C₄	#16839789	1	1	yes	characteristic
186	C₄	#16839834	1	2	yes	—
187	C₄	#16840033	1	1	yes	characteristic
188	C₄	#16840038	1	1	yes	characteristic
189	C₂	—	2	2	no	—
190	C₂	#21446543	1	1	yes	characteristic
191	C₂	#21446605	1	1	yes	characteristic
192	C₂	#21446610	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21444571

C₂

yes

maximal

C₂×#16839515

C₂

yes

maximal, characteristic

#21444111

C₂

yes

maximal, characteristic

#21443670

C₂

yes

maximal

C₄×#108×C₆₁

C₂

yes

maximal, characteristic

#16839815

C₂²

yes

characteristic

#16839815

C₂²

yes

characteristic

C₂×C₄⋊#92418

C₂²

yes

—

—

—

—

—

C₂×#108×C₆₁

C₄

yes

—

C₄×M₁₆×C₆₁

C₂²

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×#341

C₄

yes

characteristic

C₆₁×#341

C₄

yes

characteristic

C₂²×#92418

—

—

—

—

D₄

yes

characteristic

C₄⋊#92418

D₄

yes

characteristic

C₂²⋊#92418

—

C₂²⋊#92418

Q₈

yes

—

C₄×#92418

—

C₄×#92418

—

C₂×M₁₆×C₆₁

C₂×C₄

yes

—

C₂×M₁₆×C₆₁

C₂×C₄

yes

—

C₂²×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₁

C₂³

yes

characteristic

#108×C₆₁

—

#97×C₆₁

—

C₄×C₈×C₆₁

—

C₄×C₈×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

Q₈

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×Q₈

yes

characteristic

C₂²×C₄×C₆₁

#41

yes

characteristic

M₁₆×C₆₁

—

M₁₆×C₆₁

—

M₁₆×C₆₁

—

M₁₆×C₆₁

M₁₆

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄²×C₆₁

C₂×Q₈

yes

C₄²×C₆₁

yes

—

—

—

—

—

C₂³×C₆₁

C₄×Q₈

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄×D₄

yes

characteristic

C₂×C₄×C₆₁

#121

yes

characteristic

C₂×C₄×C₆₁

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₆₁

#125

yes

characteristic

C₂×C₄×C₆₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₁

C₂×M₁₆

yes

—

C₂×C₄×C₆₁

#131

yes

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

—

—

—

—

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#389

yes

characteristic

C₂²×C₆₁

#446

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

C₄×C₆₁

yes

C₄×C₆₁

yes

D₆₁

yes

—

—

—

—

C₂×C₆₁

—

C₂×C₆₁

#1940

yes

characteristic

100

C₂×C₆₁

#1952

yes

characteristic

101

C₂×C₆₁

#1953

yes

characteristic

102

#431

—

103

#431

—

104

C₂×#108

Dic₆₁

yes

—

105

C₂×C₄⋊C₈

—

106

C₄×M₁₆

C₂×D₆₁

yes

—

107

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

108

#341

Dic₆₁

yes

characteristic

109

#341

Dic₆₁

yes

characteristic

110

#335

—

122

—

111

#335

—

122

—

112

#334

—

122

—

113

#322

—

122

—

114

C₆₁

#9610

yes

characteristic, complemented, Sylow

115

C₂×M₁₆

C₂×Dic₆₁

yes

—

116

C₂×M₁₆

C₂×Dic₆₁

yes

—

117

C₂²×C₈

—

122

—

118

C₂²×C₈

C₂×Dic₆₁

yes

characteristic

119

C₂²×C₈

C₂×Dic₆₁

yes

characteristic

120

C₂×C₄²

C₂²×D₆₁

yes

characteristic

121

#108

—

122

C₄⋊C₈

—

123

C₄⋊C₈

—

124

C₄⋊C₈

—

122

—

125

C₄⋊C₈

—

122

—

126

C₄⋊C₈

—

122

—

127

C₂²⋊C₈

—

128

C₂²⋊C₈

—

122

—

129

#97

—

130

C₄×C₈

—

131

C₄×C₈

—

122

—

132

C₄×C₈

—

122

—

133

C₄×C₈

Dic₁₂₂

yes

—

134

C₄×C₈

D₂₄₄

yes

—

135

C₂²×C₄

Q₈×D₆₁

yes

characteristic

136

C₂²×C₄

#11758226

yes

characteristic

137

C₂²×C₄

C₂²×Dic₆₁

yes

characteristic

138

M₁₆

—

139

M₁₆

—

140

M₁₆

—

141

M₁₆

#11758194

yes

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

122

—

147

C₂×C₈

—

122

—

148

C₂×C₈

—

122

—

149

C₂×C₈

—

488

—

150

C₂×C₈

—

244

—

151

C₂×C₈

C₄⋊Dic₆₁

yes

—

152

C₂×C₈

C₄⋊Dic₆₁

yes

—

153

C₄²

C₂×Dic₁₂₂

yes

characteristic

154

C₄²

C₂×D₂₄₄

yes

characteristic

155

C₄²

D₄×D₆₁

yes

characteristic

156

C₄²

#11758224

yes

characteristic

157

C₂³

Q₈×Dic₆₁

yes

characteristic

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

C₂×C₄⋊Dic₆₁

yes

characteristic

162

C₂×C₄

D₄×Dic₆₁

yes

characteristic

163

C₂×C₄

#14519752

yes

characteristic

164

C₂×C₄

#14519757

yes

characteristic

165

C₂×C₄

#14519768

yes

characteristic

166

C₂×C₄

#14519776

yes

characteristic

167

C₂×C₄

C₂×#11758194

yes

—

168

C₂×C₄

#14519784

yes

—

169

C₈

—

170

C₈

—

171

C₈

—

172

C₈

—

173

C₈

—

488

—

174

C₈

—

244

—

175

C₈

—

244

—

176

C₈

—

244

—

177

C₂²

—

178

C₂²

—

179

C₂²

#16839661

yes

characteristic

180

C₂²

#16839893

yes

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

#16839786

yes

characteristic

185

C₄

#16839789

yes

characteristic

186

C₄

#16839834

yes

—

187

C₄

#16840033

yes

characteristic

188

C₄

#16840038

yes

characteristic

189

C₂

—

190

C₂

#21446543

yes

characteristic

191

C₂

#21446605

yes

characteristic

192

C₂

#21446610

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14519752
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₄×#108×C₆₁	C₂
socle	C₂²×C₆₁	#389
prime radical	C₂³×C₆₁	C₄×Q₈
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₄×#108	D₆₁
hypocenter	C₆₁	#9610
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

subgroup

G/H

center

C₂×C₄

#14519752

commutator

C₂²×C₄×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₄×#108×C₆₁

C₂

socle

C₂²×C₆₁

#389

prime radical

C₂³×C₆₁

C₄×Q₈

cosocle

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

N	H ≅ G/N	type
114. C₆₁	#9610	semidirect

H ≅ G/N

type

114. C₆₁

#9610

semidirect

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

subgroup	index
#21444571	2
C₂×#16839515	2
#21444111	2
#21443670	2
C₄×#108×C₆₁	2
#9610	61

subgroup

index

C₄×#108×C₆₁

Subgroups — #29578640

Conjugacy classes of subgroups

Complements

Maximal subgroups