Group #29688900 — subgroups

Subgroups — #29688900

Order 15616.

662 conjugacy classes of subgroups; only the 295 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 295 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	#21454754	C₂	1	2	yes	maximal
2	#21446801	C₂	1	2	yes	maximal
3	#21446568	C₂	1	4	yes	maximal
4	#21445994	C₂	1	1	yes	maximal, characteristic
5	#21445705	C₂	1	4	yes	maximal
6	#21445507	C₂	1	1	yes	maximal, characteristic
7	C₆₁×#3162	C₂	1	1	yes	maximal, characteristic
8	#16839893	—	2	4	no	—
9	Q₈×#92418	—	2	4	no	—
10	C₂×#14519713	C₂²	1	4	yes	—
11	#16839789	C₂²	1	4	yes	—
12	#16839786	C₂²	1	2	yes	—
13	#16839718	C₂²	1	4	yes	—
14	C₂×#14519687	C₂²	1	2	yes	—
15	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₆₁	C₂²	1	2	yes	—
17	#16839608	C₂²	1	1	yes	characteristic
18	#16839521	C₂²	1	8	yes	—
19	C₆₁×#533	C₂²	1	2	yes	—
20	C₆₁×#501	C₄	1	8	yes	complemented
21	C₆₁×#427	C₂²	1	4	yes	—
22	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×#11758194	C₂³	1	4	yes	—
24	C₂²×#92418	C₂³	1	1	yes	characteristic
25	C₂²×#92418	C₂³	1	1	yes	characteristic
26	C₂×C₄×Dic₆₁	C₂³	1	2	yes	—
27	#14519713	—	2	8	no	—
28	C₄⋊#92418	—	2	4	no	—
29	C₄⋊#92418	—	2	8	no	—
30	#14519688	—	2	4	no	—
31	#14519687	Q₈	1	8	yes	complemented
32	C₄×#92418	—	2	4	no	—
33	C₄×#92418	D₄	1	4	yes	—
34	C₈×Dic₆₁	—	2	4	no	—
35	#14519684	—	2	2	no	—
36	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
37	C₄⋊Q₈×C₆₁	C₂×C₄	1	4	yes	—
38	#125×C₆₁	C₂×C₄	1	4	yes	—
39	C₂²⋊Q₈×C₆₁	—	2	4	no	—
40	C₄×Q₈×C₆₁	—	2	4	no	—
41	#117×C₆₁	C₂³	1	4	yes	—
42	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
43	#107×C₆₁	C₂×C₄	1	8	yes	—
44	#106×C₆₁	C₂×C₄	1	8	yes	—
45	#97×C₆₁	C₂×C₄	1	4	yes	—
46	C₂×Q₈×C₆₁	—	2	4	no	—
47	C₂²×C₄×C₆₁	#41	1	2	yes	—
48	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
49	C₂²×Dic₆₁	—	2	2	no	—
50	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
51	C₂×C₈×C₆₁	C₂×Q₈	1	4	yes	—
52	C₂×C₈×C₆₁	#41	1	4	yes	—
53	C₄⋊C₄×C₆₁	—	2	8	no	—
54	C₄⋊C₄×C₆₁	—	2	4	no	—
55	C₄⋊C₄×C₆₁	C₂²×C₄	1	8	yes	—
56	C₂²⋊C₄×C₆₁	—	2	4	no	—
57	C₄²×C₆₁	—	2	4	no	—
58	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
59	C₄²×C₆₁	C₂×D₄	1	2	yes	—
60	C₄×Dic₆₁	—	2	2	no	—
61	C₄×Dic₆₁	C₂×Q₈	1	4	yes	—
62	#11758194	—	2	8	no	—
63	C₂×#92418	—	2	2	no	—
64	C₂×#92418	—	2	4	no	—
65	C₂×#92418	C₂×D₄	1	2	yes	—
66	C₂×#92418	C₂×D₄	1	2	yes	—
67	C₂×#92418	C₂×Q₈	1	4	yes	—
68	C₂³×C₆₁	C₂×#41	1	1	yes	characteristic
69	Q₈×C₆₁	—	4	8	no	—
70	C₂×C₄×C₆₁	—	2	8	no	—
71	C₂×C₄×C₆₁	—	4	2	no	—
72	C₂×C₄×C₆₁	C₄×D₄	1	4	yes	—
73	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
74	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
75	C₂×C₄×C₆₁	C₂²×Q₈	1	2	yes	—
76	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
77	C₂×Dic₆₁	—	2	4	no	—
78	C₂×Dic₆₁	—	4	4	no	—
79	C₈×C₆₁	C₄×Q₈	1	8	yes	—
80	#92418	—	2	2	no	—
81	#92418	—	4	4	no	—
82	#92418	C₄⋊Q₈	1	4	yes	—
83	#21737	—	61	1	no	maximal, Sylow
84	C₂²×C₆₁	—	2	2	no	—
85	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
86	C₂²×C₆₁	#533	1	2	yes	—
87	C₄×C₆₁	—	2	4	no	—
88	C₄×C₆₁	—	4	8	no	—
89	C₄×C₆₁	C₂×C₄×Q₈	1	2	yes	—
90	C₄×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
91	C₄×C₆₁	#532	1	1	yes	characteristic
92	Dic₆₁	—	4	4	no	—
93	#3162	D₆₁	1	1	yes	characteristic
94	#2960	—	61	2	no	—
95	#1995	—	61	2	no	—
96	#1945	—	61	4	no	—
97	#1849	—	61	1	no	—
98	#1773	—	61	4	no	—
99	#1726	—	61	1	no	—
100	C₂×C₆₁	—	2	2	no	—
101	C₂×C₆₁	C₄×C₄⋊Q₈	1	1	yes	characteristic
102	C₂×C₆₁	#2955	1	2	yes	—
103	#533	C₂×D₆₁	1	2	yes	—
104	#501	Dic₆₁	1	8	yes	complemented
105	#446	—	122	4	no	—
106	C₈×Q₈	—	122	4	no	—
107	C₂×#108	—	61	4	no	—
108	#427	C₂×D₆₁	1	4	yes	—
109	#424	—	61	4	no	—
110	#423	—	61	2	no	—
111	#405	—	61	4	no	—
112	C₂×#97	—	61	2	no	—
113	C₂×#97	C₂×D₆₁	1	1	yes	characteristic
114	C₂×C₄×C₈	—	61	1	no	—
115	C₂×C₄×C₈	—	61	2	no	—
116	#376	—	61	1	no	—
117	#339	—	61	8	no	—
118	C₆₁	#21737	1	1	yes	characteristic, complemented, Sylow
119	C₂×M₁₆	—	61	4	no	—
120	C₂²×C₈	—	61	1	no	—
121	C₂²×C₈	—	61	1	no	—
122	C₂²×C₈	C₂²×D₆₁	1	2	yes	—
123	C₄⋊Q₈	C₂×Dic₆₁	1	4	yes	—
124	#125	C₂×Dic₆₁	1	4	yes	—
125	C₂²⋊Q₈	—	2	4	no	—
126	C₄×Q₈	—	2	4	no	—
127	#117	C₂²×D₆₁	1	4	yes	—
128	C₂×C₄²	—	61	2	no	—
129	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
130	#108	—	122	8	no	—
131	#107	C₂×Dic₆₁	1	8	yes	—
132	#106	C₂×Dic₆₁	1	8	yes	—
133	C₄⋊C₈	—	122	8	no	—
134	C₄⋊C₈	—	122	4	no	—
135	#97	—	61	8	no	—
136	#97	—	122	4	no	—
137	#97	C₂×Dic₆₁	1	4	yes	—
138	C₄×C₈	—	61	4	no	—
139	C₄×C₈	—	122	4	no	—
140	C₄×C₈	—	122	4	no	—
141	#95	—	122	2	no	—
142	C₂×Q₈	—	2	4	no	—
143	C₂²×C₄	—	122	2	no	—
144	C₂²×C₄	#11758224	1	2	yes	—
145	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
146	M₁₆	—	122	8	no	—
147	C₂×C₈	—	61	2	no	—
148	C₂×C₈	—	61	4	no	—
149	C₂×C₈	—	61	2	no	—
150	C₂×C₈	—	122	2	no	—
151	C₂×C₈	—	122	4	no	—
152	C₂×C₈	Q₈×D₆₁	1	4	yes	—
153	C₂×C₈	#11758226	1	4	yes	—
154	C₂×C₈	C₂²×Dic₆₁	1	4	yes	—
155	C₄⋊C₄	—	2	8	no	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄⋊C₄	C₂²×Dic₆₁	1	8	yes	—
158	C₂²⋊C₄	—	2	4	no	—
159	C₄²	—	2	4	no	—
160	C₄²	—	61	4	no	—
161	C₄²	—	122	2	no	—
162	C₄²	D₄×D₆₁	1	2	yes	—
163	C₄²	C₂²×Dic₆₁	1	2	yes	—
164	C₂³	C₂×#11758224	1	1	yes	characteristic
165	Q₈	—	4	8	no	—
166	C₂×C₄	—	2	8	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	122	4	no	—
169	C₂×C₄	—	244	4	no	—
170	C₂×C₄	D₄×Dic₆₁	1	4	yes	—
171	C₂×C₄	C₂³×Dic₆₁	1	1	yes	characteristic
172	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
173	C₂×C₄	C₂×Q₈×D₆₁	1	2	yes	—
174	C₂×C₄	C₂×#11758226	1	2	yes	—
175	C₈	—	61	4	no	—
176	C₈	—	122	2	no	—
177	C₈	—	244	4	no	—
178	C₈	Q₈×Dic₆₁	1	8	yes	—
179	C₂²	—	2	2	no	—
180	C₂²	C₂×D₄×Dic₆₁	1	1	yes	characteristic
181	C₂²	#16840317	1	2	yes	—
182	C₄	—	2	4	no	—
183	C₄	—	4	8	no	—
184	C₄	—	244	4	no	—
185	C₄	C₂×Q₈×Dic₆₁	1	2	yes	—
186	C₄	C₄⋊Q₈×D₆₁	1	1	yes	characteristic
187	C₄	#16840313	1	1	yes	characteristic
188	C₂	—	2	2	no	—
189	C₂	C₄⋊Q₈×Dic₆₁	1	1	yes	characteristic
190	C₂	#21454722	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21454754

C₂

yes

maximal

#21446801

C₂

yes

maximal

#21446568

C₂

yes

maximal

#21445994

C₂

yes

maximal, characteristic

#21445705

C₂

yes

maximal

#21445507

C₂

yes

maximal, characteristic

C₆₁×#3162

C₂

yes

maximal, characteristic

#16839893

—

Q₈×#92418

—

C₂×#14519713

C₂²

yes

—

#16839789

C₂²

yes

—

#16839786

C₂²

yes

—

#16839718

C₂²

yes

—

C₂×#14519687

C₂²

yes

—

C₂×C₄×#92418

C₂²

yes

characteristic

C₂×C₈×Dic₆₁

C₂²

yes

—

#16839608

C₂²

yes

characteristic

#16839521

C₂²

yes

—

C₆₁×#533

C₂²

yes

—

C₆₁×#501

C₄

yes

complemented

C₆₁×#427

C₂²

yes

—

C₂×#97×C₆₁

C₂²

yes

characteristic

C₂×#11758194

C₂³

yes

—

C₂²×#92418

C₂³

yes

characteristic

C₂²×#92418

C₂³

yes

characteristic

C₂×C₄×Dic₆₁

C₂³

yes

—

—

—

—

—

Q₈

yes

complemented

C₄×#92418

—

C₄×#92418

D₄

yes

—

C₈×Dic₆₁

—

#14519684

—

C₂²×C₈×C₆₁

C₂³

yes

—

C₄⋊Q₈×C₆₁

C₂×C₄

yes

—

#125×C₆₁

C₂×C₄

yes

—

C₂²⋊Q₈×C₆₁

—

C₄×Q₈×C₆₁

—

#117×C₆₁

C₂³

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

#107×C₆₁

C₂×C₄

yes

—

#106×C₆₁

C₂×C₄

yes

—

#97×C₆₁

C₂×C₄

yes

—

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

#41

yes

—

C₂²×C₄×C₆₁

C₂⁴

yes

characteristic

C₂²×Dic₆₁

—

C₂×C₈×C₆₁

C₂²×C₄

yes

—

C₂×C₈×C₆₁

C₂×Q₈

yes

—

C₂×C₈×C₆₁

#41

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂²×C₄

yes

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂²×C₄

yes

—

C₄²×C₆₁

C₂×D₄

yes

—

C₄×Dic₆₁

—

C₄×Dic₆₁

C₂×Q₈

yes

—

—

—

—

C₂×D₄

yes

—

C₂×#92418

C₂×D₄

yes

—

C₂×#92418

C₂×Q₈

yes

—

C₂³×C₆₁

C₂×#41

yes

characteristic

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄×D₄

yes

—

C₂×C₄×C₆₁

C₂³×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×Q₈

yes

—

C₂×C₄×C₆₁

C₂×#41

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₈×C₆₁

C₄×Q₈

yes

—

#92418

—

#92418

—

#92418

C₄⋊Q₈

yes

—

#21737

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₄×D₄

yes

characteristic

C₂²×C₆₁

#533

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×C₄×Q₈

yes

—

C₄×C₆₁

C₂×C₄⋊Q₈

yes

characteristic

C₄×C₆₁

#532

yes

characteristic

Dic₆₁

—

#3162

D₆₁

yes

—

—

—

—

—

—

100

C₂×C₆₁

—

101

C₂×C₆₁

C₄×C₄⋊Q₈

yes

characteristic

102

C₂×C₆₁

#2955

yes

—

103

#533

C₂×D₆₁

yes

—

104

#501

Dic₆₁

yes

complemented

105

#446

—

122

—

106

C₈×Q₈

—

122

—

107

C₂×#108

—

108

#427

C₂×D₆₁

yes

—

109

#424

—

110

#423

—

111

#405

—

112

C₂×#97

—

113

C₂×#97

C₂×D₆₁

yes

characteristic

114

C₂×C₄×C₈

—

115

C₂×C₄×C₈

—

116

#376

—

117

#339

—

118

C₆₁

#21737

yes

characteristic, complemented, Sylow

119

C₂×M₁₆

—

120

C₂²×C₈

—

121

C₂²×C₈

—

122

C₂²×C₈

C₂²×D₆₁

yes

—

123

C₄⋊Q₈

C₂×Dic₆₁

yes

—

124

#125

C₂×Dic₆₁

yes

—

125

C₂²⋊Q₈

—

126

C₄×Q₈

—

127

#117

C₂²×D₆₁

yes

—

128

C₂×C₄²

—

129

C₂×C₄²

C₂²×D₆₁

yes

characteristic

130

#108

—

122

—

131

#107

C₂×Dic₆₁

yes

—

132

#106

C₂×Dic₆₁

yes

—

133

C₄⋊C₈

—

122

—

134

C₄⋊C₈

—

122

—

135

#97

—

136

#97

—

122

—

137

#97

C₂×Dic₆₁

yes

—

138

C₄×C₈

—

139

C₄×C₈

—

122

—

140

C₄×C₈

—

122

—

141

#95

—

122

—

142

C₂×Q₈

—

143

C₂²×C₄

—

122

—

144

C₂²×C₄

#11758224

yes

—

145

C₂²×C₄

C₂³×D₆₁

yes

characteristic

146

M₁₆

—

122

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

122

—

151

C₂×C₈

—

122

—

152

C₂×C₈

Q₈×D₆₁

yes

—

153

C₂×C₈

#11758226

yes

—

154

C₂×C₈

C₂²×Dic₆₁

yes

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

C₂²×Dic₆₁

yes

—

158

C₂²⋊C₄

—

159

C₄²

—

160

C₄²

—

161

C₄²

—

122

—

162

C₄²

D₄×D₆₁

yes

—

163

C₄²

C₂²×Dic₆₁

yes

—

164

C₂³

C₂×#11758224

yes

characteristic

165

Q₈

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

122

—

169

C₂×C₄

—

244

—

170

C₂×C₄

D₄×Dic₆₁

yes

—

171

C₂×C₄

C₂³×Dic₆₁

yes

characteristic

172

C₂×C₄

C₂×D₄×D₆₁

yes

characteristic

173

C₂×C₄

C₂×Q₈×D₆₁

yes

—

174

C₂×C₄

C₂×#11758226

yes

—

175

C₈

—

176

C₈

—

122

—

177

C₈

—

244

—

178

C₈

Q₈×Dic₆₁

yes

—

179

C₂²

—

180

C₂²

C₂×D₄×Dic₆₁

yes

characteristic

181

C₂²

#16840317

yes

—

182

C₄

—

183

C₄

—

184

C₄

—

244

—

185

C₄

C₂×Q₈×Dic₆₁

yes

—

186

C₄

C₄⋊Q₈×D₆₁

yes

characteristic

187

C₄

#16840313

yes

characteristic

188

C₂

—

189

C₂

C₄⋊Q₈×Dic₆₁

yes

characteristic

190

C₂

#21454722

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	C₆₁×#3162	C₂
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	C₂³×C₆₁	C₂×#41
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	#3162	D₆₁
hypocenter	C₆₁	#21737
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3162

subgroup

G/H

center

C₂×C₄

C₂×D₄×D₆₁

commutator

C₂×C₄×C₆₁

C₂³×C₄

Frattini

C₂²×C₄

C₂³×D₆₁

Fitting

C₆₁×#3162

C₂

socle

C₂²×C₆₁

C₂×C₄×D₄

prime radical

C₂³×C₆₁

C₂×#41

cosocle

C₂²×C₄×C₆₁

C₂⁴

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3162

N	H ≅ G/N	type
20. C₆₁×#501	C₄	semidirect
31. #14519687	Q₈	semidirect
104. #501	Dic₆₁	semidirect
118. C₆₁	#21737	semidirect

H ≅ G/N

type

20. C₆₁×#501

C₄

semidirect

31. #14519687

Q₈

semidirect

104. #501

Dic₆₁

semidirect

118. C₆₁

#21737

semidirect

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

subgroup	index
#21454754	2
#21446801	2
#21446568	2
#21445994	2
#21445705	2
#21445507	2
C₆₁×#3162	2
#21737	61

subgroup

index

C₆₁×#3162

Subgroups — #29688900

Conjugacy classes of subgroups

Complements

Maximal subgroups