Group #29579875 — subgroups

Subgroups — #29579875

Order 15616.

510 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 199 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	#21446781	C₂	1	2	yes	maximal
2	#21446622	C₂	1	2	yes	maximal
3	#21445979	C₂	1	1	yes	maximal, characteristic
4	C₂×#16839483	C₂	1	1	yes	maximal, characteristic
5	C₂×C₆₁×#498	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519711	—	2	2	no	—
7	C₂×#14519701	—	2	4	no	—
8	C₂×C₄×#92418	C₂²	1	2	yes	—
9	#16839632	C₂²	1	1	yes	characteristic
10	#16839632	C₂²	1	1	yes	characteristic
11	#16839524	—	2	4	no	—
12	#16839483	—	2	2	no	—
13	C₂×C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
14	C₆₁×#498	—	2	2	no	—
15	C₆₁×#498	C₄	1	4	yes	complemented
16	C₂×#106×C₆₁	—	2	2	no	—
17	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₂²×#92418	D₄	1	4	yes	—
19	C₂×C₄⋊Dic₆₁	—	2	2	no	—
20	C₂×C₄⋊Dic₆₁	—	2	2	no	—
21	C₂×C₄⋊Dic₆₁	—	2	1	no	—
22	C₂×C₄⋊Dic₆₁	—	2	1	no	—
23	#14519711	—	4	4	no	—
24	#14519701	—	4	8	no	—
25	C₄×#92418	—	2	2	no	—
26	C₄×#92418	—	2	2	no	—
27	C₂²×Q₈×C₆₁	—	2	2	no	—
28	C₂²×C₈×C₆₁	D₄	1	2	yes	—
29	C₄⋊Q₈×C₆₁	—	2	2	no	—
30	C₄⋊Q₈×C₆₁	—	2	2	no	—
31	C₄⋊Q₈×C₆₁	—	2	2	no	—
32	C₄⋊Q₈×C₆₁	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
34	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
35	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
36	#106×C₆₁	—	2	8	no	—
37	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₆₁	D₄	1	2	yes	—
39	C₂×Q₈×C₆₁	—	2	4	no	—
40	C₂×Q₈×C₆₁	—	4	2	no	—
41	C₂×Q₈×C₆₁	—	4	4	no	—
42	C₂×Q₈×C₆₁	—	4	4	no	—
43	C₂²×C₄×C₆₁	—	4	2	no	—
44	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
46	C₂²×Dic₆₁	—	4	1	no	—
47	C₂²×Dic₆₁	—	4	1	no	—
48	C₂×C₈×C₆₁	C₂²⋊C₄	1	4	yes	—
49	C₂×C₈×C₆₁	SD₁₆	1	4	yes	—
50	C₂×C₈×C₆₁	Q₁₆	1	4	yes	—
51	C₄⋊C₄×C₆₁	—	2	4	no	—
52	C₄⋊C₄×C₆₁	—	2	4	no	—
53	C₄⋊C₄×C₆₁	—	4	2	no	—
54	C₄⋊C₄×C₆₁	—	4	2	no	—
55	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₆₁	#41	1	2	yes	—
58	C₄⋊Dic₆₁	—	4	2	no	—
59	C₄⋊Dic₆₁	—	4	2	no	—
60	C₄⋊Dic₆₁	—	4	4	no	—
61	C₄⋊Dic₆₁	—	4	4	no	—
62	C₂×#92418	—	2	4	no	—
63	C₂×#92418	—	2	4	no	—
64	C₂×#92418	—	2	4	no	—
65	C₂³×C₆₁	#127	1	1	yes	characteristic
66	Q₈×C₆₁	—	4	8	no	—
67	Q₈×C₆₁	—	8	4	no	—
68	C₂×C₄×C₆₁	—	4	4	no	—
69	C₂×C₄×C₆₁	—	8	2	no	—
70	C₂×C₄×C₆₁	—	8	2	no	—
71	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
73	C₂×C₄×C₆₁	#121	1	2	yes	—
74	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
75	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
76	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
77	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	4	yes	—
78	C₂×C₄×C₆₁	#137	1	4	yes	—
79	C₂×Dic₆₁	—	8	1	no	—
80	C₂×Dic₆₁	—	8	1	no	—
81	C₂×Dic₆₁	—	8	2	no	—
82	C₂×Dic₆₁	—	8	2	no	—
83	C₈×C₆₁	#103	1	8	yes	complemented
84	#92418	—	4	4	no	—
85	#9841	—	61	1	no	maximal, Sylow
86	C₂²×C₆₁	#390	1	1	yes	characteristic
87	C₂²×C₆₁	#492	1	1	yes	characteristic
88	C₂²×C₆₁	#494	1	1	yes	characteristic
89	C₂²×C₆₁	#496	1	2	yes	—
90	C₂²×C₆₁	#497	1	2	yes	—
91	C₄×C₆₁	—	8	4	no	—
92	C₄×C₆₁	C₂×#103	1	2	yes	—
93	C₄×C₆₁	#440	1	4	yes	—
94	C₄×C₆₁	#468	1	4	yes	—
95	C₄×C₆₁	#487	1	2	yes	—
96	Dic₆₁	—	16	1	no	—
97	Dic₆₁	—	16	1	no	—
98	C₂×#498	D₆₁	1	1	yes	characteristic
99	#1990	—	61	2	no	—
100	#1956	—	61	2	no	—
101	#1845	—	61	1	no	—
102	C₂×#322	—	61	1	no	—
103	C₂×C₆₁	#1715	1	1	yes	characteristic
104	C₂×C₆₁	#1716	1	1	yes	characteristic
105	C₂×C₆₁	#1723	1	2	yes	—
106	C₂×C₆₁	#1956	1	1	yes	characteristic
107	C₂×C₆₁	#1973	1	2	yes	—
108	C₂×C₄⋊Q₈	C₂×D₆₁	1	2	yes	—
109	#498	—	2	2	no	—
110	#498	Dic₆₁	1	4	yes	complemented
111	C₂×#107	—	122	2	no	—
112	C₂×#106	—	2	2	no	—
113	C₂×#103	—	122	4	no	—
114	C₂×C₄×C₈	—	61	2	no	—
115	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
116	#384	—	61	1	no	—
117	#384	—	61	1	no	—
118	#340	—	122	4	no	—
119	#322	—	122	2	no	—
120	C₆₁	#9841	1	1	yes	characteristic, complemented, Sylow
121	C₂²×Q₈	—	2	2	no	—
122	C₂²×C₈	—	61	4	no	—
123	C₂²×C₈	C₂²⋊D₆₁	1	2	yes	—
124	C₄⋊Q₈	—	2	2	no	—
125	C₄⋊Q₈	—	2	2	no	—
126	C₄⋊Q₈	—	2	2	no	—
127	C₄⋊Q₈	C₂×Dic₆₁	1	4	yes	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	122	2	no	—
131	C₂×C₄⋊C₄	—	122	2	no	—
132	C₂×C₄⋊C₄	—	122	1	no	—
133	C₂×C₄⋊C₄	—	122	1	no	—
134	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
135	#107	—	244	4	no	—
136	#106	—	2	8	no	—
137	#103	—	244	8	no	—
138	C₄×C₈	—	122	2	no	—
139	C₄×C₈	—	122	2	no	—
140	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
141	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
142	C₂×Q₈	—	2	4	no	—
143	C₂×Q₈	—	4	2	no	—
144	C₂×Q₈	—	4	4	no	—
145	C₂×Q₈	—	4	4	no	—
146	C₂²×C₄	—	4	2	no	—
147	C₂²×C₄	—	244	1	no	—
148	C₂²×C₄	—	244	1	no	—
149	C₂²×C₄	D₄×D₆₁	1	2	yes	—
150	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
151	C₂×C₈	—	122	4	no	—
152	C₂×C₈	—	122	4	no	—
153	C₂×C₈	—	122	4	no	—
154	C₂×C₈	#11758201	1	4	yes	—
155	C₂×C₈	#11758202	1	4	yes	—
156	C₂×C₈	C₂²⋊Dic₆₁	1	4	yes	—
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	244	2	no	—
162	C₄⋊C₄	—	244	2	no	—
163	C₄⋊C₄	—	244	4	no	—
164	C₄⋊C₄	—	244	4	no	—
165	C₄²	#11758224	1	2	yes	—
166	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
167	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
168	C₂³	#14519773	1	1	yes	characteristic
169	Q₈	—	4	8	no	—
170	Q₈	—	8	4	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	—	488	1	no	—
175	C₂×C₄	—	488	1	no	—
176	C₂×C₄	—	488	2	no	—
177	C₂×C₄	—	488	2	no	—
178	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
179	C₂×C₄	D₄×Dic₆₁	1	2	yes	—
180	C₂×C₄	#14519748	1	2	yes	—
181	C₂×C₄	#14519764	1	1	yes	characteristic
182	C₂×C₄	C₂×#11758201	1	2	yes	—
183	C₂×C₄	C₂×#11758202	1	2	yes	—
184	C₂×C₄	#14519804	1	4	yes	—
185	C₂×C₄	#14519806	1	4	yes	—
186	C₈	—	244	4	no	—
187	C₈	#14519701	1	8	yes	complemented
188	C₂²	#16839665	1	1	yes	characteristic
189	C₂²	#16840136	1	1	yes	characteristic
190	C₂²	#16840141	1	1	yes	characteristic
191	C₂²	#16840148	1	2	yes	—
192	C₂²	#16840153	1	2	yes	—
193	C₄	—	8	4	no	—
194	C₄	—	976	1	no	—
195	C₄	—	976	1	no	—
196	C₄	C₂×#14519701	1	2	yes	—
197	C₄	#16839863	1	4	yes	—
198	C₄	#16840005	1	4	yes	—
199	C₄	#16840112	1	2	yes	—
200	C₂	#21445448	1	1	yes	characteristic
201	C₂	#21445453	1	1	yes	characteristic
202	C₂	#21445493	1	2	yes	—
203	C₂	#21446622	1	1	yes	characteristic
204	C₂	#21446702	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21446781

C₂

yes

maximal

#21446622

C₂

yes

maximal

#21445979

C₂

yes

maximal, characteristic

C₂×#16839483

C₂

yes

maximal, characteristic

C₂×C₆₁×#498

C₂

yes

maximal, characteristic

C₂×#14519711

—

C₂×#14519701

—

C₂×C₄×#92418

C₂²

yes

—

#16839632

C₂²

yes

characteristic

#16839632

C₂²

yes

characteristic

#16839524

—

#16839483

—

C₂×C₄⋊Q₈×C₆₁

C₂²

yes

—

C₆₁×#498

—

C₆₁×#498

C₄

yes

complemented

C₂×#106×C₆₁

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂²×#92418

D₄

yes

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

—

—

—

—

C₂²×Q₈×C₆₁

—

C₂²×C₈×C₆₁

D₄

yes

—

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

#106×C₆₁

—

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

D₄

yes

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×D₄

yes

—

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₂×C₈×C₆₁

C₂²⋊C₄

yes

—

C₂×C₈×C₆₁

SD₁₆

yes

—

C₂×C₈×C₆₁

Q₁₆

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄²×C₆₁

#41

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

—

C₂³×C₆₁

#127

yes

characteristic

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄×D₄

yes

—

C₂×C₄×C₆₁

#121

yes

—

C₂×C₄×C₆₁

#124

yes

characteristic

C₂×C₄×C₆₁

C₂×SD₁₆

yes

—

C₂×C₄×C₆₁

C₂×Q₁₆

yes

—

C₂×C₄×C₆₁

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₆₁

#137

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₈×C₆₁

yes

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

#497

yes

—

C₄×C₆₁

—

C₄×C₆₁

C₂×#103

yes

—

C₄×C₆₁

#440

yes

—

C₄×C₆₁

#468

yes

—

C₄×C₆₁

#487

yes

—

Dic₆₁

—

Dic₆₁

—

C₂×#498

D₆₁

yes

characteristic

#1990

—

100

#1956

—

101

#1845

—

102

C₂×#322

—

103

C₂×C₆₁

#1715

yes

characteristic

104

C₂×C₆₁

#1716

yes

characteristic

105

C₂×C₆₁

#1723

yes

—

106

C₂×C₆₁

#1956

yes

characteristic

107

C₂×C₆₁

#1973

yes

—

108

C₂×C₄⋊Q₈

C₂×D₆₁

yes

—

109

#498

—

110

#498

Dic₆₁

yes

complemented

111

C₂×#107

—

122

—

112

C₂×#106

—

113

C₂×#103

—

122

—

114

C₂×C₄×C₈

—

115

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

116

#384

—

117

#384

—

118

#340

—

122

—

119

#322

—

122

—

120

C₆₁

#9841

yes

characteristic, complemented, Sylow

121

C₂²×Q₈

—

122

C₂²×C₈

—

123

C₂²×C₈

C₂²⋊D₆₁

yes

—

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

C₂×Dic₆₁

yes

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

122

—

131

C₂×C₄⋊C₄

—

122

—

132

C₂×C₄⋊C₄

—

122

—

133

C₂×C₄⋊C₄

—

122

—

134

C₂×C₄²

C₂²×D₆₁

yes

characteristic

135

#107

—

244

—

136

#106

—

137

#103

—

244

—

138

C₄×C₈

—

122

—

139

C₄×C₈

—

122

—

140

C₄×C₈

C₂×Dic₆₁

yes

—

141

C₄×C₈

C₂²⋊D₆₁

yes

—

142

C₂×Q₈

—

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂×Q₈

—

146

C₂²×C₄

—

147

C₂²×C₄

—

244

—

148

C₂²×C₄

—

244

—

149

C₂²×C₄

D₄×D₆₁

yes

—

150

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

151

C₂×C₈

—

122

—

152

C₂×C₈

—

122

—

153

C₂×C₈

—

122

—

154

C₂×C₈

#11758201

yes

—

155

C₂×C₈

#11758202

yes

—

156

C₂×C₈

C₂²⋊Dic₆₁

yes

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

244

—

162

C₄⋊C₄

—

244

—

163

C₄⋊C₄

—

244

—

164

C₄⋊C₄

—

244

—

165

C₄²

#11758224

yes

—

166

C₄²

C₂²×Dic₆₁

yes

characteristic

167

C₄²

C₂×C₂²⋊D₆₁

yes

characteristic

168

C₂³

#14519773

yes

characteristic

169

Q₈

—

170

Q₈

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

488

—

175

C₂×C₄

—

488

—

176

C₂×C₄

—

488

—

177

C₂×C₄

—

488

—

178

C₂×C₄

C₂×C₂²⋊Dic₆₁

yes

characteristic

179

C₂×C₄

D₄×Dic₆₁

yes

—

180

C₂×C₄

#14519748

yes

—

181

C₂×C₄

#14519764

yes

characteristic

182

C₂×C₄

C₂×#11758201

yes

—

183

C₂×C₄

C₂×#11758202

yes

—

184

C₂×C₄

#14519804

yes

—

185

C₂×C₄

#14519806

yes

—

186

C₈

—

244

—

187

C₈

#14519701

yes

complemented

188

C₂²

#16839665

yes

characteristic

189

C₂²

#16840136

yes

characteristic

190

C₂²

#16840141

yes

characteristic

191

C₂²

#16840148

yes

—

192

C₂²

#16840153

yes

—

193

C₄

—

194

C₄

—

976

—

195

C₄

—

976

—

196

C₄

C₂×#14519701

yes

—

197

C₄

#16839863

yes

—

198

C₄

#16840005

yes

—

199

C₄

#16840112

yes

—

200

C₂

#21445448

yes

characteristic

201

C₂

#21445453

yes

characteristic

202

C₂

#21445493

yes

—

203

C₂

#21446622

yes

characteristic

204

C₂

#21446702

yes

—

	subgroup	G/H
center	C₂³	#14519773
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₆₁×#498	C₂
socle, prime radical	C₂³×C₆₁	#127
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×#498	D₆₁
hypocenter	C₆₁	#9841
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

subgroup

G/H

center

C₂³

#14519773

commutator

C₄²×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₂×C₆₁×#498

C₂

socle, prime radical

C₂³×C₆₁

#127

cosocle

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

N	H ≅ G/N	type
15. C₆₁×#498	C₄	semidirect
83. C₈×C₆₁	#103	semidirect
110. #498	Dic₆₁	semidirect
120. C₆₁	#9841	semidirect
187. C₈	#14519701	semidirect

H ≅ G/N

type

15. C₆₁×#498

C₄

semidirect

83. C₈×C₆₁

#103

semidirect

110. #498

Dic₆₁

semidirect

120. C₆₁

#9841

semidirect

187. C₈

#14519701

semidirect

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

subgroup	index
#21446781	2
#21446622	2
#21445979	2
C₂×#16839483	2
C₂×C₆₁×#498	2
#9841	61

subgroup

index

C₂×C₆₁×#498

Subgroups — #29579875

Conjugacy classes of subgroups

Complements

Maximal subgroups