Group #29580045 — subgroups

Subgroups — #29580045

Order 15616.

486 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 167 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	#21445971	C₂	1	1	yes	maximal, characteristic
2	C₄×#14519688	C₂	1	1	yes	maximal, characteristic
3	#21443660	C₂	1	4	yes	maximal
4	C₄×#125×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#92418	—	2	2	no	—
6	C₂×C₄⋊#92418	C₂²	1	2	yes	—
7	C₂×#14519688	—	2	2	no	—
8	C₂×C₄×#92418	C₂²	1	2	yes	—
9	C₂×#125×C₆₁	C₄	1	1	yes	characteristic
10	C₆₁×#389	C₄	1	2	yes	—
11	C₆₁×#387	—	2	2	no	—
12	C₆₁×#382	C₄	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₆₁	—	2	2	no	—
14	C₄×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
15	C₄³×C₆₁	C₂²	1	1	yes	characteristic
16	C₂²×#92418	—	2	2	no	—
17	C₂²×#92418	—	2	2	no	—
18	C₄⋊#92418	—	2	4	no	—
19	C₄⋊#92418	—	4	4	no	—
20	#14519688	—	2	8	no	—
21	C₄×#92418	—	2	2	no	—
22	C₄×#92418	Q₈	1	4	yes	—
23	#125×C₆₁	—	2	4	no	—
24	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
25	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
26	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
27	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₆₁	—	2	2	no	—
29	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₆₁	D₄	1	2	yes	—
32	C₂×C₄²×C₆₁	D₄	1	2	yes	—
33	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
34	#95×C₆₁	—	2	2	no	—
35	#95×C₆₁	—	2	2	no	—
36	#95×C₆₁	—	2	2	no	—
37	C₂²×C₄×C₆₁	—	2	2	no	—
38	C₂²×C₄×C₆₁	—	2	2	no	—
39	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
44	C₄⋊C₄×C₆₁	—	2	8	no	—
45	C₄⋊C₄×C₆₁	—	2	2	no	—
46	C₄⋊C₄×C₆₁	—	2	4	no	—
47	C₄⋊C₄×C₆₁	—	4	4	no	—
48	C₄⋊C₄×C₆₁	M₁₆	1	4	yes	—
49	C₄²×C₆₁	—	2	4	no	—
50	C₄²×C₆₁	—	2	2	no	—
51	C₄²×C₆₁	—	2	2	no	—
52	C₄²×C₆₁	—	2	2	no	—
53	C₄²×C₆₁	—	2	2	no	—
54	C₄²×C₆₁	—	4	2	no	—
55	C₄²×C₆₁	—	4	2	no	—
56	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
57	C₄²×C₆₁	#41	1	2	yes	—
58	C₂×#92418	—	2	8	no	—
59	C₂×#92418	—	2	4	no	—
60	C₂×#92418	—	4	2	no	—
61	C₂×#92418	—	4	4	no	—
62	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₆₁	—	2	2	no	—
64	C₂×C₄×C₆₁	—	2	2	no	—
65	C₂×C₄×C₆₁	—	2	2	no	—
66	C₂×C₄×C₆₁	—	2	2	no	—
67	C₂×C₄×C₆₁	—	2	4	no	—
68	C₂×C₄×C₆₁	—	2	2	no	—
69	C₂×C₄×C₆₁	—	2	2	no	—
70	C₂×C₄×C₆₁	—	4	2	no	—
71	C₂×C₄×C₆₁	—	4	2	no	—
72	C₂×C₄×C₆₁	—	4	2	no	—
73	C₂×C₄×C₆₁	—	4	2	no	—
74	C₂×C₄×C₆₁	—	4	2	no	—
75	C₂×C₄×C₆₁	#104	1	4	yes	—
76	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
77	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
78	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
79	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
81	C₂×C₄×C₆₁	#131	1	2	yes	—
82	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
83	C₂×C₄×C₆₁	#137	1	2	yes	—
84	#92418	—	2	8	no	—
85	#92418	—	8	2	no	—
86	#9876	—	61	1	no	maximal, Sylow
87	C₂²×C₆₁	#391	1	1	yes	characteristic
88	C₂²×C₆₁	#407	1	1	yes	characteristic
89	C₂²×C₆₁	#408	1	1	yes	characteristic
90	C₂²×C₆₁	#417	1	2	yes	—
91	C₂²×C₆₁	C₂×#104	1	2	yes	—
92	C₄×C₆₁	—	2	2	no	—
93	C₄×C₆₁	—	2	4	no	—
94	C₄×C₆₁	—	4	2	no	—
95	C₄×C₆₁	—	4	4	no	—
96	C₄×C₆₁	—	8	2	no	—
97	C₄×C₆₁	#446	1	4	yes	—
98	C₄×C₆₁	#490	1	1	yes	characteristic
99	C₄×C₆₁	#491	1	1	yes	characteristic
100	C₄×C₆₁	#501	1	2	yes	—
101	C₄×#125	D₆₁	1	1	yes	characteristic
102	#1842	—	61	1	no	—
103	C₄×#97	—	61	1	no	—
104	#1279	—	61	4	no	—
105	C₂×C₆₁	#1483	1	2	yes	—
106	C₂×C₆₁	#1489	1	2	yes	—
107	C₂×C₆₁	#1985	1	1	yes	characteristic
108	C₂×C₆₁	#1992	1	1	yes	characteristic
109	C₂×C₆₁	#1994	1	1	yes	characteristic
110	C₂×#125	Dic₆₁	1	1	yes	characteristic
111	C₂×C₄⋊C₈	—	61	2	no	—
112	C₂×C₄⋊C₈	—	122	2	no	—
113	C₂×#97	—	122	2	no	—
114	C₂×C₄×C₈	—	61	2	no	—
115	#389	Dic₆₁	1	2	yes	—
116	#387	—	2	2	no	—
117	#382	Dic₆₁	1	1	yes	characteristic
118	C₄×C₄⋊C₄	—	2	2	no	—
119	C₄×C₄⋊C₄	C₂×D₆₁	1	2	yes	—
120	C₄³	C₂×D₆₁	1	1	yes	characteristic
121	C₆₁	#9876	1	1	yes	characteristic, complemented, Sylow
122	C₂²×C₈	—	122	2	no	—
123	C₂²×C₈	—	122	2	no	—
124	#125	—	2	4	no	—
125	C₂×C₄⋊C₄	—	2	2	no	—
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₄⋊C₄	C₂×Dic₆₁	1	2	yes	—
128	C₂×C₄⋊C₄	C₂×Dic₆₁	1	2	yes	—
129	C₂×C₄²	—	2	2	no	—
130	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
131	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
132	C₂×C₄²	C₂²⋊D₆₁	1	2	yes	—
133	C₂×C₄²	C₂²⋊D₆₁	1	2	yes	—
134	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
135	C₄⋊C₈	—	122	4	no	—
136	C₄⋊C₈	—	244	4	no	—
137	#97	—	122	8	no	—
138	C₄×C₈	—	61	4	no	—
139	C₄×C₈	—	122	2	no	—
140	#95	—	2	2	no	—
141	#95	—	2	2	no	—
142	#95	—	2	2	no	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	C₂²⋊Dic₆₁	1	2	yes	—
146	C₂²×C₄	C₂²⋊Dic₆₁	1	2	yes	—
147	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
149	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
150	C₂×C₈	—	122	8	no	—
151	C₂×C₈	—	122	4	no	—
152	C₂×C₈	—	244	2	no	—
153	C₂×C₈	—	244	4	no	—
154	C₄⋊C₄	—	2	8	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄⋊C₄	—	4	4	no	—
158	C₄⋊C₄	#11758194	1	4	yes	—
159	C₄²	—	2	4	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	2	no	—
164	C₄²	—	4	2	no	—
165	C₄²	—	4	2	no	—
166	C₄²	Q₈×D₆₁	1	2	yes	—
167	C₄²	#11758226	1	2	yes	—
168	C₂³	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	#14519704	1	4	yes	—
182	C₂×C₄	Q₈×Dic₆₁	1	2	yes	—
183	C₂×C₄	#14519753	1	2	yes	—
184	C₂×C₄	#14519766	1	1	yes	characteristic
185	C₂×C₄	#14519775	1	1	yes	characteristic
186	C₂×C₄	C₂×#11758194	1	2	yes	—
187	C₂×C₄	#14519784	1	2	yes	—
188	C₂×C₄	#14519802	1	2	yes	—
189	C₂×C₄	#14519807	1	2	yes	—
190	C₈	—	122	8	no	—
191	C₈	—	488	2	no	—
192	C₂²	#16839669	1	1	yes	characteristic
193	C₂²	#16839726	1	1	yes	characteristic
194	C₂²	#16839729	1	1	yes	characteristic
195	C₂²	#16839763	1	2	yes	—
196	C₂²	C₂×#14519704	1	2	yes	—
197	C₄	—	2	4	no	—
198	C₄	—	2	2	no	—
199	C₄	—	4	2	no	—
200	C₄	—	4	4	no	—
201	C₄	—	8	2	no	—
202	C₄	#16839893	1	4	yes	—
203	C₄	#16840128	1	1	yes	characteristic
204	C₄	#16840132	1	1	yes	characteristic
205	C₄	#16840172	1	2	yes	—
206	C₂	#21444232	1	2	yes	—
207	C₂	#21444258	1	2	yes	—
208	C₂	#21446760	1	1	yes	characteristic
209	C₂	#21446789	1	1	yes	characteristic
210	C₂	#21446797	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21445971

C₂

yes

maximal, characteristic

C₄×#14519688

C₂

yes

maximal, characteristic

#21443660

C₂

yes

maximal

C₄×#125×C₆₁

C₂

yes

maximal, characteristic

C₂×C₄⋊#92418

—

C₂×C₄⋊#92418

C₂²

yes

—

C₂×#14519688

—

C₂×C₄×#92418

C₂²

yes

—

C₂×#125×C₆₁

C₄

yes

characteristic

C₆₁×#389

C₄

yes

—

C₆₁×#387

—

C₆₁×#382

C₄

yes

characteristic

C₄×C₄⋊C₄×C₆₁

—

C₄×C₄⋊C₄×C₆₁

C₂²

yes

—

C₄³×C₆₁

C₂²

yes

characteristic

C₂²×#92418

—

C₂²×#92418

—

—

—

—

—

Q₈

yes

—

#125×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

#95×C₆₁

—

#95×C₆₁

—

#95×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

M₁₆

yes

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂×Q₈

yes

—

C₄²×C₆₁

#41

yes

—

—

—

—

—

C₂³×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#104

yes

—

C₂×C₄×C₆₁

C₄×Q₈

yes

—

C₂×C₄×C₆₁

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₁

#124

yes

characteristic

C₂×C₄×C₆₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₁

C₂×M₁₆

yes

—

C₂×C₄×C₆₁

#131

yes

—

C₂×C₄×C₆₁

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₆₁

#137

yes

—

—

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

C₂×#104

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#446

yes

—

C₄×C₆₁

yes

C₄×C₆₁

yes

100

C₄×C₆₁

#501

yes

—

101

C₄×#125

D₆₁

yes

characteristic

102

#1842

—

103

C₄×#97

—

104

#1279

—

105

C₂×C₆₁

#1483

yes

—

106

C₂×C₆₁

#1489

yes

—

107

C₂×C₆₁

#1985

yes

characteristic

108

C₂×C₆₁

#1992

yes

characteristic

109

C₂×C₆₁

#1994

yes

characteristic

110

C₂×#125

Dic₆₁

yes

characteristic

111

C₂×C₄⋊C₈

—

112

C₂×C₄⋊C₈

—

122

—

113

C₂×#97

—

122

—

114

C₂×C₄×C₈

—

115

#389

Dic₆₁

yes

—

116

#387

—

117

#382

Dic₆₁

yes

characteristic

118

C₄×C₄⋊C₄

—

119

C₄×C₄⋊C₄

C₂×D₆₁

yes

—

120

C₄³

C₂×D₆₁

yes

characteristic

121

C₆₁

#9876

yes

characteristic, complemented, Sylow

122

C₂²×C₈

—

122

—

123

C₂²×C₈

—

122

—

124

#125

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄⋊C₄

C₂×Dic₆₁

yes

—

128

C₂×C₄⋊C₄

C₂×Dic₆₁

yes

—

129

C₂×C₄²

—

130

C₂×C₄²

C₂×Dic₆₁

yes

characteristic

131

C₂×C₄²

C₂×Dic₆₁

yes

characteristic

132

C₂×C₄²

C₂²⋊D₆₁

yes

—

133

C₂×C₄²

C₂²⋊D₆₁

yes

—

134

C₂×C₄²

C₂²×D₆₁

yes

characteristic

135

C₄⋊C₈

—

122

—

136

C₄⋊C₈

—

244

—

137

#97

—

122

—

138

C₄×C₈

—

139

C₄×C₈

—

122

—

140

#95

—

141

#95

—

142

#95

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

C₂²⋊Dic₆₁

yes

—

146

C₂²×C₄

C₂²⋊Dic₆₁

yes

—

147

C₂²×C₄

C₂²×Dic₆₁

yes

characteristic

148

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

149

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

150

C₂×C₈

—

122

—

151

C₂×C₈

—

122

—

152

C₂×C₈

—

244

—

153

C₂×C₈

—

244

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

#11758194

yes

—

159

C₄²

—

160

C₄²

—

161

C₄²

—

162

C₄²

—

163

C₄²

—

164

C₄²

—

165

C₄²

—

166

C₄²

Q₈×D₆₁

yes

—

167

C₄²

#11758226

yes

—

168

C₂³

C₂×C₂²⋊Dic₆₁

yes

characteristic

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

#14519704

yes

—

182

C₂×C₄

Q₈×Dic₆₁

yes

—

183

C₂×C₄

#14519753

yes

—

184

C₂×C₄

#14519766

yes

characteristic

185

C₂×C₄

#14519775

yes

characteristic

186

C₂×C₄

C₂×#11758194

yes

—

187

C₂×C₄

#14519784

yes

—

188

C₂×C₄

#14519802

yes

—

189

C₂×C₄

#14519807

yes

—

190

C₈

—

122

—

191

C₈

—

488

—

192

C₂²

#16839669

yes

characteristic

193

C₂²

#16839726

yes

characteristic

194

C₂²

#16839729

yes

characteristic

195

C₂²

#16839763

yes

—

196

C₂²

C₂×#14519704

yes

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

#16839893

yes

—

203

C₄

#16840128

yes

characteristic

204

C₄

#16840132

yes

characteristic

205

C₄

#16840172

yes

—

206

C₂

#21444232

yes

—

207

C₂

#21444258

yes

—

208

C₂

#21446760

yes

characteristic

209

C₂

#21446789

yes

characteristic

210

C₂

#21446797

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₁
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₄×#125×C₆₁	C₂
socle, prime radical	C₂³×C₆₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₄×#125	D₆₁
hypocenter	C₆₁	#9876
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

subgroup

G/H

center

C₂²×C₄

C₂×C₂²⋊D₆₁

commutator

C₂²×C₄×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₄×#125×C₆₁

C₂

socle, prime radical

C₂³×C₆₁

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

N	H ≅ G/N	type
121. C₆₁	#9876	semidirect

H ≅ G/N

type

121. C₆₁

#9876

semidirect

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

subgroup	index
#21445971	2
C₄×#14519688	2
#21443660	2
C₄×#125×C₆₁	2
#9876	61

subgroup

index

C₄×#125×C₆₁

Subgroups — #29580045

Conjugacy classes of subgroups

Complements

Maximal subgroups