Group #29572937 — subgroups

Subgroups — #29572937

Order 15616.

398 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 155 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	#21445815	C₂	1	2	yes	maximal
2	#21444575	C₂	1	2	yes	maximal
3	#21443694	C₂	1	2	yes	maximal
4	C₂×C₈×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
5	#16839786	C₂²	1	4	yes	—
6	C₂×#14519696	—	2	2	no	—
7	#16839531	—	2	2	no	—
8	#16839513	—	2	2	no	—
9	#16839513	—	2	2	no	—
10	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
11	C₈×Q₈×C₆₁	—	2	2	no	—
12	C₈×Q₈×C₆₁	—	2	2	no	—
13	C₂×C₄⋊C₈×C₆₁	C₄	1	2	yes	—
14	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₆₁	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×#11758194	—	2	4	no	—
18	C₄⋊#92418	—	2	4	no	—
19	C₄⋊#92418	—	2	4	no	—
20	#14519696	—	4	4	no	—
21	C₂²×Q₈×C₆₁	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
25	C₄×Q₈×C₆₁	—	2	2	no	—
26	C₄×Q₈×C₆₁	—	2	2	no	—
27	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₆₁	Q₈	1	2	yes	—
29	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₆₁	—	2	4	no	—
32	C₄⋊C₈×C₆₁	—	2	2	no	—
33	C₄×C₈×C₆₁	—	2	2	no	—
34	C₄×C₈×C₆₁	—	2	2	no	—
35	C₄×C₈×C₆₁	D₄	1	2	yes	—
36	C₄×C₈×C₆₁	D₄	1	2	yes	—
37	C₂×Q₈×C₆₁	—	2	2	no	—
38	C₂×Q₈×C₆₁	—	2	2	no	—
39	C₂×Q₈×C₆₁	SD₁₆	1	4	yes	—
40	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
41	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₆₁	—	2	2	no	—
46	C₂×C₈×C₆₁	—	2	2	no	—
47	C₂×C₈×C₆₁	—	4	1	no	—
48	C₂×C₈×C₆₁	—	4	2	no	—
49	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
52	C₄⋊C₄×C₆₁	—	2	2	no	—
53	C₄⋊C₄×C₆₁	—	2	4	no	—
54	C₄²×C₆₁	—	2	2	no	—
55	C₄²×C₆₁	—	2	2	no	—
56	C₄²×C₆₁	C₂×D₄	1	2	yes	—
57	C₄²×C₆₁	#41	1	2	yes	—
58	#11758194	—	4	4	no	—
59	#11758194	—	4	4	no	—
60	C₂×#92418	—	4	4	no	—
61	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
62	Q₈×C₆₁	—	2	8	no	—
63	C₂×C₄×C₆₁	—	2	2	no	—
64	C₂×C₄×C₆₁	—	2	2	no	—
65	C₂×C₄×C₆₁	—	2	1	no	—
66	C₂×C₄×C₆₁	—	2	1	no	—
67	C₂×C₄×C₆₁	—	2	2	no	—
68	C₂×C₄×C₆₁	—	4	1	no	—
69	C₂×C₄×C₆₁	—	4	2	no	—
70	C₂×C₄×C₆₁	#106	1	4	yes	—
71	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
73	C₂×C₄×C₆₁	#121	1	2	yes	—
74	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
75	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
76	C₂×C₄×C₆₁	#135	1	2	yes	—
77	C₈×C₆₁	—	2	1	no	—
78	C₈×C₆₁	—	2	2	no	—
79	C₈×C₆₁	—	4	2	no	—
80	C₈×C₆₁	#101	1	2	yes	complemented
81	#92418	—	8	4	no	—
82	#8615	—	61	1	no	maximal, Sylow
83	C₂²×C₆₁	—	2	1	no	—
84	C₂²×C₆₁	—	2	1	no	—
85	C₂²×C₆₁	#380	1	1	yes	characteristic
86	C₂²×C₆₁	C₂×#106	1	1	yes	characteristic
87	C₂²×C₆₁	#427	1	1	yes	characteristic
88	C₄×C₆₁	—	2	1	no	—
89	C₄×C₆₁	—	2	4	no	—
90	C₄×C₆₁	—	2	2	no	—
91	C₄×C₆₁	—	4	2	no	—
92	C₄×C₆₁	C₂×#101	1	1	yes	characteristic
93	C₄×C₆₁	#413	1	1	yes	characteristic
94	C₄×C₆₁	#465	1	4	yes	—
95	C₂×C₈×Q₈	D₆₁	1	1	yes	characteristic
96	#1801	—	61	2	no	—
97	#1566	—	61	2	no	—
98	#1292	—	61	2	no	—
99	C₂×C₆₁	—	2	1	no	—
100	C₂×C₆₁	—	2	1	no	—
101	C₂×C₆₁	#1801	1	1	yes	characteristic
102	C₂×C₆₁	#1807	1	1	yes	characteristic
103	C₂×C₆₁	#1813	1	1	yes	characteristic
104	C₂×C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
105	C₈×Q₈	—	2	2	no	—
106	C₈×Q₈	—	2	2	no	—
107	#423	—	61	4	no	—
108	C₂×C₄⋊C₈	Dic₆₁	1	2	yes	—
109	C₂×C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
110	C₂×#101	—	122	2	no	—
111	C₂×C₄×C₈	Dic₆₁	1	2	yes	—
112	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
113	#343	—	122	2	no	—
114	#334	—	122	2	no	—
115	#334	—	122	2	no	—
116	C₆₁	#8615	1	1	yes	characteristic, complemented, Sylow
117	C₂²×Q₈	D₂₄₄	1	1	yes	characteristic
118	C₂×M₁₆	—	122	4	no	—
119	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
120	C₂²×C₈	C₂×Dic₆₁	1	2	yes	—
121	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
122	C₄×Q₈	—	2	2	no	—
123	C₄×Q₈	—	2	2	no	—
124	C₂×C₄⋊C₄	Dic₁₂₂	1	2	yes	—
125	C₂×C₄⋊C₄	D₂₄₄	1	1	yes	characteristic
126	C₂×C₄²	C₂×Dic₆₁	1	2	yes	—
127	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
128	C₄⋊C₈	—	2	4	no	—
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	122	4	no	—
131	C₄⋊C₈	—	122	4	no	—
132	#101	—	244	4	no	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	—	2	2	no	—
135	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
136	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
137	C₂×Q₈	—	2	2	no	—
138	C₂×Q₈	—	2	2	no	—
139	C₂×Q₈	#11758190	1	4	yes	—
140	C₂²×C₄	C₄⋊Dic₆₁	1	2	yes	—
141	C₂²×C₄	C₄⋊Dic₆₁	1	2	yes	—
142	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
143	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
144	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
145	M₁₆	—	244	4	no	—
146	M₁₆	—	244	4	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	4	1	no	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	—	244	4	no	—
152	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
153	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
154	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄²	—	2	2	no	—
158	C₄²	—	2	2	no	—
159	C₄²	#11758222	1	2	yes	—
160	C₄²	C₂×C₂²⋊D₆₁	1	2	yes	—
161	C₂³	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
162	Q₈	—	2	8	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	#14519709	1	4	yes	—
171	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
172	C₂×C₄	#14519736	1	1	yes	characteristic
173	C₂×C₄	#14519751	1	2	yes	—
174	C₂×C₄	#14519756	1	2	yes	—
175	C₂×C₄	C₂×#11758190	1	2	yes	—
176	C₂×C₄	#14519799	1	2	yes	—
177	C₈	—	2	1	no	—
178	C₈	—	2	2	no	—
179	C₈	—	4	2	no	—
180	C₈	—	488	4	no	—
181	C₈	#14519696	1	2	yes	complemented
182	C₂²	—	2	1	no	—
183	C₂²	—	2	1	no	—
184	C₂²	#16839622	1	1	yes	characteristic
185	C₂²	C₂×#14519709	1	1	yes	characteristic
186	C₂²	#16839800	1	1	yes	characteristic
187	C₄	—	2	1	no	—
188	C₄	—	2	4	no	—
189	C₄	—	2	2	no	—
190	C₄	—	4	2	no	—
191	C₄	C₂×#14519696	1	1	yes	characteristic
192	C₄	#16839746	1	1	yes	characteristic
193	C₄	#16839991	1	4	yes	—
194	C₂	—	2	1	no	—
195	C₂	—	2	1	no	—
196	C₂	#21445815	1	1	yes	characteristic
197	C₂	#21445844	1	1	yes	characteristic
198	C₂	#21445869	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21445815

C₂

yes

maximal

#21444575

C₂

yes

maximal

#21443694

C₂

yes

maximal

C₂×C₈×Q₈×C₆₁

C₂

yes

maximal, characteristic

#16839786

C₂²

yes

—

—

—

—

—

C₂×C₄×Q₈×C₆₁

C₂²

yes

characteristic

C₈×Q₈×C₆₁

—

C₈×Q₈×C₆₁

—

C₂×C₄⋊C₈×C₆₁

C₄

yes

—

C₂×C₄⋊C₈×C₆₁

C₂²

yes

characteristic

C₂×C₄×C₈×C₆₁

C₄

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

—

—

—

—

C₂²×Q₈×C₆₁

D₄

yes

characteristic

C₂²×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₁

C₂×C₄

yes

—

C₂²×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄×Q₈×C₆₁

—

C₄×Q₈×C₆₁

—

C₂×C₄⋊C₄×C₆₁

D₄

yes

characteristic

C₂×C₄⋊C₄×C₆₁

Q₈

yes

—

C₂×C₄²×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊C₈×C₆₁

—

C₄⋊C₈×C₆₁

—

C₄×C₈×C₆₁

—

C₄×C₈×C₆₁

—

C₄×C₈×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

D₄

yes

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

SD₁₆

yes

—

C₂²×C₄×C₆₁

C₄⋊C₄

yes

—

C₂²×C₄×C₆₁

C₄⋊C₄

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₁

C₂²⋊C₄

yes

—

C₂×C₈×C₆₁

C₂²⋊C₄

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂×D₄

yes

—

C₄²×C₆₁

#41

yes

—

#11758194

—

#11758194

—

C₂×#92418

—

C₂³×C₆₁

C₂×C₄⋊C₄

yes

characteristic

Q₈×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

#106

yes

—

C₂×C₄×C₆₁

C₂×C₂²⋊C₄

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

—

C₂×C₄×C₆₁

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₁

C₂×SD₁₆

yes

—

C₂×C₄×C₆₁

#135

yes

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

yes

—

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

C₂×C₈×Q₈

D₆₁

yes

—

—

—

C₂×C₆₁

—

100

C₂×C₆₁

—

101

C₂×C₆₁

#1801

yes

characteristic

102

C₂×C₆₁

#1807

yes

characteristic

103

C₂×C₆₁

#1813

yes

characteristic

104

C₂×C₄×Q₈

C₂×D₆₁

yes

characteristic

105

C₈×Q₈

—

106

C₈×Q₈

—

107

#423

—

108

C₂×C₄⋊C₈

Dic₆₁

yes

—

109

C₂×C₄⋊C₈

C₂×D₆₁

yes

characteristic

110

C₂×#101

—

122

—

111

C₂×C₄×C₈

Dic₆₁

yes

—

112

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

113

#343

—

122

—

114

#334

—

122

—

115

#334

—

122

—

116

C₆₁

#8615

yes

characteristic, complemented, Sylow

117

C₂²×Q₈

D₂₄₄

yes

characteristic

118

C₂×M₁₆

—

122

—

119

C₂²×C₈

C₂×Dic₆₁

yes

characteristic

120

C₂²×C₈

C₂×Dic₆₁

yes

—

121

C₂²×C₈

C₂×Dic₆₁

yes

characteristic

122

C₄×Q₈

—

123

C₄×Q₈

—

124

C₂×C₄⋊C₄

Dic₁₂₂

yes

—

125

C₂×C₄⋊C₄

D₂₄₄

yes

characteristic

126

C₂×C₄²

C₂×Dic₆₁

yes

—

127

C₂×C₄²

C₂²×D₆₁

yes

characteristic

128

C₄⋊C₈

—

129

C₄⋊C₈

—

130

C₄⋊C₈

—

122

—

131

C₄⋊C₈

—

122

—

132

#101

—

244

—

133

C₄×C₈

—

134

C₄×C₈

—

135

C₄×C₈

C₂²⋊D₆₁

yes

—

136

C₄×C₈

C₂²⋊D₆₁

yes

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂×Q₈

#11758190

yes

—

140

C₂²×C₄

C₄⋊Dic₆₁

yes

—

141

C₂²×C₄

C₄⋊Dic₆₁

yes

—

142

C₂²×C₄

C₂×Dic₁₂₂

yes

characteristic

143

C₂²×C₄

C₂×D₂₄₄

yes

characteristic

144

C₂²×C₄

C₂²×Dic₆₁

yes

characteristic

145

M₁₆

—

244

—

146

M₁₆

—

244

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

244

—

152

C₂×C₈

C₂²⋊Dic₆₁

yes

characteristic

153

C₂×C₈

C₂²⋊Dic₆₁

yes

—

154

C₂×C₈

C₂²⋊Dic₆₁

yes

characteristic

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄²

—

158

C₄²

—

159

C₄²

#11758222

yes

—

160

C₄²

C₂×C₂²⋊D₆₁

yes

—

161

C₂³

C₂×C₄⋊Dic₆₁

yes

characteristic

162

Q₈

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

#14519709

yes

—

171

C₂×C₄

C₂×C₂²⋊Dic₆₁

yes

characteristic

172

C₂×C₄

#14519736

yes

characteristic

173

C₂×C₄

#14519751

yes

—

174

C₂×C₄

#14519756

yes

—

175

C₂×C₄

C₂×#11758190

yes

—

176

C₂×C₄

#14519799

yes

—

177

C₈

—

178

C₈

—

179

C₈

—

180

C₈

—

488

—

181

C₈

#14519696

yes

complemented

182

C₂²

—

183

C₂²

—

184

C₂²

#16839622

yes

characteristic

185

C₂²

C₂×#14519709

yes

characteristic

186

C₂²

#16839800

yes

characteristic

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

C₂×#14519696

yes

characteristic

192

C₄

#16839746

yes

characteristic

193

C₄

#16839991

yes

—

194

C₂

—

195

C₂

—

196

C₂

#21445815

yes

characteristic

197

C₂

#21445844

yes

characteristic

198

C₂

#21445869

yes

characteristic

	subgroup	G/H
center	C₂²	#16839622
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₈×Q₈×C₆₁	C₂
socle	C₂²×C₆₁	#380
prime radical	C₂³×C₆₁	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×C₈×Q₈	D₆₁
hypocenter	C₆₁	#8615
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈

subgroup

G/H

center

C₂²

#16839622

commutator

C₂²×C₄×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₂×C₈×Q₈×C₆₁

C₂

socle

C₂²×C₆₁

#380

prime radical

C₂³×C₆₁

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₆₁

C₂³

hypercenter

C₂×C₈×Q₈

D₆₁

hypocenter

C₆₁

#8615

LCS

G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈

N	H ≅ G/N	type
80. C₈×C₆₁	#101	semidirect
116. C₆₁	#8615	semidirect
181. C₈	#14519696	semidirect

H ≅ G/N

type

80. C₈×C₆₁

#101

semidirect

116. C₆₁

#8615

semidirect

181. C₈

#14519696

semidirect

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

subgroup	index
#21445815	2
#21444575	2
#21443694	2
C₂×C₈×Q₈×C₆₁	2
#8615	61

subgroup

index

C₂×C₈×Q₈×C₆₁

Subgroups — #29572937

Conjugacy classes of subgroups

Complements

Maximal subgroups