Group #29572963 — subgroups

Subgroups — #29572963

Order 15616.

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

normal subgroups only

This group has 362 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	#21444582	C₂	1	2	yes	maximal
2	#21444163	C₂	1	2	yes	maximal
3	#21443689	C₂	1	2	yes	maximal
4	C₂×C₈×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
5	#16839819	C₂²	1	4	yes	—
6	#16839515	—	2	2	no	—
7	#16839513	—	2	2	no	—
8	#16839488	—	2	4	no	—
9	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
10	C₈×Q₈×C₆₁	—	2	2	no	—
11	C₈×Q₈×C₆₁	—	2	2	no	—
12	C₂×C₄⋊C₈×C₆₁	C₄	1	2	yes	—
13	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₆₁	C₄	1	2	yes	—
15	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₄⋊#92418	—	2	4	no	—
17	C₂²⋊#92418	—	2	4	no	—
18	#14519688	—	2	4	no	—
19	C₂²×Q₈×C₆₁	Q₈	1	1	yes	characteristic
20	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₄×Q₈×C₆₁	—	2	2	no	—
24	C₄×Q₈×C₆₁	—	2	2	no	—
25	C₂×C₄⋊C₄×C₆₁	D₄	1	2	yes	—
26	C₂×C₄⋊C₄×C₆₁	Q₈	1	1	yes	characteristic
27	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₆₁	—	2	4	no	—
30	C₄⋊C₈×C₆₁	—	2	2	no	—
31	C₄×C₈×C₆₁	—	2	2	no	—
32	C₄×C₈×C₆₁	—	2	2	no	—
33	C₄×C₈×C₆₁	D₄	1	2	yes	—
34	C₄×C₈×C₆₁	D₄	1	2	yes	—
35	C₂×Q₈×C₆₁	—	2	4	no	—
36	C₂×Q₈×C₆₁	—	2	2	no	—
37	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
38	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
39	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₈×C₆₁	—	2	2	no	—
43	C₂×C₈×C₆₁	—	2	2	no	—
44	C₂×C₈×C₆₁	—	4	1	no	—
45	C₂×C₈×C₆₁	—	4	2	no	—
46	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
48	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₄⋊C₄×C₆₁	—	2	4	no	—
50	C₄⋊C₄×C₆₁	—	2	2	no	—
51	C₄²×C₆₁	—	2	2	no	—
52	C₄²×C₆₁	—	2	2	no	—
53	C₄²×C₆₁	C₂×D₄	1	2	yes	—
54	C₄²×C₆₁	#41	1	2	yes	—
55	C₂×#92418	—	4	4	no	—
56	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
57	Q₈×C₆₁	—	2	8	no	—
58	C₂×C₄×C₆₁	—	2	2	no	—
59	C₂×C₄×C₆₁	—	2	2	no	—
60	C₂×C₄×C₆₁	—	2	2	no	—
61	C₂×C₄×C₆₁	—	2	1	no	—
62	C₂×C₄×C₆₁	—	2	1	no	—
63	C₂×C₄×C₆₁	—	2	2	no	—
64	C₂×C₄×C₆₁	—	4	1	no	—
65	C₂×C₄×C₆₁	—	4	2	no	—
66	C₂×C₄×C₆₁	#108	1	4	yes	—
67	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
69	C₂×C₄×C₆₁	#121	1	2	yes	—
70	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
71	C₈×C₆₁	—	2	1	no	—
72	C₈×C₆₁	—	2	2	no	—
73	C₈×C₆₁	—	4	2	no	—
74	C₈×C₆₁	#101	1	2	yes	—
75	#92418	—	8	4	no	—
76	#8621	—	61	1	no	maximal, Sylow
77	C₂²×C₆₁	—	2	1	no	—
78	C₂²×C₆₁	—	2	1	no	—
79	C₂²×C₆₁	#380	1	1	yes	characteristic
80	C₂²×C₆₁	C₂×#108	1	2	yes	—
81	C₄×C₆₁	—	2	1	no	—
82	C₄×C₆₁	—	2	4	no	—
83	C₄×C₆₁	—	2	2	no	—
84	C₄×C₆₁	—	2	2	no	—
85	C₄×C₆₁	—	4	2	no	—
86	C₄×C₆₁	C₂×#101	1	1	yes	characteristic
87	C₄×C₆₁	#413	1	1	yes	characteristic
88	C₂×C₈×Q₈	D₆₁	1	1	yes	characteristic
89	#1568	—	61	2	no	—
90	#1467	—	61	2	no	—
91	#1290	—	61	2	no	—
92	C₂×C₆₁	—	2	2	no	—
93	C₂×C₆₁	#1801	1	1	yes	characteristic
94	C₂×C₆₁	#1810	1	1	yes	characteristic
95	C₂×C₆₁	#1812	1	1	yes	characteristic
96	C₂×C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
97	C₈×Q₈	—	2	2	no	—
98	C₈×Q₈	—	2	2	no	—
99	#432	—	61	4	no	—
100	C₂×C₄⋊C₈	Dic₆₁	1	2	yes	—
101	C₂×C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
102	C₂×C₄×C₈	Dic₆₁	1	2	yes	—
103	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
104	#335	—	122	2	no	—
105	#334	—	122	2	no	—
106	#324	—	122	4	no	—
107	C₆₁	#8621	1	1	yes	characteristic, complemented, Sylow
108	C₂²×Q₈	Dic₁₂₂	1	1	yes	characteristic
109	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
110	C₂²×C₈	C₂×Dic₆₁	1	2	yes	—
111	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
112	C₄×Q₈	—	2	2	no	—
113	C₄×Q₈	—	2	2	no	—
114	C₂×C₄⋊C₄	Dic₁₂₂	1	1	yes	characteristic
115	C₂×C₄⋊C₄	D₂₄₄	1	2	yes	—
116	C₂×C₄²	C₂×Dic₆₁	1	2	yes	—
117	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
118	C₄⋊C₈	—	2	4	no	—
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	122	4	no	—
121	C₂²⋊C₈	—	122	4	no	—
122	#97	—	122	4	no	—
123	C₄×C₈	—	2	2	no	—
124	C₄×C₈	—	2	2	no	—
125	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
126	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
127	C₂×Q₈	—	2	4	no	—
128	C₂×Q₈	—	2	2	no	—
129	C₂²×C₄	C₄⋊Dic₆₁	1	2	yes	—
130	C₂²×C₄	C₄⋊Dic₆₁	1	2	yes	—
131	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
132	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
133	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	4	1	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	—	244	4	no	—
139	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
140	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
141	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	—	2	2	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	2	2	no	—
146	C₄²	#11758222	1	2	yes	—
147	C₄²	C₂×C₂²⋊D₆₁	1	2	yes	—
148	C₂³	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
149	Q₈	—	2	8	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	#14519713	1	4	yes	—
159	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
160	C₂×C₄	#14519736	1	1	yes	characteristic
161	C₂×C₄	#14519751	1	2	yes	—
162	C₂×C₄	#14519756	1	2	yes	—
163	C₈	—	2	1	no	—
164	C₈	—	2	2	no	—
165	C₈	—	4	2	no	—
166	C₈	—	488	4	no	—
167	C₈	#14519696	1	2	yes	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	#16839622	1	1	yes	characteristic
171	C₂²	C₂×#14519713	1	2	yes	—
172	C₄	—	2	1	no	—
173	C₄	—	2	4	no	—
174	C₄	—	2	2	no	—
175	C₄	—	2	2	no	—
176	C₄	—	4	2	no	—
177	C₄	C₂×#14519696	1	1	yes	characteristic
178	C₄	#16839746	1	1	yes	characteristic
179	C₂	—	2	2	no	—
180	C₂	#21445815	1	1	yes	characteristic
181	C₂	#21445857	1	1	yes	characteristic
182	C₂	#21445865	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21444582

C₂

yes

maximal

#21444163

C₂

yes

maximal

#21443689

C₂

yes

maximal

C₂×C₈×Q₈×C₆₁

C₂

yes

maximal, characteristic

#16839819

C₂²

yes

—

#16839515

—

#16839513

—

#16839488

—

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

Q₈

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

—

C₂×C₄⋊C₄×C₆₁

Q₈

yes

characteristic

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

—

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

—

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

—

C₂×C₄×C₆₁

#108

yes

—

C₂×C₄×C₆₁

C₂×C₂²⋊C₄

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

—

C₂×C₄×C₆₁

C₂²⋊Q₈

yes

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

#101

yes

—

#92418

—

#8621

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#380

yes

characteristic

C₂²×C₆₁

C₂×#108

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₂×C₈×Q₈

D₆₁

yes

—

—

—

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₄×Q₈

C₂×D₆₁

yes

characteristic

C₈×Q₈

—

C₈×Q₈

—

#432

—

100

C₂×C₄⋊C₈

Dic₆₁

yes

—

101

C₂×C₄⋊C₈

C₂×D₆₁

yes

characteristic

102

C₂×C₄×C₈

Dic₆₁

yes

—

103

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

104

#335

—

122

—

105

#334

—

122

—

106

#324

—

122

—

107

C₆₁

#8621

yes

characteristic, complemented, Sylow

108

C₂²×Q₈

Dic₁₂₂

yes

characteristic

109

C₂²×C₈

C₂×Dic₆₁

yes

characteristic

110

C₂²×C₈

C₂×Dic₆₁

yes

—

111

C₂²×C₈

C₂×Dic₆₁

yes

characteristic

112

C₄×Q₈

—

113

C₄×Q₈

—

114

C₂×C₄⋊C₄

Dic₁₂₂

yes

characteristic

115

C₂×C₄⋊C₄

D₂₄₄

yes

—

116

C₂×C₄²

C₂×Dic₆₁

yes

—

117

C₂×C₄²

C₂²×D₆₁

yes

characteristic

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₄⋊C₈

—

122

—

121

C₂²⋊C₈

—

122

—

122

#97

—

122

—

123

C₄×C₈

—

124

C₄×C₈

—

125

C₄×C₈

C₂²⋊D₆₁

yes

—

126

C₄×C₈

C₂²⋊D₆₁

yes

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂²×C₄

C₄⋊Dic₆₁

yes

—

130

C₂²×C₄

C₄⋊Dic₆₁

yes

—

131

C₂²×C₄

C₂×Dic₁₂₂

yes

characteristic

132

C₂²×C₄

C₂×D₂₄₄

yes

characteristic

133

C₂²×C₄

C₂²×Dic₆₁

yes

characteristic

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

244

—

139

C₂×C₈

C₂²⋊Dic₆₁

yes

characteristic

140

C₂×C₈

C₂²⋊Dic₆₁

yes

—

141

C₂×C₈

C₂²⋊Dic₆₁

yes

characteristic

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄²

—

145

C₄²

—

146

C₄²

#11758222

yes

—

147

C₄²

C₂×C₂²⋊D₆₁

yes

—

148

C₂³

C₂×C₄⋊Dic₆₁

yes

characteristic

149

Q₈

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

#14519713

yes

—

159

C₂×C₄

C₂×C₂²⋊Dic₆₁

yes

characteristic

160

C₂×C₄

#14519736

yes

characteristic

161

C₂×C₄

#14519751

yes

—

162

C₂×C₄

#14519756

yes

—

163

C₈

—

164

C₈

—

165

C₈

—

166

C₈

—

488

—

167

C₈

#14519696

yes

—

168

C₂²

—

169

C₂²

—

170

C₂²

#16839622

yes

characteristic

171

C₂²

C₂×#14519713

yes

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

C₂×#14519696

yes

characteristic

178

C₄

#16839746

yes

characteristic

179

C₂

—

180

C₂

#21445815

yes

characteristic

181

C₂

#21445857

yes

characteristic

182

C₂

#21445865

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	#8621
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

#8621

LCS

G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈

N	H ≅ G/N	type
107. C₆₁	#8621	semidirect

H ≅ G/N

type

107. C₆₁

#8621

semidirect

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

subgroup	index
#21444582	2
#21444163	2
#21443689	2
C₂×C₈×Q₈×C₆₁	2
#8621	61

subgroup

index

C₂×C₈×Q₈×C₆₁

Subgroups — #29572963

Conjugacy classes of subgroups

Complements

Maximal subgroups