Group #30624460 — subgroups

Subgroups — #30624460

Order 15616.

486 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 230 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	#21462235	C₂	1	8	yes	maximal, complemented
2	#21462181	C₂	1	4	yes	maximal
3	#21462176	C₂	1	2	yes	maximal
4	C₂×#16839886	C₂	1	1	yes	maximal, characteristic
5	#16840683	C₂²	1	8	yes	—
6	C₄⋊#11758187	C₂²	1	4	yes	—
7	C₂²⋊#11758187	C₂²	1	4	yes	—
8	#16840651	C₂²	1	8	yes	—
9	C₄×#11758187	C₂²	1	4	yes	—
10	C₂×C₄×Dic₁₂₂	C₄	1	1	yes	characteristic
11	#16839886	C₄	1	4	yes	—
12	#16839886	C₂²	1	4	yes	—
13	C₂×#14519706	C₄	1	2	yes	—
14	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₆₁	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₆₁	C₄	1	1	yes	characteristic
17	C₂×#11758187	C₂³	1	8	yes	—
18	C₂²×Dic₁₂₂	C₈	1	1	yes	characteristic
19	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
20	C₂²×#92418	C₂³	1	1	yes	characteristic
21	C₄×Dic₁₂₂	C₈	1	4	yes	—
22	C₄×Dic₁₂₂	C₂×C₄	1	4	yes	—
23	C₂×C₄⋊Dic₆₁	C₈	1	1	yes	characteristic
24	C₂×#11758196	C₈	1	2	yes	—
25	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
26	#14519706	C₂×C₄	1	8	yes	—
27	C₄×#92418	C₂×C₄	1	2	yes	—
28	C₄×#92418	C₂³	1	2	yes	—
29	C₈×Dic₆₁	C₂×C₄	1	4	yes	—
30	C₈×Dic₆₁	C₂³	1	4	yes	—
31	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₄⋊C₈×C₆₁	C₂×C₄	1	4	yes	—
34	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
37	C₂²×Dic₆₁	C₂×C₈	1	2	yes	—
38	C₂²×Dic₆₁	C₂×C₈	1	2	yes	—
39	C₂×Dic₁₂₂	—	2	4	no	—
40	C₂×Dic₁₂₂	C₂×C₈	1	4	yes	—
41	C₂×C₈×C₆₁	—	4	2	no	—
42	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
43	C₄²×C₆₁	C₂×C₈	1	2	yes	—
44	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
45	C₄⋊Dic₆₁	C₂×C₈	1	4	yes	—
46	#11758196	C₂×C₈	1	8	yes	—
47	C₄×Dic₆₁	C₂×C₈	1	4	yes	—
48	C₄×Dic₆₁	C₂²×C₄	1	4	yes	—
49	C₂×#92418	—	2	2	no	—
50	C₂×#92418	—	4	1	no	—
51	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
52	C₂×#92418	C₂²×C₄	1	2	yes	—
53	C₂×#92418	C₂⁴	1	1	yes	characteristic
54	#11758187	—	2	8	no	—
55	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
56	C₂×C₄×C₆₁	—	2	1	no	—
57	C₂×C₄×C₆₁	—	2	1	no	—
58	C₂×C₄×C₆₁	—	4	1	no	—
59	C₂×C₄×C₆₁	M₃₂	1	4	yes	—
60	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
62	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
63	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
64	C₂×Dic₆₁	—	2	4	no	—
65	C₂×Dic₆₁	—	4	2	no	—
66	C₂×Dic₆₁	C₂²×C₈	1	4	yes	—
67	C₂×Dic₆₁	C₂²×C₈	1	4	yes	—
68	Dic₁₂₂	—	2	8	no	—
69	C₈×C₆₁	—	2	4	no	—
70	#92418	—	2	1	no	—
71	#92418	—	2	2	no	—
72	#92418	2₋¹⁺⁴	1	2	yes	—
73	#33497	—	61	1	no	maximal, Sylow
74	C₂²×C₆₁	—	2	1	no	—
75	C₂²×C₆₁	—	2	1	no	—
76	C₂²×C₆₁	C₂×M₃₂	1	2	yes	—
77	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
78	C₄×C₆₁	—	2	2	no	—
79	C₄×C₆₁	—	2	1	no	—
80	C₄×C₆₁	C₂×M₃₂	1	4	yes	—
81	C₄×C₆₁	#519	1	1	yes	characteristic
82	C₄×C₆₁	#568	1	1	yes	characteristic
83	Dic₆₁	—	2	4	no	—
84	Dic₆₁	—	2	4	no	—
85	C₂×C₈×Q₈	—	61	1	no	—
86	#2183	—	61	8	no	—
87	#2163	—	61	4	no	—
88	#2162	—	61	2	no	—
89	C₂×C₆₁	—	2	2	no	—
90	C₂×C₆₁	#2982	1	1	yes	characteristic
91	C₂×C₆₁	C₂²×M₃₂	1	1	yes	characteristic
92	C₂×C₆₁	#3407	1	1	yes	characteristic
93	C₂×C₄×Q₈	—	61	1	no	—
94	C₈×Q₈	—	61	4	no	—
95	C₈×Q₈	—	61	4	no	—
96	C₂×C₄⋊C₈	—	61	2	no	—
97	C₂×C₄⋊C₈	#6923	1	1	yes	characteristic
98	C₂×C₄×C₈	—	61	2	no	—
99	C₂×C₄×C₈	—	61	1	no	—
100	C₄⋊C₁₆	—	61	4	no	—
101	C₄⋊C₁₆	—	61	8	no	—
102	C₂²⋊C₁₆	—	61	4	no	—
103	#346	—	61	8	no	—
104	C₄×C₁₆	—	61	4	no	—
105	C₆₁	#33497	1	1	yes	characteristic, complemented, Sylow
106	C₂²×Q₈	—	61	1	no	—
107	C₂²×C₈	—	61	1	no	—
108	C₂²×C₈	—	61	1	no	—
109	C₂²×C₈	C₂×#6923	1	2	yes	—
110	C₄×Q₈	—	61	4	no	—
111	C₄×Q₈	—	61	4	no	—
112	C₂×C₄⋊C₄	—	61	2	no	—
113	C₂×C₄⋊C₄	—	61	1	no	—
114	C₂×C₄²	—	61	2	no	—
115	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
116	C₂×C₁₆	—	61	8	no	—
117	C₄⋊C₈	—	61	8	no	—
118	C₄⋊C₈	C₂×#6923	1	4	yes	—
119	C₄×C₈	—	61	4	no	—
120	C₄×C₈	—	61	4	no	—
121	C₄×C₈	—	61	2	no	—
122	C₄×C₈	—	61	2	no	—
123	C₂×Q₈	—	61	4	no	—
124	C₂×Q₈	—	122	4	no	—
125	C₂²×C₄	—	61	2	no	—
126	C₂²×C₄	—	61	2	no	—
127	C₂²×C₄	D₆₁⋊C₈	1	1	yes	characteristic
128	C₂²×C₄	D₆₁⋊C₈	1	1	yes	characteristic
129	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
130	C₂×C₈	—	4	2	no	—
131	C₂×C₈	—	61	1	no	—
132	C₂×C₈	—	61	1	no	—
133	C₂×C₈	—	61	2	no	—
134	C₂×C₈	—	122	2	no	—
135	C₂×C₈	—	244	1	no	—
136	C₂×C₈	C₂²×#6923	1	4	yes	—
137	C₄⋊C₄	—	61	8	no	—
138	C₄⋊C₄	—	61	4	no	—
139	C₄²	—	61	4	no	—
140	C₄²	—	61	4	no	—
141	C₄²	D₆₁⋊C₈	1	2	yes	—
142	C₄²	C₂²×#6923	1	2	yes	—
143	C₁₆	—	122	8	no	—
144	C₂³	C₂×D₆₁⋊C₈	1	1	yes	characteristic
145	Q₈	—	122	8	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	61	4	no	—
150	C₂×C₄	—	61	4	no	—
151	C₂×C₄	—	122	4	no	—
152	C₂×C₄	—	244	2	no	—
153	C₂×C₄	#14519852	1	4	yes	—
154	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
155	C₂×C₄	C₂×D₆₁⋊C₈	1	2	yes	—
156	C₂×C₄	C₂×D₆₁⋊C₈	1	2	yes	—
157	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
158	C₈	—	2	4	no	—
159	C₈	—	61	2	no	—
160	C₈	—	122	1	no	—
161	C₈	—	122	2	no	—
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	C₂×#14519852	1	2	yes	—
165	C₂²	C₂²×D₆₁⋊C₈	1	1	yes	characteristic
166	C₄	—	2	2	no	—
167	C₄	—	2	1	no	—
168	C₄	—	122	4	no	—
169	C₄	—	122	4	no	—
170	C₄	C₂×#14519852	1	4	yes	—
171	C₄	#16840830	1	1	yes	characteristic
172	C₄	#16840840	1	1	yes	characteristic
173	C₂	—	2	2	no	—
174	C₂	#21462714	1	1	yes	characteristic
175	C₂	C₂²×#14519852	1	1	yes	characteristic
176	C₂	#21462788	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21462235

C₂

yes

maximal, complemented

#21462181

C₂

yes

maximal

#21462176

C₂

yes

maximal

C₂×#16839886

C₂

yes

maximal, characteristic

#16840683

C₂²

yes

—

C₄⋊#11758187

C₂²

yes

—

C₂²⋊#11758187

C₂²

yes

—

#16840651

C₂²

yes

—

C₄×#11758187

C₂²

yes

—

C₂×C₄×Dic₁₂₂

C₄

yes

characteristic

#16839886

C₄

yes

—

#16839886

C₂²

yes

—

C₂×#14519706

C₄

yes

—

C₂×C₄×#92418

C₂²

yes

characteristic

C₂×C₈×Dic₆₁

C₂²

yes

—

C₂×C₄⋊C₈×C₆₁

C₄

yes

characteristic

C₂×#11758187

C₂³

yes

—

C₂²×Dic₁₂₂

C₈

yes

characteristic

C₂²×#92418

C₂×C₄

yes

characteristic

C₂²×#92418

C₂³

yes

characteristic

C₄×Dic₁₂₂

C₈

yes

—

C₄×Dic₁₂₂

C₂×C₄

yes

—

C₂×C₄⋊Dic₆₁

C₈

yes

characteristic

C₂×#11758196

C₈

yes

—

C₂×C₄×Dic₆₁

C₂×C₄

yes

—

#14519706

C₂×C₄

yes

—

C₄×#92418

C₂×C₄

yes

—

C₄×#92418

C₂³

yes

—

C₈×Dic₆₁

C₂×C₄

yes

—

C₈×Dic₆₁

C₂³

yes

—

C₂²×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄⋊C₈×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂×C₈

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₈

yes

characteristic

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×Dic₆₁

C₂×C₈

yes

—

C₂²×Dic₆₁

C₂×C₈

yes

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

C₂×C₈

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂²×C₄

yes

—

C₄²×C₆₁

C₂×C₈

yes

—

C₄²×C₆₁

C₂²×C₄

yes

—

C₄⋊Dic₆₁

C₂×C₈

yes

—

#11758196

C₂×C₈

yes

—

C₄×Dic₆₁

C₂×C₈

yes

—

C₄×Dic₆₁

C₂²×C₄

yes

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

C₂²×C₄

yes

characteristic

C₂×#92418

C₂²×C₄

yes

—

C₂×#92418

C₂⁴

yes

characteristic

#11758187

—

C₂³×C₆₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

M₃₂

yes

—

C₂×C₄×C₆₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₈

yes

—

C₂×C₄×C₆₁

C₂²×C₈

yes

—

C₂×C₄×C₆₁

C₂³×C₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂²×C₈

yes

—

C₂×Dic₆₁

C₂²×C₈

yes

—

Dic₁₂₂

—

C₈×C₆₁

—

#92418

—

#92418

—

#92418

2₋¹⁺⁴

yes

—

#33497

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×M₃₂

yes

—

C₂²×C₆₁

C₂³×C₈

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×M₃₂

yes

—

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

C₂×C₈×Q₈

—

#2183

—

#2163

—

#2162

—

C₂×C₆₁

—

C₂×C₆₁

#2982

yes

characteristic

C₂×C₆₁

C₂²×M₃₂

yes

characteristic

C₂×C₆₁

#3407

yes

characteristic

C₂×C₄×Q₈

—

C₈×Q₈

—

C₈×Q₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

#6923

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

—

100

C₄⋊C₁₆

—

101

C₄⋊C₁₆

—

102

C₂²⋊C₁₆

—

103

#346

—

104

C₄×C₁₆

—

105

C₆₁

#33497

yes

characteristic, complemented, Sylow

106

C₂²×Q₈

—

107

C₂²×C₈

—

108

C₂²×C₈

—

109

C₂²×C₈

C₂×#6923

yes

—

110

C₄×Q₈

—

111

C₄×Q₈

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄²

—

115

C₂×C₄²

C₂×#6923

yes

characteristic

116

C₂×C₁₆

—

117

C₄⋊C₈

—

118

C₄⋊C₈

C₂×#6923

yes

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

—

122

C₄×C₈

—

123

C₂×Q₈

—

124

C₂×Q₈

—

122

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

D₆₁⋊C₈

yes

characteristic

128

C₂²×C₄

D₆₁⋊C₈

yes

characteristic

129

C₂²×C₄

C₂²×#6923

yes

characteristic

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

122

—

135

C₂×C₈

—

244

—

136

C₂×C₈

C₂²×#6923

yes

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄²

—

140

C₄²

—

141

C₄²

D₆₁⋊C₈

yes

—

142

C₄²

C₂²×#6923

yes

—

143

C₁₆

—

122

—

144

C₂³

C₂×D₆₁⋊C₈

yes

characteristic

145

Q₈

—

122

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

122

—

152

C₂×C₄

—

244

—

153

C₂×C₄

#14519852

yes

—

154

C₂×C₄

C₂×D₆₁⋊C₈

yes

characteristic

155

C₂×C₄

C₂×D₆₁⋊C₈

yes

—

156

C₂×C₄

C₂×D₆₁⋊C₈

yes

—

157

C₂×C₄

C₂³×#6923

yes

characteristic

158

C₈

—

159

C₈

—

160

C₈

—

122

—

161

C₈

—

122

—

162

C₂²

—

163

C₂²

—

164

C₂²

C₂×#14519852

yes

—

165

C₂²

C₂²×D₆₁⋊C₈

yes

characteristic

166

C₄

—

167

C₄

—

168

C₄

—

122

—

169

C₄

—

122

—

170

C₄

C₂×#14519852

yes

—

171

C₄

#16840830

yes

characteristic

172

C₄

#16840840

yes

characteristic

173

C₂

—

174

C₂

#21462714

yes

characteristic

175

C₂

C₂²×#14519852

yes

characteristic

176

C₂

#21462788

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6923
commutator, socle	C₂²×C₆₁	C₂³×C₈
Fitting	C₂×C₄⋊C₈×C₆₁	C₄
prime radical	C₂³×C₆₁	C₂²×C₈
cosocle	C₂×#92418	C₂⁴
hypercenter	C₂×C₄⋊C₈	#6923
hypocenter	C₆₁	#33497
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×#6923

commutator, socle

C₂²×C₆₁

C₂³×C₈

Fitting

C₂×C₄⋊C₈×C₆₁

C₄

prime radical

C₂³×C₆₁

C₂²×C₈

C₂⁴

C₂×C₄⋊C₈

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
1. #21462235	C₂	semidirect
105. C₆₁	#33497	semidirect

H ≅ G/N

type

1. #21462235

C₂

semidirect

105. C₆₁

#33497

semidirect

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

subgroup	index
#21462235	2
#21462181	2
#21462176	2
C₂×#16839886	2
#33497	61

subgroup

index

Subgroups — #30624460

Conjugacy classes of subgroups

Complements

Maximal subgroups