Group #30624465 — subgroups

Subgroups — #30624465

Order 15616.

494 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 232 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	6	yes	maximal, complemented
2	Q₈×#11758187	C₂	1	2	yes	maximal, complemented
3	#21462183	C₂	1	3	yes	maximal
4	#21462178	C₂	1	3	yes	maximal
5	C₈×Q₈×D₆₁	C₂	1	1	yes	maximal, characteristic
6	#16840683	C₂²	1	6	yes	—
7	C₄⋊#11758187	C₂²	1	6	yes	—
8	#16840657	C₂²	1	1	yes	characteristic
9	#16840657	C₂²	1	3	yes	—
10	#16840651	C₂²	1	6	yes	—
11	C₄×#11758187	C₂²	1	6	yes	—
12	C₄×Q₈×D₆₁	C₄	1	1	yes	characteristic
13	Q₈×#92418	C₂²	1	1	yes	characteristic
14	C₈×Dic₁₂₂	C₄	1	3	yes	—
15	#16839886	C₂²	1	3	yes	—
16	C₄⋊C₈×D₆₁	C₄	1	3	yes	—
17	C₄×C₈×D₆₁	C₂²	1	3	yes	—
18	C₈×Q₈×C₆₁	C₄	1	1	yes	characteristic
19	C₂×#11758187	C₂³	1	2	yes	—
20	C₂×#11758187	C₂³	1	6	yes	—
21	C₂×Q₈×D₆₁	C₈	1	1	yes	characteristic
22	C₂×C₈×D₆₁	C₂×C₄	1	3	yes	—
23	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
24	Q₈×Dic₆₁	C₈	1	1	yes	characteristic
25	C₄×Dic₁₂₂	C₂×C₄	1	3	yes	—
26	#14519742	C₈	1	3	yes	—
27	C₄⋊C₄×D₆₁	C₈	1	3	yes	—
28	C₄²×D₆₁	C₂×C₄	1	3	yes	—
29	C₄⋊#92418	C₂×C₄	1	3	yes	—
30	#14519706	C₂×C₄	1	6	yes	—
31	C₄×#92418	C₂³	1	3	yes	—
32	C₈×Dic₆₁	C₂×C₄	1	3	yes	—
33	C₈×Dic₆₁	C₂³	1	3	yes	—
34	C₄×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₄⋊C₈×C₆₁	C₂×C₄	1	3	yes	—
36	C₄×C₈×C₆₁	C₂×C₄	1	3	yes	—
37	C₂×Q₈×C₆₁	C₂×C₈	1	1	yes	characteristic
38	Q₈×D₆₁	—	2	4	no	—
39	C₂×C₄×D₆₁	C₂×C₈	1	3	yes	—
40	C₂×C₄×D₆₁	C₂×C₈	1	3	yes	—
41	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×Dic₁₂₂	C₂×C₈	1	3	yes	—
43	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₆₁	C₂²×C₄	1	3	yes	—
45	C₄⋊C₄×C₆₁	C₂×C₈	1	3	yes	—
46	C₄²×C₆₁	C₂²×C₄	1	3	yes	—
47	C₄⋊Dic₆₁	C₂×C₈	1	3	yes	—
48	#11758196	C₂×C₈	1	6	yes	—
49	C₄×Dic₆₁	C₂×C₈	1	3	yes	—
50	C₄×Dic₆₁	C₂×C₈	1	3	yes	—
51	C₄×Dic₆₁	C₂²×C₄	1	3	yes	—
52	C₂×#92418	C₂²×C₄	1	3	yes	—
53	C₂×#92418	C₂⁴	1	1	yes	characteristic
54	C₈×D₆₁	—	2	2	no	—
55	C₈×D₆₁	—	4	3	no	—
56	#11758187	—	2	2	no	—
57	#11758187	—	2	6	no	—
58	C₂²×D₆₁	C₂²×C₈	1	1	yes	characteristic
59	Q₈×C₆₁	M₃₂	1	4	yes	—
60	C₂×C₄×C₆₁	C₂²×C₈	1	3	yes	—
61	C₂×C₄×C₆₁	C₂²×C₈	1	3	yes	—
62	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
63	C₂×Dic₆₁	C₂²×C₈	1	1	yes	characteristic
64	C₂×Dic₆₁	C₂²×C₈	1	3	yes	—
65	C₂×Dic₆₁	C₂²×C₈	1	3	yes	—
66	C₄×D₆₁	—	2	6	no	—
67	C₄×D₆₁	—	2	1	no	—
68	C₄×D₆₁	—	2	1	no	—
69	C₄×D₆₁	—	2	6	no	—
70	Dic₁₂₂	—	2	6	no	—
71	C₈×C₆₁	—	2	1	no	—
72	C₈×C₆₁	—	2	3	no	—
73	#92418	—	2	3	no	—
74	#92418	2₋¹⁺⁴	1	2	yes	—
75	#33498	—	61	1	no	maximal, Sylow
76	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
77	C₂×D₆₁	—	2	1	no	—
78	C₂×D₆₁	—	2	1	no	—
79	C₂×D₆₁	#504	1	2	yes	—
80	C₄×C₆₁	—	2	3	no	—
81	C₄×C₆₁	C₂×M₃₂	1	6	yes	—
82	C₄×C₆₁	#519	1	1	yes	characteristic
83	C₄×C₆₁	#568	1	1	yes	characteristic
84	Dic₆₁	—	2	3	no	—
85	Dic₆₁	—	2	3	no	—
86	Dic₆₁	—	2	1	no	—
87	C₂×C₈×Q₈	—	61	1	no	—
88	#2183	—	61	6	no	—
89	Q₈×C₁₆	—	61	2	no	—
90	#2163	—	61	3	no	—
91	#2162	—	61	3	no	—
92	C₂×C₆₁	#2982	1	1	yes	characteristic
93	C₂×C₆₁	C₂²×M₃₂	1	1	yes	characteristic
94	C₂×C₆₁	C₂×#504	1	1	yes	characteristic
95	D₆₁	—	2	2	no	—
96	C₂×C₄×Q₈	—	61	1	no	—
97	C₈×Q₈	—	61	1	no	—
98	C₈×Q₈	—	61	3	no	—
99	C₈×Q₈	—	61	3	no	—
100	C₈×Q₈	#6923	1	1	yes	characteristic
101	C₂×C₄⋊C₈	—	61	3	no	—
102	C₂×C₄×C₈	—	61	3	no	—
103	C₄⋊C₁₆	—	61	6	no	—
104	C₄⋊C₁₆	—	61	6	no	—
105	C₂²⋊C₁₆	—	61	1	no	—
106	C₂²⋊C₁₆	—	61	3	no	—
107	#346	—	61	6	no	—
108	C₄×C₁₆	—	61	6	no	—
109	C₆₁	#33498	1	1	yes	characteristic, complemented, Sylow
110	C₂²×Q₈	—	61	1	no	—
111	C₂²×C₈	—	61	1	no	—
112	C₂²×C₈	—	61	3	no	—
113	C₄×Q₈	—	61	1	no	—
114	C₄×Q₈	—	61	3	no	—
115	C₄×Q₈	—	61	3	no	—
116	C₄×Q₈	C₂×#6923	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	61	3	no	—
118	C₂×C₄²	—	61	3	no	—
119	C₂×C₁₆	—	61	2	no	—
120	C₂×C₁₆	—	61	6	no	—
121	C₄⋊C₈	—	61	6	no	—
122	C₄⋊C₈	—	61	3	no	—
123	C₄⋊C₈	C₂×#6923	1	3	yes	—
124	C₄×C₈	—	61	3	no	—
125	C₄×C₈	—	61	3	no	—
126	C₄×C₈	—	61	3	no	—
127	C₄×C₈	C₂×#6923	1	3	yes	—
128	C₂×Q₈	—	61	3	no	—
129	C₂×Q₈	—	122	4	no	—
130	C₂×Q₈	D₆₁⋊C₈	1	1	yes	characteristic
131	C₂²×C₄	—	61	3	no	—
132	C₂²×C₄	—	61	3	no	—
133	C₂²×C₄	—	61	1	no	—
134	C₂×C₈	—	61	1	no	—
135	C₂×C₈	—	61	3	no	—
136	C₂×C₈	—	122	2	no	—
137	C₂×C₈	—	244	3	no	—
138	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
139	C₂×C₈	C₂²×#6923	1	3	yes	—
140	C₄⋊C₄	—	61	6	no	—
141	C₄⋊C₄	—	61	3	no	—
142	C₄⋊C₄	D₆₁⋊C₈	1	3	yes	—
143	C₄²	—	61	3	no	—
144	C₄²	—	61	3	no	—
145	C₄²	—	61	3	no	—
146	C₄²	C₂²×#6923	1	3	yes	—
147	C₁₆	—	122	2	no	—
148	C₁₆	—	122	6	no	—
149	C₂³	—	61	1	no	—
150	Q₈	—	122	6	no	—
151	Q₈	#14519852	1	4	yes	—
152	C₂×C₄	—	61	1	no	—
153	C₂×C₄	—	61	3	no	—
154	C₂×C₄	—	61	3	no	—
155	C₂×C₄	—	122	6	no	—
156	C₂×C₄	—	122	6	no	—
157	C₂×C₄	—	122	1	no	—
158	C₂×C₄	—	122	1	no	—
159	C₂×C₄	C₂×D₆₁⋊C₈	1	3	yes	—
160	C₂×C₄	C₂×D₆₁⋊C₈	1	3	yes	—
161	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
162	C₈	—	2	1	no	—
163	C₈	—	2	3	no	—
164	C₈	—	61	2	no	—
165	C₈	—	122	3	no	—
166	C₂²	—	61	2	no	—
167	C₂²	—	122	1	no	—
168	C₂²	—	122	1	no	—
169	C₂²	C₂²×D₆₁⋊C₈	1	1	yes	characteristic
170	C₄	—	2	3	no	—
171	C₄	—	122	3	no	—
172	C₄	—	122	1	no	—
173	C₄	—	122	3	no	—
174	C₄	C₂×#14519852	1	6	yes	—
175	C₄	#16840830	1	1	yes	characteristic
176	C₄	#16840841	1	1	yes	characteristic
177	C₂	—	122	2	no	—
178	C₂	#21462713	1	1	yes	characteristic
179	C₂	C₂²×#14519852	1	1	yes	characteristic
180	C₂	D₆₁⋊#504	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21462235

C₂

yes

maximal, complemented

Q₈×#11758187

C₂

yes

maximal, complemented

#21462183

C₂

yes

maximal

#21462178

C₂

yes

maximal

C₈×Q₈×D₆₁

C₂

yes

maximal, characteristic

#16840683

C₂²

yes

—

C₄⋊#11758187

C₂²

yes

—

#16840657

C₂²

yes

characteristic

#16840657

C₂²

yes

—

#16840651

C₂²

yes

—

C₄×#11758187

C₂²

yes

—

C₄×Q₈×D₆₁

C₄

yes

characteristic

Q₈×#92418

C₂²

yes

characteristic

C₈×Dic₁₂₂

C₄

yes

—

#16839886

C₂²

yes

—

C₄⋊C₈×D₆₁

C₄

yes

—

C₄×C₈×D₆₁

C₂²

yes

—

C₈×Q₈×C₆₁

C₄

yes

characteristic

C₂×#11758187

C₂³

yes

—

C₂×#11758187

C₂³

yes

—

C₂×Q₈×D₆₁

C₈

yes

characteristic

C₂×C₈×D₆₁

C₂×C₄

yes

—

C₂×C₈×D₆₁

C₂³

yes

characteristic

Q₈×Dic₆₁

C₈

yes

characteristic

C₄×Dic₁₂₂

C₂×C₄

yes

—

#14519742

C₈

yes

—

C₄⋊C₄×D₆₁

C₈

yes

—

C₄²×D₆₁

C₂×C₄

yes

—

C₄⋊#92418

C₂×C₄

yes

—

#14519706

C₂×C₄

yes

—

C₄×#92418

C₂³

yes

—

C₈×Dic₆₁

C₂×C₄

yes

—

C₈×Dic₆₁

C₂³

yes

—

C₄×Q₈×C₆₁

C₂×C₄

yes

characteristic

C₄⋊C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₂×Q₈×C₆₁

C₂×C₈

yes

characteristic

Q₈×D₆₁

—

C₂×C₄×D₆₁

C₂×C₈

yes

—

C₂×C₄×D₆₁

C₂×C₈

yes

—

C₂×C₄×D₆₁

C₂²×C₄

yes

characteristic

C₂×Dic₁₂₂

C₂×C₈

yes

—

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₁

C₂²×C₄

yes

—

C₄⋊C₄×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

C₂×C₈

yes

—

C₄×Dic₆₁

C₂²×C₄

yes

—

C₂×#92418

C₂²×C₄

yes

—

C₂×#92418

C₂⁴

yes

characteristic

C₈×D₆₁

—

C₈×D₆₁

—

#11758187

—

#11758187

—

C₂²×D₆₁

C₂²×C₈

yes

characteristic

Q₈×C₆₁

M₃₂

yes

—

C₂×C₄×C₆₁

C₂²×C₈

yes

—

C₂×C₄×C₆₁

C₂²×C₈

yes

—

C₂×C₄×C₆₁

C₂³×C₄

yes

characteristic

C₂×Dic₆₁

C₂²×C₈

yes

characteristic

C₂×Dic₆₁

C₂²×C₈

yes

—

C₂×Dic₆₁

C₂²×C₈

yes

—

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

#92418

—

#92418

2₋¹⁺⁴

yes

—

#33498

—

maximal, Sylow

C₂²×C₆₁

C₂³×C₈

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

#504

yes

—

C₄×C₆₁

—

C₄×C₆₁

C₂×M₃₂

yes

—

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

C₂×C₈×Q₈

—

#2183

—

Q₈×C₁₆

—

#2163

—

#2162

—

C₂×C₆₁

#2982

yes

characteristic

C₂×C₆₁

C₂²×M₃₂

yes

characteristic

C₂×C₆₁

C₂×#504

yes

characteristic

D₆₁

—

C₂×C₄×Q₈

—

C₈×Q₈

—

C₈×Q₈

—

C₈×Q₈

—

100

C₈×Q₈

#6923

yes

characteristic

101

C₂×C₄⋊C₈

—

102

C₂×C₄×C₈

—

103

C₄⋊C₁₆

—

104

C₄⋊C₁₆

—

105

C₂²⋊C₁₆

—

106

C₂²⋊C₁₆

—

107

#346

—

108

C₄×C₁₆

—

109

C₆₁

#33498

yes

characteristic, complemented, Sylow

110

C₂²×Q₈

—

111

C₂²×C₈

—

112

C₂²×C₈

—

113

C₄×Q₈

—

114

C₄×Q₈

—

115

C₄×Q₈

—

116

C₄×Q₈

C₂×#6923

yes

characteristic

117

C₂×C₄⋊C₄

—

118

C₂×C₄²

—

119

C₂×C₁₆

—

120

C₂×C₁₆

—

121

C₄⋊C₈

—

122

C₄⋊C₈

—

123

C₄⋊C₈

C₂×#6923

yes

—

124

C₄×C₈

—

125

C₄×C₈

—

126

C₄×C₈

—

127

C₄×C₈

C₂×#6923

yes

—

128

C₂×Q₈

—

129

C₂×Q₈

—

122

—

130

C₂×Q₈

D₆₁⋊C₈

yes

characteristic

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

122

—

137

C₂×C₈

—

244

—

138

C₂×C₈

C₂²×#6923

yes

characteristic

139

C₂×C₈

C₂²×#6923

yes

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄⋊C₄

D₆₁⋊C₈

yes

—

143

C₄²

—

144

C₄²

—

145

C₄²

—

146

C₄²

C₂²×#6923

yes

—

147

C₁₆

—

122

—

148

C₁₆

—

122

—

149

C₂³

—

150

Q₈

—

122

—

151

Q₈

#14519852

yes

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

122

—

156

C₂×C₄

—

122

—

157

C₂×C₄

—

122

—

158

C₂×C₄

—

122

—

159

C₂×C₄

C₂×D₆₁⋊C₈

yes

—

160

C₂×C₄

C₂×D₆₁⋊C₈

yes

—

161

C₂×C₄

C₂³×#6923

yes

characteristic

162

C₈

—

163

C₈

—

164

C₈

—

165

C₈

—

122

—

166

C₂²

—

167

C₂²

—

122

—

168

C₂²

—

122

—

169

C₂²

C₂²×D₆₁⋊C₈

yes

characteristic

170

C₄

—

171

C₄

—

122

—

172

C₄

—

122

—

173

C₄

—

122

—

174

C₄

C₂×#14519852

yes

—

175

C₄

#16840830

yes

characteristic

176

C₄

#16840841

yes

characteristic

177

C₂

—

122

—

178

C₂

#21462713

yes

characteristic

179

C₂

C₂²×#14519852

yes

characteristic

180

C₂

D₆₁⋊#504

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6923
commutator, socle	C₂²×C₆₁	C₂³×C₈
Fitting	C₈×Q₈×C₆₁	C₄
prime radical	C₂²×D₆₁	C₂²×C₈
cosocle	C₂×#92418	C₂⁴
hypercenter	C₈×Q₈	#6923
hypocenter	C₆₁	#33498
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×#6923

commutator, socle

C₂²×C₆₁

C₂³×C₈

Fitting

C₈×Q₈×C₆₁

C₄

prime radical

C₂²×D₆₁

C₂²×C₈

C₂⁴

C₈×Q₈

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂×C₄ ◁ C₈×Q₈

N	H ≅ G/N	type
1. #21462235	C₂	semidirect
2. Q₈×#11758187	C₂	semidirect
109. C₆₁	#33498	semidirect

H ≅ G/N

type

1. #21462235

C₂

semidirect

2. Q₈×#11758187

C₂

semidirect

109. C₆₁

#33498

semidirect

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

subgroup	index
#21462235	2
Q₈×#11758187	2
#21462183	2
#21462178	2
C₈×Q₈×D₆₁	2
#33498	61

subgroup

index

C₈×Q₈×D₆₁

Subgroups — #30624465

Conjugacy classes of subgroups

Complements

Maximal subgroups