Group #30607380 — subgroups

Subgroups — #30607380

Order 15616.

374 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 112 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	#21462786	C₂	1	1	yes	maximal, characteristic
2	#21462156	C₂	1	2	yes	maximal, complemented
3	#21462049	C₂	1	1	yes	maximal, characteristic
4	#21460870	C₂	1	2	yes	maximal, complemented
5	#21445622	C₂	1	1	yes	maximal, characteristic
6	#16840820	—	2	4	no	—
7	C₂×#14519851	—	2	2	no	—
8	C₂×#14519851	C₂²	1	1	yes	characteristic
9	C₂×#14519851	C₂²	1	2	yes	—
10	C₂²×#11758187	C₂²	1	1	yes	characteristic
11	#16840683	—	2	4	no	—
12	C₂²⋊#11758187	—	2	2	no	—
13	C₂×#14519784	C₂²	1	1	yes	characteristic
14	#16839736	C₄	1	1	yes	characteristic
15	#16839720	C₂²	1	2	yes	—
16	#16839534	C₄	1	2	yes	—
17	C₆₁×#413	C₄	1	1	yes	characteristic
18	#14519851	—	2	4	no	—
19	#14519851	—	2	2	no	—
20	#14519851	—	2	2	no	—
21	#14519851	—	2	2	no	—
22	C₂×#11758187	—	2	4	no	—
23	C₂×#11758187	—	2	1	no	—
24	C₂×#11758187	—	2	2	no	—
25	C₂×#11758187	—	2	1	no	—
26	C₂×#11758187	D₄	1	4	yes	—
27	#14519784	—	2	2	no	—
28	#14519784	—	2	2	no	—
29	C₂×#11758194	C₂×C₄	1	1	yes	characteristic
30	C₂×#11758194	Q₈	1	2	yes	—
31	C₂²×#92418	D₄	1	2	yes	—
32	C₂²×#92418	C₂³	1	1	yes	characteristic
33	#14519736	C₂×C₄	1	2	yes	—
34	#14519692	C₈	1	4	yes	—
35	#14519687	—	2	2	no	—
36	C₈×Dic₆₁	—	2	2	no	—
37	C₂×#41×C₆₁	C₂×C₄	1	1	yes	characteristic
38	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
39	#101×C₆₁	—	2	1	no	—
40	#100×C₆₁	—	2	1	no	—
41	#41×C₆₁	—	2	2	no	—
42	#41×C₆₁	—	2	2	no	—
43	C₂×Q₈×C₆₁	M₁₆	1	1	yes	characteristic
44	C₂×D₄×C₆₁	C₂×C₈	1	2	yes	—
45	C₂×D₄×C₆₁	M₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
48	M₁₆×C₆₁	—	2	2	no	—
49	M₁₆×C₆₁	—	2	2	no	—
50	C₂×C₈×C₆₁	—	2	2	no	—
51	C₂²⋊Dic₆₁	C₂×C₈	1	4	yes	—
52	C₄⋊Dic₆₁	—	2	2	no	—
53	C₄×Dic₆₁	—	2	2	no	—
54	#11758194	—	2	2	no	—
55	#11758194	—	2	4	no	—
56	C₂×#92418	—	2	2	no	—
57	C₂×#92418	—	2	2	no	—
58	C₂×#92418	C₄⋊C₄	1	2	yes	—
59	C₂×#92418	C₂×D₄	1	2	yes	—
60	C₂×#92418	C₂×D₄	1	1	yes	characteristic
61	C₂×#92418	C₂×Q₈	1	1	yes	characteristic
62	C₂×#92418	#41	1	2	yes	—
63	#11758187	—	2	4	no	—
64	#11758187	—	4	1	no	—
65	#11758187	—	4	1	no	—
66	C₂³×C₆₁	C₄⋊C₈	1	2	yes	—
67	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
68	Q₈×C₆₁	—	2	2	no	—
69	D₄×C₆₁	—	2	4	no	—
70	D₄×C₆₁	—	2	2	no	—
71	C₂×C₄×C₆₁	—	2	2	no	—
72	C₂×C₄×C₆₁	—	2	2	no	—
73	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
74	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
76	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₆₁	#131	1	2	yes	—
78	C₂×Dic₆₁	—	2	4	no	—
79	C₈×C₆₁	—	4	2	no	—
80	#92418	—	2	2	no	—
81	#92418	#120	1	1	yes	characteristic
82	#92418	C₂²⋊Q₈	1	2	yes	—
83	#92418	#123	1	1	yes	characteristic
84	#12915	—	61	1	no	maximal, Sylow
85	C₂²×C₆₁	—	2	2	no	—
86	C₂²×C₆₁	—	4	1	no	—
87	C₂²×C₆₁	—	4	2	no	—
88	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
89	C₂²×C₆₁	C₈×D₄	1	2	yes	—
90	C₄×C₆₁	—	2	2	no	—
91	C₄×C₆₁	#385	1	1	yes	characteristic
92	C₄×C₆₁	#424	1	1	yes	characteristic
93	C₄×C₆₁	#435	1	2	yes	—
94	Dic₆₁	—	4	4	no	—
95	#3407	—	61	1	no	—
96	#2151	—	61	2	no	—
97	#2113	—	61	1	no	—
98	#1757	—	61	1	no	—
99	#1378	—	61	2	no	—
100	C₂×C₆₁	—	2	2	no	—
101	C₂×C₆₁	—	2	1	no	—
102	C₂×C₆₁	—	4	2	no	—
103	C₂×C₆₁	#1850	1	1	yes	characteristic
104	C₂×#131	—	61	1	no	—
105	#504	—	122	4	no	—
106	C₂×M₃₂	—	61	2	no	—
107	C₂×M₃₂	—	61	1	no	—
108	C₂×M₃₂	—	122	2	no	—
109	C₂²×C₁₆	—	61	1	no	—
110	#413	#6923	1	1	yes	characteristic
111	#410	—	61	1	no	—
112	#405	—	61	2	no	—
113	C₄⋊C₁₆	—	122	4	no	—
114	C₂²⋊C₁₆	—	122	2	no	—
115	#344	—	61	2	no	—
116	C₆₁	#12915	1	1	yes	characteristic, complemented, Sylow
117	C₂×#41	C₂×#6923	1	1	yes	characteristic
118	#131	—	122	2	no	—
119	#131	—	122	2	no	—
120	C₂×M₁₆	—	61	2	no	—
121	C₂×M₁₆	—	61	1	no	—
122	C₂×M₁₆	C₂×#6923	1	2	yes	—
123	C₂²×C₈	—	61	1	no	—
124	C₂²×C₈	—	61	2	no	—
125	#117	—	61	2	no	—
126	M₃₂	—	122	2	no	—
127	M₃₂	—	122	2	no	—
128	M₃₂	—	122	4	no	—
129	M₃₂	—	122	2	no	—
130	C₂×C₁₆	—	61	4	no	—
131	C₂×C₁₆	—	122	4	no	—
132	C₂×C₁₆	—	122	1	no	—
133	C₂×C₁₆	—	122	1	no	—
134	C₂×C₁₆	—	122	2	no	—
135	#101	—	2	1	no	—
136	#100	—	2	1	no	—
137	#99	—	61	4	no	—
138	#97	—	122	2	no	—
139	C₄×C₈	—	122	2	no	—
140	#41	—	2	2	no	—
141	#41	—	2	2	no	—
142	C₂×Q₈	#11758217	1	1	yes	characteristic
143	C₂×D₄	D₆₁⋊C₈	1	2	yes	—
144	C₂×D₄	#11758217	1	1	yes	characteristic
145	C₂²×C₄	C₄⋊#6923	1	2	yes	—
146	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
147	M₁₆	—	2	2	no	—
148	M₁₆	—	2	2	no	—
149	M₁₆	—	122	4	no	—
150	M₁₆	—	122	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	61	1	no	—
153	C₂×C₈	—	61	1	no	—
154	C₂×C₈	—	61	2	no	—
155	C₂×C₈	—	61	2	no	—
156	C₂×C₈	—	61	2	no	—
157	C₂×C₈	—	122	2	no	—
158	C₂×C₈	—	122	2	no	—
159	C₄⋊C₄	—	122	2	no	—
160	C₂²⋊C₄	—	61	4	no	—
161	C₄²	—	122	2	no	—
162	C₁₆	—	122	4	no	—
163	C₁₆	—	244	1	no	—
164	C₁₆	—	244	1	no	—
165	C₂³	#14519844	1	2	yes	—
166	C₂³	C₂×D₆₁⋊C₈	1	1	yes	characteristic
167	Q₈	—	2	2	no	—
168	D₄	—	2	4	no	—
169	D₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	122	4	no	—
173	C₂×C₄	#14519844	1	2	yes	—
174	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
175	C₂×C₄	D₄×#6923	1	2	yes	—
176	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
177	C₂×C₄	#14519864	1	2	yes	—
178	C₈	—	4	2	no	—
179	C₈	—	61	1	no	—
180	C₈	—	61	2	no	—
181	C₈	—	61	1	no	—
182	C₈	—	122	2	no	—
183	C₂²	—	2	2	no	—
184	C₂²	—	4	1	no	—
185	C₂²	—	4	2	no	—
186	C₂²	C₂×#14519844	1	1	yes	characteristic
187	C₂²	#16840789	1	2	yes	—
188	C₄	—	2	2	no	—
189	C₄	—	244	4	no	—
190	C₄	#16840709	1	1	yes	characteristic
191	C₄	#16840768	1	1	yes	characteristic
192	C₄	#16840793	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	—	2	1	no	—
195	C₂	—	4	2	no	—
196	C₂	#21461763	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21462786

C₂

yes

maximal, characteristic

#21462156

C₂

yes

maximal, complemented

#21462049

C₂

yes

maximal, characteristic

#21460870

C₂

yes

maximal, complemented

#21445622

C₂

yes

maximal, characteristic

#16840820

—

C₂×#14519851

—

C₂×#14519851

C₂²

yes

characteristic

C₂×#14519851

C₂²

yes

—

C₂²×#11758187

C₂²

yes

—

—

C₂²

yes

characteristic

#16839736

C₄

yes

characteristic

#16839720

C₂²

yes

—

#16839534

C₄

yes

—

C₆₁×#413

C₄

yes

—

—

—

—

—

—

—

—

D₄

yes

—

#14519784

—

#14519784

—

C₂×#11758194

C₂×C₄

yes

characteristic

C₂×#11758194

Q₈

yes

—

C₂²×#92418

D₄

yes

—

C₂²×#92418

C₂³

yes

characteristic

#14519736

C₂×C₄

yes

—

#14519692

C₈

yes

—

#14519687

—

C₈×Dic₆₁

—

C₂×#41×C₆₁

C₂×C₄

yes

characteristic

C₂×M₁₆×C₆₁

C₂×C₄

yes

—

#101×C₆₁

—

#100×C₆₁

—

#41×C₆₁

—

#41×C₆₁

—

C₂×Q₈×C₆₁

M₁₆

yes

characteristic

C₂×D₄×C₆₁

C₂×C₈

yes

—

C₂×D₄×C₆₁

M₁₆

yes

characteristic

C₂²×C₄×C₆₁

C₄⋊C₄

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

M₁₆×C₆₁

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂²⋊Dic₆₁

C₂×C₈

yes

—

C₄⋊Dic₆₁

—

C₄×Dic₆₁

—

—

—

—

—

C₄⋊C₄

yes

—

C₂×#92418

C₂×D₄

yes

—

C₂×#92418

C₂×D₄

yes

characteristic

C₂×#92418

C₂×Q₈

yes

characteristic

C₂×#92418

#41

yes

—

#11758187

—

#11758187

—

#11758187

—

C₂³×C₆₁

C₄⋊C₈

yes

—

C₂³×C₆₁

C₂²×C₈

yes

characteristic

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄⋊C₈

yes

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄×D₄

yes

—

C₂×C₄×C₆₁

C₂×M₁₆

yes

characteristic

C₂×C₄×C₆₁

#131

yes

—

C₂×Dic₆₁

—

C₈×C₆₁

—

—

yes

C₂²⋊Q₈

yes

—

yes

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₆₁

C₈×D₄

yes

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

Dic₆₁

—

—

—

—

—

—

100

C₂×C₆₁

—

101

C₂×C₆₁

—

102

C₂×C₆₁

—

103

C₂×C₆₁

#1850

yes

characteristic

104

C₂×#131

—

105

#504

—

122

—

106

C₂×M₃₂

—

107

C₂×M₃₂

—

108

C₂×M₃₂

—

122

—

109

C₂²×C₁₆

—

110

yes

111

—

112

#405

—

113

C₄⋊C₁₆

—

122

—

114

C₂²⋊C₁₆

—

122

—

115

#344

—

116

C₆₁

#12915

yes

characteristic, complemented, Sylow

117

C₂×#41

C₂×#6923

yes

characteristic

118

#131

—

122

—

119

#131

—

122

—

120

C₂×M₁₆

—

121

C₂×M₁₆

—

122

C₂×M₁₆

C₂×#6923

yes

—

123

C₂²×C₈

—

124

C₂²×C₈

—

125

#117

—

126

M₃₂

—

122

—

127

M₃₂

—

122

—

128

M₃₂

—

122

—

129

M₃₂

—

122

—

130

C₂×C₁₆

—

131

C₂×C₁₆

—

122

—

132

C₂×C₁₆

—

122

—

133

C₂×C₁₆

—

122

—

134

C₂×C₁₆

—

122

—

135

#101

—

136

#100

—

137

#99

—

138

#97

—

122

—

139

C₄×C₈

—

122

—

140

#41

—

141

#41

—

142

C₂×Q₈

#11758217

yes

characteristic

143

C₂×D₄

D₆₁⋊C₈

yes

—

144

C₂×D₄

#11758217

yes

characteristic

145

C₂²×C₄

C₄⋊#6923

yes

—

146

C₂²×C₄

C₂²×#6923

yes

characteristic

147

M₁₆

—

148

M₁₆

—

149

M₁₆

—

122

—

150

M₁₆

—

122

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

122

—

158

C₂×C₈

—

122

—

159

C₄⋊C₄

—

122

—

160

C₂²⋊C₄

—

161

C₄²

—

122

—

162

C₁₆

—

122

—

163

C₁₆

—

244

—

164

C₁₆

—

244

—

165

C₂³

#14519844

yes

—

166

C₂³

C₂×D₆₁⋊C₈

yes

characteristic

167

Q₈

—

168

D₄

—

169

D₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

122

—

173

C₂×C₄

#14519844

yes

—

174

C₂×C₄

C₂×C₄⋊#6923

yes

characteristic

175

C₂×C₄

D₄×#6923

yes

—

176

C₂×C₄

C₂×#11758217

yes

characteristic

177

C₂×C₄

#14519864

yes

—

178

C₈

—

179

C₈

—

180

C₈

—

181

C₈

—

182

C₈

—

122

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

C₂×#14519844

yes

characteristic

187

C₂²

#16840789

yes

—

188

C₄

—

189

C₄

—

244

—

190

C₄

#16840709

yes

characteristic

191

C₄

#16840768

yes

characteristic

192

C₄

#16840793

yes

—

193

C₂

—

194

C₂

—

195

C₂

—

196

C₂

#21461763

yes

characteristic

	subgroup	G/H
center	C₄	#16840709
commutator	C₂³×C₆₁	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₆₁×#413	C₄
socle	C₂×C₆₁	#1850
prime radical	C₂×#41×C₆₁	C₂×C₄
cosocle	C₂²×#92418	C₂³
hypercenter	#413	#6923
hypocenter	C₆₁	#12915
LCS	G ▷ C₂³×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

subgroup

G/H

center

C₄

#16840709

commutator

C₂³×C₆₁

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#6923

Fitting

C₆₁×#413

C₄

socle

C₂×C₆₁

#1850

prime radical

C₂×#41×C₆₁

C₂×C₄

cosocle

C₂²×#92418

C₂³

C₆₁

G ▷ C₂³×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

N	H ≅ G/N	type
2. #21462156	C₂	semidirect
4. #21460870	C₂	semidirect
116. C₆₁	#12915	semidirect

H ≅ G/N

type

2. #21462156

C₂

semidirect

4. #21460870

C₂

semidirect

116. C₆₁

#12915

semidirect

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

subgroup	index
#21462786	2
#21462156	2
#21462049	2
#21460870	2
#21445622	2
#12915	61

subgroup

index

Subgroups — #30607380

Conjugacy classes of subgroups

Complements

Maximal subgroups