Group #30594005 — subgroups

Subgroups — #30594005

Order 15616.

332 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 122 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	#21461752	C₂	1	2	yes	maximal
2	#21445970	C₂	1	1	yes	maximal, characteristic
3	C₂×#14519844	—	2	2	no	—
4	C₂×#14519844	C₄	1	4	yes	—
5	C₂×C₄⋊#92420	—	2	2	no	—
6	C₂×C₄⋊#92418	C₄	1	1	yes	characteristic
7	C₂×#14519706	—	2	2	no	—
8	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₈×C₆₁	C₄	1	1	yes	characteristic
10	C₂²×#92420	—	2	4	no	—
11	#14519844	—	4	4	no	—
12	#14519844	—	4	2	no	—
13	#14519844	—	4	2	no	—
14	C₄⋊#92420	—	4	2	no	—
15	C₄⋊#92420	—	4	2	no	—
16	C₂²×#92418	—	2	1	no	—
17	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
18	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
19	C₂×C₄×Dic₆₁	D₄	1	2	yes	—
20	C₂×C₄×Dic₆₁	Q₈	1	1	yes	characteristic
21	C₄⋊#92418	—	4	1	no	—
22	#14519706	—	4	2	no	—
23	#14519706	—	4	2	no	—
24	C₂²×C₈×C₆₁	—	2	1	no	—
25	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₄⋊C₈×C₆₁	—	4	1	no	—
27	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
28	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×Dic₆₁	C₂²⋊C₄	1	2	yes	—
31	C₂²×Dic₆₁	C₄⋊C₄	1	2	yes	—
32	C₂×#92420	—	4	4	no	—
33	C₂×#92420	—	4	4	no	—
34	C₂×#92420	—	4	4	no	—
35	C₂×C₈×C₆₁	—	4	1	no	—
36	C₂×C₈×C₆₁	—	4	2	no	—
37	C₄²×C₆₁	—	2	1	no	—
38	C₄²×C₆₁	—	2	1	no	—
39	C₄×Dic₆₁	—	2	2	no	—
40	C₄×Dic₆₁	—	2	2	no	—
41	C₄×Dic₆₁	—	4	1	no	—
42	C₄×Dic₆₁	D₈	1	1	yes	characteristic
43	C₄×Dic₆₁	D₈	1	2	yes	—
44	C₄×Dic₆₁	SD₁₆	1	2	yes	—
45	C₄×Dic₆₁	SD₁₆	1	2	yes	—
46	C₄×Dic₆₁	SD₁₆	1	2	yes	—
47	C₄×Dic₆₁	Q₁₆	1	1	yes	characteristic
48	C₄×Dic₆₁	Q₁₆	1	2	yes	—
49	C₂×#92418	—	4	1	no	—
50	C₂×#92418	—	4	2	no	—
51	C₂³×C₆₁	#95	1	1	yes	characteristic
52	C₂×C₄×C₆₁	—	2	1	no	—
53	C₂×C₄×C₆₁	—	2	1	no	—
54	C₂×C₄×C₆₁	—	2	2	no	—
55	C₂×C₄×C₆₁	—	2	1	no	—
56	C₂×C₄×C₆₁	#100	1	1	yes	characteristic
57	C₂×C₄×C₆₁	#101	1	1	yes	characteristic
58	C₂×C₄×C₆₁	#102	1	2	yes	—
59	C₂×C₄×C₆₁	#103	1	2	yes	—
60	C₂×C₄×C₆₁	#106	1	1	yes	characteristic
61	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
62	C₂×Dic₆₁	—	2	2	no	—
63	C₂×Dic₆₁	—	2	2	no	—
64	C₂×Dic₆₁	#100	1	2	yes	—
65	C₂×Dic₆₁	#101	1	2	yes	—
66	C₂×Dic₆₁	#102	1	2	yes	—
67	C₂×Dic₆₁	#102	1	2	yes	—
68	C₂×Dic₆₁	#103	1	2	yes	—
69	C₂×Dic₆₁	#103	1	2	yes	—
70	C₂×Dic₆₁	#106	1	2	yes	—
71	C₂×Dic₆₁	#107	1	2	yes	—
72	#92420	—	8	4	no	—
73	C₈×C₆₁	—	8	1	no	—
74	#92418	—	8	1	no	—
75	#7050	—	61	1	no	maximal, Sylow
76	C₂²×C₆₁	#338	1	1	yes	characteristic
77	C₂²×C₆₁	#340	1	2	yes	—
78	C₂²×C₆₁	#340	1	1	yes	characteristic
79	C₂²×C₆₁	#343	1	2	yes	—
80	C₂²×C₆₁	#343	1	1	yes	characteristic
81	C₄×C₆₁	—	2	1	no	—
82	C₄×C₆₁	—	2	1	no	—
83	C₄×C₆₁	—	4	1	no	—
84	C₄×C₆₁	#331	1	1	yes	characteristic
85	C₄×C₆₁	#332	1	2	yes	—
86	C₄×C₆₁	#333	1	1	yes	characteristic
87	Dic₆₁	—	2	2	no	—
88	Dic₆₁	—	2	2	no	—
89	Dic₆₁	#331	1	2	yes	—
90	Dic₆₁	#332	1	2	yes	—
91	Dic₆₁	#332	1	2	yes	—
92	Dic₆₁	#333	1	2	yes	—
93	#1842	—	61	2	no	—
94	#1842	—	61	1	no	—
95	C₂×C₆₁	#1292	1	2	yes	—
96	C₂×C₆₁	#1292	1	1	yes	characteristic
97	C₂×C₆₁	#1296	1	1	yes	characteristic
98	C₂×C₆₁	#1296	1	2	yes	—
99	C₂×C₆₁	#1309	1	1	yes	characteristic
100	C₂×C₄⋊C₈	—	61	4	no	—
101	C₂×C₄⋊C₈	—	61	1	no	—
102	C₂×C₄⋊C₈	—	122	2	no	—
103	C₂×C₄⋊C₈	—	122	2	no	—
104	C₂×C₄⋊C₈	—	122	2	no	—
105	C₂×C₄⋊C₈	#6923	1	1	yes	characteristic
106	C₄³	—	61	1	no	—
107	C₆₁	#7050	1	1	yes	characteristic, complemented, Sylow
108	C₂²×C₈	—	2	1	no	—
109	C₂²×C₈	—	122	4	no	—
110	C₂²×C₈	—	122	1	no	—
111	C₂×C₄²	—	61	1	no	—
112	C₂×C₄²	—	61	2	no	—
113	C₂×C₄²	—	61	2	no	—
114	C₂×C₄²	—	61	1	no	—
115	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
116	C₄⋊C₈	—	4	1	no	—
117	C₄⋊C₈	—	244	1	no	—
118	C₄⋊C₈	—	244	2	no	—
119	C₄⋊C₈	—	244	2	no	—
120	C₄⋊C₈	—	244	2	no	—
121	C₄⋊C₈	—	244	2	no	—
122	C₄⋊C₈	—	244	2	no	—
123	C₄⋊C₈	—	244	2	no	—
124	C₄⋊C₈	—	244	4	no	—
125	C₂²×C₄	—	61	2	no	—
126	C₂²×C₄	—	61	2	no	—
127	C₂²×C₄	C₄×#6923	1	1	yes	characteristic
128	C₂²×C₄	C₄⋊#6923	1	1	yes	characteristic
129	C₂²×C₄	C₂²⋊#6923	1	1	yes	characteristic
130	C₂×C₈	—	4	1	no	—
131	C₂×C₈	—	4	2	no	—
132	C₂×C₈	—	244	4	no	—
133	C₂×C₈	—	244	4	no	—
134	C₂×C₈	—	244	4	no	—
135	C₂×C₈	—	244	1	no	—
136	C₂×C₈	—	244	2	no	—
137	C₄²	—	2	1	no	—
138	C₄²	—	2	1	no	—
139	C₄²	—	61	2	no	—
140	C₄²	—	61	2	no	—
141	C₄²	—	61	2	no	—
142	C₄²	—	61	2	no	—
143	C₄²	—	61	2	no	—
144	C₄²	—	61	1	no	—
145	C₄²	—	61	1	no	—
146	C₄²	—	122	2	no	—
147	C₄²	—	122	2	no	—
148	C₄²	—	244	1	no	—
149	C₂³	(C₂×#6923)⋊C₄	1	1	yes	characteristic
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	61	2	no	—
155	C₂×C₄	—	61	2	no	—
156	C₂×C₄	—	61	2	no	—
157	C₂×C₄	—	61	2	no	—
158	C₂×C₄	—	61	2	no	—
159	C₂×C₄	—	61	2	no	—
160	C₂×C₄	—	61	2	no	—
161	C₂×C₄	—	61	2	no	—
162	C₂×C₄	—	122	2	no	—
163	C₂×C₄	—	122	2	no	—
164	C₂×C₄	#14519837	1	1	yes	characteristic
165	C₂×C₄	#14519838	1	1	yes	characteristic
166	C₂×C₄	#14519839	1	2	yes	—
167	C₂×C₄	#14519840	1	2	yes	—
168	C₂×C₄	#14519845	1	1	yes	characteristic
169	C₂×C₄	#14519846	1	1	yes	characteristic
170	C₈	—	8	1	no	—
171	C₈	—	488	4	no	—
172	C₈	—	488	1	no	—
173	C₂²	#16840631	1	1	yes	characteristic
174	C₂²	#16840635	1	2	yes	—
175	C₂²	#16840636	1	1	yes	characteristic
176	C₂²	#16840641	1	2	yes	—
177	C₂²	#16840642	1	1	yes	characteristic
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	—	4	1	no	—
181	C₄	—	61	2	no	—
182	C₄	—	61	2	no	—
183	C₄	—	61	2	no	—
184	C₄	—	61	2	no	—
185	C₄	—	122	2	no	—
186	C₄	—	122	2	no	—
187	C₄	#16840620	1	1	yes	characteristic
188	C₄	#16840621	1	2	yes	—
189	C₄	#16840622	1	1	yes	characteristic
190	C₂	#21460688	1	2	yes	—
191	C₂	#21460689	1	1	yes	characteristic
192	C₂	#21460697	1	1	yes	characteristic
193	C₂	#21460698	1	2	yes	—
194	C₂	#21460727	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21461752

C₂

yes

maximal

#21445970

C₂

yes

maximal, characteristic

C₂×#14519844

—

C₂×#14519844

C₄

yes

—

C₂×C₄⋊#92420

—

C₂×C₄⋊#92418

C₄

yes

characteristic

C₂×#14519706

—

C₄²×Dic₆₁

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₆₁

C₄

yes

characteristic

C₂²×#92420

—

—

—

—

—

—

C₂²×#92418

—

C₂×C₄×Dic₆₁

C₂×C₄

yes

—

C₂×C₄×Dic₆₁

D₄

yes

characteristic

C₂×C₄×Dic₆₁

D₄

yes

—

C₂×C₄×Dic₆₁

Q₈

yes

—

—

—

C₂²×C₈×C₆₁

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄⋊C₈×C₆₁

—

C₂²×C₄×C₆₁

C₄²

yes

characteristic

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂²×Dic₆₁

C₂²⋊C₄

yes

—

C₂²×Dic₆₁

C₄⋊C₄

yes

—

C₂×#92420

—

C₂×#92420

—

C₂×#92420

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

D₈

yes

characteristic

C₄×Dic₆₁

D₈

yes

—

C₄×Dic₆₁

SD₁₆

yes

—

C₄×Dic₆₁

SD₁₆

yes

—

C₄×Dic₆₁

SD₁₆

yes

—

C₄×Dic₆₁

Q₁₆

yes

characteristic

C₄×Dic₆₁

Q₁₆

yes

—

C₂×#92418

—

C₂×#92418

—

C₂³×C₆₁

#95

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

—

C₂×C₄×C₆₁

#103

yes

—

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

#100

yes

—

C₂×Dic₆₁

#101

yes

—

C₂×Dic₆₁

#102

yes

—

C₂×Dic₆₁

#102

yes

—

C₂×Dic₆₁

#103

yes

—

C₂×Dic₆₁

#103

yes

—

C₂×Dic₆₁

#106

yes

—

C₂×Dic₆₁

#107

yes

—

#92420

—

C₈×C₆₁

—

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

#340

yes

characteristic

C₂²×C₆₁

#343

yes

—

C₂²×C₆₁

#343

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#331

yes

characteristic

C₄×C₆₁

#332

yes

—

C₄×C₆₁

#333

yes

characteristic

Dic₆₁

—

Dic₆₁

—

Dic₆₁

#331

yes

—

Dic₆₁

#332

yes

—

Dic₆₁

#332

yes

—

Dic₆₁

#333

yes

—

#1842

—

#1842

—

C₂×C₆₁

#1292

yes

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×C₆₁

#1309

yes

characteristic

100

C₂×C₄⋊C₈

—

101

C₂×C₄⋊C₈

—

102

C₂×C₄⋊C₈

—

122

—

103

C₂×C₄⋊C₈

—

122

—

104

C₂×C₄⋊C₈

—

122

—

105

C₂×C₄⋊C₈

#6923

yes

characteristic

106

C₄³

—

107

C₆₁

#7050

yes

characteristic, complemented, Sylow

108

C₂²×C₈

—

109

C₂²×C₈

—

122

—

110

C₂²×C₈

—

122

—

111

C₂×C₄²

—

112

C₂×C₄²

—

113

C₂×C₄²

—

114

C₂×C₄²

—

115

C₂×C₄²

C₂×#6923

yes

characteristic

116

C₄⋊C₈

—

117

C₄⋊C₈

—

244

—

118

C₄⋊C₈

—

244

—

119

C₄⋊C₈

—

244

—

120

C₄⋊C₈

—

244

—

121

C₄⋊C₈

—

244

—

122

C₄⋊C₈

—

244

—

123

C₄⋊C₈

—

244

—

124

C₄⋊C₈

—

244

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

C₄×#6923

yes

characteristic

128

C₂²×C₄

C₄⋊#6923

yes

characteristic

129

C₂²×C₄

C₂²⋊#6923

yes

characteristic

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

244

—

133

C₂×C₈

—

244

—

134

C₂×C₈

—

244

—

135

C₂×C₈

—

244

—

136

C₂×C₈

—

244

—

137

C₄²

—

138

C₄²

—

139

C₄²

—

140

C₄²

—

141

C₄²

—

142

C₄²

—

143

C₄²

—

144

C₄²

—

145

C₄²

—

146

C₄²

—

122

—

147

C₄²

—

122

—

148

C₄²

—

244

—

149

C₂³

(C₂×#6923)⋊C₄

yes

characteristic

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

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

122

—

163

C₂×C₄

—

122

—

164

C₂×C₄

#14519837

yes

characteristic

165

C₂×C₄

#14519838

yes

characteristic

166

C₂×C₄

#14519839

yes

—

167

C₂×C₄

#14519840

yes

—

168

C₂×C₄

#14519845

yes

characteristic

169

C₂×C₄

#14519846

yes

characteristic

170

C₈

—

171

C₈

—

488

—

172

C₈

—

488

—

173

C₂²

#16840631

yes

characteristic

174

C₂²

#16840635

yes

—

175

C₂²

#16840636

yes

characteristic

176

C₂²

#16840641

yes

—

177

C₂²

#16840642

yes

characteristic

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

122

—

186

C₄

—

122

—

187

C₄

#16840620

yes

characteristic

188

C₄

#16840621

yes

—

189

C₄

#16840622

yes

characteristic

190

C₂

#21460688

yes

—

191

C₂

#21460689

yes

characteristic

192

C₂

#21460697

yes

characteristic

193

C₂

#21460698

yes

—

194

C₂

#21460727

yes

characteristic

	subgroup	G/H
center	C₂³	(C₂×#6923)⋊C₄
commutator	C₂²×C₄×C₆₁	C₄²
Frattini	C₂×C₄²	C₂×#6923
Fitting	C₂×C₄⋊C₈×C₆₁	C₄
socle, prime radical	C₂³×C₆₁	#95
cosocle	C₄²×Dic₆₁	C₂²
hypercenter	C₂×C₄⋊C₈	#6923
hypocenter	C₆₁	#7050
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

(C₂×#6923)⋊C₄

commutator

C₂²×C₄×C₆₁

C₄²

Frattini

C₂×C₄²

C₂×#6923

Fitting

C₂×C₄⋊C₈×C₆₁

C₄

socle, prime radical

C₂³×C₆₁

#95

cosocle

C₄²×Dic₆₁

C₂²

C₂×C₄⋊C₈

C₆₁

G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
107. C₆₁	#7050	semidirect

H ≅ G/N

type

107. C₆₁

#7050

semidirect

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

subgroup	index
#21461752	2
#21445970	2
#7050	61

subgroup

index

#21461752

#21445970

#7050

Subgroups — #30594005

Conjugacy classes of subgroups

Complements

Maximal subgroups