Group #29585026 — subgroups

Subgroups — #29585026

Order 15616.

354 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 99 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	#21445973	C₂	1	1	yes	maximal, characteristic, complemented
2	#21444591	C₂	1	2	yes	maximal, complemented
3	#21444218	C₂	1	2	yes	maximal
4	#21443648	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1963	C₂	1	1	yes	maximal, characteristic
6	#16839815	C₂²	1	2	yes	—
7	C₂×C₄⋊#92418	—	2	2	no	—
8	C₂×C₄⋊#92418	C₂²	1	2	yes	—
9	#16839512	—	2	4	no	—
10	D₄⋊#92418	—	2	4	no	—
11	C₆₁×#532	C₂²	1	1	yes	characteristic
12	C₆₁×#423	C₂²	1	1	yes	characteristic
13	C₂×#104×C₆₁	—	2	2	no	—
14	C₆₁×#376	C₂²	1	1	yes	characteristic
15	C₂²×#92418	—	2	2	no	—
16	C₄⋊#92418	—	2	2	no	—
17	C₄⋊#92418	—	2	2	no	—
18	C₄⋊#92418	—	2	2	no	—
19	C₄⋊#92418	—	4	4	no	—
20	C₂²⋊#92418	—	2	2	no	—
21	C₄×#92418	D₄	1	4	yes	—
22	C₂×#41×C₆₁	—	2	1	no	—
23	C₂×M₁₆×C₆₁	—	2	1	no	—
24	C₄⋊Q₈×C₆₁	Q₈	1	1	yes	characteristic
25	#127×C₆₁	Q₈	1	1	yes	characteristic
26	#124×C₆₁	—	2	1	no	—
27	#121×C₆₁	—	2	2	no	—
28	C₄×D₄×C₆₁	—	2	2	no	—
29	C₂×C₄²×C₆₁	D₄	1	2	yes	—
30	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₆₁	—	2	2	no	—
32	#104×C₆₁	—	4	4	no	—
33	#95×C₆₁	—	2	2	no	—
34	#41×C₆₁	—	4	2	no	—
35	#41×C₆₁	—	4	2	no	—
36	C₂×Q₈×C₆₁	—	2	1	no	—
37	C₂×D₄×C₆₁	—	2	2	no	—
38	C₂×D₄×C₆₁	—	2	1	no	—
39	C₂×D₄×C₆₁	—	4	1	no	—
40	C₂²×C₄×C₆₁	—	2	2	no	—
41	C₂²×C₄×C₆₁	—	4	1	no	—
42	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₆₁	#41	1	2	yes	—
44	M₁₆×C₆₁	—	4	2	no	—
45	C₂×C₈×C₆₁	—	4	1	no	—
46	C₄⋊C₄×C₆₁	—	2	2	no	—
47	C₂²⋊C₄×C₆₁	—	4	2	no	—
48	C₄²×C₆₁	—	2	2	no	—
49	C₄²×C₆₁	SD₁₆	1	4	yes	—
50	C₄²×C₆₁	C₂×D₄	1	2	yes	—
51	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
52	C₄²×C₆₁	#41	1	1	yes	characteristic
53	C₂×#92418	—	2	4	no	—
54	C₂×#92418	—	4	2	no	—
55	C₂×#92418	—	4	4	no	—
56	C₂³×C₆₁	—	4	1	no	—
57	C₂³×C₆₁	#123	1	1	yes	characteristic
58	Q₈×C₆₁	—	4	2	no	—
59	D₄×C₆₁	—	4	4	no	—
60	D₄×C₆₁	—	4	2	no	—
61	D₄×C₆₁	—	8	2	no	—
62	C₂×C₄×C₆₁	—	2	4	no	—
63	C₂×C₄×C₆₁	—	4	2	no	—
64	C₂×C₄×C₆₁	—	4	4	no	—
65	C₂×C₄×C₆₁	—	4	1	no	—
66	C₂×C₄×C₆₁	—	8	1	no	—
67	C₂×C₄×C₆₁	—	8	1	no	—
68	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
69	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₆₁	#123	1	2	yes	—
71	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
72	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
73	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
74	C₈×C₆₁	—	8	1	no	—
75	#92418	—	4	4	no	—
76	#92418	—	8	2	no	—
77	#10666	—	61	1	no	maximal, Sylow
78	C₂²×C₆₁	—	2	2	no	—
79	C₂²×C₆₁	—	8	1	no	—
80	C₂²×C₆₁	—	8	2	no	—
81	C₂²×C₆₁	#399	1	1	yes	characteristic
82	C₂²×C₆₁	#481	1	2	yes	—
83	C₄×C₆₁	—	2	2	no	—
84	C₄×C₆₁	—	2	2	no	—
85	C₄×C₆₁	—	4	4	no	—
86	C₄×C₆₁	—	8	1	no	—
87	C₄×C₆₁	#476	1	2	yes	—
88	C₄×C₆₁	#492	1	1	yes	characteristic
89	C₄×C₆₁	#496	1	1	yes	characteristic
90	#1963	D₆₁	1	1	yes	characteristic
91	#1843	—	61	1	no	—
92	#1570	—	61	2	no	—
93	#1480	—	61	2	no	—
94	#1274	—	61	1	no	—
95	C₂×C₆₁	—	2	2	no	—
96	C₂×C₆₁	—	16	1	no	—
97	C₂×C₆₁	#2094	1	1	yes	characteristic
98	C₂×C₆₁	#2098	1	2	yes	—
99	#532	C₂×D₆₁	1	1	yes	characteristic
100	#431	—	61	2	no	—
101	#423	C₂×D₆₁	1	1	yes	characteristic
102	C₂×C₄⋊C₈	—	61	2	no	—
103	C₂×C₄⋊C₈	—	122	2	no	—
104	C₂×#104	—	2	2	no	—
105	#376	C₂×D₆₁	1	1	yes	characteristic
106	#334	—	122	4	no	—
107	D₄⋊C₈	—	122	4	no	—
108	C₆₁	#10666	1	1	yes	characteristic, complemented, Sylow
109	C₂×#41	—	2	1	no	—
110	C₂×M₁₆	—	2	1	no	—
111	C₂²×C₈	—	122	2	no	—
112	C₄⋊Q₈	Dic₁₂₂	1	1	yes	characteristic
113	#127	Dic₁₂₂	1	1	yes	characteristic
114	#124	—	2	1	no	—
115	#121	—	2	2	no	—
116	C₄×D₄	—	2	2	no	—
117	C₂×C₄²	C₂²⋊D₆₁	1	2	yes	—
118	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	122	2	no	—
121	C₄⋊C₈	—	122	2	no	—
122	C₄⋊C₈	—	122	2	no	—
123	C₄⋊C₈	—	244	4	no	—
124	#104	—	4	4	no	—
125	C₂²⋊C₈	—	122	2	no	—
126	C₄×C₈	—	61	4	no	—
127	#95	—	2	2	no	—
128	#41	—	4	2	no	—
129	#41	—	4	2	no	—
130	C₂×Q₈	—	2	1	no	—
131	C₂×D₄	—	2	2	no	—
132	C₂×D₄	—	2	1	no	—
133	C₂×D₄	—	4	1	no	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	—	4	1	no	—
136	C₂²×C₄	#11758224	1	2	yes	—
137	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
138	M₁₆	—	4	2	no	—
139	C₂×C₈	—	4	1	no	—
140	C₂×C₈	—	122	4	no	—
141	C₂×C₈	—	244	2	no	—
142	C₂×C₈	—	244	4	no	—
143	C₄⋊C₄	—	2	2	no	—
144	C₂²⋊C₄	—	4	2	no	—
145	C₄²	—	2	2	no	—
146	C₄²	#11758200	1	4	yes	—
147	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
148	C₄²	#11758222	1	1	yes	characteristic
149	C₄²	D₄×D₆₁	1	2	yes	—
150	C₂³	—	4	1	no	—
151	C₂³	#14519758	1	1	yes	characteristic
152	Q₈	—	4	2	no	—
153	D₄	—	4	4	no	—
154	D₄	—	4	2	no	—
155	D₄	—	8	2	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	—	8	1	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	#14519754	1	2	yes	—
163	C₂×C₄	#14519756	1	1	yes	characteristic
164	C₂×C₄	#14519759	1	2	yes	—
165	C₂×C₄	#14519773	1	1	yes	characteristic
166	C₂×C₄	C₂×#11758200	1	2	yes	—
167	C₂×C₄	#14519802	1	2	yes	—
168	C₈	—	8	1	no	—
169	C₈	—	488	2	no	—
170	C₈	—	244	4	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	8	1	no	—
173	C₂²	—	8	2	no	—
174	C₂²	#16839698	1	1	yes	characteristic
175	C₂²	#16840078	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	4	4	no	—
179	C₄	—	8	1	no	—
180	C₄	#16840052	1	2	yes	—
181	C₄	#16840134	1	1	yes	characteristic
182	C₄	#16840152	1	1	yes	characteristic
183	C₂	—	2	2	no	—
184	C₂	—	16	1	no	—
185	C₂	#21447352	1	1	yes	characteristic
186	C₂	#21447374	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21445973

C₂

yes

maximal, characteristic, complemented

#21444591

C₂

yes

maximal, complemented

#21444218

C₂

yes

maximal

#21443648

C₂

yes

maximal, characteristic, complemented

C₆₁×#1963

C₂

yes

maximal, characteristic

#16839815

C₂²

yes

—

C₂×C₄⋊#92418

—

C₂×C₄⋊#92418

C₂²

yes

—

#16839512

—

D₄⋊#92418

—

C₆₁×#532

C₂²

yes

characteristic

C₆₁×#423

C₂²

yes

characteristic

C₂×#104×C₆₁

—

C₆₁×#376

C₂²

yes

characteristic

C₂²×#92418

—

—

—

—

—

—

D₄

yes

—

C₂×#41×C₆₁

—

C₂×M₁₆×C₆₁

—

C₄⋊Q₈×C₆₁

Q₈

yes

characteristic

#127×C₆₁

Q₈

yes

characteristic

#124×C₆₁

—

#121×C₆₁

—

C₄×D₄×C₆₁

—

C₂×C₄²×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊C₈×C₆₁

—

#104×C₆₁

—

#95×C₆₁

—

#41×C₆₁

—

#41×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

#41

yes

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₄⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

SD₁₆

yes

—

C₄²×C₆₁

C₂×D₄

yes

—

C₄²×C₆₁

C₂×Q₈

yes

C₄²×C₆₁

yes

—

—

—

C₂³×C₆₁

—

C₂³×C₆₁

#123

yes

characteristic

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₁

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₆₁

#123

yes

—

C₂×C₄×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

C₂×SD₁₆

yes

—

C₂×C₄×C₆₁

AGL(1,ℤ/8)

yes

—

C₈×C₆₁

—

—

—

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#399

yes

characteristic

C₂²×C₆₁

#481

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#476

yes

—

C₄×C₆₁

yes

C₄×C₆₁

yes

D₆₁

yes

—

—

—

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#2094

yes

characteristic

C₂×C₆₁

#2098

yes

—

#532

C₂×D₆₁

yes

characteristic

100

#431

—

101

#423

C₂×D₆₁

yes

characteristic

102

C₂×C₄⋊C₈

—

103

C₂×C₄⋊C₈

—

122

—

104

C₂×#104

—

105

#376

C₂×D₆₁

yes

characteristic

106

#334

—

122

—

107

D₄⋊C₈

—

122

—

108

C₆₁

#10666

yes

characteristic, complemented, Sylow

109

C₂×#41

—

110

C₂×M₁₆

—

111

C₂²×C₈

—

122

—

112

C₄⋊Q₈

Dic₁₂₂

yes

characteristic

113

#127

Dic₁₂₂

yes

characteristic

114

#124

—

115

#121

—

116

C₄×D₄

—

117

C₂×C₄²

C₂²⋊D₆₁

yes

—

118

C₂×C₄²

C₂²×D₆₁

yes

characteristic

119

C₄⋊C₈

—

120

C₄⋊C₈

—

122

—

121

C₄⋊C₈

—

122

—

122

C₄⋊C₈

—

122

—

123

C₄⋊C₈

—

244

—

124

#104

—

125

C₂²⋊C₈

—

122

—

126

C₄×C₈

—

127

#95

—

128

#41

—

129

#41

—

130

C₂×Q₈

—

131

C₂×D₄

—

132

C₂×D₄

—

133

C₂×D₄

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

#11758224

yes

—

137

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

138

M₁₆

—

139

C₂×C₈

—

140

C₂×C₈

—

122

—

141

C₂×C₈

—

244

—

142

C₂×C₈

—

244

—

143

C₄⋊C₄

—

144

C₂²⋊C₄

—

145

C₄²

—

146

C₄²

#11758200

yes

—

147

C₄²

C₂×Dic₁₂₂

yes

characteristic

148

C₄²

#11758222

yes

characteristic

149

C₄²

D₄×D₆₁

yes

—

150

C₂³

—

151

C₂³

#14519758

yes

characteristic

152

Q₈

—

153

D₄

—

154

D₄

—

155

D₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

#14519754

yes

—

163

C₂×C₄

#14519756

yes

characteristic

164

C₂×C₄

#14519759

yes

—

165

C₂×C₄

#14519773

yes

characteristic

166

C₂×C₄

C₂×#11758200

yes

—

167

C₂×C₄

#14519802

yes

—

168

C₈

—

169

C₈

—

488

—

170

C₈

—

244

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

#16839698

yes

characteristic

175

C₂²

#16840078

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

#16840052

yes

—

181

C₄

#16840134

yes

characteristic

182

C₄

#16840152

yes

characteristic

183

C₂

—

184

C₂

—

185

C₂

#21447352

yes

characteristic

186

C₂

#21447374

yes

—

	subgroup	G/H
center	C₂×C₄	#14519773
commutator, cosocle	C₂×C₄²×C₆₁	C₂³
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1963	C₂
socle	C₂²×C₆₁	#399
prime radical	C₆₁×#532	C₂²
hypercenter	#1963	D₆₁
hypocenter	C₆₁	#10666
LCS	G ▷ C₂×C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1963

subgroup

G/H

center

C₂×C₄

#14519773

commutator, cosocle

C₂×C₄²×C₆₁

C₂³

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#1963

C₂

socle

C₂²×C₆₁

#399

prime radical

C₆₁×#532

C₂²

D₆₁

C₆₁

G ▷ C₂×C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1963

N	H ≅ G/N	type
1. #21445973	C₂	semidirect
2. #21444591	C₂	semidirect
4. #21443648	C₂	semidirect
108. C₆₁	#10666	semidirect

H ≅ G/N

type

1. #21445973

C₂

semidirect

2. #21444591

C₂

semidirect

4. #21443648

C₂

semidirect

108. C₆₁

#10666

semidirect

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

subgroup	index
#21445973	2
#21444591	2
#21444218	2
#21443648	2
C₆₁×#1963	2
#10666	61

subgroup

index

C₆₁×#1963

Subgroups — #29585026

Conjugacy classes of subgroups

Complements

Maximal subgroups