Group #29579410 — subgroups

Subgroups — #29579410

Order 15616.

478 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 119 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	#21445950	C₂	1	1	yes	maximal, characteristic, complemented
2	#21444379	C₂	1	4	yes	maximal, complemented
3	#21444111	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#3147	C₂	1	1	yes	maximal, characteristic
5	#16839815	C₂²	1	2	yes	—
6	#16839628	C₂²	1	2	yes	—
7	#16839525	—	2	2	no	—
8	#16839496	—	2	8	no	—
9	#16839484	—	2	2	no	—
10	C₂×#135×C₆₁	—	2	2	no	—
11	C₆₁×#532	C₂²	1	2	yes	—
12	C₆₁×#495	—	2	2	no	—
13	C₆₁×#494	C₄	1	1	yes	characteristic, complemented
14	C₆₁×#493	C₄	1	1	yes	characteristic, complemented
15	C₆₁×#492	C₄	1	2	yes	complemented
16	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₆₁	—	2	2	no	—
18	C₄⋊#92418	—	2	2	no	—
19	C₂²⋊#92418	D₄	1	4	yes	—
20	C₄×#92418	—	2	2	no	—
21	#14519684	—	2	4	no	—
22	C₂×#41×C₆₁	—	2	2	no	—
23	#135×C₆₁	—	4	8	no	—
24	C₂×Q₁₆×C₆₁	—	2	2	no	—
25	C₂×SD₁₆×C₆₁	—	2	4	no	—
26	C₂×D₈×C₆₁	—	2	2	no	—
27	C₂²×C₈×C₆₁	D₄	1	2	yes	—
28	C₄⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
29	#127×C₆₁	C₂×C₄	1	2	yes	—
30	#124×C₆₁	—	2	2	no	—
31	#121×C₆₁	—	2	4	no	—
32	C₄×D₄×C₆₁	—	2	4	no	—
33	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
34	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₆₁	D₄	1	2	yes	—
37	#41×C₆₁	—	4	8	no	—
38	C₂×Q₈×C₆₁	—	2	2	no	—
39	C₂×D₄×C₆₁	—	2	2	no	—
40	C₂×D₄×C₆₁	—	2	4	no	—
41	C₂×D₄×C₆₁	—	4	2	no	—
42	C₂²×C₄×C₆₁	—	4	2	no	—
43	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₆₁	—	4	2	no	—
46	Q₁₆×C₆₁	—	4	4	no	—
47	SD₁₆×C₆₁	—	4	8	no	—
48	D₈×C₆₁	—	4	4	no	—
49	C₂×C₈×C₆₁	—	2	4	no	—
50	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
52	C₄⋊C₄×C₆₁	—	2	4	no	—
53	C₂²⋊C₄×C₆₁	—	4	4	no	—
54	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
55	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₆₁	#41	1	2	yes	—
57	C₄⋊Dic₆₁	—	4	4	no	—
58	C₂×#92418	—	2	4	no	—
59	C₂³×C₆₁	—	4	2	no	—
60	C₂³×C₆₁	#127	1	1	yes	characteristic
61	Q₈×C₆₁	—	4	4	no	—
62	D₄×C₆₁	—	4	4	no	—
63	D₄×C₆₁	—	8	4	no	—
64	D₄×C₆₁	—	8	4	no	—
65	C₂×C₄×C₆₁	—	4	2	no	—
66	C₂×C₄×C₆₁	—	4	2	no	—
67	C₂×C₄×C₆₁	—	8	4	no	—
68	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
70	C₂×C₄×C₆₁	#121	1	2	yes	—
71	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
72	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	4	yes	—
73	C₂×Dic₆₁	—	8	4	no	—
74	C₂×Dic₆₁	—	8	2	no	—
75	C₈×C₆₁	—	2	2	no	—
76	C₈×C₆₁	—	2	2	no	—
77	#92418	—	4	4	no	—
78	#9760	—	61	1	no	maximal, Sylow
79	C₂²×C₆₁	—	2	2	no	—
80	C₂²×C₆₁	—	8	2	no	—
81	C₂²×C₆₁	—	8	4	no	—
82	C₂²×C₆₁	#390	1	1	yes	characteristic
83	C₂²×C₆₁	#496	1	2	yes	—
84	C₄×C₆₁	—	2	2	no	—
85	C₄×C₆₁	—	2	2	no	—
86	C₄×C₆₁	—	8	2	no	—
87	C₄×C₆₁	#418	1	2	yes	—
88	C₄×C₆₁	#490	1	2	yes	—
89	Dic₆₁	—	16	2	no	—
90	#3147	D₆₁	1	1	yes	characteristic
91	#1835	—	61	1	no	—
92	#1517	—	61	4	no	—
93	#1452	—	61	1	no	—
94	C₂×C₆₁	—	2	2	no	—
95	C₂×C₆₁	—	16	2	no	—
96	C₂×C₆₁	#1961	1	1	yes	characteristic
97	C₂×C₆₁	#1976	1	2	yes	—
98	C₂×#135	—	2	2	no	—
99	#532	C₂×D₆₁	1	2	yes	—
100	#495	—	2	2	no	—
101	#494	Dic₆₁	1	1	yes	characteristic, complemented
102	#493	Dic₆₁	1	1	yes	characteristic, complemented
103	#492	Dic₆₁	1	2	yes	complemented
104	#431	—	61	2	no	—
105	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
106	#382	—	61	2	no	—
107	#340	—	122	2	no	—
108	#327	—	122	8	no	—
109	#322	—	122	2	no	—
110	C₆₁	#9760	1	1	yes	characteristic, complemented, Sylow
111	C₂×#41	—	2	2	no	—
112	#135	—	4	8	no	—
113	C₂×Q₁₆	—	2	2	no	—
114	C₂×SD₁₆	—	2	4	no	—
115	C₂×D₈	—	2	2	no	—
116	C₂²×C₈	C₂²⋊D₆₁	1	2	yes	—
117	C₄⋊Q₈	C₂×Dic₆₁	1	2	yes	—
118	#127	C₂×Dic₆₁	1	2	yes	—
119	#124	—	2	2	no	—
120	#121	—	2	4	no	—
121	C₄×D₄	—	2	4	no	—
122	C₂×C₄⋊C₄	—	122	2	no	—
123	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
124	C₄⋊C₈	—	122	2	no	—
125	C₂²⋊C₈	—	61	4	no	—
126	C₄×C₈	—	122	2	no	—
127	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
128	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
129	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
130	#95	—	122	4	no	—
131	#41	—	4	8	no	—
132	C₂×Q₈	—	2	2	no	—
133	C₂×D₄	—	2	2	no	—
134	C₂×D₄	—	2	4	no	—
135	C₂×D₄	—	4	2	no	—
136	C₂²×C₄	—	4	2	no	—
137	C₂²×C₄	—	244	2	no	—
138	C₂²×C₄	D₄×D₆₁	1	2	yes	—
139	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
140	Q₁₆	—	4	4	no	—
141	SD₁₆	—	4	8	no	—
142	D₈	—	4	4	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	—	122	4	no	—
145	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
146	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
147	C₄⋊C₄	—	2	4	no	—
148	C₄⋊C₄	—	244	4	no	—
149	C₂²⋊C₄	—	4	4	no	—
150	C₄²	#11758224	1	2	yes	—
151	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
152	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
153	C₂³	—	4	2	no	—
154	C₂³	#14519773	1	1	yes	characteristic
155	Q₈	—	4	4	no	—
156	D₄	—	4	4	no	—
157	D₄	—	8	4	no	—
158	D₄	—	8	4	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	8	4	no	—
162	C₂×C₄	—	488	4	no	—
163	C₂×C₄	—	488	2	no	—
164	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
165	C₂×C₄	D₄×Dic₆₁	1	2	yes	—
166	C₂×C₄	#14519748	1	2	yes	—
167	C₂×C₄	#14519764	1	1	yes	characteristic
168	C₂×C₄	#14519803	1	4	yes	—
169	C₈	—	2	2	no	—
170	C₈	—	2	2	no	—
171	C₈	—	244	4	no	—
172	C₂²	—	2	2	no	—
173	C₂²	—	8	4	no	—
174	C₂²	—	8	2	no	—
175	C₂²	#16839665	1	1	yes	characteristic
176	C₂²	#16840149	1	2	yes	—
177	C₄	—	2	2	no	—
178	C₄	—	2	2	no	—
179	C₄	—	8	2	no	—
180	C₄	—	976	2	no	—
181	C₄	#16839767	1	2	yes	—
182	C₄	#16840126	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	—	16	2	no	—
185	C₂	#21446644	1	1	yes	characteristic
186	C₂	#21446718	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21445950

C₂

yes

maximal, characteristic, complemented

#21444379

C₂

yes

maximal, complemented

#21444111

C₂

yes

maximal, characteristic, complemented

C₆₁×#3147

C₂

yes

maximal, characteristic

#16839815

C₂²

yes

—

#16839628

C₂²

yes

—

#16839525

—

#16839496

—

#16839484

—

C₂×#135×C₆₁

—

C₆₁×#532

C₂²

yes

—

C₆₁×#495

—

C₆₁×#494

C₄

yes

characteristic, complemented

C₆₁×#493

C₄

yes

characteristic, complemented

C₆₁×#492

C₄

yes

complemented

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₁

—

C₄⋊#92418

—

C₂²⋊#92418

D₄

yes

—

C₄×#92418

—

#14519684

—

C₂×#41×C₆₁

—

#135×C₆₁

—

C₂×Q₁₆×C₆₁

—

C₂×SD₁₆×C₆₁

—

C₂×D₈×C₆₁

—

C₂²×C₈×C₆₁

D₄

yes

—

C₄⋊Q₈×C₆₁

C₂×C₄

yes

—

#127×C₆₁

C₂×C₄

yes

—

#124×C₆₁

—

#121×C₆₁

—

C₄×D₄×C₆₁

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

D₄

yes

—

#41×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×Dic₆₁

—

Q₁₆×C₆₁

—

SD₁₆×C₆₁

—

D₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂²⋊C₄

yes

—

C₂×C₈×C₆₁

C₂²⋊C₄

yes

—

C₄⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄²×C₆₁

#41

yes

—

C₄⋊Dic₆₁

—

C₂×#92418

—

C₂³×C₆₁

—

C₂³×C₆₁

#127

yes

characteristic

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄×D₄

yes

—

C₂×C₄×C₆₁

#121

yes

—

C₂×C₄×C₆₁

#124

yes

characteristic

C₂×C₄×C₆₁

AGL(1,ℤ/8)

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

#92418

—

#9760

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#390

yes

characteristic

C₂²×C₆₁

#496

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#418

yes

—

C₄×C₆₁

#490

yes

—

Dic₆₁

—

#3147

D₆₁

yes

—

—

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#1961

yes

characteristic

C₂×C₆₁

#1976

yes

—

C₂×#135

—

#532

C₂×D₆₁

yes

—

100

#495

—

101

#494

Dic₆₁

yes

characteristic, complemented

102

#493

Dic₆₁

yes

characteristic, complemented

103

#492

Dic₆₁

yes

complemented

104

#431

—

105

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

106

#382

—

107

#340

—

122

—

108

#327

—

122

—

109

#322

—

122

—

110

C₆₁

#9760

yes

characteristic, complemented, Sylow

111

C₂×#41

—

112

#135

—

113

C₂×Q₁₆

—

114

C₂×SD₁₆

—

115

C₂×D₈

—

116

C₂²×C₈

C₂²⋊D₆₁

yes

—

117

C₄⋊Q₈

C₂×Dic₆₁

yes

—

118

#127

C₂×Dic₆₁

yes

—

119

#124

—

120

#121

—

121

C₄×D₄

—

122

C₂×C₄⋊C₄

—

122

—

123

C₂×C₄²

C₂²×D₆₁

yes

characteristic

124

C₄⋊C₈

—

122

—

125

C₂²⋊C₈

—

126

C₄×C₈

—

122

—

127

C₄×C₈

C₂×Dic₆₁

yes

characteristic

128

C₄×C₈

C₂×Dic₆₁

yes

characteristic

129

C₄×C₈

C₂²⋊D₆₁

yes

—

130

#95

—

122

—

131

#41

—

132

C₂×Q₈

—

133

C₂×D₄

—

134

C₂×D₄

—

135

C₂×D₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

244

—

138

C₂²×C₄

D₄×D₆₁

yes

—

139

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

140

Q₁₆

—

141

SD₁₆

—

142

D₈

—

143

C₂×C₈

—

144

C₂×C₈

—

122

—

145

C₂×C₈

C₂²⋊Dic₆₁

yes

—

146

C₂×C₈

C₂²⋊Dic₆₁

yes

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

244

—

149

C₂²⋊C₄

—

150

C₄²

#11758224

yes

—

151

C₄²

C₂²×Dic₆₁

yes

characteristic

152

C₄²

C₂×C₂²⋊D₆₁

yes

characteristic

153

C₂³

—

154

C₂³

#14519773

yes

characteristic

155

Q₈

—

156

D₄

—

157

D₄

—

158

D₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

488

—

163

C₂×C₄

—

488

—

164

C₂×C₄

C₂×C₂²⋊Dic₆₁

yes

characteristic

165

C₂×C₄

D₄×Dic₆₁

yes

—

166

C₂×C₄

#14519748

yes

—

167

C₂×C₄

#14519764

yes

characteristic

168

C₂×C₄

#14519803

yes

—

169

C₈

—

170

C₈

—

171

C₈

—

244

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

#16839665

yes

characteristic

176

C₂²

#16840149

yes

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

976

—

181

C₄

#16839767

yes

—

182

C₄

#16840126

yes

—

183

C₂

—

184

C₂

—

185

C₂

#21446644

yes

characteristic

186

C₂

#21446718

yes

—

	subgroup	G/H
center	C₂²	#16839665
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting, prime radical	C₆₁×#3147	C₂
socle	C₂²×C₆₁	#390
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#3147	D₆₁
hypocenter	C₆₁	#9760
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3147

subgroup

G/H

center

C₂²

#16839665

commutator

C₄²×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting, prime radical

C₆₁×#3147

C₂

socle

C₂²×C₆₁

#390

cosocle

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3147

N	H ≅ G/N	type
1. #21445950	C₂	semidirect
2. #21444379	C₂	semidirect
3. #21444111	C₂	semidirect
13. C₆₁×#494	C₄	semidirect
14. C₆₁×#493	C₄	semidirect
15. C₆₁×#492	C₄	semidirect
101. #494	Dic₆₁	semidirect
102. #493	Dic₆₁	semidirect
103. #492	Dic₆₁	semidirect
110. C₆₁	#9760	semidirect

H ≅ G/N

type

1. #21445950

C₂

semidirect

2. #21444379

C₂

semidirect

3. #21444111

C₂

semidirect

13. C₆₁×#494

C₄

semidirect

14. C₆₁×#493

C₄

semidirect

15. C₆₁×#492

C₄

semidirect

101. #494

Dic₆₁

semidirect

102. #493

Dic₆₁

semidirect

103. #492

Dic₆₁

semidirect

110. C₆₁

#9760

semidirect

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

subgroup	index
#21445950	2
#21444379	2
#21444111	2
C₆₁×#3147	2
#9760	61

subgroup

index

C₆₁×#3147

Subgroups — #29579410

Conjugacy classes of subgroups

Complements

Maximal subgroups