Group #29589045 — subgroups

Subgroups — #29589045

Order 15616.

290 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 143 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	#21447616	C₂	1	2	yes	maximal
2	C₈²×D₆₁	C₂	1	1	yes	maximal, characteristic
3	#21443738	C₂	1	2	yes	maximal, complemented
4	#21443736	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1311	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×C₈×D₆₁	C₄	1	1	yes	characteristic
7	C₄×C₈×D₆₁	C₄	1	1	yes	characteristic
8	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
9	#16839580	C₄	1	4	yes	—
10	#16839541	C₄	1	4	yes	—
11	C₄×#11758185	C₂²	1	2	yes	—
12	C₈×#92418	C₄	1	1	yes	characteristic
13	C₈×#92418	C₄	1	1	yes	characteristic
14	C₈×#92418	C₂²	1	1	yes	characteristic
15	C₄×C₁₆×C₆₁	C₂²	1	2	yes	—
16	C₈²×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₆₁	C₈	1	2	yes	—
18	C₂×C₈×D₆₁	C₈	1	2	yes	—
19	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
20	C₄²×D₆₁	C₂×C₄	1	1	yes	characteristic
21	C₂×#11758185	C₂×C₄	1	4	yes	—
22	C₄×#92418	C₂×C₄	1	1	yes	characteristic
23	C₄×#92418	C₂×C₄	1	1	yes	characteristic
24	C₄×#92418	C₂×C₄	1	1	yes	characteristic
25	C₈×Dic₆₁	C₈	1	2	yes	—
26	C₈×Dic₆₁	C₈	1	2	yes	—
27	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
28	C₂×C₁₆×C₆₁	C₂×C₄	1	4	yes	—
29	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
32	C₂×C₄×D₆₁	C₄²	1	1	yes	characteristic
33	C₂×C₄×D₆₁	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×D₆₁	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
36	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
37	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
38	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
39	C₄×Dic₆₁	C₄²	1	1	yes	characteristic
40	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
41	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
42	C₂×#92418	C₄²	1	2	yes	—
43	C₂×#92418	C₂×C₈	1	2	yes	—
44	C₂×#92418	C₂×C₈	1	2	yes	—
45	C₈×D₆₁	—	2	4	no	—
46	C₈×D₆₁	—	2	4	no	—
47	C₈×D₆₁	—	2	4	no	—
48	C₁₆×C₆₁	—	2	4	no	—
49	#11758185	—	2	4	no	—
50	C₂²×D₆₁	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
53	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
54	C₂×Dic₆₁	C₄×C₈	1	1	yes	characteristic
55	C₂×Dic₆₁	C₄×C₈	1	1	yes	characteristic
56	C₂×Dic₆₁	C₄×C₈	1	1	yes	characteristic
57	C₄×D₆₁	—	2	2	no	—
58	C₄×D₆₁	—	2	2	no	—
59	C₄×D₆₁	M₃₂	1	4	yes	—
60	C₈×C₆₁	—	2	2	no	—
61	C₈×C₆₁	—	2	2	no	—
62	C₈×C₆₁	#131	1	4	yes	—
63	#92418	—	2	2	no	—
64	#92418	—	2	2	no	—
65	#92418	M₃₂	1	4	yes	—
66	#11411	—	61	1	no	maximal, Sylow
67	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
68	C₂×D₆₁	—	2	1	no	—
69	C₂×D₆₁	—	2	1	no	—
70	C₂×D₆₁	#346	1	2	yes	—
71	C₄×C₆₁	#405	1	2	yes	—
72	C₄×C₆₁	C₂×M₃₂	1	2	yes	—
73	C₄×C₆₁	C₂×M₃₂	1	2	yes	—
74	Dic₆₁	—	2	1	no	—
75	Dic₆₁	—	2	1	no	—
76	Dic₆₁	#346	1	2	yes	—
77	#2162	—	61	2	no	—
78	C₂×C₈²	—	61	1	no	—
79	#1312	—	61	2	no	—
80	#1311	—	61	1	no	—
81	#1311	D₆₁	1	1	yes	characteristic, complemented
82	C₂×C₆₁	#1450	1	1	yes	characteristic
83	C₂×C₆₁	C₂×#346	1	1	yes	characteristic
84	C₂×C₆₁	C₄×M₃₂	1	1	yes	characteristic
85	D₆₁	—	2	2	no	—
86	C₂×C₄×C₈	—	61	1	no	—
87	C₂×C₄×C₈	—	61	1	no	—
88	C₂×C₄×C₈	—	61	1	no	—
89	C₄⋊C₁₆	—	61	4	no	—
90	C₂²⋊C₁₆	—	61	4	no	—
91	C₄×C₁₆	—	61	2	no	—
92	C₄×C₁₆	C₂×D₆₁	1	2	yes	—
93	C₈²	—	61	1	no	—
94	C₈²	—	61	1	no	—
95	C₈²	—	61	1	no	—
96	C₈²	C₂×D₆₁	1	1	yes	characteristic
97	C₆₁	#11411	1	1	yes	characteristic, complemented, Sylow
98	C₂²×C₈	—	61	2	no	—
99	C₂²×C₈	—	61	2	no	—
100	C₂²×C₈	—	61	2	no	—
101	C₂×C₄²	—	61	1	no	—
102	C₂×C₁₆	—	61	4	no	—
103	C₂×C₁₆	C₄×D₆₁	1	4	yes	—
104	C₄×C₈	—	61	2	no	—
105	C₄×C₈	—	61	2	no	—
106	C₄×C₈	—	61	2	no	—
107	C₄×C₈	—	61	1	no	—
108	C₄×C₈	—	61	1	no	—
109	C₄×C₈	—	61	1	no	—
110	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
111	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
112	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
113	C₂²×C₄	—	61	1	no	—
114	C₂²×C₄	—	61	1	no	—
115	C₂²×C₄	—	61	1	no	—
116	C₂×C₈	—	61	2	no	—
117	C₂×C₈	—	61	2	no	—
118	C₂×C₈	—	61	2	no	—
119	C₂×C₈	—	122	4	no	—
120	C₂×C₈	—	122	4	no	—
121	C₂×C₈	—	122	4	no	—
122	C₂×C₈	C₈×D₆₁	1	2	yes	—
123	C₂×C₈	C₈×D₆₁	1	2	yes	—
124	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
125	C₄²	—	61	1	no	—
126	C₄²	—	61	1	no	—
127	C₄²	—	61	1	no	—
128	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
129	C₁₆	—	2	4	no	—
130	C₁₆	—	122	4	no	—
131	C₂³	—	61	1	no	—
132	C₂×C₄	—	61	1	no	—
133	C₂×C₄	—	61	4	no	—
134	C₂×C₄	—	61	1	no	—
135	C₂×C₄	—	61	1	no	—
136	C₂×C₄	—	122	2	no	—
137	C₂×C₄	—	122	2	no	—
138	C₂×C₄	C₄²×D₆₁	1	1	yes	characteristic
139	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
140	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
141	C₈	—	2	2	no	—
142	C₈	—	2	2	no	—
143	C₈	—	61	4	no	—
144	C₈	—	122	2	no	—
145	C₈	—	122	2	no	—
146	C₈	#14519783	1	4	yes	—
147	C₂²	—	61	2	no	—
148	C₂²	—	122	1	no	—
149	C₂²	—	122	1	no	—
150	C₂²	C₄×C₈×D₆₁	1	1	yes	characteristic
151	C₄	—	61	2	no	—
152	C₄	—	122	1	no	—
153	C₄	—	122	1	no	—
154	C₄	#16839716	1	2	yes	—
155	C₄	M₃₂×D₆₁	1	2	yes	—
156	C₄	M₃₂×D₆₁	1	2	yes	—
157	C₂	—	122	2	no	—
158	C₂	#21444101	1	1	yes	characteristic
159	C₂	#346×D₆₁	1	1	yes	characteristic
160	C₂	#21447412	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21447616

C₂

yes

maximal

C₈²×D₆₁

C₂

yes

maximal, characteristic

#21443738

C₂

yes

maximal, complemented

#21443736

C₂

yes

maximal, characteristic, complemented

C₆₁×#1311

C₂

yes

maximal, characteristic, complemented

C₄×C₈×D₆₁

C₄

yes

characteristic

C₄×C₈×D₆₁

C₄

yes

characteristic

C₄×C₈×D₆₁

C₂²

yes

characteristic

#16839580

C₄

yes

—

#16839541

C₄

yes

—

C₄×#11758185

C₂²

yes

—

C₈×#92418

C₄

yes

characteristic

C₈×#92418

C₄

yes

characteristic

C₈×#92418

C₂²

yes

characteristic

C₄×C₁₆×C₆₁

C₂²

yes

—

C₈²×C₆₁

C₂²

yes

characteristic

C₂×C₈×D₆₁

C₈

yes

—

C₂×C₈×D₆₁

C₈

yes

—

C₂×C₈×D₆₁

C₂×C₄

yes

—

C₄²×D₆₁

C₂×C₄

yes

characteristic

C₂×#11758185

C₂×C₄

yes

—

C₄×#92418

C₂×C₄

yes

characteristic

C₄×#92418

C₂×C₄

yes

characteristic

C₄×#92418

C₂×C₄

yes

characteristic

C₈×Dic₆₁

C₈

yes

—

C₈×Dic₆₁

C₈

yes

—

C₈×Dic₆₁

C₂×C₄

yes

—

C₂×C₁₆×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₄×D₆₁

C₄²

yes

characteristic

C₂×C₄×D₆₁

C₂×C₈

yes

characteristic

C₂×C₄×D₆₁

C₂×C₈

yes

characteristic

C₂×C₈×C₆₁

C₂×C₈

yes

—

C₂×C₈×C₆₁

C₂×C₈

yes

—

C₂×C₈×C₆₁

C₂²×C₄

yes

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄×Dic₆₁

C₄²

yes

characteristic

C₄×Dic₆₁

C₂×C₈

yes

characteristic

C₄×Dic₆₁

C₂×C₈

yes

characteristic

C₂×#92418

C₄²

yes

—

C₂×#92418

C₂×C₈

yes

—

C₂×#92418

C₂×C₈

yes

—

C₈×D₆₁

—

C₈×D₆₁

—

C₈×D₆₁

—

C₁₆×C₆₁

—

#11758185

—

C₂²×D₆₁

C₄×C₈

yes

characteristic

C₂×C₄×C₆₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₈

yes

characteristic

C₂×Dic₆₁

C₄×C₈

yes

characteristic

C₂×Dic₆₁

C₄×C₈

yes

characteristic

C₂×Dic₆₁

C₄×C₈

yes

characteristic

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

M₃₂

yes

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

#131

yes

—

#92418

—

#92418

—

#92418

M₃₂

yes

—

#11411

—

maximal, Sylow

C₂²×C₆₁

C₂×C₄×C₈

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

#346

yes

—

C₄×C₆₁

#405

yes

—

C₄×C₆₁

C₂×M₃₂

yes

—

C₄×C₆₁

C₂×M₃₂

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

#346

yes

—

#2162

—

C₂×C₈²

—

#1312

—

#1311

—

#1311

D₆₁

yes

characteristic, complemented

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

C₄×M₃₂

yes

characteristic

D₆₁

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₆₁

yes

—

C₈²

—

C₈²

—

C₈²

—

C₈²

C₂×D₆₁

yes

characteristic

C₆₁

#11411

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

100

C₂²×C₈

—

101

C₂×C₄²

—

102

C₂×C₁₆

—

103

C₂×C₁₆

C₄×D₆₁

yes

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

C₄×D₆₁

yes

characteristic

111

C₄×C₈

C₄×D₆₁

yes

characteristic

112

C₄×C₈

C₂²×D₆₁

yes

characteristic

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

122

—

120

C₂×C₈

—

122

—

121

C₂×C₈

—

122

—

122

C₂×C₈

C₈×D₆₁

yes

—

123

C₂×C₈

C₈×D₆₁

yes

—

124

C₂×C₈

C₂×C₄×D₆₁

yes

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

C₂×C₄×D₆₁

yes

characteristic

129

C₁₆

—

130

C₁₆

—

122

—

131

C₂³

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

122

—

137

C₂×C₄

—

122

—

138

C₂×C₄

C₄²×D₆₁

yes

characteristic

139

C₂×C₄

C₂×C₈×D₆₁

yes

characteristic

140

C₂×C₄

C₂×C₈×D₆₁

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

122

—

145

C₈

—

122

—

146

C₈

#14519783

yes

—

147

C₂²

—

148

C₂²

—

122

—

149

C₂²

—

122

—

150

C₂²

C₄×C₈×D₆₁

yes

characteristic

151

C₄

—

152

C₄

—

122

—

153

C₄

—

122

—

154

C₄

#16839716

yes

—

155

C₄

M₃₂×D₆₁

yes

—

156

C₄

M₃₂×D₆₁

yes

—

157

C₂

—

122

—

158

C₂

#21444101

yes

characteristic

159

C₂

#346×D₆₁

yes

characteristic

160

C₂

#21447412

yes

characteristic

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₆₁
commutator, socle	C₂²×C₆₁	C₂×C₄×C₈
Fitting	C₆₁×#1311	C₂
prime radical	C₂²×D₆₁	C₄×C₈
cosocle	C₄×C₈×C₆₁	C₂³
hypercenter	#1311	D₆₁
hypocenter	C₆₁	#11411
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄×C₈ ◁ #1311

subgroup

G/H

center, Frattini

C₄×C₈

C₂²×D₆₁

commutator, socle

C₂²×C₆₁

C₂×C₄×C₈

Fitting

C₆₁×#1311

C₂

prime radical

C₂²×D₆₁

C₄×C₈

cosocle

C₄×C₈×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₄×C₈ ◁ #1311

N	H ≅ G/N	type
3. #21443738	C₂	semidirect
4. #21443736	C₂	semidirect
5. C₆₁×#1311	C₂	semidirect
81. #1311	D₆₁	semidirect
97. C₆₁	#11411	semidirect

H ≅ G/N

type

3. #21443738

C₂

semidirect

4. #21443736

C₂

semidirect

5. C₆₁×#1311

C₂

semidirect

81. #1311

D₆₁

semidirect

97. C₆₁

#11411

semidirect

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

subgroup	index
#21447616	2
C₈²×D₆₁	2
#21443738	2
#21443736	2
C₆₁×#1311	2
#11411	61

subgroup

index

C₈²×D₆₁

C₆₁×#1311

Subgroups — #29589045

Conjugacy classes of subgroups

Complements

Maximal subgroups