Group #29571176 — subgroups

Subgroups — #29571176

Order 15616.

390 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 155 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	C₄×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
2	#21444565	C₂	1	2	yes	maximal
3	#21444111	C₂	1	1	yes	maximal, characteristic
4	#21443651	C₂	1	2	yes	maximal
5	C₆₁×#1452	C₂	1	1	yes	maximal, characteristic
6	#16839815	C₂²	1	2	yes	—
7	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₆₁	C₄	1	2	yes	—
9	C₂×C₈×Dic₆₁	C₄	1	2	yes	—
10	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
11	#16839514	C₄	1	4	yes	complemented
12	#16839512	C₄	1	4	yes	complemented
13	#16839484	—	2	2	no	—
14	C₆₁×#431	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×#322	—	2	2	no	—
17	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
18	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₆₁	—	2	2	no	—
20	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
21	C₄⋊#92418	C₂×C₄	1	4	yes	—
22	C₂²⋊#92418	—	2	2	no	—
23	C₄×#92418	—	2	2	no	—
24	C₄×#92418	C₂×C₄	1	2	yes	—
25	C₄×#92418	C₂×C₄	1	2	yes	—
26	C₈×Dic₆₁	—	2	4	no	—
27	C₈×Dic₆₁	—	2	4	no	—
28	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₆₁	C₂×C₄	1	4	yes	—
31	C₂²⋊C₈×C₆₁	—	2	2	no	—
32	C₄×C₈×C₆₁	—	2	2	no	—
33	C₄×C₈×C₆₁	D₄	1	2	yes	—
34	C₄×C₈×C₆₁	Q₈	1	2	yes	—
35	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₆₁	#41	1	2	yes	—
37	C₂²×Dic₆₁	—	2	2	no	—
38	C₂×C₈×C₆₁	—	2	2	no	—
39	C₂×C₈×C₆₁	—	2	2	no	—
40	C₂×C₈×C₆₁	—	2	4	no	—
41	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
42	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
43	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
44	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
45	C₄×Dic₆₁	—	2	8	no	—
46	C₄×Dic₆₁	—	2	2	no	—
47	C₄×Dic₆₁	M₁₆	1	4	yes	—
48	C₂×#92418	—	2	2	no	—
49	C₂×#92418	—	2	4	no	—
50	C₂×#92418	C₄²	1	2	yes	—
51	C₂×#92418	C₄²	1	1	yes	characteristic
52	C₂×#92418	C₄²	1	1	yes	characteristic
53	C₂×#92418	M₁₆	1	4	yes	—
54	C₂³×C₆₁	#117	1	1	yes	characteristic
55	C₂×C₄×C₆₁	—	2	4	no	—
56	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
57	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
59	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
60	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
61	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
62	C₂×Dic₆₁	—	2	8	no	—
63	C₂×Dic₆₁	—	2	4	no	—
64	C₈×C₆₁	—	2	4	no	—
65	C₈×C₆₁	—	4	4	no	—
66	#92418	—	2	2	no	—
67	#92418	—	4	4	no	—
68	#92418	#97	1	2	yes	—
69	#92418	#97	1	2	yes	—
70	#8267	—	61	1	no	maximal, Sylow
71	C₂²×C₆₁	—	2	2	no	—
72	C₂²×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₆₁	#432	1	2	yes	—
74	C₄×C₆₁	—	2	2	no	—
75	C₄×C₆₁	C₂×#97	1	1	yes	characteristic
76	C₄×C₆₁	C₄×M₁₆	1	1	yes	characteristic
77	C₄×C₆₁	#423	1	2	yes	—
78	C₄×C₆₁	#443	1	4	yes	—
79	Dic₆₁	—	2	8	no	—
80	C₄²×C₈	—	61	1	no	—
81	#1564	—	61	2	no	—
82	#1452	—	61	1	no	—
83	#1452	D₆₁	1	1	yes	characteristic
84	#1276	—	61	2	no	—
85	C₂×C₆₁	—	2	2	no	—
86	C₂×C₆₁	#1767	1	1	yes	characteristic
87	C₂×C₆₁	#1772	1	2	yes	—
88	#431	—	61	2	no	—
89	#431	C₂×D₆₁	1	2	yes	—
90	C₂×C₄×C₈	—	61	1	no	—
91	C₂×C₄×C₈	—	61	2	no	—
92	C₂×C₄×C₈	—	61	2	no	—
93	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
94	C₄³	—	61	1	no	—
95	#335	—	61	4	no	—
96	#334	—	61	4	no	—
97	#322	—	2	2	no	—
98	#322	—	122	2	no	—
99	C₆₁	#8267	1	1	yes	characteristic, complemented, Sylow
100	C₂²×C₈	—	61	1	no	—
101	C₂²×C₈	—	61	1	no	—
102	C₂²×C₈	C₄×D₆₁	1	2	yes	—
103	C₂×C₄²	—	61	2	no	—
104	C₂×C₄²	—	122	2	no	—
105	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
106	C₄⋊C₈	—	61	4	no	—
107	C₄⋊C₈	C₄×D₆₁	1	4	yes	—
108	C₂²⋊C₈	—	2	2	no	—
109	C₂²⋊C₈	—	122	2	no	—
110	C₄×C₈	—	2	2	no	—
111	C₄×C₈	—	61	2	no	—
112	C₄×C₈	—	61	2	no	—
113	C₄×C₈	—	122	4	no	—
114	C₄×C₈	—	122	4	no	—
115	C₄×C₈	—	122	2	no	—
116	C₄×C₈	Dic₁₂₂	1	2	yes	—
117	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
118	C₂²×C₄	—	122	2	no	—
119	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
120	C₂²×C₄	#11758222	1	2	yes	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	4	no	—
124	C₂×C₈	—	61	1	no	—
125	C₂×C₈	—	61	1	no	—
126	C₂×C₈	—	61	4	no	—
127	C₂×C₈	—	61	2	no	—
128	C₂×C₈	—	122	2	no	—
129	C₂×C₈	—	122	4	no	—
130	C₂×C₈	#11758196	1	4	yes	—
131	C₄²	—	61	4	no	—
132	C₄²	—	122	8	no	—
133	C₄²	—	122	2	no	—
134	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
135	C₄²	C₂×C₄×D₆₁	1	2	yes	—
136	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
137	C₂³	#14519734	1	1	yes	characteristic
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	122	8	no	—
140	C₂×C₄	—	122	4	no	—
141	C₂×C₄	C₄²×D₆₁	1	1	yes	characteristic
142	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
143	C₂×C₄	C₄×C₂²⋊D₆₁	1	2	yes	—
144	C₂×C₄	C₄×Dic₁₂₂	1	2	yes	—
145	C₂×C₄	M₁₆×D₆₁	1	2	yes	—
146	C₂×C₄	M₁₆×D₆₁	1	2	yes	—
147	C₈	—	2	4	no	—
148	C₈	—	4	4	no	—
149	C₈	—	61	2	no	—
150	C₈	—	61	2	no	—
151	C₈	—	122	2	no	—
152	C₈	—	244	4	no	—
153	C₂²	—	2	2	no	—
154	C₂²	C₄×#11758196	1	1	yes	characteristic
155	C₂²	#16839817	1	2	yes	—
156	C₄	—	2	2	no	—
157	C₄	—	122	8	no	—
158	C₄	#97×D₆₁	1	1	yes	characteristic
159	C₄	#16839714	1	1	yes	characteristic
160	C₄	#16839785	1	2	yes	—
161	C₄	#16839877	1	4	yes	—
162	C₂	—	2	2	no	—
163	C₂	#21445673	1	1	yes	characteristic
164	C₂	#21445699	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₈×Dic₆₁

C₂

yes

maximal, characteristic

#21444565

C₂

yes

maximal

#21444111

C₂

yes

maximal, characteristic

#21443651

C₂

yes

maximal

C₆₁×#1452

C₂

yes

maximal, characteristic

#16839815

C₂²

yes

—

C₂×C₄×#92418

C₂²

yes

characteristic

C₂×C₈×Dic₆₁

C₄

yes

—

C₂×C₈×Dic₆₁

C₄

yes

—

C₄²×Dic₆₁

C₂²

yes

characteristic

#16839514

C₄

yes

complemented

#16839512

C₄

yes

complemented

#16839484

—

C₆₁×#431

C₂²

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×#322

—

C₂²×#92418

C₂×C₄

yes

characteristic

C₂²×#92418

C₂×C₄

yes

characteristic

C₂×C₄×Dic₆₁

—

C₂×C₄×Dic₆₁

C₂×C₄

yes

—

C₄⋊#92418

C₂×C₄

yes

—

C₂²⋊#92418

—

C₄×#92418

—

C₄×#92418

C₂×C₄

yes

—

C₄×#92418

C₂×C₄

yes

—

C₈×Dic₆₁

—

C₈×Dic₆₁

—

C₂²×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊C₈×C₆₁

C₂×C₄

yes

—

C₂²⋊C₈×C₆₁

—

C₄×C₈×C₆₁

—

C₄×C₈×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

Q₈

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₁

#41

yes

—

C₂²×Dic₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₄²×C₆₁

C₂²×C₄

yes

—

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄²×C₆₁

C₂×Q₈

yes

characteristic

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

M₁₆

yes

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

C₄²

yes

—

C₂×#92418

C₄²

yes

characteristic

C₂×#92418

C₄²

yes

characteristic

C₂×#92418

M₁₆

yes

—

C₂³×C₆₁

#117

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄×D₄

yes

—

C₂×C₄×C₆₁

C₄×Q₈

yes

—

C₂×C₄×C₆₁

C₂×M₁₆

yes

—

C₂×C₄×C₆₁

C₂×M₁₆

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

—

—

yes

—

#92418

#97

yes

—

#8267

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

C₄×C₄⋊C₄

yes

characteristic

C₂²×C₆₁

#432

yes

—

C₄×C₆₁

—

C₄×C₆₁

C₂×#97

yes

characteristic

C₄×C₆₁

C₄×M₁₆

yes

characteristic

C₄×C₆₁

#423

yes

—

C₄×C₆₁

#443

yes

—

Dic₆₁

—

C₄²×C₈

—

#1564

—

#1452

—

#1452

D₆₁

yes

characteristic

#1276

—

C₂×C₆₁

—

C₂×C₆₁

#1767

yes

characteristic

C₂×C₆₁

#1772

yes

—

#431

—

#431

C₂×D₆₁

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

C₄³

—

—

—

—

—

122

—

C₆₁

#8267

yes

characteristic, complemented, Sylow

100

C₂²×C₈

—

101

C₂²×C₈

—

102

C₂²×C₈

C₄×D₆₁

yes

—

103

C₂×C₄²

—

104

C₂×C₄²

—

122

—

105

C₂×C₄²

C₂²×D₆₁

yes

characteristic

106

C₄⋊C₈

—

107

C₄⋊C₈

C₄×D₆₁

yes

—

108

C₂²⋊C₈

—

109

C₂²⋊C₈

—

122

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

122

—

114

C₄×C₈

—

122

—

115

C₄×C₈

—

122

—

116

C₄×C₈

Dic₁₂₂

yes

—

117

C₄×C₈

C₂²⋊D₆₁

yes

—

118

C₂²×C₄

—

122

—

119

C₂²×C₄

C₂×C₄×D₆₁

yes

characteristic

120

C₂²×C₄

#11758222

yes

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

122

—

129

C₂×C₈

—

122

—

130

C₂×C₈

#11758196

yes

—

131

C₄²

—

132

C₄²

—

122

—

133

C₄²

—

122

—

134

C₄²

C₂×Dic₁₂₂

yes

characteristic

135

C₄²

C₂×C₄×D₆₁

yes

—

136

C₄²

C₂×C₂²⋊D₆₁

yes

characteristic

137

C₂³

#14519734

yes

characteristic

138

C₂×C₄

—

139

C₂×C₄

—

122

—

140

C₂×C₄

—

122

—

141

C₂×C₄

C₄²×D₆₁

yes

characteristic

142

C₂×C₄

C₂×#11758196

yes

characteristic

143

C₂×C₄

C₄×C₂²⋊D₆₁

yes

—

144

C₂×C₄

C₄×Dic₁₂₂

yes

—

145

C₂×C₄

M₁₆×D₆₁

yes

—

146

C₂×C₄

M₁₆×D₆₁

yes

—

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

122

—

152

C₈

—

244

—

153

C₂²

—

154

C₂²

C₄×#11758196

yes

characteristic

155

C₂²

#16839817

yes

—

156

C₄

—

157

C₄

—

122

—

158

C₄

#97×D₆₁

yes

characteristic

159

C₄

#16839714

yes

characteristic

160

C₄

#16839785

yes

—

161

C₄

#16839877

yes

—

162

C₂

—

163

C₂

#21445673

yes

characteristic

164

C₂

#21445699

yes

—

	subgroup	G/H
center	C₄²	C₂×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1452	C₂
socle	C₂²×C₆₁	C₄×C₄⋊C₄
prime radical	C₂³×C₆₁	#117
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1452	D₆₁
hypocenter	C₆₁	#8267
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ #1452

subgroup

G/H

center

C₄²

C₂×C₂²⋊D₆₁

commutator

C₂×C₄×C₆₁

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#1452

C₂

socle

C₂²×C₆₁

C₄×C₄⋊C₄

prime radical

C₂³×C₆₁

#117

cosocle

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ #1452

N	H ≅ G/N	type
11. #16839514	C₄	semidirect
12. #16839512	C₄	semidirect
99. C₆₁	#8267	semidirect

H ≅ G/N

type

11. #16839514

C₄

semidirect

12. #16839512

C₄

semidirect

99. C₆₁

#8267

semidirect

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

subgroup	index
C₄×C₈×Dic₆₁	2
#21444565	2
#21444111	2
#21443651	2
C₆₁×#1452	2
#8267	61

subgroup

index

C₄×C₈×Dic₆₁

C₆₁×#1452

Subgroups — #29571176

Conjugacy classes of subgroups

Complements

Maximal subgroups