Group #29595014 — subgroups

Subgroups — #29595014

Order 15616.

390 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 203 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	C₂×#16839538	C₂	1	1	yes	maximal, characteristic
2	C₄×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
3	#21443896	C₂	1	4	yes	maximal
4	C₂×C₆₁×#346	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11758185	C₂²	1	2	yes	—
6	C₂×C₄×#92418	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₆₁	C₄	1	2	yes	—
8	C₂×C₈×Dic₆₁	C₂²	1	2	yes	—
9	C₄²×Dic₆₁	C₄	1	1	yes	characteristic
10	#16839538	C₄	1	4	yes	complemented
11	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₆₁×#346	C₄	1	4	yes	complemented
14	C₂²×#92418	C₂×C₄	1	2	yes	—
15	C₂×C₄×Dic₆₁	C₈	1	4	yes	—
16	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
17	C₂×#11758185	—	2	4	no	—
18	C₂×#11758185	C₂×C₄	1	4	yes	—
19	C₄×#92418	C₈	1	4	yes	—
20	C₈×Dic₆₁	—	2	4	no	—
21	C₈×Dic₆₁	—	2	4	no	—
22	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₆₁	—	2	4	no	—
26	C₂×C₁₆×C₆₁	C₂×C₄	1	4	yes	—
27	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
30	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
31	C₂²×Dic₆₁	C₂×C₈	1	4	yes	—
32	C₂×C₈×C₆₁	C₄²	1	2	yes	—
33	C₂×C₈×C₆₁	M₁₆	1	4	yes	—
34	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₆₁	#41	1	4	yes	—
37	C₄²×C₆₁	C₄²	1	2	yes	—
38	C₄²×C₆₁	C₂×C₈	1	2	yes	—
39	C₄×Dic₆₁	—	2	2	no	—
40	C₄×Dic₆₁	—	2	2	no	—
41	C₄×Dic₆₁	—	2	8	no	—
42	C₂×#92418	—	2	2	no	—
43	C₂×#92418	—	2	2	no	—
44	C₂×#92418	C₂×C₈	1	4	yes	—
45	C₁₆×C₆₁	—	2	4	no	—
46	#11758185	—	2	4	no	—
47	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
48	C₂×C₄×C₆₁	—	2	4	no	—
49	C₂×C₄×C₆₁	C₄×C₈	1	4	yes	—
50	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
51	C₂×C₄×C₆₁	#117	1	2	yes	—
52	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
53	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
54	C₂×Dic₆₁	—	2	4	no	—
55	C₂×Dic₆₁	—	2	4	no	—
56	C₂×Dic₆₁	M₃₂	1	8	yes	—
57	C₈×C₆₁	#97	1	4	yes	—
58	C₈×C₆₁	#117	1	2	yes	—
59	C₈×C₆₁	#117	1	2	yes	—
60	#92418	—	2	4	no	—
61	#12591	—	61	1	no	maximal, Sylow
62	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
63	C₂²×C₆₁	#431	1	2	yes	—
64	C₂²×C₆₁	C₂×M₃₂	1	4	yes	—
65	C₄×C₆₁	—	2	4	no	—
66	C₄×C₆₁	#376	1	1	yes	characteristic
67	C₄×C₆₁	C₂×#97	1	1	yes	characteristic
68	C₄×C₆₁	#431	1	2	yes	—
69	Dic₆₁	—	2	8	no	—
70	C₂×#346	—	61	1	no	—
71	C₂×#346	D₆₁	1	1	yes	characteristic
72	C₄²×C₈	—	61	1	no	—
73	#1374	—	61	4	no	—
74	C₂×C₆₁	#1744	1	1	yes	characteristic
75	C₂×C₆₁	C₄×M₃₂	1	2	yes	—
76	C₂×C₆₁	#2163	1	4	yes	—
77	C₂²×C₁₆	—	61	2	no	—
78	C₂²×C₁₆	C₂×D₆₁	1	2	yes	—
79	C₂×C₄×C₈	—	61	1	no	—
80	C₂×C₄×C₈	—	61	2	no	—
81	C₂×C₄×C₈	—	61	2	no	—
82	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
83	C₄³	—	61	1	no	—
84	#346	—	61	4	no	—
85	#346	Dic₆₁	1	4	yes	complemented
86	C₆₁	#12591	1	1	yes	characteristic, complemented, Sylow
87	C₂²×C₈	—	61	2	no	—
88	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
89	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
90	C₂×C₄²	—	61	4	no	—
91	C₂×C₄²	—	61	2	no	—
92	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
93	C₂×C₁₆	—	2	4	no	—
94	C₂×C₁₆	—	61	4	no	—
95	C₂×C₁₆	—	122	4	no	—
96	C₂×C₁₆	C₂×Dic₆₁	1	4	yes	—
97	C₄×C₈	—	61	4	no	—
98	C₄×C₈	—	122	4	no	—
99	C₄×C₈	—	122	4	no	—
100	C₄×C₈	C₄×D₆₁	1	2	yes	—
101	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
102	C₂²×C₄	—	61	4	no	—
103	C₂²×C₄	C₈×D₆₁	1	2	yes	—
104	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
105	C₂×C₈	—	61	4	no	—
106	C₂×C₈	—	122	2	no	—
107	C₂×C₈	—	122	2	no	—
108	C₂×C₈	#11758189	1	4	yes	—
109	C₂×C₈	C₄×Dic₆₁	1	2	yes	—
110	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
111	C₂×C₈	#11758222	1	4	yes	—
112	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
113	C₄²	—	122	8	no	—
114	C₄²	—	122	2	no	—
115	C₄²	—	122	2	no	—
116	C₄²	C₈×D₆₁	1	2	yes	—
117	C₄²	C₄×Dic₆₁	1	2	yes	—
118	C₁₆	—	2	4	no	—
119	C₁₆	—	122	4	no	—
120	C₂³	C₂×C₈×D₆₁	1	1	yes	characteristic
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	61	8	no	—
123	C₂×C₄	—	122	4	no	—
124	C₂×C₄	—	122	4	no	—
125	C₂×C₄	C₈×Dic₆₁	1	4	yes	—
126	C₂×C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
127	C₂×C₄	#14519734	1	2	yes	—
128	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
129	C₂×C₄	C₂×#11758189	1	2	yes	—
130	C₈	—	122	4	no	—
131	C₈	#14519687	1	4	yes	—
132	C₈	#14519734	1	2	yes	—
133	C₈	#14519736	1	2	yes	—
134	C₂²	C₂×C₈×Dic₆₁	1	1	yes	characteristic
135	C₂²	#16839813	1	2	yes	—
136	C₂²	M₃₂×D₆₁	1	4	yes	—
137	C₄	—	2	4	no	—
138	C₄	—	122	8	no	—
139	C₄	#16839608	1	1	yes	characteristic
140	C₄	C₂×#14519687	1	1	yes	characteristic
141	C₄	#16839813	1	2	yes	—
142	C₂	#21445564	1	1	yes	characteristic
143	C₂	M₃₂×Dic₆₁	1	2	yes	—
144	C₂	#21447620	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16839538

C₂

yes

maximal, characteristic

C₄×C₈×Dic₆₁

C₂

yes

maximal, characteristic

#21443896

C₂

yes

maximal

C₂×C₆₁×#346

C₂

yes

maximal, characteristic

C₂²×#11758185

C₂²

yes

—

C₂×C₄×#92418

C₄

yes

characteristic

C₂×C₈×Dic₆₁

C₄

yes

—

C₂×C₈×Dic₆₁

C₂²

yes

—

C₄²×Dic₆₁

C₄

yes

characteristic

#16839538

C₄

yes

complemented

C₂²×C₁₆×C₆₁

C₂²

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×#346

C₄

yes

complemented

C₂²×#92418

C₂×C₄

yes

—

C₂×C₄×Dic₆₁

C₈

yes

—

C₂×C₄×Dic₆₁

C₂×C₄

yes

—

C₂×#11758185

—

C₂×#11758185

C₂×C₄

yes

—

C₄×#92418

C₈

yes

—

C₈×Dic₆₁

—

C₈×Dic₆₁

—

C₂²×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₁

—

C₂×C₁₆×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂×C₈

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×Dic₆₁

C₂×C₈

yes

—

C₂×C₈×C₆₁

C₄²

yes

—

C₂×C₈×C₆₁

M₁₆

yes

—

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₁

#41

yes

—

C₄²×C₆₁

C₄²

yes

—

C₄²×C₆₁

C₂×C₈

yes

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

C₂×C₈

yes

—

C₁₆×C₆₁

—

#11758185

—

C₂³×C₆₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄×C₈

yes

—

C₂×C₄×C₆₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₁

#117

yes

—

C₂×C₄×C₆₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₁

C₂×M₁₆

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

M₃₂

yes

—

C₈×C₆₁

#97

yes

—

C₈×C₆₁

#117

yes

—

C₈×C₆₁

#117

yes

—

#92418

—

#12591

—

maximal, Sylow

C₂²×C₆₁

C₂×C₄×C₈

yes

characteristic

C₂²×C₆₁

#431

yes

—

C₂²×C₆₁

C₂×M₃₂

yes

—

C₄×C₆₁

—

C₄×C₆₁

#376

yes

characteristic

C₄×C₆₁

C₂×#97

yes

characteristic

C₄×C₆₁

#431

yes

—

Dic₆₁

—

C₂×#346

—

C₂×#346

D₆₁

yes

characteristic

C₄²×C₈

—

#1374

—

C₂×C₆₁

#1744

yes

characteristic

C₂×C₆₁

C₄×M₃₂

yes

—

C₂×C₆₁

#2163

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₆₁

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

C₄³

—

#346

—

#346

Dic₆₁

yes

complemented

C₆₁

#12591

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₄×D₆₁

yes

characteristic

C₂²×C₈

C₂²×D₆₁

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₄×D₆₁

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

122

—

C₂×C₁₆

C₂×Dic₆₁

yes

—

C₄×C₈

—

C₄×C₈

—

122

—

C₄×C₈

—

122

—

100

C₄×C₈

C₄×D₆₁

yes

—

101

C₄×C₈

C₂×Dic₆₁

yes

—

102

C₂²×C₄

—

103

C₂²×C₄

C₈×D₆₁

yes

—

104

C₂²×C₄

C₂×C₄×D₆₁

yes

characteristic

105

C₂×C₈

—

106

C₂×C₈

—

122

—

107

C₂×C₈

—

122

—

108

C₂×C₈

#11758189

yes

—

109

C₂×C₈

C₄×Dic₆₁

yes

—

110

C₂×C₈

C₂×C₄×D₆₁

yes

characteristic

111

C₂×C₈

#11758222

yes

—

112

C₂×C₈

C₂²×Dic₆₁

yes

characteristic

113

C₄²

—

122

—

114

C₄²

—

122

—

115

C₄²

—

122

—

116

C₄²

C₈×D₆₁

yes

—

117

C₄²

C₄×Dic₆₁

yes

—

118

C₁₆

—

119

C₁₆

—

122

—

120

C₂³

C₂×C₈×D₆₁

yes

characteristic

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

122

—

124

C₂×C₄

—

122

—

125

C₂×C₄

C₈×Dic₆₁

yes

—

126

C₂×C₄

C₂×C₄×Dic₆₁

yes

characteristic

127

C₂×C₄

#14519734

yes

—

128

C₂×C₄

C₂×C₈×D₆₁

yes

characteristic

129

C₂×C₄

C₂×#11758189

yes

—

130

C₈

—

122

—

131

C₈

#14519687

yes

—

132

C₈

#14519734

yes

—

133

C₈

#14519736

yes

—

134

C₂²

C₂×C₈×Dic₆₁

yes

characteristic

135

C₂²

#16839813

yes

—

136

C₂²

M₃₂×D₆₁

yes

—

137

C₄

—

138

C₄

—

122

—

139

C₄

#16839608

yes

characteristic

140

C₄

C₂×#14519687

yes

characteristic

141

C₄

#16839813

yes

—

142

C₂

#21445564

yes

characteristic

143

C₂

M₃₂×Dic₆₁

yes

—

144

C₂

#21447620

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₆₁
commutator	C₂²×C₆₁	C₂×C₄×C₈
Fitting	C₂×C₆₁×#346	C₂
socle, prime radical	C₂³×C₆₁	C₂²×C₈
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	C₂×#346	D₆₁
hypocenter	C₆₁	#12591
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₆₁

commutator

C₂²×C₆₁

C₂×C₄×C₈

Fitting

C₂×C₆₁×#346

C₂

socle, prime radical

C₂³×C₆₁

C₂²×C₈

cosocle

C₂²×C₈×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂²×C₈ ◁ C₂×#346

N	H ≅ G/N	type
10. #16839538	C₄	semidirect
13. C₆₁×#346	C₄	semidirect
85. #346	Dic₆₁	semidirect
86. C₆₁	#12591	semidirect

H ≅ G/N

type

10. #16839538

C₄

semidirect

13. C₆₁×#346

C₄

semidirect

85. #346

Dic₆₁

semidirect

86. C₆₁

#12591

semidirect

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

subgroup	index
C₂×#16839538	2
C₄×C₈×Dic₆₁	2
#21443896	2
C₂×C₆₁×#346	2
#12591	61

subgroup

index

C₂×#16839538

C₄×C₈×Dic₆₁

#21443896

C₂×C₆₁×#346

#12591

Subgroups — #29595014

Conjugacy classes of subgroups

Complements

Maximal subgroups