Group #20276554 — subgroups

Subgroups — #20276554

Order 6784.

144 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 63 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	#16811041	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811040	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810738	C₂	1	1	yes	maximal, characteristic
4	#16810443	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810406	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810398	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#350	C₂	1	1	yes	maximal, characteristic, complemented
8	#13133237	C₄	1	1	yes	characteristic
9	#13133223	C₂²	1	1	yes	characteristic
10	#13133157	—	2	1	no	—
11	#13133157	C₂²	1	1	yes	characteristic
12	#13133156	—	2	1	no	—
13	C₂×#11719908	—	2	1	no	—
14	C₂×#11719908	C₂²	1	1	yes	characteristic
15	C₁₆×D₅₃	—	2	1	no	—
16	#13133152	C₄	1	1	yes	characteristic
17	#13133144	C₄	1	1	yes	characteristic
18	#13133143	C₄	1	1	yes	characteristic
19	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
20	#131×C₅₃	C₂²	1	1	yes	characteristic
21	M₃₂×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
23	#41×C₅₃	C₂×C₄	1	1	yes	characteristic
24	#11719945	C₂×C₄	1	1	yes	characteristic
25	M₁₆×C₅₃	—	2	1	no	—
26	M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₅₃	—	2	1	no	—
28	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	#11719925	C₈	1	1	yes	characteristic
30	#11719924	C₈	1	1	yes	characteristic
31	#11719923	C₈	1	1	yes	characteristic
32	#11719922	C₈	1	1	yes	characteristic
33	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
34	#11719917	C₂×C₄	1	1	yes	characteristic
35	C₂×#88322	C₂×C₄	1	1	yes	characteristic
36	#11719912	—	2	1	no	—
37	C₈×D₅₃	—	2	1	no	—
38	C₁₆×C₅₃	—	2	1	no	—
39	C₁₆×C₅₃	—	2	1	no	—
40	#11719908	—	2	1	no	—
41	#11719908	D₄	1	1	yes	characteristic
42	#11719908	D₄	1	1	yes	characteristic
43	Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
44	D₄×C₅₃	—	2	1	no	—
45	D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
46	C₂×C₄×C₅₃	—	2	1	no	—
47	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂²⋊D₅₃	—	2	1	no	—
49	C₂×Dic₅₃	—	2	1	no	—
50	D₂₁₂	C₂×C₈	1	1	yes	characteristic
51	C₄×D₅₃	—	2	1	no	—
52	Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
53	C₈×C₅₃	—	2	1	no	—
54	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₈×C₅₃	#41	1	1	yes	characteristic
56	#88322	—	2	1	no	—
57	#88322	C₂×C₈	1	1	yes	characteristic
58	#88322	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₅₃	—	2	1	no	—
60	C₂²×C₅₃	#131	1	1	yes	characteristic
61	C₂×D₅₃	—	2	1	no	—
62	C₄×C₅₃	—	2	1	no	—
63	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
64	C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
65	Dic₅₃	—	2	1	no	—
66	Dic₅₃	—	4	1	no	—
67	#2171	—	53	1	no	maximal, Sylow
68	C₂×C₅₃	—	2	1	no	—
69	C₂×C₅₃	—	4	1	no	—
70	C₂×C₅₃	C₈×D₄	1	1	yes	characteristic
71	D₅₃	—	4	1	no	—
72	#504	—	53	1	no	—
73	#504	—	53	1	no	—
74	#443	—	53	1	no	—
75	#364	—	53	1	no	—
76	#350	—	53	1	no	—
77	#350	D₅₃	1	1	yes	characteristic, complemented
78	#346	—	53	1	no	—
79	C₅₃	#2171	1	1	yes	characteristic, complemented, Sylow
80	#135	—	53	1	no	—
81	#131	—	53	1	no	—
82	#131	C₂×D₅₃	1	1	yes	characteristic
83	M₃₂	—	53	1	no	—
84	M₃₂	—	106	1	no	—
85	M₃₂	—	106	1	no	—
86	M₃₂	C₂×D₅₃	1	1	yes	characteristic
87	C₂×C₁₆	—	53	1	no	—
88	C₂×C₁₆	—	106	1	no	—
89	C₂×C₁₆	—	106	1	no	—
90	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
91	#108	—	53	1	no	—
92	#104	—	53	1	no	—
93	#104	—	53	1	no	—
94	C₄×C₈	—	53	1	no	—
95	#41	—	53	1	no	—
96	#41	C₄×D₅₃	1	1	yes	characteristic
97	Q₁₆	—	53	1	no	—
98	SD₁₆	—	53	1	no	—
99	SD₁₆	—	53	1	no	—
100	D₈	—	53	1	no	—
101	M₁₆	—	2	1	no	—
102	M₁₆	—	53	1	no	—
103	M₁₆	—	106	1	no	—
104	M₁₆	C₄×D₅₃	1	1	yes	characteristic
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	53	1	no	—
107	C₂×C₈	—	106	1	no	—
108	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
109	C₄²	—	53	1	no	—
110	C₁₆	—	2	1	no	—
111	C₁₆	—	2	1	no	—
112	C₁₆	—	53	1	no	—
113	C₁₆	—	53	1	no	—
114	C₁₆	—	106	1	no	—
115	Q₈	—	53	1	no	—
116	Q₈	C₈×D₅₃	1	1	yes	characteristic
117	D₄	—	2	1	no	—
118	D₄	—	53	1	no	—
119	D₄	—	106	1	no	—
120	D₄	C₈×D₅₃	1	1	yes	characteristic
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	106	1	no	—
123	C₂×C₄	—	106	1	no	—
124	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	53	1	no	—
127	C₈	—	53	1	no	—
128	C₈	—	106	1	no	—
129	C₈	D₄×D₅₃	1	1	yes	characteristic
130	C₈	#11719947	1	1	yes	characteristic
131	C₂²	—	2	1	no	—
132	C₂²	—	106	1	no	—
133	C₂²	#13133225	1	1	yes	characteristic
134	C₄	—	2	1	no	—
135	C₄	—	106	1	no	—
136	C₄	—	212	1	no	—
137	C₄	#13133177	1	1	yes	characteristic
138	C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
139	C₂	—	2	1	no	—
140	C₂	—	4	1	no	—
141	C₂	—	212	1	no	—
142	C₂	#16810686	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811041

C₂

yes

maximal, characteristic, complemented

#16811040

C₂

yes

maximal, characteristic, complemented

#16810738

C₂

yes

maximal, characteristic

#16810443

C₂

yes

maximal, characteristic, complemented

#16810406

C₂

yes

maximal, characteristic, complemented

#16810398

C₂

yes

maximal, characteristic, complemented

C₅₃×#350

C₂

yes

maximal, characteristic, complemented

#13133237

C₄

yes

characteristic

#13133223

C₂²

yes

characteristic

#13133157

—

#13133157

C₂²

yes

—

—

C₂²

yes

characteristic

C₁₆×D₅₃

—

#13133152

C₄

yes

characteristic

#13133144

C₄

yes

characteristic

#13133143

C₄

yes

characteristic

C₈×Dic₅₃

C₂²

yes

characteristic

#131×C₅₃

C₂²

yes

characteristic

M₃₂×C₅₃

C₂²

yes

characteristic

C₂×C₁₆×C₅₃

C₂²

yes

characteristic

#41×C₅₃

C₂×C₄

yes

characteristic

#11719945

C₂×C₄

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

#11719925

C₈

yes

characteristic

#11719924

C₈

yes

characteristic

#11719923

C₈

yes

characteristic

#11719922

C₈

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

characteristic

#11719917

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

#11719912

—

C₈×D₅₃

—

C₁₆×C₅₃

—

C₁₆×C₅₃

—

#11719908

—

#11719908

D₄

yes

characteristic

#11719908

D₄

yes

characteristic

Q₈×C₅₃

C₂×C₈

yes

characteristic

D₄×C₅₃

—

D₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

D₂₁₂

C₂×C₈

yes

characteristic

C₄×D₅₃

—

Dic₁₀₆

C₂×C₈

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

C₂×D₄

yes

C₈×C₅₃

yes

—

C₂×C₈

yes

characteristic

#88322

C₂×C₈

yes

characteristic

C₂²×C₅₃

—

C₂²×C₅₃

#131

yes

characteristic

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×D₄

yes

characteristic

C₄×C₅₃

C₂²×C₈

yes

characteristic

Dic₅₃

—

Dic₅₃

—

#2171

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₈×D₄

yes

characteristic

D₅₃

—

—

—

—

—

—

D₅₃

yes

characteristic, complemented

#346

—

C₅₃

#2171

yes

characteristic, complemented, Sylow

#135

—

#131

—

#131

C₂×D₅₃

yes

characteristic

M₃₂

—

M₃₂

—

106

—

M₃₂

—

106

—

M₃₂

C₂×D₅₃

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₂×C₁₆

—

106

—

C₂×C₁₆

C₂×D₅₃

yes

—

—

—

C₄×C₈

—

#41

—

#41

C₄×D₅₃

yes

characteristic

Q₁₆

—

SD₁₆

—

SD₁₆

—

100

D₈

—

101

M₁₆

—

102

M₁₆

—

103

M₁₆

—

106

—

104

M₁₆

C₄×D₅₃

yes

characteristic

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

106

—

108

C₂×C₈

C₂²×D₅₃

yes

characteristic

109

C₄²

—

110

C₁₆

—

111

C₁₆

—

112

C₁₆

—

113

C₁₆

—

114

C₁₆

—

106

—

115

Q₈

—

116

Q₈

C₈×D₅₃

yes

characteristic

117

D₄

—

118

D₄

—

119

D₄

—

106

—

120

D₄

C₈×D₅₃

yes

characteristic

121

C₂×C₄

—

122

C₂×C₄

—

106

—

123

C₂×C₄

—

106

—

124

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

106

—

129

C₈

D₄×D₅₃

yes

characteristic

130

C₈

#11719947

yes

characteristic

131

C₂²

—

132

C₂²

—

106

—

133

C₂²

#13133225

yes

characteristic

134

C₄

—

135

C₄

—

106

—

136

C₄

—

212

—

137

C₄

#13133177

yes

characteristic

138

C₄

C₂×C₈×D₅₃

yes

characteristic

139

C₂

—

140

C₂

—

141

C₂

—

212

—

142

C₂

#16810686

yes

characteristic

	subgroup	G/H
center	C₈	D₄×D₅₃
commutator	C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#350	C₂
socle	C₂×C₅₃	C₈×D₄
hypercenter	#350	D₅₃
hypocenter	C₅₃	#2171
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #350

subgroup

G/H

center

C₈

D₄×D₅₃

commutator

C₄×C₅₃

C₂²×C₈

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#350

C₂

socle

C₂×C₅₃

C₈×D₄

D₅₃

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ #350

N	H ≅ G/N	type
1. #16811041	C₂	semidirect
2. #16811040	C₂	semidirect
4. #16810443	C₂	semidirect
5. #16810406	C₂	semidirect
6. #16810398	C₂	semidirect
7. C₅₃×#350	C₂	semidirect
77. #350	D₅₃	semidirect
79. C₅₃	#2171	semidirect

H ≅ G/N

type

1. #16811041

C₂

semidirect

2. #16811040

C₂

semidirect

4. #16810443

C₂

semidirect

5. #16810406

C₂

semidirect

6. #16810398

C₂

semidirect

7. C₅₃×#350

C₂

semidirect

77. #350

D₅₃

semidirect

79. C₅₃

#2171

semidirect

Subgroups — #20276554

Conjugacy classes of subgroups

Complements