Group #27271332 — subgroups

Subgroups — #27271332

Order 13568.

180 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 83 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	#20276518	C₂	1	1	yes	maximal, characteristic
2	#20273478	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic
4	#16810680	—	2	1	no	—
5	C₄⋊#11719908	—	2	1	no	—
6	C₂²⋊#11719908	C₄	1	2	yes	—
7	#16810399	—	2	1	no	—
8	#16810376	—	2	1	no	—
9	#16810374	C₄	1	2	yes	—
10	C₅₃×#431	—	2	1	no	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#335	—	2	1	no	—
13	C₅₃×#334	C₄	1	2	yes	—
14	C₂×#11719908	—	2	2	no	—
15	C₄⋊#88322	—	2	2	no	—
16	C₂²⋊#88322	—	4	1	no	—
17	#13133128	—	4	1	no	—
18	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
20	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
21	C₄⋊C₈×C₅₃	—	2	2	no	—
22	C₂²⋊C₈×C₅₃	—	4	1	no	—
23	C₄×C₈×C₅₃	—	4	1	no	—
24	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
27	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
31	C₂×C₈×C₅₃	—	2	2	no	—
32	C₂×C₈×C₅₃	—	4	1	no	—
33	C₂×C₈×C₅₃	—	4	2	no	—
34	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
35	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
38	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
41	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
42	C₂×#88322	—	4	2	no	—
43	#11719908	—	4	2	no	—
44	C₂³×C₅₃	#107	1	1	yes	characteristic
45	C₂×C₄×C₅₃	—	2	1	no	—
46	C₂×C₄×C₅₃	—	2	1	no	—
47	C₂×C₄×C₅₃	—	4	1	no	—
48	C₂×C₄×C₅₃	—	4	1	no	—
49	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
50	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
51	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
55	C₈×C₅₃	—	2	1	no	—
56	C₈×C₅₃	—	2	1	no	—
57	C₈×C₅₃	—	2	1	no	—
58	C₈×C₅₃	—	8	2	no	—
59	C₈×C₅₃	#108	1	2	yes	—
60	#88322	—	8	2	no	—
61	#7293	—	53	1	no	maximal, Sylow
62	C₂²×C₅₃	—	2	1	no	—
63	C₂²×C₅₃	—	2	1	no	—
64	C₂²×C₅₃	—	2	1	no	—
65	C₂²×C₅₃	#340	1	1	yes	characteristic
66	C₄×C₅₃	—	2	1	no	—
67	C₄×C₅₃	—	2	1	no	—
68	C₄×C₅₃	—	2	1	no	—
69	C₄×C₅₃	—	2	1	no	—
70	C₄×C₅₃	—	2	1	no	—
71	C₄×C₅₃	#341	1	1	yes	characteristic
72	C₄×C₅₃	#344	1	1	yes	characteristic
73	#2163	—	53	1	no	—
74	#1568	—	53	1	no	—
75	#1564	D₅₃	1	1	yes	characteristic
76	C₂×C₅₃	—	4	1	no	—
77	C₂×C₅₃	#1301	1	1	yes	characteristic
78	C₂×C₅₃	#1384	1	1	yes	characteristic
79	C₂×C₅₃	#1388	1	1	yes	characteristic
80	#432	—	106	1	no	—
81	#431	—	2	1	no	—
82	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
83	C₄⋊C₁₆	—	106	1	no	—
84	C₂²⋊C₁₆	—	53	2	no	—
85	#346	—	106	1	no	—
86	#335	—	2	1	no	—
87	#335	—	106	1	no	—
88	#334	—	53	2	no	—
89	#334	Dic₅₃	1	2	yes	—
90	C₅₃	#7293	1	1	yes	characteristic, complemented, Sylow
91	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
92	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
93	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
94	C₂×C₁₆	—	106	2	no	—
95	C₄⋊C₈	—	2	2	no	—
96	C₄⋊C₈	—	106	2	no	—
97	C₂²⋊C₈	—	4	1	no	—
98	C₂²⋊C₈	—	212	1	no	—
99	#97	—	212	1	no	—
100	C₄×C₈	—	4	1	no	—
101	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
102	C₄×C₈	D₂₁₂	1	1	yes	characteristic
103	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
104	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
105	C₂²×C₄	#11719919	1	1	yes	characteristic
106	C₂²×C₄	#11719922	1	1	yes	characteristic
107	C₂²×C₄	#11719925	1	1	yes	characteristic
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	4	1	no	—
110	C₂×C₈	—	4	2	no	—
111	C₂×C₈	—	212	2	no	—
112	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
113	C₂×C₈	#11719919	1	1	yes	characteristic
114	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
115	C₂×C₈	#11719921	1	1	yes	characteristic
116	C₄²	#11719921	1	1	yes	characteristic
117	C₄²	#11719923	1	1	yes	characteristic
118	C₄²	#11719924	1	1	yes	characteristic
119	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
120	C₁₆	—	212	2	no	—
121	C₂³	#13133150	1	1	yes	characteristic
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	#13133124	1	1	yes	characteristic
127	C₂×C₄	#13133137	1	1	yes	characteristic
128	C₂×C₄	#13133138	1	1	yes	characteristic
129	C₂×C₄	#13133140	1	1	yes	characteristic
130	C₂×C₄	#13133141	1	1	yes	characteristic
131	C₂×C₄	#13133148	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	2	1	no	—
134	C₈	—	2	1	no	—
135	C₈	—	8	2	no	—
136	C₈	—	424	2	no	—
137	C₈	#13133153	1	2	yes	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	#16810385	1	1	yes	characteristic
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	#16810388	1	1	yes	characteristic
148	C₄	#16810395	1	1	yes	characteristic
149	C₂	—	4	1	no	—
150	C₂	#20272612	1	1	yes	characteristic
151	C₂	#20272816	1	1	yes	characteristic
152	C₂	#20272826	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276518

C₂

yes

maximal, characteristic

#20273478

C₂

yes

maximal, characteristic

C₅₃×#1564

C₂

yes

maximal, characteristic

#16810680

—

C₄⋊#11719908

—

C₂²⋊#11719908

C₄

yes

—

#16810399

—

#16810376

—

#16810374

C₄

yes

—

C₅₃×#431

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#335

—

C₅₃×#334

C₄

yes

—

—

—

—

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

Q₈

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₅₃

D₈

yes

characteristic

C₂²×C₄×C₅₃

Q₁₆

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₄²×C₅₃

C₂²⋊C₄

yes

characteristic

C₄²×C₅₃

C₂²⋊C₄

yes

characteristic

C₄²×C₅₃

SD₁₆

yes

characteristic

C₄²×C₅₃

SD₁₆

yes

characteristic

C₂×#88322

—

#11719908

—

C₂³×C₅₃

#107

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#108

yes

—

#88322

—

#7293

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#340

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

—

—

D₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

106

—

#431

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

C₄⋊C₁₆

—

106

—

C₂²⋊C₁₆

—

#346

—

106

—

#335

—

#335

—

106

—

#334

—

#334

Dic₅₃

yes

—

C₅₃

#7293

yes

characteristic, complemented, Sylow

C₂²×C₈

Dic₁₀₆

yes

characteristic

C₂²×C₈

C₄×D₅₃

yes

characteristic

C₂×C₄²

C₂²⋊D₅₃

yes

characteristic

C₂×C₁₆

—

106

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

C₂²⋊C₈

—

212

—

#97

—

212

—

100

C₄×C₈

—

101

C₄×C₈

C₄×D₅₃

yes

characteristic

102

C₄×C₈

D₂₁₂

yes

characteristic

103

C₄×C₈

C₂×Dic₅₃

yes

characteristic

104

C₄×C₈

C₂²⋊D₅₃

yes

characteristic

105

C₂²×C₄

#11719919

yes

characteristic

106

C₂²×C₄

#11719922

yes

characteristic

107

C₂²×C₄

#11719925

yes

characteristic

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

212

—

112

C₂×C₈

C₄×Dic₅₃

yes

characteristic

113

C₂×C₈

#11719919

yes

characteristic

114

C₂×C₈

C₄⋊Dic₅₃

yes

characteristic

115

C₂×C₈

#11719921

yes

characteristic

116

C₄²

#11719921

yes

characteristic

117

C₄²

#11719923

yes

characteristic

118

C₄²

#11719924

yes

characteristic

119

C₄²

C₂²⋊Dic₅₃

yes

characteristic

120

C₁₆

—

212

—

121

C₂³

#13133150

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

#13133124

yes

characteristic

127

C₂×C₄

#13133137

yes

characteristic

128

C₂×C₄

#13133138

yes

characteristic

129

C₂×C₄

#13133140

yes

characteristic

130

C₂×C₄

#13133141

yes

characteristic

131

C₂×C₄

#13133148

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

424

—

137

C₈

#13133153

yes

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

#16810385

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

#16810388

yes

characteristic

148

C₄

#16810395

yes

characteristic

149

C₂

—

150

C₂

#20272612

yes

characteristic

151

C₂

#20272816

yes

characteristic

152

C₂

#20272826

yes

characteristic

	subgroup	G/H
center	C₂²	#16810385
commutator	C₂×C₈×C₅₃	C₄²
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₅₃×#1564	C₂
socle	C₂²×C₅₃	#340
prime radical	C₂³×C₅₃	#107
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	#1564	D₅₃
hypocenter	C₅₃	#7293
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1564

subgroup

G/H

center

C₂²

#16810385

commutator

C₂×C₈×C₅₃

C₄²

Frattini

C₂×C₄×C₈

C₂×D₅₃

Fitting

C₅₃×#1564

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄×C₈×C₅₃

C₂²

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1564

N	H ≅ G/N	type
90. C₅₃	#7293	semidirect

H ≅ G/N

type

90. C₅₃

#7293

semidirect

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

subgroup	index
#20276518	2
#20273478	2
C₅₃×#1564	2
#7293	53

subgroup

index

#20276518

#20273478

C₅₃×#1564

#7293

Subgroups — #27271332

Conjugacy classes of subgroups

Complements

Maximal subgroups