Group #27271351 — subgroups

Subgroups — #27271351

Order 13568.

212 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 87 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	C₄×#13133153	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1566	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133153	—	2	1	no	—
5	C₄×#11719917	—	2	1	no	—
6	C₄⋊#11719908	C₄	1	2	yes	—
7	C₂²⋊#11719908	—	2	1	no	—
8	#16810399	—	2	1	no	—
9	#16810388	C₄	1	2	yes	—
10	C₅₃×#423	—	2	1	no	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#334	—	2	1	no	—
13	C₅₃×#334	C₄	1	2	yes	—
14	C₂×#11719917	—	2	2	no	—
15	C₂×#11719908	—	2	2	no	—
16	#13133153	—	2	4	no	—
17	#13133128	—	4	1	no	—
18	C₄×#88322	—	4	1	no	—
19	C₂×M₁₆×C₅₃	—	4	1	no	—
20	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
22	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
23	C₄⋊C₈×C₅₃	—	2	2	no	—
24	C₄⋊C₈×C₅₃	—	4	1	no	—
25	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
28	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
32	M₁₆×C₅₃	—	8	2	no	—
33	C₂×C₈×C₅₃	—	2	2	no	—
34	C₂×C₈×C₅₃	—	4	1	no	—
35	C₂×C₈×C₅₃	—	4	2	no	—
36	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
40	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₄²×C₅₃	D₈	1	1	yes	characteristic
43	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
44	#11719917	—	2	4	no	—
45	#11719917	—	4	2	no	—
46	C₂×#88322	—	4	2	no	—
47	#11719908	—	4	2	no	—
48	C₂³×C₅₃	#103	1	1	yes	characteristic
49	C₂×C₄×C₅₃	—	2	1	no	—
50	C₂×C₄×C₅₃	—	2	1	no	—
51	C₂×C₄×C₅₃	—	2	2	no	—
52	C₂×C₄×C₅₃	—	4	1	no	—
53	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
55	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
57	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
58	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
59	C₈×C₅₃	—	2	1	no	—
60	C₈×C₅₃	—	2	1	no	—
61	C₈×C₅₃	—	2	1	no	—
62	C₈×C₅₃	—	8	2	no	—
63	C₈×C₅₃	#104	1	2	yes	—
64	#88322	—	4	4	no	—
65	#7300	—	53	1	no	maximal, Sylow
66	C₂²×C₅₃	—	2	1	no	—
67	C₂²×C₅₃	—	2	1	no	—
68	C₂²×C₅₃	—	2	1	no	—
69	C₂²×C₅₃	#340	1	1	yes	characteristic
70	C₄×C₅₃	—	2	1	no	—
71	C₄×C₅₃	—	2	1	no	—
72	C₄×C₅₃	—	2	1	no	—
73	C₄×C₅₃	—	2	1	no	—
74	C₄×C₅₃	#339	1	1	yes	characteristic
75	C₄×C₅₃	#344	1	1	yes	characteristic
76	C₄×C₅₃	#365	1	2	yes	—
77	#2163	—	53	1	no	—
78	C₄×#108	—	53	1	no	—
79	#1566	D₅₃	1	1	yes	characteristic
80	C₂×C₅₃	—	4	1	no	—
81	C₂×C₅₃	#1303	1	1	yes	characteristic
82	C₂×C₅₃	#1383	1	1	yes	characteristic
83	C₂×C₅₃	#1388	1	1	yes	characteristic
84	C₂×#108	—	106	1	no	—
85	#423	—	2	1	no	—
86	C₄×M₁₆	—	106	1	no	—
87	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
88	C₄⋊C₁₆	—	53	2	no	—
89	C₂²⋊C₁₆	—	106	1	no	—
90	#346	—	106	1	no	—
91	#341	—	53	2	no	—
92	#334	—	2	1	no	—
93	#334	Dic₅₃	1	2	yes	—
94	C₅₃	#7300	1	1	yes	characteristic, complemented, Sylow
95	C₂×M₁₆	—	4	1	no	—
96	C₂×M₁₆	—	106	2	no	—
97	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
98	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
99	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
100	C₂×C₁₆	—	106	2	no	—
101	#108	—	106	4	no	—
102	C₄⋊C₈	—	2	2	no	—
103	C₄⋊C₈	—	4	1	no	—
104	#97	—	212	1	no	—
105	C₄×C₈	—	212	1	no	—
106	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
107	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
108	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
109	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
110	C₂²×C₄	#11719921	1	1	yes	characteristic
111	C₂²×C₄	#11719923	1	1	yes	characteristic
112	C₂²×C₄	#11719925	1	1	yes	characteristic
113	M₁₆	—	8	2	no	—
114	M₁₆	—	106	4	no	—
115	M₁₆	—	212	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	4	1	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	—	212	2	no	—
120	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
121	C₂×C₈	#11719919	1	1	yes	characteristic
122	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
123	C₂×C₈	#11719921	1	1	yes	characteristic
124	C₄²	#11719919	1	1	yes	characteristic
125	C₄²	#11719922	1	1	yes	characteristic
126	C₄²	#11719924	1	1	yes	characteristic
127	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
128	C₁₆	—	212	2	no	—
129	C₂³	#13133140	1	1	yes	characteristic
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	#13133124	1	1	yes	characteristic
135	C₂×C₄	#13133137	1	1	yes	characteristic
136	C₂×C₄	#13133138	1	1	yes	characteristic
137	C₂×C₄	#13133141	1	1	yes	characteristic
138	C₂×C₄	#13133148	1	1	yes	characteristic
139	C₂×C₄	#13133150	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	—	8	2	no	—
144	C₈	—	212	4	no	—
145	C₈	#13133143	1	2	yes	—
146	C₂²	—	2	1	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	#16810385	1	1	yes	characteristic
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	#16810384	1	1	yes	characteristic
155	C₄	#16810394	1	1	yes	characteristic
156	C₄	#16810445	1	2	yes	—
157	C₂	—	4	1	no	—
158	C₂	#20272616	1	1	yes	characteristic
159	C₂	#20272813	1	1	yes	characteristic
160	C₂	#20272828	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276518

C₂

yes

maximal, characteristic

C₄×#13133153

C₂

yes

maximal, characteristic

C₅₃×#1566

C₂

yes

maximal, characteristic

C₂×#13133153

—

C₄×#11719917

—

C₄⋊#11719908

C₄

yes

—

C₂²⋊#11719908

—

#16810399

—

#16810388

C₄

yes

—

C₅₃×#423

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#334

—

C₅₃×#334

C₄

yes

—

—

—

—

—

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₄⋊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

Q₈

yes

characteristic

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅₃

SD₁₆

yes

characteristic

C₂²×C₄×C₅₃

Q₁₆

yes

characteristic

M₁₆×C₅₃

—

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

C₄⋊C₄

yes

characteristic

C₄²×C₅₃

D₈

yes

characteristic

C₄²×C₅₃

SD₁₆

yes

—

—

—

—

C₂³×C₅₃

#103

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

#104

yes

—

#88322

—

#7300

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#340

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

#2163

—

C₄×#108

—

#1566

D₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

106

—

#423

—

C₄×M₁₆

—

106

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

106

—

#346

—

106

—

#341

—

#334

—

#334

Dic₅₃

yes

—

C₅₃

#7300

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

106

—

C₂²×C₈

C₄×D₅₃

yes

characteristic

C₂²×C₈

D₂₁₂

yes

characteristic

C₂×C₄²

C₂²⋊D₅₃

yes

characteristic

100

C₂×C₁₆

—

106

—

101

#108

—

106

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

#97

—

212

—

105

C₄×C₈

—

212

—

106

C₄×C₈

Dic₁₀₆

yes

characteristic

107

C₄×C₈

C₄×D₅₃

yes

characteristic

108

C₄×C₈

C₂×Dic₅₃

yes

characteristic

109

C₄×C₈

C₂²⋊D₅₃

yes

characteristic

110

C₂²×C₄

#11719921

yes

characteristic

111

C₂²×C₄

#11719923

yes

characteristic

112

C₂²×C₄

#11719925

yes

characteristic

113

M₁₆

—

114

M₁₆

—

106

—

115

M₁₆

—

212

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

212

—

120

C₂×C₈

C₄×Dic₅₃

yes

characteristic

121

C₂×C₈

#11719919

yes

characteristic

122

C₂×C₈

C₄⋊Dic₅₃

yes

characteristic

123

C₂×C₈

#11719921

yes

characteristic

124

C₄²

#11719919

yes

characteristic

125

C₄²

#11719922

yes

characteristic

126

C₄²

#11719924

yes

characteristic

127

C₄²

C₂²⋊Dic₅₃

yes

characteristic

128

C₁₆

—

212

—

129

C₂³

#13133140

yes

characteristic

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

#13133124

yes

characteristic

135

C₂×C₄

#13133137

yes

characteristic

136

C₂×C₄

#13133138

yes

characteristic

137

C₂×C₄

#13133141

yes

characteristic

138

C₂×C₄

#13133148

yes

characteristic

139

C₂×C₄

#13133150

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

212

—

145

C₈

#13133143

yes

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

#16810385

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

#16810384

yes

characteristic

155

C₄

#16810394

yes

characteristic

156

C₄

#16810445

yes

—

157

C₂

—

158

C₂

#20272616

yes

characteristic

159

C₂

#20272813

yes

characteristic

160

C₂

#20272828

yes

characteristic

	subgroup	G/H
center	C₂²	#16810385
commutator	C₂×C₈×C₅₃	C₄²
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₅₃×#1566	C₂
socle	C₂²×C₅₃	#340
prime radical	C₂³×C₅₃	#103
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	#1566	D₅₃
hypocenter	C₅₃	#7300
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1566

subgroup

G/H

center

C₂²

#16810385

commutator

C₂×C₈×C₅₃

C₄²

Frattini

C₂×C₄×C₈

C₂×D₅₃

Fitting

C₅₃×#1566

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 ◁ #1566

N	H ≅ G/N	type
94. C₅₃	#7300	semidirect

H ≅ G/N

type

94. C₅₃

#7300

semidirect

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

subgroup	index
#20276518	2
C₄×#13133153	2
C₅₃×#1566	2
#7300	53

subgroup

index

#20276518

C₄×#13133153

C₅₃×#1566

#7300

Subgroups — #27271351

Conjugacy classes of subgroups

Complements

Maximal subgroups