Group #27270944 — subgroups

Subgroups — #27270944

Order 13568.

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

normal subgroups only

This group has 188 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	#20273007	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2162	C₂	1	1	yes	maximal, characteristic
4	#16810676	C₄	1	1	yes	characteristic
5	#16810676	C₄	1	1	yes	characteristic
6	C₄⋊#11719908	—	2	1	no	—
7	C₂²⋊#11719908	—	2	1	no	—
8	#16810399	C₄	1	2	yes	—
9	#16810345	—	2	1	no	—
10	#16810345	—	2	1	no	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×C₄⋊C₁₆	—	2	1	no	—
13	C₅₃×C₂²⋊C₁₆	—	2	1	no	—
14	C₄×C₁₆×C₅₃	C₄	1	2	yes	—
15	C₂×#11719908	—	2	2	no	—
16	C₄⋊#88322	—	2	1	no	—
17	C₄⋊#88322	C₈	1	2	yes	—
18	C₂²⋊#88322	—	2	1	no	—
19	C₂²⋊#88322	C₈	1	2	yes	—
20	C₄×#88322	—	2	1	no	—
21	C₄×#88322	—	2	1	no	—
22	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
24	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₅₃	—	2	2	no	—
26	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
33	C₂×C₈×C₅₃	—	4	1	no	—
34	C₂×C₈×C₅₃	—	4	1	no	—
35	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₄²×C₅₃	C₄²	1	1	yes	characteristic
40	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
42	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
43	C₂×#88322	—	2	2	no	—
44	C₂×#88322	—	2	2	no	—
45	C₁₆×C₅₃	—	4	2	no	—
46	#11719908	—	4	2	no	—
47	C₂³×C₅₃	C₄⋊C₈	1	1	yes	characteristic
48	C₂×C₄×C₅₃	—	2	1	no	—
49	C₂×C₄×C₅₃	—	2	1	no	—
50	C₂×C₄×C₅₃	—	2	2	no	—
51	C₂×C₄×C₅₃	—	4	1	no	—
52	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
58	C₈×C₅₃	—	2	1	no	—
59	C₈×C₅₃	—	2	1	no	—
60	C₈×C₅₃	—	2	1	no	—
61	C₈×C₅₃	—	2	1	no	—
62	#88322	—	4	2	no	—
63	#88322	—	4	2	no	—
64	#7147	—	53	1	no	maximal, Sylow
65	C₂²×C₅₃	—	2	1	no	—
66	C₂²×C₅₃	—	2	1	no	—
67	C₂²×C₅₃	—	2	1	no	—
68	C₂²×C₅₃	#336	1	1	yes	characteristic
69	C₄×C₅₃	—	2	1	no	—
70	C₄×C₅₃	—	2	1	no	—
71	C₄×C₅₃	—	2	1	no	—
72	C₄×C₅₃	—	2	1	no	—
73	C₄×C₅₃	H(3,ℤ/4)	1	1	yes	characteristic
74	C₄×C₅₃	#338	1	1	yes	characteristic
75	C₄×C₅₃	#350	1	2	yes	—
76	#2163	—	53	1	no	—
77	#2162	D₅₃	1	1	yes	characteristic
78	#1452	—	53	1	no	—
79	C₂×C₅₃	—	4	1	no	—
80	C₂×C₅₃	#1275	1	1	yes	characteristic
81	C₂×C₅₃	#1378	1	1	yes	characteristic
82	C₂×C₅₃	#1379	1	1	yes	characteristic
83	#431	—	53	1	no	—
84	#431	—	53	1	no	—
85	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
86	C₄⋊C₁₆	—	2	1	no	—
87	C₄⋊C₁₆	—	106	1	no	—
88	C₂²⋊C₁₆	—	2	1	no	—
89	C₂²⋊C₁₆	—	106	1	no	—
90	#346	—	53	2	no	—
91	C₄×C₁₆	Dic₅₃	1	2	yes	—
92	#322	—	106	1	no	—
93	#322	—	106	1	no	—
94	C₅₃	#7147	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
96	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
97	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
98	C₂×C₁₆	—	2	2	no	—
99	C₂×C₁₆	—	106	2	no	—
100	C₄⋊C₈	—	53	2	no	—
101	C₄⋊C₈	—	106	1	no	—
102	C₂²⋊C₈	—	53	2	no	—
103	C₂²⋊C₈	—	106	1	no	—
104	C₄×C₈	—	106	1	no	—
105	C₄×C₈	—	106	1	no	—
106	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
107	C₄×C₈	D₂₁₂	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	C₈×D₅₃	1	1	yes	characteristic
111	C₂²×C₄	#11719912	1	1	yes	characteristic
112	C₂²×C₄	#11719919	1	1	yes	characteristic
113	C₂×C₈	—	4	1	no	—
114	C₂×C₈	—	4	1	no	—
115	C₂×C₈	—	106	2	no	—
116	C₂×C₈	—	106	2	no	—
117	C₂×C₈	#11719919	1	1	yes	characteristic
118	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
119	C₂×C₈	#11719921	1	1	yes	characteristic
120	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
121	C₄²	C₈×D₅₃	1	1	yes	characteristic
122	C₄²	#11719912	1	1	yes	characteristic
123	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
124	C₄²	#11719921	1	1	yes	characteristic
125	C₁₆	—	4	2	no	—
126	C₁₆	—	212	2	no	—
127	C₂³	#13133146	1	1	yes	characteristic
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	#13133124	1	1	yes	characteristic
133	C₂×C₄	C₈×Dic₅₃	1	1	yes	characteristic
134	C₂×C₄	#13133127	1	1	yes	characteristic
135	C₂×C₄	#13133129	1	1	yes	characteristic
136	C₂×C₄	#13133129	1	1	yes	characteristic
137	C₂×C₄	#13133146	1	1	yes	characteristic
138	C₈	—	2	1	no	—
139	C₈	—	2	1	no	—
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	—	212	2	no	—
143	C₈	—	212	2	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	#16810377	1	1	yes	characteristic
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	#16810379	1	1	yes	characteristic
153	C₄	#16810381	1	1	yes	characteristic
154	C₄	#16810406	1	2	yes	—
155	C₂	—	4	1	no	—
156	C₂	#20272545	1	1	yes	characteristic
157	C₂	#20272800	1	1	yes	characteristic
158	C₂	#20272803	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276518

C₂

yes

maximal, characteristic

#20273007

C₂

yes

maximal, characteristic

C₅₃×#2162

C₂

yes

maximal, characteristic

#16810676

C₄

yes

characteristic

#16810676

C₄

yes

—

—

C₄

yes

—

#16810345

—

#16810345

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×C₄⋊C₁₆

—

C₅₃×C₂²⋊C₁₆

—

C₄×C₁₆×C₅₃

C₄

yes

—

C₂×#11719908

—

C₄⋊#88322

—

C₄⋊#88322

C₈

yes

—

C₂²⋊#88322

—

C₂²⋊#88322

C₈

yes

—

C₄×#88322

—

C₄×#88322

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

Q₈

yes

characteristic

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×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

C₂×C₈

yes

characteristic

C₂²×C₄×C₅₃

M₁₆

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

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

C₄²

yes

characteristic

C₄²×C₅₃

C₂²⋊C₄

yes

characteristic

C₄²×C₅₃

C₂×C₈

yes

characteristic

C₄²×C₅₃

M₁₆

yes

characteristic

C₂×#88322

—

C₂×#88322

—

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

C₄⋊C₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#95

yes

characteristic

C₂×C₄×C₅₃

C₄×C₈

yes

characteristic

C₂×C₄×C₅₃

#97

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₈

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#336

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

#2163

—

#2162

D₅₃

yes

characteristic

#1452

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

106

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

—

106

—

#346

—

C₄×C₁₆

Dic₅₃

yes

—

#322

—

106

—

#322

—

106

—

C₅₃

#7147

yes

characteristic, complemented, Sylow

C₂²×C₈

Dic₁₀₆

yes

characteristic

C₂²×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₄²

C₄×D₅₃

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

106

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

106

—

102

C₂²⋊C₈

—

103

C₂²⋊C₈

—

106

—

104

C₄×C₈

—

106

—

105

C₄×C₈

—

106

—

106

C₄×C₈

C₄×D₅₃

yes

characteristic

107

C₄×C₈

D₂₁₂

yes

characteristic

108

C₄×C₈

C₂×Dic₅₃

yes

characteristic

109

C₄×C₈

C₂²⋊D₅₃

yes

characteristic

110

C₂²×C₄

C₈×D₅₃

yes

characteristic

111

C₂²×C₄

#11719912

yes

characteristic

112

C₂²×C₄

#11719919

yes

characteristic

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

106

—

116

C₂×C₈

—

106

—

117

C₂×C₈

#11719919

yes

characteristic

118

C₂×C₈

C₄⋊Dic₅₃

yes

characteristic

119

C₂×C₈

#11719921

yes

characteristic

120

C₂×C₈

C₂²⋊Dic₅₃

yes

characteristic

121

C₄²

C₈×D₅₃

yes

characteristic

122

C₄²

#11719912

yes

characteristic

123

C₄²

C₄×Dic₅₃

yes

characteristic

124

C₄²

#11719921

yes

characteristic

125

C₁₆

—

126

C₁₆

—

212

—

127

C₂³

#13133146

yes

characteristic

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

#13133124

yes

characteristic

133

C₂×C₄

C₈×Dic₅₃

yes

characteristic

134

C₂×C₄

#13133127

yes

characteristic

135

C₂×C₄

#13133129

yes

characteristic

136

C₂×C₄

#13133129

yes

characteristic

137

C₂×C₄

#13133146

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

212

—

143

C₈

—

212

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

#16810377

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

#16810379

yes

characteristic

153

C₄

#16810381

yes

characteristic

154

C₄

#16810406

yes

—

155

C₂

—

156

C₂

#20272545

yes

characteristic

157

C₂

#20272800

yes

characteristic

158

C₂

#20272803

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133124
commutator	C₂×C₄×C₅₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₅₃×#2162	C₂
socle	C₂²×C₅₃	#336
prime radical	C₂³×C₅₃	C₄⋊C₈
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	#2162	D₅₃
hypocenter	C₅₃	#7147
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2162

subgroup

G/H

center

C₂×C₄

#13133124

commutator

C₂×C₄×C₅₃

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₅₃

Fitting

C₅₃×#2162

C₂

socle

C₂²×C₅₃

#336

prime radical

C₂³×C₅₃

C₄⋊C₈

cosocle

C₂×C₄×C₈×C₅₃

C₂²

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2162

N	H ≅ G/N	type
94. C₅₃	#7147	semidirect

H ≅ G/N

type

94. C₅₃

#7147

semidirect

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

subgroup	index
#20276518	2
#20273007	2
C₅₃×#2162	2
#7147	53

subgroup

index

#20276518

#20273007

C₅₃×#2162

#7147

Subgroups — #27270944

Conjugacy classes of subgroups

Complements

Maximal subgroups