Group #27271326 — subgroups

Subgroups — #27271326

Order 13568.

192 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 91 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	#20276514	C₂	1	1	yes	maximal, characteristic
2	#20273474	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic
4	#16810647	—	2	1	no	—
5	C₄⋊#11719908	—	2	1	no	—
6	C₂²⋊#11719908	C₄	1	2	yes	—
7	C₄×#11719908	—	2	1	no	—
8	#16810376	—	2	1	no	—
9	#16810376	C₄	1	2	yes	—
10	C₅₃×#431	—	2	1	no	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#335	C₄	1	2	yes	—
13	C₅₃×#334	—	2	1	no	—
14	C₂×#11719917	—	4	1	no	—
15	C₂×#11719908	—	2	2	no	—
16	C₄⋊#88322	—	2	2	no	—
17	C₄⋊#88322	—	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	SD₁₆	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	SD₁₆	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	D₈	1	1	yes	characteristic
41	C₄²×C₅₃	Q₁₆	1	1	yes	characteristic
42	#11719917	—	8	2	no	—
43	C₂×#88322	—	4	2	no	—
44	#11719908	—	4	2	no	—
45	C₂³×C₅₃	#106	1	1	yes	characteristic
46	C₂×C₄×C₅₃	—	2	1	no	—
47	C₂×C₄×C₅₃	—	2	1	no	—
48	C₂×C₄×C₅₃	—	2	2	no	—
49	C₂×C₄×C₅₃	—	2	1	no	—
50	C₂×C₄×C₅₃	—	2	1	no	—
51	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
55	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
57	C₈×C₅₃	—	2	1	no	—
58	C₈×C₅₃	—	2	1	no	—
59	C₈×C₅₃	—	2	1	no	—
60	C₈×C₅₃	—	8	2	no	—
61	C₈×C₅₃	#108	1	2	yes	—
62	#88322	—	8	2	no	—
63	#7291	—	53	1	no	maximal, Sylow
64	C₂²×C₅₃	—	2	1	no	—
65	C₂²×C₅₃	—	2	1	no	—
66	C₂²×C₅₃	—	2	1	no	—
67	C₂²×C₅₃	#340	1	1	yes	characteristic
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	C₄×C₅₃	#359	1	2	yes	—
74	C₄×C₅₃	#361	1	1	yes	characteristic
75	C₄×C₅₃	#362	1	1	yes	characteristic
76	#2162	—	53	1	no	—
77	#1567	—	53	1	no	—
78	#1564	D₅₃	1	1	yes	characteristic
79	C₂×C₅₃	—	4	1	no	—
80	C₂×C₅₃	#1301	1	1	yes	characteristic
81	C₂×C₅₃	#1382	1	1	yes	characteristic
82	C₂×C₅₃	#1387	1	1	yes	characteristic
83	#431	—	2	1	no	—
84	#423	—	106	1	no	—
85	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
86	C₄⋊C₁₆	—	106	1	no	—
87	C₂²⋊C₁₆	—	53	2	no	—
88	C₄×C₁₆	—	106	1	no	—
89	#335	—	53	2	no	—
90	#335	—	106	1	no	—
91	#335	Dic₅₃	1	2	yes	—
92	#334	—	2	1	no	—
93	C₅₃	#7291	1	1	yes	characteristic, complemented, Sylow
94	C₂×M₁₆	—	212	1	no	—
95	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
96	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
97	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
98	C₂×C₁₆	—	106	2	no	—
99	C₄⋊C₈	—	2	2	no	—
100	C₄⋊C₈	—	106	2	no	—
101	C₄⋊C₈	—	212	1	no	—
102	C₂²⋊C₈	—	4	1	no	—
103	C₄×C₈	—	4	1	no	—
104	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
105	C₄×C₈	D₂₁₂	1	1	yes	characteristic
106	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
107	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
108	C₂²×C₄	#11719919	1	1	yes	characteristic
109	C₂²×C₄	#11719923	1	1	yes	characteristic
110	C₂²×C₄	#11719924	1	1	yes	characteristic
111	M₁₆	—	424	2	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	4	1	no	—
114	C₂×C₈	—	4	2	no	—
115	C₂×C₈	—	212	2	no	—
116	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
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	C₄²	#11719921	1	1	yes	characteristic
121	C₄²	#11719922	1	1	yes	characteristic
122	C₄²	#11719925	1	1	yes	characteristic
123	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
124	C₁₆	—	212	2	no	—
125	C₂³	#13133148	1	1	yes	characteristic
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	#13133124	1	1	yes	characteristic
132	C₂×C₄	#13133137	1	1	yes	characteristic
133	C₂×C₄	#13133138	1	1	yes	characteristic
134	C₂×C₄	#13133140	1	1	yes	characteristic
135	C₂×C₄	#13133141	1	1	yes	characteristic
136	C₂×C₄	#13133150	1	1	yes	characteristic
137	C₈	—	2	1	no	—
138	C₈	—	2	1	no	—
139	C₈	—	2	1	no	—
140	C₈	—	8	2	no	—
141	C₈	—	424	2	no	—
142	C₈	#13133153	1	2	yes	—
143	C₂²	—	2	1	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	#16810385	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	#16810388	1	1	yes	characteristic
151	C₄	#16810395	1	1	yes	characteristic
152	C₄	#16810430	1	2	yes	—
153	C₄	#16810435	1	1	yes	characteristic
154	C₄	#16810438	1	1	yes	characteristic
155	C₂	—	4	1	no	—
156	C₂	#20272612	1	1	yes	characteristic
157	C₂	#20272810	1	1	yes	characteristic
158	C₂	#20272825	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276514

C₂

yes

maximal, characteristic

#20273474

C₂

yes

maximal, characteristic

C₅₃×#1564

C₂

yes

maximal, characteristic

#16810647

—

C₄⋊#11719908

—

C₂²⋊#11719908

C₄

yes

—

C₄×#11719908

—

#16810376

—

#16810376

C₄

yes

—

C₅₃×#431

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#335

C₄

yes

—

C₅₃×#334

—

—

—

—

—

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

SD₁₆

yes

characteristic

C₂²×C₄×C₅₃

SD₁₆

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

D₈

yes

characteristic

C₄²×C₅₃

Q₁₆

yes

—

—

—

C₂³×C₅₃

#106

yes

characteristic

C₂×C₄×C₅₃

—

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

—

#7291

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#340

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

—

—

D₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

106

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

C₄⋊C₁₆

—

106

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

106

—

#335

—

#335

—

106

—

#335

Dic₅₃

yes

—

#334

—

C₅₃

#7291

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

212

—

C₂²×C₈

Dic₁₀₆

yes

characteristic

C₂²×C₈

C₄×D₅₃

yes

characteristic

C₂×C₄²

C₂²⋊D₅₃

yes

characteristic

C₂×C₁₆

—

106

—

C₄⋊C₈

—

100

C₄⋊C₈

—

106

—

101

C₄⋊C₈

—

212

—

102

C₂²⋊C₈

—

103

C₄×C₈

—

104

C₄×C₈

C₄×D₅₃

yes

characteristic

105

C₄×C₈

D₂₁₂

yes

characteristic

106

C₄×C₈

C₂×Dic₅₃

yes

characteristic

107

C₄×C₈

C₂²⋊D₅₃

yes

characteristic

108

C₂²×C₄

#11719919

yes

characteristic

109

C₂²×C₄

#11719923

yes

characteristic

110

C₂²×C₄

#11719924

yes

characteristic

111

M₁₆

—

424

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

212

—

116

C₂×C₈

C₄×Dic₅₃

yes

characteristic

117

C₂×C₈

#11719919

yes

characteristic

118

C₂×C₈

C₄⋊Dic₅₃

yes

characteristic

119

C₂×C₈

#11719921

yes

characteristic

120

C₄²

#11719921

yes

characteristic

121

C₄²

#11719922

yes

characteristic

122

C₄²

#11719925

yes

characteristic

123

C₄²

C₂²⋊Dic₅₃

yes

characteristic

124

C₁₆

—

212

—

125

C₂³

#13133148

yes

characteristic

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

#13133124

yes

characteristic

132

C₂×C₄

#13133137

yes

characteristic

133

C₂×C₄

#13133138

yes

characteristic

134

C₂×C₄

#13133140

yes

characteristic

135

C₂×C₄

#13133141

yes

characteristic

136

C₂×C₄

#13133150

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

424

—

142

C₈

#13133153

yes

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

#16810385

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

#16810388

yes

characteristic

151

C₄

#16810395

yes

characteristic

152

C₄

#16810430

yes

—

153

C₄

#16810435

yes

characteristic

154

C₄

#16810438

yes

characteristic

155

C₂

—

156

C₂

#20272612

yes

characteristic

157

C₂

#20272810

yes

characteristic

158

C₂

#20272825

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133137
commutator	C₂×C₈×C₅₃	C₄²
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₅₃×#1564	C₂
socle	C₂²×C₅₃	#340
prime radical	C₂³×C₅₃	#106
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	#1564	D₅₃
hypocenter	C₅₃	#7291
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1564

subgroup

G/H

center

C₂×C₄

#13133137

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 ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1564

N	H ≅ G/N	type
93. C₅₃	#7291	semidirect

H ≅ G/N

type

93. C₅₃

#7291

semidirect

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

subgroup	index
#20276514	2
#20273474	2
C₅₃×#1564	2
#7291	53

subgroup

index

#20276514

#20273474

C₅₃×#1564

#7291

Subgroups — #27271326

Conjugacy classes of subgroups

Complements

Maximal subgroups