Group #27271334 — subgroups

Subgroups — #27271334

Order 13568.

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

normal subgroups only

This group has 190 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	#20273478	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810374	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2163	C₂	1	1	yes	maximal, characteristic
4	#16810680	—	2	1	no	—
5	C₂×C₄⋊#88322	—	2	1	no	—
6	#16810376	—	2	1	no	—
7	#16810374	—	2	1	no	—
8	#16810374	C₄	1	2	yes	—
9	#16810374	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×C₄⋊C₁₆	—	2	1	no	—
12	C₅₃×C₂²⋊C₁₆	C₄	1	2	yes	—
13	C₅₃×#346	—	2	1	no	—
14	C₂²×#88322	—	4	1	no	—
15	C₄⋊#88322	—	2	2	no	—
16	C₄⋊#88322	—	2	2	no	—
17	C₄⋊#88322	—	4	1	no	—
18	C₂²⋊#88322	—	4	1	no	—
19	#13133128	—	4	1	no	—
20	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
22	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
23	C₂×C₁₆×C₅₃	—	2	2	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	C₄²	1	1	yes	characteristic
34	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
37	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₄²×C₅₃	D₈	1	1	yes	characteristic
40	C₄²×C₅₃	Q₁₆	1	1	yes	characteristic
41	C₂×#88322	—	4	2	no	—
42	C₂×#88322	—	4	2	no	—
43	C₂×#88322	—	8	2	no	—
44	C₁₆×C₅₃	—	4	2	no	—
45	C₂³×C₅₃	#106	1	1	yes	characteristic
46	C₂×C₄×C₅₃	—	2	2	no	—
47	C₂×C₄×C₅₃	—	2	1	no	—
48	C₂×C₄×C₅₃	—	2	1	no	—
49	C₂×C₄×C₅₃	—	4	1	no	—
50	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
51	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
55	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
56	C₈×C₅₃	—	2	1	no	—
57	C₈×C₅₃	—	2	1	no	—
58	C₈×C₅₃	—	2	1	no	—
59	C₈×C₅₃	#108	1	2	yes	—
60	#88322	—	8	2	no	—
61	#88322	—	8	2	no	—
62	#7294	—	53	1	no	maximal, Sylow
63	C₂²×C₅₃	—	2	1	no	—
64	C₂²×C₅₃	—	2	1	no	—
65	C₂²×C₅₃	#340	1	1	yes	characteristic
66	C₂²×C₅₃	#365	1	2	yes	—
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	—	2	1	no	—
72	C₄×C₅₃	#341	1	1	yes	characteristic
73	C₄×C₅₃	#344	1	1	yes	characteristic
74	#2163	D₅₃	1	1	yes	characteristic
75	#1568	—	53	1	no	—
76	C₂×#334	—	53	1	no	—
77	C₂×C₅₃	—	2	2	no	—
78	C₂×C₅₃	#1301	1	1	yes	characteristic
79	C₂×C₅₃	#1383	1	1	yes	characteristic
80	C₂×C₅₃	#1389	1	1	yes	characteristic
81	#432	—	106	1	no	—
82	C₂×C₄⋊C₈	—	106	1	no	—
83	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
84	C₄⋊C₁₆	—	2	1	no	—
85	C₂²⋊C₁₆	Dic₅₃	1	2	yes	—
86	#346	—	2	1	no	—
87	#335	—	106	1	no	—
88	#334	—	53	2	no	—
89	#334	—	53	2	no	—
90	#334	—	106	1	no	—
91	C₅₃	#7294	1	1	yes	characteristic, complemented, Sylow
92	C₂²×C₈	—	212	1	no	—
93	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
94	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
95	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
96	C₂×C₁₆	—	2	2	no	—
97	C₄⋊C₈	—	106	2	no	—
98	C₄⋊C₈	—	106	2	no	—
99	C₄⋊C₈	—	212	1	no	—
100	C₂²⋊C₈	—	212	1	no	—
101	#97	—	212	1	no	—
102	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
103	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
104	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
105	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
106	C₂²×C₄	#11719913	1	1	yes	characteristic
107	C₂²×C₄	#11719913	1	1	yes	characteristic
108	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	4	1	no	—
111	C₂×C₈	—	424	2	no	—
112	C₂×C₈	—	212	2	no	—
113	C₂×C₈	—	212	2	no	—
114	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
115	C₂×C₈	#11719919	1	1	yes	characteristic
116	C₂×C₈	#11719919	1	1	yes	characteristic
117	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
118	C₄²	D₄₂₄	1	1	yes	characteristic
119	C₄²	Dic₂₁₂	1	1	yes	characteristic
120	C₄²	#11719921	1	1	yes	characteristic
121	C₄²	#11719921	1	1	yes	characteristic
122	C₁₆	—	4	2	no	—
123	C₂³	#13133149	1	1	yes	characteristic
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	#13133124	1	1	yes	characteristic
129	C₂×C₄	#13133139	1	1	yes	characteristic
130	C₂×C₄	#13133139	1	1	yes	characteristic
131	C₂×C₄	#13133142	1	1	yes	characteristic
132	C₂×C₄	#13133142	1	1	yes	characteristic
133	C₂×C₄	#13133151	1	1	yes	characteristic
134	C₈	—	2	1	no	—
135	C₈	—	2	1	no	—
136	C₈	—	2	1	no	—
137	C₈	—	424	2	no	—
138	C₈	—	424	2	no	—
139	C₈	#13133152	1	2	yes	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	#16810386	1	1	yes	characteristic
143	C₂²	#16810446	1	2	yes	—
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	#16810387	1	1	yes	characteristic
150	C₄	#16810393	1	1	yes	characteristic
151	C₂	—	2	2	no	—
152	C₂	#20272611	1	1	yes	characteristic
153	C₂	#20272814	1	1	yes	characteristic
154	C₂	#20272830	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20273478

C₂

yes

maximal, characteristic

C₂×#16810374

C₂

yes

maximal, characteristic

C₅₃×#2163

C₂

yes

maximal, characteristic

#16810680

—

C₂×C₄⋊#88322

—

#16810376

—

#16810374

—

#16810374

C₄

yes

—

#16810374

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×C₄⋊C₁₆

—

C₅₃×C₂²⋊C₁₆

C₄

yes

—

C₅₃×#346

—

C₂²×#88322

—

—

—

—

—

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

Q₈

yes

characteristic

C₂×C₄²×C₅₃

D₄

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

SD₁₆

yes

characteristic

C₂²×C₄×C₅₃

SD₁₆

yes

characteristic

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

—

—

—

C₁₆×C₅₃

—

C₂³×C₅₃

#106

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

#108

yes

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#340

yes

characteristic

C₂²×C₅₃

#365

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

D₅₃

yes

characteristic

#1568

—

C₂×#334

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

106

—

C₂×C₄⋊C₈

—

106

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

Dic₅₃

yes

—

#346

—

#335

—

106

—

#334

—

#334

—

#334

—

106

—

C₅₃

#7294

yes

characteristic, complemented, Sylow

C₂²×C₈

—

212

—

C₂²×C₈

Dic₁₀₆

yes

characteristic

C₂²×C₈

C₂×Dic₅₃

yes

characteristic

C₂×C₄²

D₂₁₂

yes

characteristic

C₂×C₁₆

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

212

—

100

C₂²⋊C₈

—

212

—

101

#97

—

212

—

102

C₄×C₈

C₄×D₅₃

yes

characteristic

103

C₄×C₈

C₄×D₅₃

yes

characteristic

104

C₄×C₈

C₂²⋊D₅₃

yes

characteristic

105

C₄×C₈

C₂²⋊D₅₃

yes

characteristic

106

C₂²×C₄

#11719913

yes

characteristic

107

C₂²×C₄

#11719913

yes

characteristic

108

C₂²×C₄

C₄⋊Dic₅₃

yes

characteristic

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

424

—

112

C₂×C₈

—

212

—

113

C₂×C₈

—

212

—

114

C₂×C₈

C₄×Dic₅₃

yes

characteristic

115

C₂×C₈

#11719919

yes

characteristic

116

C₂×C₈

#11719919

yes

characteristic

117

C₂×C₈

C₂²⋊Dic₅₃

yes

characteristic

118

C₄²

D₄₂₄

yes

characteristic

119

C₄²

Dic₂₁₂

yes

characteristic

120

C₄²

#11719921

yes

characteristic

121

C₄²

#11719921

yes

characteristic

122

C₁₆

—

123

C₂³

#13133149

yes

characteristic

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

#13133124

yes

characteristic

129

C₂×C₄

#13133139

yes

characteristic

130

C₂×C₄

#13133139

yes

characteristic

131

C₂×C₄

#13133142

yes

characteristic

132

C₂×C₄

#13133142

yes

characteristic

133

C₂×C₄

#13133151

yes

characteristic

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

424

—

138

C₈

—

424

—

139

C₈

#13133152

yes

—

140

C₂²

—

141

C₂²

—

142

C₂²

#16810386

yes

characteristic

143

C₂²

#16810446

yes

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

#16810387

yes

characteristic

150

C₄

#16810393

yes

characteristic

151

C₂

—

152

C₂

#20272611

yes

characteristic

153

C₂

#20272814

yes

characteristic

154

C₂

#20272830

yes

characteristic

	subgroup	G/H
center	C₂²	#16810386
commutator	C₂×C₈×C₅₃	C₄²
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₅₃×#2163	C₂
socle	C₂²×C₅₃	#340
prime radical	C₂³×C₅₃	#106
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	#2163	D₅₃
hypocenter	C₅₃	#7294
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2163

subgroup

G/H

center

C₂²

#16810386

commutator

C₂×C₈×C₅₃

C₄²

Frattini

C₂×C₄×C₈

C₂×D₅₃

Fitting

C₅₃×#2163

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 ◁ #2163

N	H ≅ G/N	type
91. C₅₃	#7294	semidirect

H ≅ G/N

type

91. C₅₃

#7294

semidirect

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

subgroup	index
#20273478	2
C₂×#16810374	2
C₅₃×#2163	2
#7294	53

subgroup

index

#20273478

C₂×#16810374

C₅₃×#2163

#7294

Subgroups — #27271334

Conjugacy classes of subgroups

Complements

Maximal subgroups