Group #27271328 — subgroups

Subgroups — #27271328

Order 13568.

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

normal subgroups only

This group has 198 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	#20273471	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810376	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2163	C₂	1	1	yes	maximal, characteristic
4	#16810647	—	2	1	no	—
5	C₂×C₄⋊#88322	—	2	1	no	—
6	#16810376	—	2	1	no	—
7	#16810376	C₄	1	2	yes	—
8	#16810374	—	2	1	no	—
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	C₂×#11719917	—	4	1	no	—
15	C₂²×#88322	—	4	1	no	—
16	C₄⋊#88322	—	2	2	no	—
17	C₄⋊#88322	—	2	2	no	—
18	C₄⋊#88322	—	4	1	no	—
19	C₄⋊#88322	—	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	#11719917	—	8	2	no	—
42	C₂×#88322	—	4	2	no	—
43	C₂×#88322	—	4	2	no	—
44	C₂×#88322	—	8	2	no	—
45	C₁₆×C₅₃	—	4	2	no	—
46	C₂³×C₅₃	#106	1	1	yes	characteristic
47	C₂×C₄×C₅₃	—	2	2	no	—
48	C₂×C₄×C₅₃	—	2	1	no	—
49	C₂×C₄×C₅₃	—	2	1	no	—
50	C₂×C₄×C₅₃	—	2	1	no	—
51	C₂×C₄×C₅₃	—	2	1	no	—
52	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
55	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
57	C₂×C₄×C₅₃	#107	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	#108	1	2	yes	—
62	#88322	—	8	2	no	—
63	#88322	—	8	2	no	—
64	#7292	—	53	1	no	maximal, Sylow
65	C₂²×C₅₃	—	2	1	no	—
66	C₂²×C₅₃	—	2	1	no	—
67	C₂²×C₅₃	#340	1	1	yes	characteristic
68	C₂²×C₅₃	#365	1	2	yes	—
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	#341	1	1	yes	characteristic
74	C₄×C₅₃	#344	1	1	yes	characteristic
75	C₄×C₅₃	#361	1	1	yes	characteristic
76	C₄×C₅₃	#362	1	1	yes	characteristic
77	#2163	D₅₃	1	1	yes	characteristic
78	#1566	—	53	1	no	—
79	C₂×#335	—	53	1	no	—
80	C₂×C₅₃	—	2	2	no	—
81	C₂×C₅₃	#1301	1	1	yes	characteristic
82	C₂×C₅₃	#1383	1	1	yes	characteristic
83	C₂×C₅₃	#1387	1	1	yes	characteristic
84	#423	—	106	1	no	—
85	C₂×C₄⋊C₈	—	106	1	no	—
86	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
87	C₄⋊C₁₆	—	2	1	no	—
88	C₂²⋊C₁₆	Dic₅₃	1	2	yes	—
89	#346	—	2	1	no	—
90	#335	—	53	2	no	—
91	#335	—	106	1	no	—
92	#334	—	53	2	no	—
93	#334	—	106	1	no	—
94	C₅₃	#7292	1	1	yes	characteristic, complemented, Sylow
95	C₂×M₁₆	—	212	1	no	—
96	C₂²×C₈	—	212	1	no	—
97	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
98	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
99	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
100	C₂×C₁₆	—	2	2	no	—
101	C₄⋊C₈	—	106	2	no	—
102	C₄⋊C₈	—	106	2	no	—
103	C₄⋊C₈	—	212	1	no	—
104	C₄⋊C₈	—	212	1	no	—
105	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
106	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
107	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
108	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
109	C₂²×C₄	#11719913	1	1	yes	characteristic
110	C₂²×C₄	#11719913	1	1	yes	characteristic
111	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
112	M₁₆	—	424	2	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	4	1	no	—
115	C₂×C₈	—	424	2	no	—
116	C₂×C₈	—	212	2	no	—
117	C₂×C₈	—	212	2	no	—
118	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
119	C₂×C₈	#11719919	1	1	yes	characteristic
120	C₂×C₈	#11719919	1	1	yes	characteristic
121	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
122	C₄²	D₄₂₄	1	1	yes	characteristic
123	C₄²	Dic₂₁₂	1	1	yes	characteristic
124	C₄²	#11719921	1	1	yes	characteristic
125	C₄²	#11719921	1	1	yes	characteristic
126	C₁₆	—	4	2	no	—
127	C₂³	#13133149	1	1	yes	characteristic
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	#13133124	1	1	yes	characteristic
134	C₂×C₄	#13133139	1	1	yes	characteristic
135	C₂×C₄	#13133139	1	1	yes	characteristic
136	C₂×C₄	#13133142	1	1	yes	characteristic
137	C₂×C₄	#13133142	1	1	yes	characteristic
138	C₂×C₄	#13133151	1	1	yes	characteristic
139	C₈	—	2	1	no	—
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	—	424	2	no	—
143	C₈	—	424	2	no	—
144	C₈	#13133152	1	2	yes	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	#16810386	1	1	yes	characteristic
148	C₂²	#16810446	1	2	yes	—
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	#16810387	1	1	yes	characteristic
154	C₄	#16810393	1	1	yes	characteristic
155	C₄	#16810437	1	1	yes	characteristic
156	C₄	#16810440	1	1	yes	characteristic
157	C₂	—	2	2	no	—
158	C₂	#20272611	1	1	yes	characteristic
159	C₂	#20272814	1	1	yes	characteristic
160	C₂	#20272824	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20273471

C₂

yes

maximal, characteristic

C₂×#16810376

C₂

yes

maximal, characteristic

C₅₃×#2163

C₂

yes

maximal, characteristic

#16810647

—

C₂×C₄⋊#88322

—

#16810376

—

#16810376

C₄

yes

—

#16810374

—

#16810374

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×C₄⋊C₁₆

—

C₅₃×C₂²⋊C₁₆

C₄

yes

—

C₅₃×#346

—

C₂×#11719917

—

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

—

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

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

D₅₃

yes

characteristic

#1566

—

C₂×#335

—

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

—

#335

—

106

—

#334

—

#334

—

106

—

C₅₃

#7292

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

212

—

C₂²×C₈

—

212

—

C₂²×C₈

Dic₁₀₆

yes

characteristic

C₂²×C₈

C₂×Dic₅₃

yes

characteristic

C₂×C₄²

D₂₁₂

yes

characteristic

100

C₂×C₁₆

—

101

C₄⋊C₈

—

106

—

102

C₄⋊C₈

—

106

—

103

C₄⋊C₈

—

212

—

104

C₄⋊C₈

—

212

—

105

C₄×C₈

C₄×D₅₃

yes

characteristic

106

C₄×C₈

C₄×D₅₃

yes

characteristic

107

C₄×C₈

C₂²⋊D₅₃

yes

characteristic

108

C₄×C₈

C₂²⋊D₅₃

yes

characteristic

109

C₂²×C₄

#11719913

yes

characteristic

110

C₂²×C₄

#11719913

yes

characteristic

111

C₂²×C₄

C₄⋊Dic₅₃

yes

characteristic

112

M₁₆

—

424

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

424

—

116

C₂×C₈

—

212

—

117

C₂×C₈

—

212

—

118

C₂×C₈

C₄×Dic₅₃

yes

characteristic

119

C₂×C₈

#11719919

yes

characteristic

120

C₂×C₈

#11719919

yes

characteristic

121

C₂×C₈

C₂²⋊Dic₅₃

yes

characteristic

122

C₄²

D₄₂₄

yes

characteristic

123

C₄²

Dic₂₁₂

yes

characteristic

124

C₄²

#11719921

yes

characteristic

125

C₄²

#11719921

yes

characteristic

126

C₁₆

—

127

C₂³

#13133149

yes

characteristic

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

#13133124

yes

characteristic

134

C₂×C₄

#13133139

yes

characteristic

135

C₂×C₄

#13133139

yes

characteristic

136

C₂×C₄

#13133142

yes

characteristic

137

C₂×C₄

#13133142

yes

characteristic

138

C₂×C₄

#13133151

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

424

—

143

C₈

—

424

—

144

C₈

#13133152

yes

—

145

C₂²

—

146

C₂²

—

147

C₂²

#16810386

yes

characteristic

148

C₂²

#16810446

yes

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

#16810387

yes

characteristic

154

C₄

#16810393

yes

characteristic

155

C₄

#16810437

yes

characteristic

156

C₄

#16810440

yes

characteristic

157

C₂

—

158

C₂

#20272611

yes

characteristic

159

C₂

#20272814

yes

characteristic

160

C₂

#20272824

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	#7292
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
94. C₅₃	#7292	semidirect

H ≅ G/N

type

94. C₅₃

#7292

semidirect

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

subgroup	index
#20273471	2
C₂×#16810376	2
C₅₃×#2163	2
#7292	53

subgroup

index

#20273471

C₂×#16810376

C₅₃×#2163

#7292

Subgroups — #27271328

Conjugacy classes of subgroups

Complements

Maximal subgroups