Group #20272539 — subgroups

Subgroups — #20272539

Order 6784.

158 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 61 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	#16810521	C₂	1	1	yes	maximal, characteristic
2	#16810378	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#336	C₂	1	1	yes	maximal, characteristic
4	C₂²×#88322	—	2	1	no	—
5	C₂×C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
6	C₂×C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
7	C₂×#11719919	—	2	1	no	—
8	#13133124	—	2	1	no	—
9	C₂²×C₈×C₅₃	—	2	1	no	—
10	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
13	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
14	C₂²×Dic₅₃	—	2	1	no	—
15	C₂²×Dic₅₃	—	2	1	no	—
16	C₂×C₈×C₅₃	—	2	2	no	—
17	C₂×C₈×C₅₃	—	4	1	no	—
18	C₂×C₈×C₅₃	—	4	1	no	—
19	C₄²×C₅₃	—	4	1	no	—
20	C₄⋊Dic₅₃	—	2	1	no	—
21	C₄⋊Dic₅₃	—	4	1	no	—
22	C₄⋊Dic₅₃	C₈	1	2	yes	complemented
23	#11719919	—	4	1	no	—
24	#11719919	—	4	1	no	—
25	C₂×#88322	—	2	2	no	—
26	C₂×#88322	—	4	1	no	—
27	C₂×#88322	—	4	1	no	—
28	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₅₃	—	2	1	no	—
30	C₂×C₄×C₅₃	—	2	1	no	—
31	C₂×C₄×C₅₃	—	2	1	no	—
32	C₂×C₄×C₅₃	—	2	1	no	—
33	C₂×C₄×C₅₃	—	2	1	no	—
34	C₂×C₄×C₅₃	—	2	1	no	—
35	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
36	C₂×C₄×C₅₃	M₁₆	1	1	yes	characteristic
37	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
38	C₂×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
39	C₂×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
40	C₂×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
41	C₂×Dic₅₃	—	2	2	no	—
42	C₂×Dic₅₃	—	4	1	no	—
43	C₂×Dic₅₃	—	4	1	no	—
44	C₂×Dic₅₃	—	4	1	no	—
45	C₂×Dic₅₃	—	4	1	no	—
46	C₂×Dic₅₃	—	4	1	no	—
47	C₈×C₅₃	—	4	2	no	—
48	#88322	—	4	2	no	—
49	C₂²×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
50	C₂²×C₅₃	#99	1	1	yes	characteristic
51	C₂²×C₅₃	#101	1	1	yes	characteristic
52	C₂²×C₅₃	#102	1	1	yes	characteristic
53	C₂²×C₅₃	#103	1	1	yes	characteristic
54	C₂²×C₅₃	#104	1	1	yes	characteristic
55	C₂²×C₅₃	#104	1	1	yes	characteristic
56	C₄×C₅₃	—	2	1	no	—
57	C₄×C₅₃	—	2	1	no	—
58	C₄×C₅₃	—	2	1	no	—
59	C₄×C₅₃	—	2	1	no	—
60	C₄×C₅₃	—	2	1	no	—
61	C₄×C₅₃	—	2	1	no	—
62	Dic₅₃	—	4	2	no	—
63	Dic₅₃	—	8	1	no	—
64	#1271	—	53	1	no	maximal, Sylow
65	C₂×C₅₃	#324	1	1	yes	characteristic
66	C₂×C₅₃	D₄⋊C₈	1	1	yes	characteristic
67	C₂×C₅₃	Q₈⋊C₈	1	1	yes	characteristic
68	C₂×C₅₃	#327	1	1	yes	characteristic
69	C₂×C₅₃	#328	1	1	yes	characteristic
70	C₂×C₅₃	#330	1	1	yes	characteristic
71	C₂×C₅₃	#332	1	1	yes	characteristic
72	#389	—	53	1	no	—
73	#336	—	53	1	no	—
74	#336	D₅₃	1	1	yes	characteristic
75	C₅₃	#1271	1	1	yes	characteristic, complemented, Sylow
76	C₂²×C₈	—	2	1	no	—
77	C₂²×C₈	—	106	1	no	—
78	C₂×C₄⋊C₄	—	53	1	no	—
79	C₂×C₄⋊C₄	—	53	1	no	—
80	C₂×C₄⋊C₄	—	106	1	no	—
81	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
82	#95	—	106	1	no	—
83	C₂²×C₄	—	106	1	no	—
84	C₂²×C₄	—	106	1	no	—
85	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
86	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
87	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	4	1	no	—
90	C₂×C₈	—	4	1	no	—
91	C₂×C₈	—	106	2	no	—
92	C₂×C₈	—	212	1	no	—
93	C₂×C₈	—	212	1	no	—
94	C₄⋊C₄	—	53	2	no	—
95	C₄⋊C₄	—	106	1	no	—
96	C₄⋊C₄	—	212	1	no	—
97	C₄⋊C₄	—	212	1	no	—
98	C₄⋊C₄	—	212	1	no	—
99	C₄²	—	4	1	no	—
100	C₂³	#11719921	1	1	yes	characteristic
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	106	2	no	—
108	C₂×C₄	—	212	1	no	—
109	C₂×C₄	—	212	1	no	—
110	C₂×C₄	—	212	1	no	—
111	C₂×C₄	—	212	1	no	—
112	C₂×C₄	—	212	1	no	—
113	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
114	C₂×C₄	#11719912	1	1	yes	characteristic
115	C₂×C₄	#11719913	1	1	yes	characteristic
116	C₂×C₄	Dic₂₁₂	1	1	yes	characteristic
117	C₂×C₄	#11719922	1	1	yes	characteristic
118	C₂×C₄	#11719924	1	1	yes	characteristic
119	C₈	—	4	2	no	—
120	C₈	—	212	2	no	—
121	C₂²	#13133129	1	1	yes	characteristic
122	C₂²	#13133131	1	1	yes	characteristic
123	C₂²	#13133135	1	1	yes	characteristic
124	C₂²	#13133137	1	1	yes	characteristic
125	C₂²	#13133142	1	1	yes	characteristic
126	C₂²	#13133143	1	1	yes	characteristic
127	C₂²	#13133145	1	1	yes	characteristic
128	C₄	—	2	1	no	—
129	C₄	—	2	1	no	—
130	C₄	—	2	1	no	—
131	C₄	—	2	1	no	—
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	424	1	no	—
135	C₄	—	212	2	no	—
136	C₂	#16810348	1	1	yes	characteristic
137	C₂	#16810350	1	1	yes	characteristic
138	C₂	#16810355	1	1	yes	characteristic
139	C₂	#16810356	1	1	yes	characteristic
140	C₂	#16810361	1	1	yes	characteristic
141	C₂	#16810364	1	1	yes	characteristic
142	C₂	#16810370	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810521

C₂

yes

maximal, characteristic

#16810378

C₂

yes

maximal, characteristic

C₅₃×#336

C₂

yes

maximal, characteristic

C₂²×#88322

—

C₂×C₄⋊Dic₅₃

C₄

yes

characteristic

C₂×C₄⋊Dic₅₃

C₄

yes

characteristic

C₂×#11719919

—

#13133124

—

C₂²×C₈×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₈

yes

—

—

—

—

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

M₁₆

yes

characteristic

C₂×C₄×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

SD₁₆

yes

characteristic

C₂×C₄×C₅₃

SD₁₆

yes

characteristic

C₂×C₄×C₅₃

Q₁₆

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

C₂²⋊C₈

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

D₄⋊C₈

yes

characteristic

C₂×C₅₃

Q₈⋊C₈

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

D₅₃

yes

characteristic

C₅₃

#1271

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#95

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

D₂₁₂

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

212

—

C₂×C₈

—

212

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₄⋊C₄

—

212

—

C₄⋊C₄

—

212

—

C₄²

—

100

C₂³

#11719921

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

106

—

108

C₂×C₄

—

212

—

109

C₂×C₄

—

212

—

110

C₂×C₄

—

212

—

111

C₂×C₄

—

212

—

112

C₂×C₄

—

212

—

113

C₂×C₄

C₈×D₅₃

yes

characteristic

114

C₂×C₄

#11719912

yes

characteristic

115

C₂×C₄

#11719913

yes

characteristic

116

C₂×C₄

Dic₂₁₂

yes

characteristic

117

C₂×C₄

#11719922

yes

characteristic

118

C₂×C₄

#11719924

yes

characteristic

119

C₈

—

120

C₈

—

212

—

121

C₂²

#13133129

yes

characteristic

122

C₂²

#13133131

yes

characteristic

123

C₂²

#13133135

yes

characteristic

124

C₂²

#13133137

yes

characteristic

125

C₂²

#13133142

yes

characteristic

126

C₂²

#13133143

yes

characteristic

127

C₂²

#13133145

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

424

—

135

C₄

—

212

—

136

C₂

#16810348

yes

characteristic

137

C₂

#16810350

yes

characteristic

138

C₂

#16810355

yes

characteristic

139

C₂

#16810356

yes

characteristic

140

C₂

#16810361

yes

characteristic

141

C₂

#16810364

yes

characteristic

142

C₂

#16810370

yes

characteristic

	subgroup	G/H
center	C₂³	#11719921
commutator	C₂×C₄×C₅₃	C₂×C₈
Frattini	C₂×C₄²	C₂×D₅₃
Fitting	C₅₃×#336	C₂
socle	C₂³×C₅₃	C₂²⋊C₄
hypercenter	#336	D₅₃
hypocenter	C₅₃	#1271
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

#11719921

commutator

C₂×C₄×C₅₃

C₂×C₈

Frattini

C₂×C₄²

C₂×D₅₃

Fitting

C₅₃×#336

C₂

socle

C₂³×C₅₃

C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
22. C₄⋊Dic₅₃	C₈	semidirect
75. C₅₃	#1271	semidirect

H ≅ G/N

type

22. C₄⋊Dic₅₃

C₈

semidirect

75. C₅₃

#1271

semidirect

Subgroups — #20272539

Conjugacy classes of subgroups

Complements