Group #20272573 — subgroups

Subgroups — #20272573

Order 6784.

156 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 73 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	#16810676	C₂	1	1	yes	maximal, characteristic
2	#16810647	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#382	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719917	—	2	1	no	—
5	C₄⋊#88322	—	2	1	no	—
6	C₄⋊#88322	—	2	1	no	—
7	C₄⋊#88322	C₄	1	2	yes	complemented
8	C₂²⋊#88322	—	2	1	no	—
9	C₄×#88322	C₄	1	2	yes	complemented
10	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
11	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	#95×C₅₃	—	2	1	no	—
13	#95×C₅₃	C₄	1	2	yes	—
14	C₂²×C₄×C₅₃	—	2	2	no	—
15	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
18	C₄⋊C₄×C₅₃	—	4	1	no	—
19	C₄⋊C₄×C₅₃	—	4	1	no	—
20	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₅₃	D₄	1	1	yes	characteristic
23	C₄²×C₅₃	Q₈	1	1	yes	characteristic
24	#11719917	—	4	1	no	—
25	#11719917	—	4	1	no	—
26	C₂×#88322	—	2	2	no	—
27	C₂×#88322	—	2	2	no	—
28	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₅₃	—	4	1	no	—
30	C₂×C₄×C₅₃	—	4	1	no	—
31	C₂×C₄×C₅₃	—	4	2	no	—
32	C₂×C₄×C₅₃	—	4	2	no	—
33	C₂×C₄×C₅₃	—	4	2	no	—
34	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
41	C₂×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
42	C₂×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
43	C₂×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
44	#88322	—	4	2	no	—
45	#88322	—	4	2	no	—
46	C₂²×C₅₃	—	2	1	no	—
47	C₂²×C₅₃	—	2	1	no	—
48	C₂²×C₅₃	#95	1	1	yes	characteristic
49	C₂²×C₅₃	#102	1	1	yes	characteristic
50	C₂²×C₅₃	#103	1	1	yes	characteristic
51	C₄×C₅₃	—	2	1	no	—
52	C₄×C₅₃	—	2	1	no	—
53	C₄×C₅₃	—	2	1	no	—
54	C₄×C₅₃	—	2	1	no	—
55	C₄×C₅₃	—	8	2	no	—
56	C₄×C₅₃	#102	1	1	yes	characteristic
57	C₄×C₅₃	#103	1	1	yes	characteristic
58	C₄×C₅₃	#106	1	1	yes	characteristic
59	C₄×C₅₃	#107	1	1	yes	characteristic
60	#1286	—	53	1	no	maximal, Sylow
61	C₂×C₅₃	—	2	1	no	—
62	C₂×C₅₃	—	2	1	no	—
63	C₂×C₅₃	H(3,ℤ/4)	1	1	yes	characteristic
64	C₂×C₅₃	#340	1	1	yes	characteristic
65	C₂×C₅₃	#344	1	1	yes	characteristic
66	#431	—	53	1	no	—
67	#423	—	53	1	no	—
68	#382	D₅₃	1	1	yes	characteristic
69	C₅₃	#1286	1	1	yes	characteristic, complemented, Sylow
70	C₂×M₁₆	—	106	1	no	—
71	C₂×C₄⋊C₄	—	2	1	no	—
72	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
73	C₄⋊C₈	—	53	2	no	—
74	C₄⋊C₈	—	106	1	no	—
75	C₄⋊C₈	—	106	1	no	—
76	C₂²⋊C₈	—	106	1	no	—
77	C₄×C₈	—	53	2	no	—
78	#95	—	2	1	no	—
79	#95	Dic₅₃	1	2	yes	—
80	C₂²×C₄	—	2	2	no	—
81	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
82	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
83	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
84	M₁₆	—	212	1	no	—
85	M₁₆	—	212	1	no	—
86	C₂×C₈	—	106	2	no	—
87	C₂×C₈	—	106	2	no	—
88	C₄⋊C₄	—	4	1	no	—
89	C₄⋊C₄	—	4	1	no	—
90	C₄²	Dic₁₀₆	1	1	yes	characteristic
91	C₄²	C₄×D₅₃	1	1	yes	characteristic
92	C₄²	C₄×D₅₃	1	1	yes	characteristic
93	C₄²	D₂₁₂	1	1	yes	characteristic
94	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
95	C₂×C₄	—	4	1	no	—
96	C₂×C₄	—	4	1	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
101	C₂×C₄	#11719919	1	1	yes	characteristic
102	C₂×C₄	#11719919	1	1	yes	characteristic
103	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
104	C₂×C₄	#11719921	1	1	yes	characteristic
105	C₂×C₄	#11719921	1	1	yes	characteristic
106	C₂×C₄	#11719922	1	1	yes	characteristic
107	C₂×C₄	#11719923	1	1	yes	characteristic
108	C₂×C₄	#11719924	1	1	yes	characteristic
109	C₂×C₄	#11719925	1	1	yes	characteristic
110	C₈	—	212	2	no	—
111	C₈	—	212	2	no	—
112	C₂²	—	2	1	no	—
113	C₂²	—	2	1	no	—
114	C₂²	#13133124	1	1	yes	characteristic
115	C₂²	#13133138	1	1	yes	characteristic
116	C₂²	#13133141	1	1	yes	characteristic
117	C₄	—	2	1	no	—
118	C₄	—	2	1	no	—
119	C₄	—	2	1	no	—
120	C₄	—	2	1	no	—
121	C₄	—	8	2	no	—
122	C₄	#13133137	1	1	yes	characteristic
123	C₄	#13133140	1	1	yes	characteristic
124	C₄	#13133148	1	1	yes	characteristic
125	C₄	#13133150	1	1	yes	characteristic
126	C₂	—	2	1	no	—
127	C₂	—	2	1	no	—
128	C₂	#16810379	1	1	yes	characteristic
129	C₂	#16810385	1	1	yes	characteristic
130	C₂	#16810393	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810676

C₂

yes

maximal, characteristic

#16810647

C₂

yes

maximal, characteristic

C₅₃×#382

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

complemented

C₂²⋊#88322

—

C₄×#88322

C₄

yes

complemented

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#95×C₅₃

—

#95×C₅₃

C₄

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

Q₈

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

C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

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

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

D₅₃

yes

characteristic

C₅₃

#1286

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

106

—

C₄×C₈

—

#95

—

#95

Dic₅₃

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

M₁₆

—

212

—

M₁₆

—

212

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Dic₁₀₆

yes

characteristic

C₄²

C₄×D₅₃

yes

characteristic

C₄²

C₄×D₅₃

yes

characteristic

C₄²

D₂₁₂

yes

characteristic

C₂³

C₂²⋊Dic₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

C₄×Dic₅₃

yes

characteristic

101

C₂×C₄

#11719919

yes

characteristic

102

C₂×C₄

#11719919

yes

characteristic

103

C₂×C₄

C₄⋊Dic₅₃

yes

characteristic

104

C₂×C₄

#11719921

yes

characteristic

105

C₂×C₄

#11719921

yes

characteristic

106

C₂×C₄

#11719922

yes

characteristic

107

C₂×C₄

#11719923

yes

characteristic

108

C₂×C₄

#11719924

yes

characteristic

109

C₂×C₄

#11719925

yes

characteristic

110

C₈

—

212

—

111

C₈

—

212

—

112

C₂²

—

113

C₂²

—

114

C₂²

#13133124

yes

characteristic

115

C₂²

#13133138

yes

characteristic

116

C₂²

#13133141

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

#13133137

yes

characteristic

123

C₄

#13133140

yes

characteristic

124

C₄

#13133148

yes

characteristic

125

C₄

#13133150

yes

characteristic

126

C₂

—

127

C₂

—

128

C₂

#16810379

yes

characteristic

129

C₂

#16810385

yes

characteristic

130

C₂

#16810393

yes

characteristic

	subgroup	G/H
center	C₂²	#13133124
commutator	C₂×C₄×C₅₃	C₄²
Frattini	C₂×C₄²	C₂×D₅₃
Fitting	C₅₃×#382	C₂
socle	C₂²×C₅₃	#95
hypercenter	#382	D₅₃
hypocenter	C₅₃	#1286
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂²

#13133124

commutator

C₂×C₄×C₅₃

C₄²

Frattini

C₂×C₄²

C₂×D₅₃

Fitting

C₅₃×#382

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
7. C₄⋊#88322	C₄	semidirect
9. C₄×#88322	C₄	semidirect
69. C₅₃	#1286	semidirect

H ≅ G/N

type

7. C₄⋊#88322

C₄

semidirect

9. C₄×#88322

C₄

semidirect

69. C₅₃

#1286

semidirect

Subgroups — #20272573

Conjugacy classes of subgroups

Complements