Group #20270795 — subgroups

Subgroups — C₅₃×#1858 #20270795

Order 6784.

252 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 108 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	C₅₃×#568	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#430	C₂	1	2	yes	maximal, complemented
3	C₅₃×#413	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#408	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#341	C₂	1	2	yes	maximal, complemented
6	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
7	#131×C₅₃	—	2	4	no	—
8	C₂×M₁₆×C₅₃	—	2	2	no	—
9	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
11	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
12	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	#108×C₅₃	—	2	4	no	—
14	#101×C₅₃	C₄	1	2	yes	—
15	#100×C₅₃	C₄	1	2	yes	—
16	C₂²⋊C₈×C₅₃	—	2	2	no	—
17	#41×C₅₃	—	2	2	no	—
18	#41×C₅₃	—	2	2	no	—
19	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×D₄×C₅₃	Q₈	1	2	yes	—
22	C₂²×C₄×C₅₃	D₄	1	2	yes	—
23	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
24	M₁₆×C₅₃	—	2	4	no	—
25	M₁₆×C₅₃	—	2	2	no	—
26	M₁₆×C₅₃	—	2	2	no	—
27	M₁₆×C₅₃	—	2	2	no	—
28	M₁₆×C₅₃	C₂×C₄	1	4	yes	—
29	M₁₆×C₅₃	D₄	1	4	yes	—
30	C₂×C₈×C₅₃	—	2	4	no	—
31	C₂×C₈×C₅₃	—	2	2	no	—
32	C₂×C₈×C₅₃	—	2	2	no	—
33	C₂×C₈×C₅₃	—	2	1	no	—
34	C₂×C₈×C₅₃	—	2	1	no	—
35	C₂×C₈×C₅₃	—	2	2	no	—
36	C₂³×C₅₃	C₄⋊C₄	1	2	yes	—
37	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
38	Q₈×C₅₃	—	2	2	no	—
39	D₄×C₅₃	—	2	4	no	—
40	D₄×C₅₃	—	2	2	no	—
41	C₂×C₄×C₅₃	—	2	2	no	—
42	C₂×C₄×C₅₃	—	2	2	no	—
43	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
44	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	#41	1	2	yes	—
48	C₈×C₅₃	—	2	4	no	—
49	C₈×C₅₃	—	2	4	no	—
50	C₈×C₅₃	—	4	1	no	—
51	C₈×C₅₃	—	4	1	no	—
52	C₂²×C₅₃	—	2	2	no	—
53	C₂²×C₅₃	—	4	1	no	—
54	C₂²×C₅₃	—	4	2	no	—
55	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₅₃	C₂²⋊Q₈	1	2	yes	—
57	C₄×C₅₃	—	2	2	no	—
58	C₄×C₅₃	C₄×D₄	1	2	yes	—
59	C₄×C₅₃	#120	1	1	yes	characteristic
60	C₄×C₅₃	#123	1	1	yes	characteristic
61	#1858	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂×C₅₃	—	2	1	no	—
63	C₂×C₅₃	—	2	2	no	—
64	C₂×C₅₃	—	4	2	no	—
65	C₂×C₅₃	#385	1	1	yes	characteristic
66	#568	C₂×C₅₃	1	1	yes	characteristic
67	#430	C₂×C₅₃	1	2	yes	complemented
68	#413	C₂×C₅₃	1	1	yes	characteristic
69	#408	C₂×C₅₃	1	1	yes	characteristic
70	#341	C₂×C₅₃	1	2	yes	complemented
71	C₅₃	#1858	1	1	yes	characteristic, direct product factor, Sylow
72	C₂×#41	C₂²×C₅₃	1	1	yes	characteristic
73	#131	—	2	4	no	—
74	C₂×M₁₆	—	2	2	no	—
75	C₂×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
76	C₂×M₁₆	C₂²×C₅₃	1	2	yes	—
77	C₂×M₁₆	C₂²×C₅₃	1	2	yes	—
78	C₂²×C₈	C₂²×C₅₃	1	1	yes	characteristic
79	#108	—	2	4	no	—
80	#101	C₄×C₅₃	1	2	yes	—
81	#100	C₄×C₅₃	1	2	yes	—
82	C₂²⋊C₈	—	2	2	no	—
83	#41	—	2	2	no	—
84	#41	—	2	2	no	—
85	C₂×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
86	C₂×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
87	C₂×D₄	Q₈×C₅₃	1	2	yes	—
88	C₂²×C₄	D₄×C₅₃	1	2	yes	—
89	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
90	M₁₆	—	2	4	no	—
91	M₁₆	—	2	2	no	—
92	M₁₆	—	2	2	no	—
93	M₁₆	—	2	2	no	—
94	M₁₆	C₂×C₄×C₅₃	1	4	yes	—
95	M₁₆	D₄×C₅₃	1	4	yes	—
96	C₂×C₈	—	2	4	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂³	C₄⋊C₄×C₅₃	1	2	yes	—
103	C₂³	C₂×Q₈×C₅₃	1	1	yes	characteristic
104	Q₈	—	2	2	no	—
105	D₄	—	2	4	no	—
106	D₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
110	C₂×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
111	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
112	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
113	C₂×C₄	#41×C₅₃	1	2	yes	—
114	C₈	—	2	4	no	—
115	C₈	—	2	4	no	—
116	C₈	—	4	1	no	—
117	C₈	—	4	1	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	4	1	no	—
120	C₂²	—	4	2	no	—
121	C₂²	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
122	C₂²	C₂²⋊Q₈×C₅₃	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	C₄×D₄×C₅₃	1	2	yes	—
125	C₄	#120×C₅₃	1	1	yes	characteristic
126	C₄	#123×C₅₃	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	2	no	—
129	C₂	—	4	2	no	—
130	C₂	C₅₃×#385	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#568

C₂

yes

maximal, characteristic

C₅₃×#430

C₂

yes

maximal, complemented

C₅₃×#413

C₂

yes

maximal, characteristic

C₅₃×#408

C₂

yes

maximal, characteristic

C₅₃×#341

C₂

yes

maximal, complemented

C₂×#41×C₅₃

C₂²

yes

characteristic

#131×C₅₃

—

C₂×M₁₆×C₅₃

—

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂×M₁₆×C₅₃

C₂²

yes

—

C₂×M₁₆×C₅₃

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

#108×C₅₃

—

#101×C₅₃

C₄

yes

—

#100×C₅₃

C₄

yes

—

C₂²⋊C₈×C₅₃

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₄

yes

characteristic

C₂×D₄×C₅₃

C₂×C₄

yes

characteristic

C₂×D₄×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

C₂×C₄

yes

—

M₁₆×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂³×C₅₃

C₄⋊C₄

yes

—

C₂³×C₅₃

C₂×Q₈

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₄×D₄

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#385

yes

characteristic

#568

C₂×C₅₃

yes

characteristic

#430

C₂×C₅₃

yes

complemented

#413

C₂×C₅₃

yes

characteristic

#408

C₂×C₅₃

yes

characteristic

#341

C₂×C₅₃

yes

complemented

C₅₃

#1858

yes

characteristic, direct product factor, Sylow

C₂×#41

C₂²×C₅₃

yes

characteristic

#131

—

C₂×M₁₆

—

C₂×M₁₆

C₂²×C₅₃

yes

characteristic

C₂×M₁₆

C₂²×C₅₃

yes

—

C₂×M₁₆

C₂²×C₅₃

yes

—

C₂²×C₈

C₂²×C₅₃

yes

characteristic

#108

—

#101

C₄×C₅₃

yes

—

#100

C₄×C₅₃

yes

—

C₂²⋊C₈

—

#41

—

#41

—

C₂×Q₈

C₂×C₄×C₅₃

yes

characteristic

C₂×D₄

C₂×C₄×C₅₃

yes

characteristic

C₂×D₄

Q₈×C₅₃

yes

—

C₂²×C₄

D₄×C₅₃

yes

—

C₂²×C₄

C₂³×C₅₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

C₂×C₄×C₅₃

yes

—

M₁₆

D₄×C₅₃

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂³

C₄⋊C₄×C₅₃

yes

—

103

C₂³

C₂×Q₈×C₅₃

yes

characteristic

104

Q₈

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₄⋊C₄×C₅₃

yes

—

110

C₂×C₄

C₂²×C₄×C₅₃

yes

characteristic

111

C₂×C₄

C₂×D₄×C₅₃

yes

—

112

C₂×C₄

C₂×D₄×C₅₃

yes

characteristic

113

C₂×C₄

#41×C₅₃

yes

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂×C₄⋊C₄×C₅₃

yes

characteristic

122

C₂²

C₂²⋊Q₈×C₅₃

yes

—

123

C₄

—

124

C₄

C₄×D₄×C₅₃

yes

—

125

C₄

#120×C₅₃

yes

characteristic

126

C₄

#123×C₅₃

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

C₅₃×#385

yes

characteristic

	subgroup	G/H
center	C₄×C₅₃	#120
commutator	C₂×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂³×C₅₃
socle	C₂×C₅₃	#385
prime radical	C₂×#41×C₅₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ G

subgroup

G/H

center

C₄×C₅₃

#120

commutator

C₂×C₄

C₂²×C₄×C₅₃

Frattini, cosocle

C₂²×C₄

C₂³×C₅₃

socle

C₂×C₅₃

#385

prime radical

C₂×#41×C₅₃

C₂²

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ G

N	H ≅ G/N	type
2. C₅₃×#430	C₂	semidirect
5. C₅₃×#341	C₂	semidirect
61. #1858	C₅₃	direct (N × H)
67. #430	C₂×C₅₃	semidirect
70. #341	C₂×C₅₃	semidirect
71. C₅₃	#1858	direct (N × H)

H ≅ G/N

type

2. C₅₃×#430

C₂

semidirect

5. C₅₃×#341

C₂

semidirect

61. #1858

C₅₃

direct (N × H)

67. #430

C₂×C₅₃

semidirect

70. #341

C₂×C₅₃

semidirect

71. C₅₃

#1858

direct (N × H)

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

subgroup	index
C₅₃×#568	2
C₅₃×#430	2
C₅₃×#413	2
C₅₃×#408	2
C₅₃×#341	2
#1858	53

subgroup

index

C₅₃×#568

C₅₃×#430

C₅₃×#413

C₅₃×#408

C₅₃×#341

#1858

Subgroups — C₅₃×#1858 #20270795

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#1858 #20270795

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#1858 #20270795