Group #20270982 — subgroups

Subgroups — C₅₃×#2045 #20270982

Order 6784.

296 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 84 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	C₂²×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#424	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#418	C₂	1	2	yes	maximal, complemented
4	C₂×#100×C₅₃	C₂	1	2	yes	maximal, complemented
5	C₅₃×#341	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄×C₅₃	C₂²	1	2	yes	—
7	C₂×D₈×C₅₃	—	2	2	no	—
8	C₂×D₈×C₅₃	—	2	4	no	—
9	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
10	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	#117×C₅₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₅₃	—	2	2	no	—
14	#102×C₅₃	—	2	4	no	—
15	#100×C₅₃	—	2	4	no	—
16	C₂⁴×C₅₃	—	2	2	no	—
17	C₂×D₄×C₅₃	—	2	2	no	—
18	C₂×D₄×C₅₃	—	2	2	no	—
19	C₂×D₄×C₅₃	—	2	2	no	—
20	C₂×D₄×C₅₃	—	2	2	no	—
21	C₂×D₄×C₅₃	—	2	2	no	—
22	C₂×D₄×C₅₃	D₄	1	4	yes	—
23	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
24	D₈×C₅₃	—	2	8	no	—
25	M₁₆×C₅₃	—	2	2	no	—
26	M₁₆×C₅₃	—	2	2	no	—
27	M₁₆×C₅₃	—	2	2	no	—
28	C₂×C₈×C₅₃	—	2	2	no	—
29	C₂×C₈×C₅₃	—	2	2	no	—
30	C₂×C₈×C₅₃	—	2	1	no	—
31	C₂×C₈×C₅₃	D₄	1	2	yes	—
32	C₄⋊C₄×C₅₃	D₄	1	2	yes	—
33	C₂²⋊C₄×C₅₃	—	2	1	no	—
34	C₄²×C₅₃	—	2	1	no	—
35	C₂³×C₅₃	—	2	4	no	—
36	C₂³×C₅₃	—	4	2	no	—
37	C₂³×C₅₃	—	4	2	no	—
38	C₂³×C₅₃	—	8	4	no	—
39	C₂³×C₅₃	#41	1	1	yes	characteristic
40	D₄×C₅₃	—	2	4	no	—
41	D₄×C₅₃	—	2	4	no	—
42	D₄×C₅₃	—	4	2	no	—
43	D₄×C₅₃	—	4	2	no	—
44	C₂×C₄×C₅₃	—	2	2	no	—
45	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	#41	1	2	yes	—
49	C₈×C₅₃	—	2	2	no	—
50	C₈×C₅₃	—	4	1	no	—
51	C₈×C₅₃	—	4	2	no	—
52	C₂²×C₅₃	—	2	2	no	—
53	C₂²×C₅₃	—	4	4	no	—
54	C₂²×C₅₃	—	8	4	no	—
55	C₂²×C₅₃	—	8	4	no	—
56	C₂²×C₅₃	—	8	4	no	—
57	C₂²×C₅₃	#121	1	1	yes	characteristic
58	C₂²×C₅₃	#123	1	2	yes	—
59	C₄×C₅₃	—	4	2	no	—
60	C₄×C₅₃	#120	1	1	yes	characteristic
61	C₄×C₅₃	#121	1	2	yes	—
62	C₄×C₅₃	#124	1	1	yes	characteristic
63	#2045	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂×C₅₃	—	2	2	no	—
65	C₂×C₅₃	—	8	4	no	—
66	C₂×C₅₃	#394	1	1	yes	characteristic
67	C₂×C₅₃	#472	1	2	yes	—
68	C₂²×D₈	C₂×C₅₃	1	1	yes	characteristic
69	#424	C₂×C₅₃	1	1	yes	characteristic, complemented
70	#418	C₂×C₅₃	1	2	yes	complemented
71	C₂×#100	C₂×C₅₃	1	2	yes	complemented
72	#341	C₂×C₅₃	1	1	yes	characteristic, complemented
73	C₅₃	#2045	1	1	yes	characteristic, direct product factor, Sylow
74	C₂²×D₄	C₂²×C₅₃	1	2	yes	—
75	C₂×D₈	—	2	4	no	—
76	C₂×D₈	—	2	2	no	—
77	C₂×M₁₆	C₂²×C₅₃	1	2	yes	—
78	C₂×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
79	C₂²×C₈	C₂²×C₅₃	1	1	yes	characteristic
80	#117	C₂²×C₅₃	1	1	yes	characteristic
81	C₄⋊C₈	—	2	2	no	—
82	#102	—	2	4	no	—
83	#100	—	2	4	no	—
84	C₂⁴	—	2	2	no	—
85	C₂×D₄	—	2	2	no	—
86	C₂×D₄	—	2	2	no	—
87	C₂×D₄	—	2	2	no	—
88	C₂×D₄	—	2	2	no	—
89	C₂×D₄	—	2	2	no	—
90	C₂×D₄	D₄×C₅₃	1	4	yes	—
91	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
92	D₈	—	2	8	no	—
93	M₁₆	—	2	2	no	—
94	M₁₆	—	2	2	no	—
95	M₁₆	—	2	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	D₄×C₅₃	1	2	yes	—
100	C₄⋊C₄	D₄×C₅₃	1	2	yes	—
101	C₂²⋊C₄	—	2	1	no	—
102	C₄²	—	2	1	no	—
103	C₂³	—	2	4	no	—
104	C₂³	—	4	2	no	—
105	C₂³	—	4	2	no	—
106	C₂³	—	8	4	no	—
107	C₂³	#41×C₅₃	1	1	yes	characteristic
108	D₄	—	2	4	no	—
109	D₄	—	2	4	no	—
110	D₄	—	4	2	no	—
111	D₄	—	4	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
114	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
115	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
116	C₂×C₄	#41×C₅₃	1	2	yes	—
117	C₈	—	2	2	no	—
118	C₈	—	4	1	no	—
119	C₈	—	4	2	no	—
120	C₂²	—	2	2	no	—
121	C₂²	—	4	4	no	—
122	C₂²	—	8	4	no	—
123	C₂²	—	8	4	no	—
124	C₂²	—	8	4	no	—
125	C₂²	#121×C₅₃	1	1	yes	characteristic
126	C₂²	#123×C₅₃	1	2	yes	—
127	C₄	—	4	2	no	—
128	C₄	#120×C₅₃	1	1	yes	characteristic
129	C₄	#121×C₅₃	1	2	yes	—
130	C₄	#124×C₅₃	1	1	yes	characteristic
131	C₂	—	2	2	no	—
132	C₂	—	8	4	no	—
133	C₂	C₅₃×#394	1	1	yes	characteristic
134	C₂	C₅₃×#472	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₈×C₅₃

C₂

yes

maximal, characteristic

C₅₃×#424

C₂

yes

maximal, characteristic, complemented

C₅₃×#418

C₂

yes

maximal, complemented

C₂×#100×C₅₃

C₂

yes

maximal, complemented

C₅₃×#341

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₅₃

C₂²

yes

—

C₂×D₈×C₅₃

—

C₂×D₈×C₅₃

—

C₂×M₁₆×C₅₃

C₂²

yes

—

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

#117×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

#102×C₅₃

—

#100×C₅₃

—

C₂⁴×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

D₈×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

D₄

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

#41

yes

characteristic

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#121

yes

characteristic

C₂²×C₅₃

#123

yes

—

C₄×C₅₃

—

C₄×C₅₃

#120

yes

characteristic

C₄×C₅₃

#121

yes

—

C₄×C₅₃

#124

yes

characteristic

#2045

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#394

yes

characteristic

C₂×C₅₃

#472

yes

—

C₂²×D₈

C₂×C₅₃

yes

characteristic

#424

C₂×C₅₃

yes

characteristic, complemented

#418

C₂×C₅₃

yes

complemented

C₂×#100

C₂×C₅₃

yes

complemented

#341

C₂×C₅₃

yes

characteristic, complemented

C₅₃

#2045

yes

characteristic, direct product factor, Sylow

C₂²×D₄

C₂²×C₅₃

yes

—

C₂×D₈

—

C₂×D₈

—

C₂×M₁₆

C₂²×C₅₃

yes

—

C₂×M₁₆

C₂²×C₅₃

yes

characteristic

C₂²×C₈

C₂²×C₅₃

yes

characteristic

#117

C₂²×C₅₃

yes

characteristic

C₄⋊C₈

—

#102

—

#100

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

D₄×C₅₃

yes

—

C₂²×C₄

C₂³×C₅₃

yes

characteristic

D₈

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×C₅₃

yes

—

100

C₄⋊C₄

D₄×C₅₃

yes

—

101

C₂²⋊C₄

—

102

C₄²

—

103

C₂³

—

104

C₂³

—

105

C₂³

—

106

C₂³

—

107

C₂³

#41×C₅₃

yes

characteristic

108

D₄

—

109

D₄

—

110

D₄

—

111

D₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂×D₄×C₅₃

yes

—

114

C₂×C₄

C₂×D₄×C₅₃

yes

characteristic

115

C₂×C₄

C₂×D₄×C₅₃

yes

characteristic

116

C₂×C₄

#41×C₅₃

yes

—

117

C₈

—

118

C₈

—

119

C₈

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

#121×C₅₃

yes

characteristic

126

C₂²

#123×C₅₃

yes

—

127

C₄

—

128

C₄

#120×C₅₃

yes

characteristic

129

C₄

#121×C₅₃

yes

—

130

C₄

#124×C₅₃

yes

characteristic

131

C₂

—

132

C₂

—

133

C₂

C₅₃×#394

yes

characteristic

134

C₂

C₅₃×#472

yes

—

	subgroup	G/H
center, socle	C₂²×C₅₃	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₅₃
prime radical	C₂²×D₈×C₅₃	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

#121

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₅₃

prime radical

C₂²×D₈×C₅₃

C₂

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ G

N	H ≅ G/N	type
2. C₅₃×#424	C₂	semidirect
3. C₅₃×#418	C₂	semidirect
4. C₂×#100×C₅₃	C₂	semidirect
5. C₅₃×#341	C₂	semidirect
63. #2045	C₅₃	direct (N × H)
69. #424	C₂×C₅₃	semidirect
70. #418	C₂×C₅₃	semidirect
71. C₂×#100	C₂×C₅₃	semidirect
72. #341	C₂×C₅₃	semidirect
73. C₅₃	#2045	direct (N × H)

H ≅ G/N

type

2. C₅₃×#424

C₂

semidirect

3. C₅₃×#418

C₂

semidirect

4. C₂×#100×C₅₃

C₂

semidirect

5. C₅₃×#341

C₂

semidirect

63. #2045

C₅₃

direct (N × H)

69. #424

C₂×C₅₃

semidirect

70. #418

C₂×C₅₃

semidirect

71. C₂×#100

C₂×C₅₃

semidirect

72. #341

C₂×C₅₃

semidirect

73. C₅₃

#2045

direct (N × H)

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

subgroup	index
C₂²×D₈×C₅₃	2
C₅₃×#424	2
C₅₃×#418	2
C₂×#100×C₅₃	2
C₅₃×#341	2
#2045	53

subgroup

index

C₂²×D₈×C₅₃

C₅₃×#424

C₅₃×#418

C₂×#100×C₅₃

C₅₃×#341

#2045

Subgroups — C₅₃×#2045 #20270982

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#2045 #20270982

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2045 #20270982