Group #20270620 — subgroups

Subgroups — C₅₃×#1683 #20270620

Order 6784.

160 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 68 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×#487	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#476	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#474	C₂	1	1	yes	maximal, characteristic
4	C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#333	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#332	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×C₅₃	—	2	1	no	—
9	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
12	#107×C₅₃	—	2	1	no	—
13	#106×C₅₃	—	2	1	no	—
14	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
17	#103×C₅₃	—	2	1	no	—
18	#103×C₅₃	—	2	1	no	—
19	#102×C₅₃	—	2	1	no	—
20	#102×C₅₃	—	2	1	no	—
21	C₂²⋊C₈×C₅₃	—	2	1	no	—
22	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₅₃	—	2	1	no	—
24	C₂×Q₈×C₅₃	—	2	1	no	—
25	C₂×D₄×C₅₃	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	—	2	1	no	—
27	C₂×C₈×C₅₃	—	2	1	no	—
28	C₂×C₈×C₅₃	—	2	1	no	—
29	C₂×C₈×C₅₃	—	2	1	no	—
30	C₂×C₈×C₅₃	—	4	2	no	—
31	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₄⋊C₄×C₅₃	—	2	1	no	—
34	C₄⋊C₄×C₅₃	—	2	1	no	—
35	C₄⋊C₄×C₅₃	—	2	1	no	—
36	C₄⋊C₄×C₅₃	—	2	1	no	—
37	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
38	C₂²⋊C₄×C₅₃	—	2	1	no	—
39	C₄²×C₅₃	C₂³	1	1	yes	characteristic
40	C₂³×C₅₃	—	2	1	no	—
41	Q₈×C₅₃	—	4	1	no	—
42	Q₈×C₅₃	—	4	1	no	—
43	Q₈×C₅₃	—	8	1	no	—
44	D₄×C₅₃	—	2	1	no	—
45	D₄×C₅₃	Q₁₆	1	2	yes	—
46	C₂×C₄×C₅₃	—	2	1	no	—
47	C₂×C₄×C₅₃	—	4	1	no	—
48	C₂×C₄×C₅₃	—	4	1	no	—
49	C₂×C₄×C₅₃	—	4	1	no	—
50	C₂×C₄×C₅₃	—	4	1	no	—
51	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
54	C₈×C₅₃	—	2	1	no	—
55	C₈×C₅₃	—	2	1	no	—
56	C₈×C₅₃	—	4	1	no	—
57	C₈×C₅₃	—	4	1	no	—
58	C₈×C₅₃	—	4	1	no	—
59	C₂²×C₅₃	—	2	2	no	—
60	C₂²×C₅₃	—	4	1	no	—
61	C₂²×C₅₃	—	4	1	no	—
62	C₂²×C₅₃	#121	1	1	yes	characteristic
63	C₄×C₅₃	—	2	1	no	—
64	C₄×C₅₃	—	4	1	no	—
65	C₄×C₅₃	—	8	1	no	—
66	C₄×C₅₃	—	8	1	no	—
67	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
68	C₄×C₅₃	#135	1	1	yes	characteristic
69	C₄×C₅₃	#135	1	1	yes	characteristic
70	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
71	#1683	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂×C₅₃	—	4	2	no	—
73	C₂×C₅₃	#463	1	1	yes	characteristic
74	C₂×C₅₃	#467	1	1	yes	characteristic
75	C₂×C₅₃	#471	1	1	yes	characteristic
76	#487	C₂×C₅₃	1	1	yes	characteristic, complemented
77	#476	C₂×C₅₃	1	1	yes	characteristic
78	#474	C₂×C₅₃	1	1	yes	characteristic
79	C₈×D₄	C₂×C₅₃	1	1	yes	characteristic
80	#335	C₂×C₅₃	1	1	yes	characteristic, complemented
81	#333	C₂×C₅₃	1	1	yes	characteristic, complemented
82	#332	C₂×C₅₃	1	1	yes	characteristic, complemented
83	C₅₃	#1683	1	1	yes	characteristic, direct product factor, Sylow
84	C₂²×C₈	—	2	1	no	—
85	C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
86	C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
87	C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
88	#107	—	2	1	no	—
89	#106	—	2	1	no	—
90	C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
91	C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
92	C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
93	#103	—	2	1	no	—
94	#103	—	2	1	no	—
95	#102	—	2	1	no	—
96	#102	—	2	1	no	—
97	C₂²⋊C₈	—	2	1	no	—
98	C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
99	C₂×Q₈	—	2	1	no	—
100	C₂×Q₈	—	2	1	no	—
101	C₂×D₄	D₄×C₅₃	1	1	yes	characteristic
102	C₂²×C₄	—	2	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	4	2	no	—
107	C₂×C₈	D₄×C₅₃	1	1	yes	characteristic
108	C₂×C₈	D₄×C₅₃	1	1	yes	characteristic
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
114	C₂²⋊C₄	—	2	1	no	—
115	C₄²	C₂³×C₅₃	1	1	yes	characteristic
116	C₂³	—	2	1	no	—
117	Q₈	—	4	1	no	—
118	Q₈	—	4	1	no	—
119	Q₈	—	8	1	no	—
120	D₄	—	2	1	no	—
121	D₄	Q₁₆×C₅₃	1	2	yes	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
128	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
129	C₂×C₄	#41×C₅₃	1	1	yes	characteristic
130	C₈	—	2	1	no	—
131	C₈	—	2	1	no	—
132	C₈	—	4	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	4	1	no	—
135	C₂²	—	2	2	no	—
136	C₂²	—	4	1	no	—
137	C₂²	—	4	1	no	—
138	C₂²	#121×C₅₃	1	1	yes	characteristic
139	C₄	—	2	1	no	—
140	C₄	—	4	1	no	—
141	C₄	—	8	1	no	—
142	C₄	—	8	1	no	—
143	C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
144	C₄	#135×C₅₃	1	1	yes	characteristic
145	C₄	#135×C₅₃	1	1	yes	characteristic
146	C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
147	C₂	—	4	2	no	—
148	C₂	C₅₃×#463	1	1	yes	characteristic
149	C₂	C₅₃×#467	1	1	yes	characteristic
150	C₂	C₅₃×#471	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#487

C₂

yes

maximal, characteristic, complemented

C₅₃×#476

C₂

yes

maximal, characteristic

C₅₃×#474

C₂

yes

maximal, characteristic

C₈×D₄×C₅₃

C₂

yes

maximal, characteristic

C₅₃×#335

C₂

yes

maximal, characteristic, complemented

C₅₃×#333

C₂

yes

maximal, characteristic, complemented

C₅₃×#332

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×C₅₃

—

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₄×D₄×C₅₃

C₂²

yes

characteristic

#107×C₅₃

—

#106×C₅₃

—

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

#103×C₅₃

—

#103×C₅₃

—

#102×C₅₃

—

#102×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

C₂³×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

Q₁₆

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#121

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×Q₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

characteristic, complemented

#476

C₂×C₅₃

yes

characteristic

#474

C₂×C₅₃

yes

characteristic

C₈×D₄

C₂×C₅₃

yes

characteristic

#335

C₂×C₅₃

yes

characteristic, complemented

#333

C₂×C₅₃

yes

characteristic, complemented

#332

C₂×C₅₃

yes

characteristic, complemented

C₅₃

#1683

yes

characteristic, direct product factor, Sylow

C₂²×C₈

—

C₄⋊Q₈

C₂²×C₅₃

yes

characteristic

C₄⋊Q₈

C₂²×C₅₃

yes

characteristic

C₄×D₄

C₂²×C₅₃

yes

characteristic

#107

—

#106

—

C₄⋊C₈

C₂²×C₅₃

yes

characteristic

C₄⋊C₈

C₂²×C₅₃

yes

characteristic

C₄⋊C₈

C₂²×C₅₃

yes

—

—

—

—

C₂²⋊C₈

—

C₄×C₈

C₂²×C₅₃

yes

characteristic

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×D₄

D₄×C₅₃

yes

characteristic

102

C₂²×C₄

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

D₄×C₅₃

yes

characteristic

108

C₂×C₈

D₄×C₅₃

yes

characteristic

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

D₄×C₅₃

yes

characteristic

114

C₂²⋊C₄

—

115

C₄²

C₂³×C₅₃

yes

characteristic

116

C₂³

—

117

Q₈

—

118

Q₈

—

119

Q₈

—

120

D₄

—

121

D₄

Q₁₆×C₅₃

yes

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

C₂×D₄×C₅₃

yes

characteristic

128

C₂×C₄

C₂×D₄×C₅₃

yes

characteristic

129

C₂×C₄

#41×C₅₃

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

#121×C₅₃

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

C₂×Q₁₆×C₅₃

yes

characteristic

144

C₄

#135×C₅₃

yes

characteristic

145

C₄

#135×C₅₃

yes

characteristic

146

C₄

AGL(1,ℤ/8)×C₅₃

yes

characteristic

147

C₂

—

148

C₂

C₅₃×#463

yes

characteristic

149

C₂

C₅₃×#467

yes

characteristic

150

C₂

C₅₃×#471

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₅₃
prime radical	C₂×D₄×C₅₃	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄²×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

#121

commutator, Frattini, cosocle

C₄²

C₂³×C₅₃

prime radical

C₂×D₄×C₅₃

D₄

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₅₃ ◁ C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
1. C₅₃×#487	C₂	semidirect
5. C₅₃×#335	C₂	semidirect
6. C₅₃×#333	C₂	semidirect
7. C₅₃×#332	C₂	semidirect
71. #1683	C₅₃	direct (N × H)
76. #487	C₂×C₅₃	semidirect
80. #335	C₂×C₅₃	semidirect
81. #333	C₂×C₅₃	semidirect
82. #332	C₂×C₅₃	semidirect
83. C₅₃	#1683	direct (N × H)

H ≅ G/N

type

1. C₅₃×#487

C₂

semidirect

5. C₅₃×#335

C₂

semidirect

6. C₅₃×#333

C₂

semidirect

7. C₅₃×#332

C₂

semidirect

71. #1683

C₅₃

direct (N × H)

76. #487

C₂×C₅₃

semidirect

80. #335

C₂×C₅₃

semidirect

81. #333

C₂×C₅₃

semidirect

82. #332

C₂×C₅₃

semidirect

83. C₅₃

#1683

direct (N × H)

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

subgroup	index
C₅₃×#487	2
C₅₃×#476	2
C₅₃×#474	2
C₈×D₄×C₅₃	2
C₅₃×#335	2
C₅₃×#333	2
C₅₃×#332	2
#1683	53

subgroup

index

C₅₃×#487

C₅₃×#476

C₅₃×#474

C₈×D₄×C₅₃

C₅₃×#335

C₅₃×#333

C₅₃×#332

#1683

Subgroups — C₅₃×#1683 #20270620

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#1683 #20270620

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#1683 #20270620