Group #20271037 — subgroups

Subgroups — C₅₃×#2100 #20271037

Order 6784.

214 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 100 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×#389	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#384	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#340	C₂	1	2	yes	maximal
4	C₅₃×#340	C₂	1	2	yes	maximal
5	C₅₃×#336	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
7	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
8	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
9	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
10	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
11	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	#95×C₅₃	—	2	1	no	—
15	C₂²×C₄×C₅₃	—	2	1	no	—
16	C₂²×C₄×C₅₃	—	2	1	no	—
17	C₂²×C₄×C₅₃	—	2	2	no	—
18	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
21	C₂×C₈×C₅₃	—	2	2	no	—
22	C₂×C₈×C₅₃	—	2	2	no	—
23	C₂×C₈×C₅₃	—	2	2	no	—
24	C₄⋊C₄×C₅₃	—	2	1	no	—
25	C₄⋊C₄×C₅₃	—	2	2	no	—
26	C₄⋊C₄×C₅₃	—	2	4	no	—
27	C₄⋊C₄×C₅₃	—	4	2	no	—
28	C₄⋊C₄×C₅₃	—	4	2	no	—
29	C₄⋊C₄×C₅₃	—	4	2	no	—
30	C₄⋊C₄×C₅₃	Q₈	1	2	yes	—
31	C₄²×C₅₃	—	2	2	no	—
32	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₅₃	—	2	1	no	—
34	C₂×C₄×C₅₃	—	2	1	no	—
35	C₂×C₄×C₅₃	—	2	1	no	—
36	C₂×C₄×C₅₃	—	2	1	no	—
37	C₂×C₄×C₅₃	—	2	2	no	—
38	C₂×C₄×C₅₃	—	4	1	no	—
39	C₂×C₄×C₅₃	—	4	1	no	—
40	C₂×C₄×C₅₃	—	4	1	no	—
41	C₂×C₄×C₅₃	—	4	2	no	—
42	C₂×C₄×C₅₃	—	4	1	no	—
43	C₂×C₄×C₅₃	—	4	1	no	—
44	C₂×C₄×C₅₃	—	4	2	no	—
45	C₂×C₄×C₅₃	—	4	2	no	—
46	C₂×C₄×C₅₃	SD₁₆	1	2	yes	—
47	C₂×C₄×C₅₃	Q₁₆	1	2	yes	—
48	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₄×C₅₃	#41	1	2	yes	—
50	C₂×C₄×C₅₃	#41	1	2	yes	—
51	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
52	C₈×C₅₃	—	4	2	no	—
53	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
54	C₂²×C₅₃	#123	1	1	yes	characteristic
55	C₂²×C₅₃	#124	1	1	yes	characteristic
56	C₂²×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
57	C₂²×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
58	C₂²×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
59	C₂²×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₄×C₅₃	—	2	2	no	—
61	C₄×C₅₃	—	2	2	no	—
62	C₄×C₅₃	—	4	2	no	—
63	C₄×C₅₃	—	8	1	no	—
64	C₄×C₅₃	—	8	2	no	—
65	C₄×C₅₃	#125	1	2	yes	—
66	C₄×C₅₃	#126	1	2	yes	—
67	#2100	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂×C₅₃	#400	1	1	yes	characteristic
69	C₂×C₅₃	#476	1	1	yes	characteristic
70	C₂×C₅₃	#477	1	1	yes	characteristic
71	C₂×C₅₃	#481	1	1	yes	characteristic
72	C₂×C₅₃	#484	1	1	yes	characteristic
73	C₂×C₅₃	#487	1	1	yes	characteristic
74	C₂×C₅₃	#490	1	1	yes	characteristic
75	#389	C₂×C₅₃	1	1	yes	characteristic
76	#384	C₂×C₅₃	1	1	yes	characteristic
77	#340	C₂×C₅₃	1	2	yes	—
78	#340	C₂×C₅₃	1	2	yes	—
79	#336	C₂×C₅₃	1	1	yes	characteristic
80	C₅₃	#2100	1	1	yes	characteristic, direct product factor, Sylow
81	C₂²×C₈	C₂²×C₅₃	1	2	yes	—
82	C₂×C₄⋊C₄	—	2	1	no	—
83	C₂×C₄⋊C₄	—	2	1	no	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
86	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
87	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
88	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
89	#95	—	2	1	no	—
90	C₂²×C₄	—	2	1	no	—
91	C₂²×C₄	—	2	1	no	—
92	C₂²×C₄	—	2	2	no	—
93	C₂²×C₄	D₄×C₅₃	1	1	yes	characteristic
94	C₂²×C₄	D₄×C₅₃	1	1	yes	characteristic
95	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₄⋊C₄	—	2	1	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	2	4	no	—
102	C₄⋊C₄	—	4	2	no	—
103	C₄⋊C₄	—	4	2	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	Q₈×C₅₃	1	2	yes	—
106	C₄²	—	2	2	no	—
107	C₂³	C₂×D₄×C₅₃	1	1	yes	characteristic
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	4	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	SD₁₆×C₅₃	1	2	yes	—
122	C₂×C₄	Q₁₆×C₅₃	1	2	yes	—
123	C₂×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
124	C₂×C₄	#41×C₅₃	1	2	yes	—
125	C₂×C₄	#41×C₅₃	1	2	yes	—
126	C₂×C₄	#41×C₅₃	1	1	yes	characteristic
127	C₈	—	4	2	no	—
128	C₂²	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
129	C₂²	#123×C₅₃	1	1	yes	characteristic
130	C₂²	#124×C₅₃	1	1	yes	characteristic
131	C₂²	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
132	C₂²	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
133	C₂²	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
134	C₂²	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	2	2	no	—
137	C₄	—	4	2	no	—
138	C₄	—	8	2	no	—
139	C₄	—	8	1	no	—
140	C₄	#125×C₅₃	1	2	yes	—
141	C₄	#126×C₅₃	1	2	yes	—
142	C₂	C₅₃×#400	1	1	yes	characteristic
143	C₂	C₅₃×#476	1	1	yes	characteristic
144	C₂	C₅₃×#477	1	1	yes	characteristic
145	C₂	C₅₃×#481	1	1	yes	characteristic
146	C₂	C₅₃×#484	1	1	yes	characteristic
147	C₂	C₅₃×#487	1	1	yes	characteristic
148	C₂	C₅₃×#490	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#389

C₂

yes

maximal, characteristic

C₅₃×#384

C₂

yes

maximal, characteristic

C₅₃×#340

C₂

yes

maximal

C₅₃×#340

C₂

yes

maximal

C₅₃×#336

C₂

yes

maximal, characteristic

C₂²×C₈×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#95×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

Q₈

yes

—

C₄²×C₅₃

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

SD₁₆

yes

—

C₂×C₄×C₅₃

Q₁₆

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

#41

yes

characteristic

C₈×C₅₃

—

C₂²×C₅₃

C₂²⋊Q₈

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

C₂×SD₁₆

yes

characteristic

C₂²×C₅₃

C₂×Q₁₆

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#125

yes

—

C₄×C₅₃

#126

yes

—

#2100

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

characteristic

#384

C₂×C₅₃

yes

characteristic

#340

C₂×C₅₃

yes

—

#340

C₂×C₅₃

yes

—

#336

C₂×C₅₃

yes

characteristic

C₅₃

#2100

yes

characteristic, direct product factor, Sylow

C₂²×C₈

C₂²×C₅₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₅₃

yes

—

C₂×C₄⋊C₄

C₂²×C₅₃

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₅₃

yes

characteristic

C₂×C₄²

C₂²×C₅₃

yes

characteristic

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₅₃

yes

characteristic

C₂²×C₄

D₄×C₅₃

yes

characteristic

C₂²×C₄

C₂³×C₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

Q₈×C₅₃

yes

—

106

C₄²

—

107

C₂³

C₂×D₄×C₅₃

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

SD₁₆×C₅₃

yes

—

122

C₂×C₄

Q₁₆×C₅₃

yes

—

123

C₂×C₄

C₂×Q₈×C₅₃

yes

characteristic

124

C₂×C₄

#41×C₅₃

yes

—

125

C₂×C₄

#41×C₅₃

yes

—

126

C₂×C₄

#41×C₅₃

yes

characteristic

127

C₈

—

128

C₂²

C₂²⋊Q₈×C₅₃

yes

characteristic

129

C₂²

#123×C₅₃

yes

characteristic

130

C₂²

#124×C₅₃

yes

characteristic

131

C₂²

C₂×SD₁₆×C₅₃

yes

characteristic

132

C₂²

C₂×Q₁₆×C₅₃

yes

characteristic

133

C₂²

AGL(1,ℤ/8)×C₅₃

yes

characteristic

134

C₂²

AGL(1,ℤ/8)×C₅₃

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

#125×C₅₃

yes

—

141

C₄

#126×C₅₃

yes

—

142

C₂

C₅₃×#400

yes

characteristic

143

C₂

C₅₃×#476

yes

characteristic

144

C₂

C₅₃×#477

yes

characteristic

145

C₂

C₅₃×#481

yes

characteristic

146

C₂

C₅₃×#484

yes

characteristic

147

C₂

C₅₃×#487

yes

characteristic

148

C₂

C₅₃×#490

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₅₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₅₃

C₂×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₅₃

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
67. #2100	C₅₃	direct (N × H)
80. C₅₃	#2100	direct (N × H)

H ≅ G/N

type

67. #2100

C₅₃

direct (N × H)

80. C₅₃

#2100

direct (N × H)

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

subgroup	index
C₅₃×#389	2
C₅₃×#384	2
C₅₃×#340	2
C₅₃×#336	2
#2100	53

subgroup

index

C₅₃×#389

C₅₃×#384

C₅₃×#340

C₅₃×#336

#2100

Subgroups — C₅₃×#2100 #20271037

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#2100 #20271037

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2100 #20271037