Group #27221031 — subgroups

Subgroups — C₅₃×#13519 #27221031

Order 13568.

252 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2161	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2150	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1945	C₂	1	2	yes	maximal
4	C₅₃×#1389	C₂	1	2	yes	maximal
5	C₅₃×#1384	C₂	1	1	yes	maximal, characteristic
6	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×#108×C₅₃	C₂²	1	2	yes	—
9	C₅₃×#427	C₂²	1	2	yes	—
10	C₅₃×#424	—	2	2	no	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#368	—	2	2	no	—
13	C₅₃×C₄⋊C₁₆	—	2	2	no	—
14	C₅₃×#339	—	2	2	no	—
15	C₂×M₁₆×C₅₃	—	2	2	no	—
16	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
18	#117×C₅₃	—	2	2	no	—
19	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
20	M₃₂×C₅₃	—	2	2	no	—
21	M₃₂×C₅₃	—	2	2	no	—
22	C₂×C₁₆×C₅₃	—	2	1	no	—
23	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
24	#108×C₅₃	—	2	2	no	—
25	#108×C₅₃	—	2	2	no	—
26	#107×C₅₃	—	2	2	no	—
27	#106×C₅₃	Q₈	1	4	yes	—
28	C₄⋊C₈×C₅₃	—	4	4	no	—
29	C₄×C₈×C₅₃	—	2	1	no	—
30	C₄×C₈×C₅₃	—	2	1	no	—
31	C₂²×C₄×C₅₃	—	2	1	no	—
32	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
33	M₁₆×C₅₃	—	4	2	no	—
34	M₁₆×C₅₃	—	4	2	no	—
35	C₂×C₈×C₅₃	—	2	1	no	—
36	C₂×C₈×C₅₃	—	2	2	no	—
37	C₂×C₈×C₅₃	—	4	2	no	—
38	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
40	C₂×C₈×C₅₃	#41	1	2	yes	—
41	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
42	C₄⋊C₄×C₅₃	—	2	4	no	—
43	C₂²⋊C₄×C₅₃	—	4	2	no	—
44	C₄²×C₅₃	—	2	1	no	—
45	C₄²×C₅₃	—	4	2	no	—
46	C₄²×C₅₃	Q₁₆	1	1	yes	characteristic
47	C₄²×C₅₃	Q₁₆	1	1	yes	characteristic
48	C₁₆×C₅₃	—	2	2	no	—
49	C₁₆×C₅₃	—	4	1	no	—
50	C₂³×C₅₃	#135	1	1	yes	characteristic
51	C₂×C₄×C₅₃	—	2	1	no	—
52	C₂×C₄×C₅₃	—	2	1	no	—
53	C₂×C₄×C₅₃	—	4	2	no	—
54	C₂×C₄×C₅₃	—	4	4	no	—
55	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
57	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
58	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
59	C₈×C₅₃	—	4	1	no	—
60	C₈×C₅₃	—	8	2	no	—
61	C₈×C₅₃	#123	1	1	yes	characteristic
62	C₈×C₅₃	#125	1	2	yes	—
63	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
64	#13519	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂²×C₅₃	—	4	1	no	—
66	C₂²×C₅₃	#467	1	1	yes	characteristic
67	C₂²×C₅₃	#478	1	2	yes	—
68	C₄×C₅₃	—	2	2	no	—
69	C₄×C₅₃	—	4	1	no	—
70	C₄×C₅₃	—	8	4	no	—
71	C₄×C₅₃	#398	1	1	yes	characteristic
72	C₄×C₅₃	#469	1	1	yes	characteristic
73	C₄×C₅₃	#477	1	2	yes	—
74	#2161	C₂×C₅₃	1	1	yes	characteristic
75	#2150	C₂×C₅₃	1	1	yes	characteristic
76	#1945	C₂×C₅₃	1	2	yes	—
77	#1389	C₂×C₅₃	1	2	yes	—
78	#1384	C₂×C₅₃	1	1	yes	characteristic
79	C₂×C₅₃	—	2	1	no	—
80	C₂×C₅₃	—	2	2	no	—
81	C₂×C₅₃	#2087	1	1	yes	characteristic
82	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic
83	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic
84	C₂×#108	C₂²×C₅₃	1	2	yes	—
85	#427	C₂²×C₅₃	1	2	yes	—
86	#424	—	2	2	no	—
87	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
88	#368	—	2	2	no	—
89	C₄⋊C₁₆	—	2	2	no	—
90	#339	—	2	2	no	—
91	C₅₃	#13519	1	1	yes	characteristic, direct product factor, Sylow
92	C₂×M₁₆	—	2	2	no	—
93	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
94	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
95	#117	—	2	2	no	—
96	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
97	M₃₂	—	2	2	no	—
98	M₃₂	—	2	2	no	—
99	C₂×C₁₆	—	2	1	no	—
100	C₂×C₁₆	D₄×C₅₃	1	2	yes	—
101	#108	—	2	2	no	—
102	#108	—	2	2	no	—
103	#107	—	2	2	no	—
104	#106	Q₈×C₅₃	1	4	yes	—
105	C₄⋊C₈	—	4	4	no	—
106	C₄×C₈	—	2	1	no	—
107	C₄×C₈	—	2	1	no	—
108	C₂²×C₄	—	2	1	no	—
109	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
110	M₁₆	—	4	2	no	—
111	M₁₆	—	4	2	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	4	2	no	—
115	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
116	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
117	C₂×C₈	#41×C₅₃	1	2	yes	—
118	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
119	C₄⋊C₄	—	2	4	no	—
120	C₂²⋊C₄	—	4	2	no	—
121	C₄²	—	2	1	no	—
122	C₄²	—	4	2	no	—
123	C₄²	Q₁₆×C₅₃	1	1	yes	characteristic
124	C₄²	Q₁₆×C₅₃	1	1	yes	characteristic
125	C₁₆	—	2	2	no	—
126	C₁₆	—	4	1	no	—
127	C₂³	#135×C₅₃	1	1	yes	characteristic
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	4	no	—
132	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
133	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
134	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
135	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
136	C₈	—	4	1	no	—
137	C₈	—	8	2	no	—
138	C₈	#123×C₅₃	1	1	yes	characteristic
139	C₈	#125×C₅₃	1	2	yes	—
140	C₈	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
141	C₂²	—	4	1	no	—
142	C₂²	C₅₃×#467	1	1	yes	characteristic
143	C₂²	C₅₃×#478	1	2	yes	—
144	C₄	—	2	2	no	—
145	C₄	—	4	1	no	—
146	C₄	—	8	4	no	—
147	C₄	C₅₃×#398	1	1	yes	characteristic
148	C₄	C₅₃×#469	1	1	yes	characteristic
149	C₄	C₅₃×#477	1	2	yes	—
150	C₂	—	2	1	no	—
151	C₂	—	2	2	no	—
152	C₂	C₅₃×#2087	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#2161

C₂

yes

maximal, characteristic

C₅₃×#2150

C₂

yes

maximal, characteristic

C₅₃×#1945

C₂

yes

maximal

C₅₃×#1389

C₂

yes

maximal

C₅₃×#1384

C₂

yes

maximal, characteristic

C₂×M₃₂×C₅₃

C₂²

yes

characteristic

C₂×M₃₂×C₅₃

C₂²

yes

characteristic

C₂×#108×C₅₃

C₂²

yes

—

C₅₃×#427

C₂²

yes

—

C₅₃×#424

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#368

—

C₅₃×C₄⋊C₁₆

—

C₅₃×#339

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

#117×C₅₃

—

C₂×C₄²×C₅₃

D₄

yes

characteristic

M₃₂×C₅₃

—

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

D₄

yes

—

#108×C₅₃

—

#108×C₅₃

—

#107×C₅₃

—

#106×C₅₃

Q₈

yes

—

C₄⋊C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×Q₈

yes

—

C₂×C₈×C₅₃

#41

yes

—

C₂×C₈×C₅₃

#41

yes

characteristic

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

Q₁₆

yes

characteristic

C₄²×C₅₃

Q₁₆

yes

characteristic

C₁₆×C₅₃

—

C₁₆×C₅₃

—

C₂³×C₅₃

#135

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#121

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#123

yes

characteristic

C₈×C₅₃

#125

yes

—

C₈×C₅₃

C₄⋊Q₈

yes

characteristic

#13519

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#467

yes

characteristic

C₂²×C₅₃

#478

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

#2161

C₂×C₅₃

yes

characteristic

#2150

C₂×C₅₃

yes

characteristic

#1945

C₂×C₅₃

yes

—

#1389

C₂×C₅₃

yes

—

#1384

C₂×C₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#2087

yes

characteristic

C₂×M₃₂

C₂²×C₅₃

yes

characteristic

C₂×M₃₂

C₂²×C₅₃

yes

characteristic

C₂×#108

C₂²×C₅₃

yes

—

#427

C₂²×C₅₃

yes

—

#424

—

C₂×C₄×C₈

C₂²×C₅₃

yes

characteristic

#368

—

C₄⋊C₁₆

—

#339

—

C₅₃

#13519

yes

characteristic, direct product factor, Sylow

C₂×M₁₆

—

C₂²×C₈

D₄×C₅₃

yes

characteristic

C₂²×C₈

C₂³×C₅₃

yes

characteristic

#117

—

C₂×C₄²

D₄×C₅₃

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

100

C₂×C₁₆

D₄×C₅₃

yes

—

101

#108

—

102

#108

—

103

#107

—

104

#106

Q₈×C₅₃

yes

—

105

C₄⋊C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₂²×C₄

—

109

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

110

M₁₆

—

111

M₁₆

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

C₂×D₄×C₅₃

yes

characteristic

116

C₂×C₈

C₂×Q₈×C₅₃

yes

—

117

C₂×C₈

#41×C₅₃

yes

—

118

C₂×C₈

#41×C₅₃

yes

characteristic

119

C₄⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₄²

—

123

C₄²

Q₁₆×C₅₃

yes

characteristic

124

C₄²

Q₁₆×C₅₃

yes

characteristic

125

C₁₆

—

126

C₁₆

—

127

C₂³

#135×C₅₃

yes

characteristic

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

#121×C₅₃

yes

characteristic

133

C₂×C₄

C₂²⋊Q₈×C₅₃

yes

—

134

C₂×C₄

C₂×Q₁₆×C₅₃

yes

characteristic

135

C₂×C₄

AGL(1,ℤ/8)×C₅₃

yes

—

136

C₈

—

137

C₈

—

138

C₈

#123×C₅₃

yes

characteristic

139

C₈

#125×C₅₃

yes

—

140

C₈

C₄⋊Q₈×C₅₃

yes

characteristic

141

C₂²

—

142

C₂²

C₅₃×#467

yes

characteristic

143

C₂²

C₅₃×#478

yes

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

C₅₃×#398

yes

characteristic

148

C₄

C₅₃×#469

yes

characteristic

149

C₄

C₅₃×#477

yes

—

150

C₂

—

151

C₂

—

152

C₂

C₅₃×#2087

yes

characteristic

	subgroup	G/H
center	C₄×C₅₃	#469
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂×C₅₃	#2087
prime radical	C₂³×C₅₃	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

subgroup

G/H

center

C₄×C₅₃

#469

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₅₃

C₂×C₅₃

C₂³×C₅₃

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

N	H ≅ G/N	type
64. #13519	C₅₃	direct (N × H)
91. C₅₃	#13519	direct (N × H)

H ≅ G/N

type

64. #13519

C₅₃

direct (N × H)

91. C₅₃

#13519

direct (N × H)

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

subgroup	index
C₅₃×#2161	2
C₅₃×#2150	2
C₅₃×#1945	2
C₅₃×#1389	2
C₅₃×#1384	2
#13519	53

subgroup

index

C₅₃×#2161

C₅₃×#2150

C₅₃×#1945

C₅₃×#1389

C₅₃×#1384

#13519

Subgroups — C₅₃×#13519 #27221031

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#13519 #27221031

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#13519 #27221031