Group #27484889 — subgroups

Subgroups — #27484889

Order 13568.

404 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 167 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	#20276290	C₂	1	4	yes	maximal
2	#20275506	C₂	1	4	yes	maximal
3	#20274840	C₂	1	4	yes	maximal
4	#20274599	C₂	1	2	yes	maximal
5	C₅₃×#1849	C₂	1	1	yes	maximal, characteristic
6	#16810672	C₂²	1	4	yes	—
7	C₂×#13133153	C₂²	1	2	yes	—
8	#16810661	C₂²	1	2	yes	—
9	#16810578	C₂²	1	4	yes	—
10	#16810395	C₂²	1	8	yes	—
11	#16810383	C₂²	1	4	yes	—
12	#16810380	C₂²	1	4	yes	—
13	C₂×#108×C₅₃	C₂²	1	4	yes	—
14	C₅₃×#423	C₂²	1	2	yes	—
15	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#11719917	C₂³	1	4	yes	—
17	#13133176	C₂³	1	4	yes	—
18	#13133153	—	2	4	no	—
19	#13133153	—	4	4	no	—
20	#13133151	—	2	2	no	—
21	#13133149	—	2	2	no	—
22	#13133127	—	2	4	no	—
23	C₈×Dic₅₃	—	2	4	no	—
24	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
25	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
26	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
27	#108×C₅₃	—	2	4	no	—
28	#108×C₅₃	—	2	4	no	—
29	C₄⋊C₈×C₅₃	—	4	2	no	—
30	#97×C₅₃	Q₈	1	4	yes	—
31	C₂²×C₄×C₅₃	—	2	1	no	—
32	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
33	M₁₆×C₅₃	—	2	8	no	—
34	C₂×C₈×C₅₃	—	2	4	no	—
35	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
36	C₂×C₈×C₅₃	#41	1	2	yes	—
37	C₂×C₈×C₅₃	#41	1	2	yes	—
38	C₂×C₈×C₅₃	#41	1	4	yes	—
39	C₄²×C₅₃	—	2	1	no	—
40	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
41	C₂²⋊Dic₅₃	—	2	4	no	—
42	C₄⋊Dic₅₃	—	2	4	no	—
43	C₄×Dic₅₃	—	2	4	no	—
44	#11719917	—	2	4	no	—
45	#11719917	—	2	4	no	—
46	C₂×#88322	—	2	4	no	—
47	C₂³×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
48	C₂×C₄×C₅₃	—	2	2	no	—
49	C₂×C₄×C₅₃	—	8	1	no	—
50	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
52	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
53	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
54	C₂×Dic₅₃	—	4	4	no	—
55	C₈×C₅₃	—	4	4	no	—
56	C₈×C₅₃	#125	1	4	yes	—
57	C₈×C₅₃	#125	1	4	yes	—
58	#88322	—	4	4	no	—
59	#29231	—	53	1	no	maximal, Sylow
60	C₂²×C₅₃	—	4	1	no	—
61	C₂²×C₅₃	#544	1	1	yes	characteristic
62	C₂²×C₅₃	#556	1	2	yes	—
63	C₄×C₅₃	—	4	2	no	—
64	C₄×C₅₃	C₂×#125	1	2	yes	—
65	C₄×C₅₃	#555	1	1	yes	characteristic
66	C₄×C₅₃	#559	1	1	yes	characteristic
67	Dic₅₃	—	8	4	no	—
68	#2102	—	53	4	no	—
69	#1953	—	53	4	no	—
70	#1849	D₅₃	1	1	yes	characteristic
71	#1833	—	53	4	no	—
72	#1773	—	53	2	no	—
73	C₂×C₅₃	—	2	2	no	—
74	C₂×C₅₃	—	2	1	no	—
75	C₂×C₅₃	#2571	1	1	yes	characteristic
76	#430	—	53	4	no	—
77	C₂×#108	—	53	2	no	—
78	C₂×#108	C₂×D₅₃	1	4	yes	—
79	#427	—	53	2	no	—
80	#423	C₂×D₅₃	1	2	yes	—
81	#405	—	53	4	no	—
82	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
83	#344	—	53	8	no	—
84	#339	—	53	4	no	—
85	H(3,ℤ/4)	—	53	4	no	—
86	C₅₃	#29231	1	1	yes	characteristic, complemented, Sylow
87	C₂×M₁₆	—	53	4	no	—
88	C₂×M₁₆	C₂²×D₅₃	1	4	yes	—
89	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
90	#117	—	53	4	no	—
91	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
92	#108	—	2	4	no	—
93	#108	—	2	4	no	—
94	#108	—	106	4	no	—
95	#108	—	212	4	no	—
96	#107	—	106	2	no	—
97	#106	—	106	2	no	—
98	C₄⋊C₈	—	4	2	no	—
99	#97	—	106	4	no	—
100	#97	Dic₁₀₆	1	4	yes	—
101	C₄×C₈	—	106	4	no	—
102	C₂²×C₄	—	2	1	no	—
103	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
104	M₁₆	—	2	8	no	—
105	M₁₆	—	106	4	no	—
106	M₁₆	—	106	4	no	—
107	C₂×C₈	—	2	4	no	—
108	C₂×C₈	—	106	4	no	—
109	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
110	C₂×C₈	#11719947	1	2	yes	—
111	C₂×C₈	#11719947	1	2	yes	—
112	C₂×C₈	#11719949	1	4	yes	—
113	C₄⋊C₄	—	106	4	no	—
114	C₂²⋊C₄	—	106	4	no	—
115	C₄²	—	2	1	no	—
116	C₄²	—	106	4	no	—
117	C₄²	C₂×Dic₁₀₆	1	2	yes	—
118	C₂³	#13133265	1	1	yes	characteristic
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	8	1	no	—
121	C₂×C₄	—	212	4	no	—
122	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
123	C₂×C₄	C₂×#11719947	1	2	yes	—
124	C₂×C₄	C₂×#11719949	1	2	yes	—
125	C₂×C₄	#13133262	1	1	yes	characteristic
126	C₈	—	4	4	no	—
127	C₈	—	212	4	no	—
128	C₈	#13133208	1	4	yes	—
129	C₈	#13133208	1	4	yes	—
130	C₂²	—	4	1	no	—
131	C₂²	#16811254	1	1	yes	characteristic
132	C₂²	#16811347	1	2	yes	—
133	C₄	—	4	2	no	—
134	C₄	—	424	4	no	—
135	C₄	C₂×#13133208	1	2	yes	—
136	C₄	#16811339	1	1	yes	characteristic
137	C₄	#16811359	1	1	yes	characteristic
138	C₂	—	2	1	no	—
139	C₂	—	2	2	no	—
140	C₂	#20279957	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276290

C₂

yes

maximal

#20275506

C₂

yes

maximal

#20274840

C₂

yes

maximal

#20274599

C₂

yes

maximal

C₅₃×#1849

C₂

yes

maximal, characteristic

#16810672

C₂²

yes

—

C₂×#13133153

C₂²

yes

—

#16810661

C₂²

yes

—

#16810578

C₂²

yes

—

#16810395

C₂²

yes

—

#16810383

C₂²

yes

—

#16810380

C₂²

yes

—

C₂×#108×C₅₃

C₂²

yes

—

C₅₃×#423

C₂²

yes

—

C₂×#97×C₅₃

C₂²

yes

characteristic

C₂×#11719917

C₂³

yes

—

#13133176

C₂³

yes

—

—

—

—

—

—

C₈×Dic₅₃

—

C₂×M₁₆×C₅₃

C₂³

yes

—

C₂²×C₈×C₅₃

C₂³

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#108×C₅₃

—

#108×C₅₃

—

C₄⋊C₈×C₅₃

—

#97×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂⁴

yes

characteristic

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×Q₈

yes

—

C₂×C₈×C₅₃

#41

yes

—

C₂×C₈×C₅₃

#41

yes

—

C₂×C₈×C₅₃

#41

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×Q₈

yes

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

#11719917

—

#11719917

—

C₂×#88322

—

C₂³×C₅₃

2₋¹⁺⁴

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×#41

yes

—

C₂×C₄×C₅₃

C₂×#41

yes

—

C₂×C₄×C₅₃

H(5,𝔽₂)

yes

characteristic

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#125

yes

—

C₈×C₅₃

#125

yes

—

#88322

—

#29231

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#544

yes

characteristic

C₂²×C₅₃

#556

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#125

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#2102

—

#1953

—

#1849

D₅₃

yes

characteristic

#1833

—

#1773

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

—

—

C₂×D₅₃

yes

—

#427

—

#423

C₂×D₅₃

yes

—

#405

—

C₂×#97

C₂×D₅₃

yes

—

—

—

C₅₃

#29231

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₂²×D₅₃

yes

—

C₂²×C₈

C₂²×D₅₃

yes

—

#117

—

C₂×C₄²

C₂²×D₅₃

yes

—

—

—

106

—

#108

—

212

—

#107

—

106

—

#106

—

106

—

C₄⋊C₈

—

#97

—

106

—

100

#97

Dic₁₀₆

yes

—

101

C₄×C₈

—

106

—

102

C₂²×C₄

—

103

C₂²×C₄

C₂³×D₅₃

yes

characteristic

104

M₁₆

—

105

M₁₆

—

106

—

106

M₁₆

—

106

—

107

C₂×C₈

—

108

C₂×C₈

—

106

—

109

C₂×C₈

C₂×Dic₁₀₆

yes

—

110

C₂×C₈

#11719947

yes

—

111

C₂×C₈

#11719947

yes

—

112

C₂×C₈

#11719949

yes

—

113

C₄⋊C₄

—

106

—

114

C₂²⋊C₄

—

106

—

115

C₄²

—

116

C₄²

—

106

—

117

C₄²

C₂×Dic₁₀₆

yes

—

118

C₂³

#13133265

yes

characteristic

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

212

—

122

C₂×C₄

C₂²×Dic₁₀₆

yes

characteristic

123

C₂×C₄

C₂×#11719947

yes

—

124

C₂×C₄

C₂×#11719949

yes

—

125

C₂×C₄

#13133262

yes

characteristic

126

C₈

—

127

C₈

—

212

—

128

C₈

#13133208

yes

—

129

C₈

#13133208

yes

—

130

C₂²

—

131

C₂²

#16811254

yes

characteristic

132

C₂²

#16811347

yes

—

133

C₄

—

134

C₄

—

424

—

135

C₄

C₂×#13133208

yes

—

136

C₄

#16811339

yes

characteristic

137

C₄

#16811359

yes

characteristic

138

C₂

—

139

C₂

—

140

C₂

#20279957

yes

characteristic

	subgroup	G/H
center	C₄	#16811339
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1849	C₂
socle	C₂×C₅₃	#2571
prime radical	C₂³×C₅₃	2₋¹⁺⁴
hypercenter	#1849	D₅₃
hypocenter	C₅₃	#29231
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1849

subgroup

G/H

center

C₄

#16811339

commutator, cosocle

C₂²×C₄×C₅₃

C₂⁴

Frattini

C₂²×C₄

C₂³×D₅₃

Fitting

C₅₃×#1849

C₂

socle

C₂×C₅₃

#2571

prime radical

C₂³×C₅₃

2₋¹⁺⁴

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1849

N	H ≅ G/N	type
86. C₅₃	#29231	semidirect

H ≅ G/N

type

86. C₅₃

#29231

semidirect

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

subgroup	index
#20276290	2
#20275506	2
#20274840	2
#20274599	2
C₅₃×#1849	2
#29231	53

subgroup

index

C₅₃×#1849

Subgroups — #27484889

Conjugacy classes of subgroups

Complements

Maximal subgroups