Group #27496241 — subgroups

Subgroups — #27496241

Order 13568.

436 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 135 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	#20276184	C₂	1	6	yes	maximal
2	#20276117	C₂	1	2	yes	maximal
3	#20275630	C₂	1	3	yes	maximal
4	#20275505	C₂	1	3	yes	maximal
5	C₅₃×#1747	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₁₂	C₂²	1	1	yes	characteristic
7	#16810671	C₂²	1	6	yes	—
8	C₂×#13133153	C₂²	1	3	yes	—
9	#16810631	C₂²	1	6	yes	—
10	C₂×#13133135	C₂²	1	6	yes	—
11	#16810393	C₂²	1	6	yes	—
12	C₅₃×#405	C₂²	1	3	yes	—
13	C₅₃×#338	C₂²	1	1	yes	characteristic
14	C₅₃×H(3,ℤ/4)	C₂²	1	3	yes	—
15	C₂²×Dic₁₀₆	C₂³	1	2	yes	—
16	C₂×Dic₂₁₂	—	2	3	no	—
17	C₂×Dic₂₁₂	—	2	3	no	—
18	C₂×#11719917	C₂³	1	6	yes	—
19	#13133153	—	2	3	no	—
20	#13133153	—	2	3	no	—
21	#13133152	—	4	6	no	—
22	#13133140	—	4	6	no	—
23	#13133135	—	4	6	no	—
24	C₂×M₁₆×C₅₃	C₂³	1	3	yes	—
25	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
26	#117×C₅₃	C₂³	1	3	yes	—
27	#97×C₅₃	—	2	3	no	—
28	C₄×C₈×C₅₃	—	2	3	no	—
29	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
30	C₂²×Dic₅₃	—	2	2	no	—
31	C₂×Dic₁₀₆	—	2	6	no	—
32	C₂×Dic₁₀₆	—	2	6	no	—
33	M₁₆×C₅₃	—	2	3	no	—
34	M₁₆×C₅₃	—	2	3	no	—
35	C₂×C₈×C₅₃	—	2	3	no	—
36	C₂×C₈×C₅₃	#41	1	3	yes	—
37	C₂×C₈×C₅₃	#41	1	3	yes	—
38	C₄⋊C₄×C₅₃	—	2	3	no	—
39	C₂²⋊C₄×C₅₃	—	2	3	no	—
40	C₄²×C₅₃	—	2	3	no	—
41	#11719917	—	2	6	no	—
42	#11719917	—	2	6	no	—
43	C₂×#88322	—	2	6	no	—
44	Dic₂₁₂	—	4	1	no	—
45	Dic₂₁₂	—	4	3	no	—
46	C₂³×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
47	C₂×C₄×C₅₃	—	4	3	no	—
48	C₂×C₄×C₅₃	C₂×#41	1	3	yes	—
49	C₂×C₄×C₅₃	H(5,𝔽₂)	1	3	yes	—
50	C₂×Dic₅₃	—	4	6	no	—
51	Dic₁₀₆	—	4	2	no	—
52	Dic₁₀₆	—	4	6	no	—
53	C₈×C₅₃	—	4	3	no	—
54	C₈×C₅₃	#126	1	1	yes	characteristic
55	C₈×C₅₃	#126	1	3	yes	—
56	#88322	—	4	6	no	—
57	#30238	—	53	1	no	maximal, Sylow
58	C₂²×C₅₃	—	4	1	no	—
59	C₂²×C₅₃	#542	1	3	yes	—
60	C₄×C₅₃	—	8	3	no	—
61	C₄×C₅₃	C₂×#126	1	1	yes	characteristic
62	C₄×C₅₃	#540	1	3	yes	—
63	Dic₅₃	—	8	2	no	—
64	#2082	—	53	6	no	—
65	#2070	—	53	2	no	—
66	#1979	—	53	3	no	—
67	#1953	—	53	3	no	—
68	#1747	D₅₃	1	1	yes	characteristic
69	C₂×C₅₃	—	2	3	no	—
70	C₂×C₅₃	#2644	1	1	yes	characteristic
71	C₂²×Q₁₆	—	53	1	no	—
72	#430	—	53	6	no	—
73	C₂×#108	—	53	3	no	—
74	#419	—	53	6	no	—
75	C₂×#101	—	53	6	no	—
76	#405	C₂×D₅₃	1	3	yes	—
77	#344	—	53	6	no	—
78	#338	C₂×D₅₃	1	1	yes	characteristic
79	H(3,ℤ/4)	C₂×D₅₃	1	3	yes	—
80	C₅₃	#30238	1	1	yes	characteristic, complemented, Sylow
81	C₂²×Q₈	—	53	2	no	—
82	C₂×Q₁₆	—	106	3	no	—
83	C₂×Q₁₆	—	106	3	no	—
84	C₂×M₁₆	—	53	6	no	—
85	C₂×M₁₆	C₂²×D₅₃	1	3	yes	—
86	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
87	#117	C₂²×D₅₃	1	3	yes	—
88	#108	—	106	3	no	—
89	#108	—	106	3	no	—
90	#108	—	212	6	no	—
91	#103	—	212	6	no	—
92	#101	—	212	6	no	—
93	#97	—	2	3	no	—
94	C₄×C₈	—	2	3	no	—
95	C₂×Q₈	—	106	6	no	—
96	C₂×Q₈	—	106	6	no	—
97	C₂²×C₄	—	106	2	no	—
98	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
99	Q₁₆	—	212	1	no	—
100	Q₁₆	—	212	3	no	—
101	M₁₆	—	2	3	no	—
102	M₁₆	—	2	3	no	—
103	M₁₆	—	106	6	no	—
104	M₁₆	—	106	6	no	—
105	C₂×C₈	—	2	3	no	—
106	C₂×C₈	—	106	6	no	—
107	C₂×C₈	#11719945	1	3	yes	—
108	C₂×C₈	#11719945	1	3	yes	—
109	C₄⋊C₄	—	2	3	no	—
110	C₂²⋊C₄	—	2	3	no	—
111	C₄²	—	2	3	no	—
112	C₂³	#13133264	1	1	yes	characteristic
113	Q₈	—	212	2	no	—
114	Q₈	—	212	6	no	—
115	C₂×C₄	—	4	3	no	—
116	C₂×C₄	—	212	6	no	—
117	C₂×C₄	C₂×#11719945	1	3	yes	—
118	C₂×C₄	#13133263	1	3	yes	—
119	C₈	—	4	3	no	—
120	C₈	—	212	6	no	—
121	C₈	#13133211	1	1	yes	characteristic
122	C₈	#13133211	1	3	yes	—
123	C₂²	—	4	1	no	—
124	C₂²	#16811244	1	3	yes	—
125	C₄	—	8	3	no	—
126	C₄	—	424	2	no	—
127	C₄	C₂×#13133211	1	1	yes	characteristic
128	C₄	#16811228	1	3	yes	—
129	C₂	—	2	3	no	—
130	C₂	#20280675	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276184

C₂

yes

maximal

#20276117

C₂

yes

maximal

#20275630

C₂

yes

maximal

#20275505

C₂

yes

maximal

C₅₃×#1747

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₂

C₂²

yes

characteristic

#16810671

C₂²

yes

—

C₂×#13133153

C₂²

yes

—

#16810631

C₂²

yes

—

C₂×#13133135

C₂²

yes

—

#16810393

C₂²

yes

—

C₅₃×#405

C₂²

yes

—

C₅₃×#338

C₂²

yes

characteristic

C₅₃×H(3,ℤ/4)

C₂²

yes

—

C₂²×Dic₁₀₆

C₂³

yes

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×#11719917

C₂³

yes

—

—

—

—

—

—

C₂×M₁₆×C₅₃

C₂³

yes

—

C₂²×C₈×C₅₃

C₂³

yes

characteristic

#117×C₅₃

C₂³

yes

—

#97×C₅₃

—

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

C₂⁴

yes

characteristic

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

#41

yes

—

C₂×C₈×C₅₃

#41

yes

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

#11719917

—

#11719917

—

C₂×#88322

—

Dic₂₁₂

—

Dic₂₁₂

—

C₂³×C₅₃

2₋¹⁺⁴

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×#41

yes

—

C₂×C₄×C₅₃

H(5,𝔽₂)

yes

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

#126

yes

characteristic

C₈×C₅₃

#126

yes

—

#88322

—

#30238

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#542

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#126

yes

characteristic

C₄×C₅₃

#540

yes

—

Dic₅₃

—

—

—

—

—

D₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

#2644

yes

characteristic

C₂²×Q₁₆

—

—

—

—

—

C₂×D₅₃

yes

—

#344

—

#338

C₂×D₅₃

yes

characteristic

H(3,ℤ/4)

C₂×D₅₃

yes

—

C₅₃

#30238

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×Q₁₆

—

106

—

C₂×Q₁₆

—

106

—

C₂×M₁₆

—

C₂×M₁₆

C₂²×D₅₃

yes

—

C₂²×C₈

C₂²×D₅₃

yes

characteristic

#117

C₂²×D₅₃

yes

—

#108

—

106

—

#108

—

106

—

#108

—

212

—

#103

—

212

—

#101

—

212

—

#97

—

C₄×C₈

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂³×D₅₃

yes

characteristic

Q₁₆

—

212

—

100

Q₁₆

—

212

—

101

M₁₆

—

102

M₁₆

—

103

M₁₆

—

106

—

104

M₁₆

—

106

—

105

C₂×C₈

—

106

C₂×C₈

—

106

—

107

C₂×C₈

#11719945

yes

—

108

C₂×C₈

#11719945

yes

—

109

C₄⋊C₄

—

110

C₂²⋊C₄

—

111

C₄²

—

112

C₂³

#13133264

yes

characteristic

113

Q₈

—

212

—

114

Q₈

—

212

—

115

C₂×C₄

—

116

C₂×C₄

—

212

—

117

C₂×C₄

C₂×#11719945

yes

—

118

C₂×C₄

#13133263

yes

—

119

C₈

—

120

C₈

—

212

—

121

C₈

#13133211

yes

characteristic

122

C₈

#13133211

yes

—

123

C₂²

—

124

C₂²

#16811244

yes

—

125

C₄

—

126

C₄

—

424

—

127

C₄

C₂×#13133211

yes

characteristic

128

C₄

#16811228

yes

—

129

C₂

—

130

C₂

#20280675

yes

characteristic

	subgroup	G/H
center	C₂	#20280675
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1747	C₂
socle	C₂×C₅₃	#2644
prime radical	C₂³×C₅₃	2₋¹⁺⁴
hypercenter	#1747	D₅₃
hypocenter	C₅₃	#30238
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1747

subgroup

G/H

center

C₂

#20280675

commutator, cosocle

C₂²×C₄×C₅₃

C₂⁴

Frattini

C₂²×C₄

C₂³×D₅₃

Fitting

C₅₃×#1747

C₂

socle

C₂×C₅₃

#2644

prime radical

C₂³×C₅₃

2₋¹⁺⁴

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1747

N	H ≅ G/N	type
80. C₅₃	#30238	semidirect

H ≅ G/N

type

80. C₅₃

#30238

semidirect

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

subgroup	index
#20276184	2
#20276117	2
#20275630	2
#20275505	2
C₅₃×#1747	2
#30238	53

subgroup

index

C₅₃×#1747

Subgroups — #27496241

Conjugacy classes of subgroups

Complements

Maximal subgroups