Group #27496236 — subgroups

Subgroups — #27496236

Order 13568.

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

normal subgroups only

This group has 456 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	#20276177	C₂	1	6	yes	maximal, complemented
2	#20276114	C₂	1	2	yes	maximal, complemented
3	#20275625	C₂	1	3	yes	maximal
4	#20275505	C₂	1	3	yes	maximal, complemented
5	C₅₃×#1747	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄₂₄	C₂²	1	1	yes	characteristic
7	#16810671	C₂²	1	6	yes	—
8	C₂×#13133153	C₂²	1	3	yes	—
9	#16810627	C₂²	1	6	yes	—
10	C₂×#13133133	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×D₂₁₂	C₂³	1	2	yes	—
16	C₂×D₄₂₄	—	2	3	no	—
17	C₂×D₄₂₄	—	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	#13133137	—	4	6	no	—
23	#13133133	—	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	C₂³×D₅₃	—	2	2	no	—
30	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
31	C₂×D₂₁₂	—	2	6	no	—
32	C₂×D₂₁₂	—	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	D₄₂₄	—	4	1	no	—
45	D₄₂₄	—	4	3	no	—
46	C₂³×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
47	C₂²×D₅₃	—	4	6	no	—
48	C₂²×D₅₃	—	16	2	no	—
49	C₂×C₄×C₅₃	—	4	3	no	—
50	C₂×C₄×C₅₃	C₂×#41	1	3	yes	—
51	C₂×C₄×C₅₃	H(5,𝔽₂)	1	3	yes	—
52	D₂₁₂	—	4	2	no	—
53	D₂₁₂	—	4	6	no	—
54	C₈×C₅₃	—	4	3	no	—
55	C₈×C₅₃	#126	1	1	yes	characteristic
56	C₈×C₅₃	#126	1	3	yes	—
57	#88322	—	4	6	no	—
58	#30237	—	53	1	no	maximal, Sylow
59	C₂²×C₅₃	—	4	1	no	—
60	C₂²×C₅₃	#542	1	3	yes	—
61	C₂×D₅₃	—	8	2	no	—
62	C₂×D₅₃	—	16	6	no	—
63	C₄×C₅₃	—	8	3	no	—
64	C₄×C₅₃	C₂×#126	1	1	yes	characteristic
65	C₄×C₅₃	#540	1	3	yes	—
66	#2081	—	53	6	no	—
67	#2069	—	53	2	no	—
68	#1978	—	53	3	no	—
69	#1953	—	53	3	no	—
70	#1747	D₅₃	1	1	yes	characteristic, complemented
71	C₂×C₅₃	—	2	3	no	—
72	C₂×C₅₃	#2644	1	1	yes	characteristic
73	D₅₃	—	16	2	no	—
74	C₂²×D₈	—	53	1	no	—
75	#430	—	53	6	no	—
76	C₂×#108	—	53	3	no	—
77	#418	—	53	6	no	—
78	C₂×#100	—	53	6	no	—
79	#405	C₂×D₅₃	1	3	yes	—
80	#344	—	53	6	no	—
81	#338	C₂×D₅₃	1	1	yes	characteristic
82	H(3,ℤ/4)	C₂×D₅₃	1	3	yes	—
83	C₅₃	#30237	1	1	yes	characteristic, complemented, Sylow
84	C₂²×D₄	—	53	2	no	—
85	C₂×D₈	—	106	3	no	—
86	C₂×D₈	—	106	3	no	—
87	C₂×M₁₆	—	53	6	no	—
88	C₂×M₁₆	C₂²×D₅₃	1	3	yes	—
89	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
90	#117	C₂²×D₅₃	1	3	yes	—
91	#108	—	106	3	no	—
92	#108	—	106	3	no	—
93	#108	—	212	6	no	—
94	#102	—	212	6	no	—
95	#100	—	212	6	no	—
96	#97	—	2	3	no	—
97	C₄×C₈	—	2	3	no	—
98	C₂⁴	—	106	2	no	—
99	C₂×D₄	—	106	6	no	—
100	C₂×D₄	—	106	6	no	—
101	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
102	D₈	—	212	1	no	—
103	D₈	—	212	3	no	—
104	M₁₆	—	2	3	no	—
105	M₁₆	—	2	3	no	—
106	M₁₆	—	106	6	no	—
107	M₁₆	—	106	6	no	—
108	C₂×C₈	—	2	3	no	—
109	C₂×C₈	—	106	6	no	—
110	C₂×C₈	#11719945	1	3	yes	—
111	C₂×C₈	#11719945	1	3	yes	—
112	C₄⋊C₄	—	2	3	no	—
113	C₂²⋊C₄	—	2	3	no	—
114	C₄²	—	2	3	no	—
115	C₂³	—	848	2	no	—
116	C₂³	—	212	6	no	—
117	C₂³	#13133264	1	1	yes	characteristic
118	D₄	—	212	2	no	—
119	D₄	—	212	6	no	—
120	C₂×C₄	—	4	3	no	—
121	C₂×C₄	C₂×#11719945	1	3	yes	—
122	C₂×C₄	#13133263	1	3	yes	—
123	C₈	—	4	3	no	—
124	C₈	—	212	6	no	—
125	C₈	#13133211	1	1	yes	characteristic
126	C₈	#13133211	1	3	yes	—
127	C₂²	—	4	1	no	—
128	C₂²	—	848	6	no	—
129	C₂²	—	424	2	no	—
130	C₂²	#16811244	1	3	yes	—
131	C₄	—	8	3	no	—
132	C₄	C₂×#13133211	1	1	yes	characteristic
133	C₄	#16811228	1	3	yes	—
134	C₂	—	2	3	no	—
135	C₂	—	848	2	no	—
136	C₂	#20280675	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276177

C₂

yes

maximal, complemented

#20276114

C₂

yes

maximal, complemented

#20275625

C₂

yes

maximal

#20275505

C₂

yes

maximal, complemented

C₅₃×#1747

C₂

yes

maximal, characteristic, complemented

C₂²×D₄₂₄

C₂²

yes

characteristic

#16810671

C₂²

yes

—

C₂×#13133153

C₂²

yes

—

#16810627

C₂²

yes

—

C₂×#13133133

C₂²

yes

—

#16810393

C₂²

yes

—

C₅₃×#405

C₂²

yes

—

C₅₃×#338

C₂²

yes

characteristic

C₅₃×H(3,ℤ/4)

C₂²

yes

—

C₂²×D₂₁₂

C₂³

yes

—

C₂×D₄₂₄

—

C₂×D₄₂₄

—

C₂×#11719917

C₂³

yes

—

—

—

—

—

—

C₂×M₁₆×C₅₃

C₂³

yes

—

C₂²×C₈×C₅₃

C₂³

yes

characteristic

#117×C₅₃

C₂³

yes

—

#97×C₅₃

—

C₄×C₈×C₅₃

—

C₂³×D₅₃

—

C₂²×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×D₂₁₂

—

C₂×D₂₁₂

—

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

—

D₄₂₄

—

D₄₂₄

—

C₂³×C₅₃

2₋¹⁺⁴

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×#41

yes

—

C₂×C₄×C₅₃

H(5,𝔽₂)

yes

—

D₂₁₂

—

D₂₁₂

—

C₈×C₅₃

—

C₈×C₅₃

#126

yes

characteristic

C₈×C₅₃

#126

yes

—

#88322

—

#30237

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#542

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#126

yes

characteristic

C₄×C₅₃

#540

yes

—

—

—

—

—

D₅₃

yes

characteristic, complemented

C₂×C₅₃

—

C₂×C₅₃

#2644

yes

characteristic

D₅₃

—

C₂²×D₈

—

—

—

—

—

C₂×D₅₃

yes

—

#344

—

#338

C₂×D₅₃

yes

characteristic

H(3,ℤ/4)

C₂×D₅₃

yes

—

C₅₃

#30237

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂×D₈

—

106

—

C₂×D₈

—

106

—

C₂×M₁₆

—

C₂×M₁₆

C₂²×D₅₃

yes

—

C₂²×C₈

C₂²×D₅₃

yes

characteristic

#117

C₂²×D₅₃

yes

—

#108

—

106

—

#108

—

106

—

#108

—

212

—

#102

—

212

—

#100

—

212

—

#97

—

C₄×C₈

—

C₂⁴

—

106

—

C₂×D₄

—

106

—

100

C₂×D₄

—

106

—

101

C₂²×C₄

C₂³×D₅₃

yes

characteristic

102

D₈

—

212

—

103

D₈

—

212

—

104

M₁₆

—

105

M₁₆

—

106

M₁₆

—

106

—

107

M₁₆

—

106

—

108

C₂×C₈

—

109

C₂×C₈

—

106

—

110

C₂×C₈

#11719945

yes

—

111

C₂×C₈

#11719945

yes

—

112

C₄⋊C₄

—

113

C₂²⋊C₄

—

114

C₄²

—

115

C₂³

—

848

—

116

C₂³

—

212

—

117

C₂³

#13133264

yes

characteristic

118

D₄

—

212

—

119

D₄

—

212

—

120

C₂×C₄

—

121

C₂×C₄

C₂×#11719945

yes

—

122

C₂×C₄

#13133263

yes

—

123

C₈

—

124

C₈

—

212

—

125

C₈

#13133211

yes

characteristic

126

C₈

#13133211

yes

—

127

C₂²

—

128

C₂²

—

848

—

129

C₂²

—

424

—

130

C₂²

#16811244

yes

—

131

C₄

—

132

C₄

C₂×#13133211

yes

characteristic

133

C₄

#16811228

yes

—

134

C₂

—

135

C₂

—

848

—

136

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	C₂²×D₄₂₄	C₂²
hypercenter	#1747	D₅₃
hypocenter	C₅₃	#30237
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

C₂²×D₄₂₄

C₂²

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1747

N	H ≅ G/N	type
1. #20276177	C₂	semidirect
2. #20276114	C₂	semidirect
4. #20275505	C₂	semidirect
5. C₅₃×#1747	C₂	semidirect
70. #1747	D₅₃	semidirect
83. C₅₃	#30237	semidirect

H ≅ G/N

type

1. #20276177

C₂

semidirect

2. #20276114

C₂

semidirect

4. #20275505

C₂

semidirect

5. C₅₃×#1747

C₂

semidirect

70. #1747

D₅₃

semidirect

83. C₅₃

#30237

semidirect

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

subgroup	index
#20276177	2
#20276114	2
#20275625	2
#20275505	2
C₅₃×#1747	2
#30237	53

subgroup

index

C₅₃×#1747

Subgroups — #27496236

Conjugacy classes of subgroups

Complements

Maximal subgroups