Group #27277904 — subgroups

Subgroups — #27277904

Order 13568.

338 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 163 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	#20274887	C₂	1	1	yes	maximal, characteristic
2	#20273473	C₂	1	4	yes	maximal
3	C₂×#16810344	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133153	—	2	2	no	—
6	#16810647	C₂²	1	2	yes	—
7	C₂×C₄×#88322	C₂²	1	2	yes	—
8	#16810375	—	2	8	no	—
9	#16810344	C₄	1	4	yes	—
10	C₂×#108×C₅₃	—	2	2	no	—
11	C₅₃×#423	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#11719917	—	2	2	no	—
14	C₂²×#88322	Q₈	1	4	yes	—
15	#13133153	—	4	4	no	—
16	C₄⋊#88322	—	2	4	no	—
17	C₄×#88322	C₂×C₄	1	4	yes	—
18	C₄×#88322	D₄	1	4	yes	—
19	C₂×M₁₆×C₅₃	—	2	2	no	—
20	C₂²×C₈×C₅₃	D₄	1	2	yes	—
21	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
22	#108×C₅₃	—	4	4	no	—
23	C₄⋊C₈×C₅₃	—	2	4	no	—
24	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₅₃	Q₈	1	2	yes	—
26	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
28	M₁₆×C₅₃	—	4	4	no	—
29	C₂×C₈×C₅₃	—	4	2	no	—
30	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
31	C₂×C₈×C₅₃	D₈	1	4	yes	—
32	C₂×C₈×C₅₃	Q₁₆	1	4	yes	—
33	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₄²×C₅₃	C₂×D₄	1	2	yes	—
35	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
36	#11719917	—	4	4	no	—
37	C₂×#88322	—	4	4	no	—
38	C₂×#88322	—	4	2	no	—
39	C₂×#88322	C₄⋊C₄	1	8	yes	—
40	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
41	C₂×C₄×C₅₃	—	2	4	no	—
42	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
46	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
47	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
48	C₈×C₅₃	—	8	2	no	—
49	C₈×C₅₃	#107	1	8	yes	—
50	#88322	—	2	8	no	—
51	#88322	—	8	2	no	—
52	#8826	—	53	1	no	maximal, Sylow
53	C₂²×C₅₃	—	2	2	no	—
54	C₂²×C₅₃	#384	1	1	yes	characteristic
55	C₂²×C₅₃	#500	1	2	yes	—
56	C₄×C₅₃	—	2	2	no	—
57	C₄×C₅₃	C₂×#107	1	2	yes	—
58	C₄×C₅₃	#430	1	4	yes	—
59	C₄×C₅₃	#493	1	1	yes	characteristic
60	C₄×C₅₃	#494	1	1	yes	characteristic
61	#1848	—	53	1	no	—
62	#1848	D₅₃	1	1	yes	characteristic
63	#1567	—	53	4	no	—
64	C₂×#322	—	53	1	no	—
65	C₂×C₅₃	—	2	2	no	—
66	C₂×C₅₃	#1845	1	1	yes	characteristic
67	C₂×C₅₃	#1849	1	2	yes	—
68	C₂×#108	—	2	2	no	—
69	C₂×#108	—	106	2	no	—
70	#423	—	53	2	no	—
71	#423	C₂×D₅₃	1	2	yes	—
72	C₂×C₄×C₈	—	53	2	no	—
73	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
74	#335	—	106	8	no	—
75	#322	—	53	4	no	—
76	C₅₃	#8826	1	1	yes	characteristic, complemented, Sylow
77	C₂×M₁₆	—	2	2	no	—
78	C₂×M₁₆	—	106	2	no	—
79	C₂²×C₈	—	53	4	no	—
80	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
81	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
82	#108	—	4	4	no	—
83	#108	—	212	4	no	—
84	C₄⋊C₈	—	2	4	no	—
85	C₄⋊C₈	—	106	4	no	—
86	C₄×C₈	—	53	4	no	—
87	C₄×C₈	—	53	4	no	—
88	C₄×C₈	Dic₁₀₆	1	2	yes	—
89	C₄×C₈	C₄×D₅₃	1	2	yes	—
90	C₂²×C₄	Q₈×D₅₃	1	2	yes	—
91	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
92	M₁₆	—	4	4	no	—
93	M₁₆	—	212	4	no	—
94	C₂×C₈	—	4	2	no	—
95	C₂×C₈	—	53	8	no	—
96	C₂×C₈	—	212	4	no	—
97	C₂×C₈	—	212	2	no	—
98	C₂×C₈	#11719919	1	4	yes	—
99	C₂×C₈	#11719922	1	4	yes	—
100	C₂×C₈	#11719925	1	4	yes	—
101	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
102	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
103	C₄²	D₄×D₅₃	1	2	yes	—
104	C₂³	#13133215	1	1	yes	characteristic
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
107	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
108	C₂×C₄	#13133213	1	1	yes	characteristic
109	C₂×C₄	#13133216	1	2	yes	—
110	C₂×C₄	C₂×#11719922	1	2	yes	—
111	C₂×C₄	C₂×#11719925	1	2	yes	—
112	C₈	—	8	2	no	—
113	C₈	—	106	8	no	—
114	C₈	—	424	2	no	—
115	C₈	#13133150	1	8	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	#16810492	1	1	yes	characteristic
118	C₂²	#16811026	1	2	yes	—
119	C₄	—	2	2	no	—
120	C₄	C₂×#13133150	1	2	yes	—
121	C₄	#16810670	1	4	yes	—
122	C₄	#16810999	1	1	yes	characteristic
123	C₄	#16811002	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	#20274874	1	1	yes	characteristic
126	C₂	#20274888	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20274887

C₂

yes

maximal, characteristic

#20273473

C₂

yes

maximal

C₂×#16810344

C₂

yes

maximal, characteristic

C₅₃×#1848

C₂

yes

maximal, characteristic

C₂×#13133153

—

#16810647

C₂²

yes

—

C₂×C₄×#88322

C₂²

yes

—

#16810375

—

#16810344

C₄

yes

—

C₂×#108×C₅₃

—

C₅₃×#423

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×#11719917

—

C₂²×#88322

Q₈

yes

—

#13133153

—

C₄⋊#88322

—

C₄×#88322

C₂×C₄

yes

—

C₄×#88322

D₄

yes

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#108×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

Q₁₆

yes

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×Q₈

yes

—

—

—

C₄⋊C₄

yes

—

C₂³×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

—

C₂×C₄×C₅₃

C₂×Q₁₆

yes

—

C₈×C₅₃

—

C₈×C₅₃

#107

yes

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

#384

yes

characteristic

C₂²×C₅₃

#500

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#107

yes

—

C₄×C₅₃

#430

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

—

D₅₃

yes

characteristic

#1567

—

C₂×#322

—

C₂×C₅₃

—

C₂×C₅₃

#1845

yes

characteristic

C₂×C₅₃

#1849

yes

—

C₂×#108

—

C₂×#108

—

106

—

#423

—

#423

C₂×D₅₃

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#335

—

106

—

#322

—

C₅₃

#8826

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

106

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₅₃

yes

—

C₂×C₄²

C₂²×D₅₃

yes

characteristic

#108

—

#108

—

212

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₁₀₆

yes

—

C₄×C₈

C₄×D₅₃

yes

—

C₂²×C₄

Q₈×D₅₃

yes

—

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

212

—

C₂×C₈

—

212

—

C₂×C₈

#11719919

yes

—

C₂×C₈

#11719922

yes

—

100

C₂×C₈

#11719925

yes

—

101

C₄²

C₂×Dic₁₀₆

yes

characteristic

102

C₄²

C₂×C₄×D₅₃

yes

characteristic

103

C₄²

D₄×D₅₃

yes

—

104

C₂³

#13133215

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

C₄⋊C₄×D₅₃

yes

—

107

C₂×C₄

C₂×#11719919

yes

characteristic

108

C₂×C₄

#13133213

yes

characteristic

109

C₂×C₄

#13133216

yes

—

110

C₂×C₄

C₂×#11719922

yes

—

111

C₂×C₄

C₂×#11719925

yes

—

112

C₈

—

113

C₈

—

106

—

114

C₈

—

424

—

115

C₈

#13133150

yes

—

116

C₂²

—

117

C₂²

#16810492

yes

characteristic

118

C₂²

#16811026

yes

—

119

C₄

—

120

C₄

C₂×#13133150

yes

—

121

C₄

#16810670

yes

—

122

C₄

#16810999

yes

characteristic

123

C₄

#16811002

yes

characteristic

124

C₂

—

125

C₂

#20274874

yes

characteristic

126

C₂

#20274888

yes

—

	subgroup	G/H
center	C₂×C₄	#13133213
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1848	C₂
socle	C₂²×C₅₃	#384
prime radical	C₂³×C₅₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1848	D₅₃
hypocenter	C₅₃	#8826
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

subgroup

G/H

center

C₂×C₄

#13133213

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1848

C₂

socle

C₂²×C₅₃

#384

prime radical

C₂³×C₅₃

C₄⋊Q₈

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

N	H ≅ G/N	type
76. C₅₃	#8826	semidirect

H ≅ G/N

type

76. C₅₃

#8826

semidirect

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

subgroup	index
#20274887	2
#20273473	2
C₂×#16810344	2
C₅₃×#1848	2
#8826	53

subgroup

index

C₅₃×#1848

Subgroups — #27277904

Conjugacy classes of subgroups

Complements

Maximal subgroups