Group #27277790 — subgroups

Subgroups — #27277790

Order 13568.

270 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 115 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	#20274846	C₂	1	1	yes	maximal, characteristic
3	#20272574	C₂	1	4	yes	maximal
4	C₅₃×#1452	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133153	—	2	2	no	—
6	#16810647	C₂²	1	2	yes	—
7	#16810489	C₂²	1	2	yes	—
8	#16810386	—	2	2	no	—
9	C₅₃×#431	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×#322	C₄	1	4	yes	complemented
12	C₂×#11719917	—	2	2	no	—
13	C₂×C₄⋊Dic₅₃	—	2	2	no	—
14	#13133153	—	4	4	no	—
15	C₄⋊#88322	—	2	4	no	—
16	#13133124	—	2	4	no	—
17	C₂²×C₈×C₅₃	D₄	1	2	yes	—
18	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
19	C₄⋊C₈×C₅₃	—	2	2	no	—
20	C₂²⋊C₈×C₅₃	—	2	2	no	—
21	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
22	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₅₃	Q₈	1	2	yes	—
24	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	#41	1	2	yes	—
26	C₂²×Dic₅₃	—	4	2	no	—
27	C₂×C₈×C₅₃	—	2	2	no	—
28	C₂×C₈×C₅₃	—	2	2	no	—
29	C₂×C₈×C₅₃	—	2	4	no	—
30	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
31	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
33	C₄²×C₅₃	#41	1	2	yes	—
34	C₄⋊Dic₅₃	—	4	2	no	—
35	C₄⋊Dic₅₃	—	4	2	no	—
36	#11719917	—	4	4	no	—
37	C₂×#88322	—	4	2	no	—
38	C₂³×C₅₃	#124	1	1	yes	characteristic
39	C₂×C₄×C₅₃	—	4	2	no	—
40	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₅₃	#117	1	2	yes	—
42	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
43	C₂×C₄×C₅₃	#125	1	2	yes	—
44	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
45	C₂×C₄×C₅₃	#137	1	2	yes	—
46	C₂×Dic₅₃	—	8	4	no	—
47	C₂×Dic₅₃	—	8	2	no	—
48	C₈×C₅₃	—	2	4	no	—
49	C₈×C₅₃	—	4	4	no	—
50	#88322	—	8	2	no	—
51	#8794	—	53	1	no	maximal, Sylow
52	C₂²×C₅₃	—	2	2	no	—
53	C₂²×C₅₃	#382	1	1	yes	characteristic
54	C₂²×C₅₃	#489	1	2	yes	—
55	C₄×C₅₃	—	2	2	no	—
56	C₄×C₅₃	—	2	2	no	—
57	C₄×C₅₃	#428	1	2	yes	—
58	C₄×C₅₃	#490	1	1	yes	characteristic
59	C₄×C₅₃	#491	1	1	yes	characteristic
60	Dic₅₃	—	16	2	no	—
61	#1848	—	53	1	no	—
62	#1835	—	53	1	no	—
63	#1452	D₅₃	1	1	yes	characteristic
64	#1286	—	53	4	no	—
65	C₂×C₅₃	—	2	2	no	—
66	C₂×C₅₃	#1833	1	2	yes	—
67	C₂×C₅₃	#1836	1	1	yes	characteristic
68	#431	C₂×D₅₃	1	2	yes	—
69	C₂×#108	—	106	2	no	—
70	#423	—	53	2	no	—
71	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
72	#382	—	53	2	no	—
73	#340	—	106	2	no	—
74	#322	Dic₅₃	1	4	yes	complemented
75	C₅₃	#8794	1	1	yes	characteristic, complemented, Sylow
76	C₂×M₁₆	—	106	2	no	—
77	C₂²×C₈	D₂₁₂	1	2	yes	—
78	C₂×C₄⋊C₄	—	106	2	no	—
79	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
80	#108	—	212	4	no	—
81	C₄⋊C₈	—	2	2	no	—
82	C₄⋊C₈	—	106	4	no	—
83	C₂²⋊C₈	—	2	2	no	—
84	C₄×C₈	Dic₁₀₆	1	2	yes	—
85	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
86	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
87	#95	—	106	4	no	—
88	C₂²×C₄	—	212	2	no	—
89	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
90	C₂²×C₄	#11719945	1	2	yes	—
91	M₁₆	—	212	4	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	4	no	—
95	C₂×C₈	—	212	2	no	—
96	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
97	C₄⋊C₄	—	212	2	no	—
98	C₄⋊C₄	—	212	2	no	—
99	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
100	C₄²	#11719945	1	2	yes	—
101	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
102	C₂³	#13133205	1	1	yes	characteristic
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	424	4	no	—
105	C₂×C₄	—	424	2	no	—
106	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
107	C₂×C₄	#13133176	1	2	yes	—
108	C₂×C₄	#13133205	1	1	yes	characteristic
109	C₂×C₄	#13133209	1	2	yes	—
110	C₂×C₄	#13133245	1	2	yes	—
111	C₂×C₄	#13133250	1	2	yes	—
112	C₈	—	2	4	no	—
113	C₈	—	4	4	no	—
114	C₈	—	424	2	no	—
115	C₂²	—	2	2	no	—
116	C₂²	#16810489	1	1	yes	characteristic
117	C₂²	#16810983	1	2	yes	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	848	2	no	—
121	C₄	#16810665	1	2	yes	—
122	C₄	#16810988	1	1	yes	characteristic
123	C₄	#16810992	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	#20274839	1	2	yes	—
126	C₂	#20274849	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274887

C₂

yes

maximal, characteristic

#20274846

C₂

yes

maximal, characteristic

#20272574

C₂

yes

maximal

C₅₃×#1452

C₂

yes

maximal, characteristic

C₂×#13133153

—

#16810647

C₂²

yes

—

#16810489

C₂²

yes

—

#16810386

—

C₅₃×#431

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#322

C₄

yes

complemented

C₂×#11719917

—

C₂×C₄⋊Dic₅₃

—

#13133153

—

C₄⋊#88322

—

#13133124

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₄²×C₅₃

#41

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719917

—

C₂×#88322

—

C₂³×C₅₃

#124

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

#124

yes

characteristic

C₂×C₄×C₅₃

#125

yes

—

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#8794

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#382

yes

characteristic

C₂²×C₅₃

#489

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#428

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#1848

—

#1835

—

#1452

D₅₃

yes

characteristic

#1286

—

C₂×C₅₃

—

C₂×C₅₃

#1833

yes

—

C₂×C₅₃

#1836

yes

characteristic

#431

C₂×D₅₃

yes

—

C₂×#108

—

106

—

#423

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#382

—

#340

—

106

—

#322

Dic₅₃

yes

complemented

C₅₃

#8794

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₂²×C₈

D₂₁₂

yes

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

C₂²×D₅₃

yes

characteristic

#108

—

212

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

C₄×C₈

Dic₁₀₆

yes

—

C₄×C₈

C₂×Dic₅₃

yes

—

C₄×C₈

C₂×Dic₅₃

yes

—

#95

—

106

—

C₂²×C₄

—

212

—

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

C₂²×C₄

#11719945

yes

—

M₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

212

—

C₂×C₈

C₄⋊Dic₅₃

yes

—

C₄⋊C₄

—

212

—

C₄⋊C₄

—

212

—

C₄²

C₂×Dic₁₀₆

yes

characteristic

100

C₄²

#11719945

yes

—

101

C₄²

C₂²×Dic₅₃

yes

characteristic

102

C₂³

#13133205

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

424

—

105

C₂×C₄

—

424

—

106

C₂×C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

107

C₂×C₄

#13133176

yes

—

108

C₂×C₄

#13133205

yes

characteristic

109

C₂×C₄

#13133209

yes

—

110

C₂×C₄

#13133245

yes

—

111

C₂×C₄

#13133250

yes

—

112

C₈

—

113

C₈

—

114

C₈

—

424

—

115

C₂²

—

116

C₂²

#16810489

yes

characteristic

117

C₂²

#16810983

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

848

—

121

C₄

#16810665

yes

—

122

C₄

#16810988

yes

characteristic

123

C₄

#16810992

yes

characteristic

124

C₂

—

125

C₂

#20274839

yes

—

126

C₂

#20274849

yes

characteristic

	subgroup	G/H
center	C₂²	#16810489
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1452	C₂
socle	C₂²×C₅₃	#382
prime radical	C₂³×C₅₃	#124
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1452	D₅₃
hypocenter	C₅₃	#8794
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

subgroup

G/H

center

C₂²

#16810489

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1452

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

N	H ≅ G/N	type
11. C₅₃×#322	C₄	semidirect
74. #322	Dic₅₃	semidirect
75. C₅₃	#8794	semidirect

H ≅ G/N

type

11. C₅₃×#322

C₄

semidirect

74. #322

Dic₅₃

semidirect

75. C₅₃

#8794

semidirect

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

subgroup	index
#20274887	2
#20274846	2
#20272574	2
C₅₃×#1452	2
#8794	53

subgroup

index

C₅₃×#1452

Subgroups — #27277790

Conjugacy classes of subgroups

Complements

Maximal subgroups