Group #27277869 — subgroups

Subgroups — #27277869

Order 13568.

422 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 227 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	#20274875	C₂	1	2	yes	maximal
2	#20274843	C₂	1	4	yes	maximal
3	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133151	—	2	4	no	—
5	#16810493	C₂²	1	4	yes	—
6	#16810386	—	2	8	no	—
7	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₈²×C₅₃	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₅₃	—	2	8	no	—
11	C₂×C₄⋊Dic₅₃	—	2	4	no	—
12	#13133151	—	4	8	no	—
13	C₂²×C₈×C₅₃	D₄	1	4	yes	—
14	C₂²×C₈×C₅₃	D₄	1	2	yes	—
15	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
17	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₅₃	Q₈	1	4	yes	—
19	C₄×C₈×C₅₃	Q₈	1	2	yes	—
20	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
22	C₂²×Dic₅₃	—	4	4	no	—
23	C₂×C₈×C₅₃	—	2	8	no	—
24	C₂×C₈×C₅₃	C₄⋊C₄	1	8	yes	—
25	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
26	C₂×C₈×C₅₃	D₈	1	4	yes	—
27	C₂×C₈×C₅₃	Q₁₆	1	4	yes	—
28	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
29	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
30	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
31	C₄⋊Dic₅₃	—	4	8	no	—
32	C₄⋊Dic₅₃	—	4	16	no	—
33	C₂³×C₅₃	#127	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
36	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
37	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
38	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
39	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
40	C₂×C₄×C₅₃	#135	1	8	yes	—
41	C₂×Dic₅₃	—	8	4	no	—
42	C₂×Dic₅₃	—	8	8	no	—
43	C₈×C₅₃	—	2	8	no	—
44	C₈×C₅₃	#107	1	8	yes	—
45	#8816	—	53	1	no	maximal, Sylow
46	C₂²×C₅₃	#384	1	1	yes	characteristic
47	C₂²×C₅₃	#493	1	1	yes	characteristic
48	C₂²×C₅₃	#494	1	1	yes	characteristic
49	C₂²×C₅₃	#495	1	4	yes	—
50	C₄×C₅₃	C₂×#107	1	2	yes	—
51	C₄×C₅₃	#427	1	4	yes	—
52	C₄×C₅₃	#499	1	4	yes	—
53	C₄×C₅₃	#500	1	2	yes	—
54	Dic₅₃	—	16	4	no	—
55	#1845	—	53	2	no	—
56	#1834	—	53	4	no	—
57	C₂×C₈²	D₅₃	1	1	yes	characteristic
58	C₂×C₅₃	#1709	1	2	yes	—
59	C₂×C₅₃	#1710	1	2	yes	—
60	C₂×C₅₃	#1845	1	1	yes	characteristic
61	C₂×C₅₃	#1846	1	2	yes	—
62	C₂×#107	—	106	4	no	—
63	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
64	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
65	#384	—	53	4	no	—
66	#340	—	106	8	no	—
67	C₈²	Dic₅₃	1	4	yes	complemented
68	C₅₃	#8816	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₈	D₂₁₂	1	4	yes	—
70	C₂²×C₈	D₂₁₂	1	2	yes	—
71	C₂×C₄⋊C₄	—	106	8	no	—
72	C₂×C₄⋊C₄	—	106	4	no	—
73	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
74	#107	—	212	8	no	—
75	C₄×C₈	Dic₁₀₆	1	4	yes	—
76	C₄×C₈	Dic₁₀₆	1	2	yes	—
77	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
78	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
79	C₂²×C₄	—	212	4	no	—
80	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
81	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
82	C₂×C₈	—	2	8	no	—
83	C₂×C₈	D₄₂₄	1	4	yes	—
84	C₂×C₈	Dic₂₁₂	1	4	yes	—
85	C₂×C₈	C₄⋊Dic₅₃	1	8	yes	—
86	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
87	C₄⋊C₄	—	212	8	no	—
88	C₄⋊C₄	—	212	16	no	—
89	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
90	C₄²	C₂×Dic₁₀₆	1	2	yes	—
91	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
92	C₂³	#13133214	1	1	yes	characteristic
93	C₂×C₄	—	424	4	no	—
94	C₂×C₄	—	424	8	no	—
95	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
96	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
97	C₂×C₄	#13133217	1	2	yes	—
98	C₂×C₄	#13133217	1	1	yes	characteristic
99	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
100	C₂×C₄	C₂×Dic₂₁₂	1	2	yes	—
101	C₂×C₄	#13133239	1	8	yes	—
102	C₈	—	2	8	no	—
103	C₈	#13133151	1	8	yes	—
104	C₂²	#16810493	1	1	yes	characteristic
105	C₂²	#16811000	1	1	yes	characteristic
106	C₂²	#16811003	1	1	yes	characteristic
107	C₂²	#16811007	1	4	yes	—
108	C₄	—	848	4	no	—
109	C₄	C₂×#13133151	1	2	yes	—
110	C₄	#16810661	1	4	yes	—
111	C₄	#16811025	1	4	yes	—
112	C₄	#16811028	1	2	yes	—
113	C₂	#20274321	1	2	yes	—
114	C₂	#20274325	1	2	yes	—
115	C₂	#20274875	1	1	yes	characteristic
116	C₂	#20274879	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20274875

C₂

yes

maximal

#20274843

C₂

yes

maximal

C₂×C₈²×C₅₃

C₂

yes

maximal, characteristic

C₂×#13133151

—

#16810493

C₂²

yes

—

#16810386

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₈²×C₅₃

C₄

yes

complemented

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

#13133151

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

Q₈

yes

—

C₄×C₈×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

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×Q₈

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂³×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₄⋊Q₈

yes

—

C₂×C₄×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

—

C₂×C₄×C₅₃

C₂×Q₁₆

yes

—

C₂×C₄×C₅₃

#135

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#107

yes

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₄×C₅₃

C₂×#107

yes

—

C₄×C₅₃

#427

yes

—

C₄×C₅₃

#499

yes

—

C₄×C₅₃

#500

yes

—

Dic₅₃

—

#1845

—

#1834

—

C₂×C₈²

D₅₃

yes

characteristic

C₂×C₅₃

#1709

yes

—

C₂×C₅₃

#1710

yes

—

C₂×C₅₃

#1845

yes

characteristic

C₂×C₅₃

#1846

yes

—

C₂×#107

—

106

—

C₂×C₄×C₈

C₂×D₅₃

yes

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#384

—

#340

—

106

—

C₈²

Dic₅₃

yes

complemented

C₅₃

#8816

yes

characteristic, complemented, Sylow

C₂²×C₈

D₂₁₂

yes

—

C₂²×C₈

D₂₁₂

yes

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

C₂²×D₅₃

yes

characteristic

#107

—

212

—

C₄×C₈

Dic₁₀₆

yes

—

C₄×C₈

Dic₁₀₆

yes

—

C₄×C₈

C₂×Dic₅₃

yes

—

C₄×C₈

C₂×Dic₅₃

yes

—

C₂²×C₄

—

212

—

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

C₂²×C₄

C₂×D₂₁₂

yes

—

C₂×C₈

—

C₂×C₈

D₄₂₄

yes

—

C₂×C₈

Dic₂₁₂

yes

—

C₂×C₈

C₄⋊Dic₅₃

yes

—

C₂×C₈

C₄⋊Dic₅₃

yes

—

C₄⋊C₄

—

212

—

C₄⋊C₄

—

212

—

C₄²

C₂×Dic₁₀₆

yes

characteristic

C₄²

C₂×Dic₁₀₆

yes

—

C₄²

C₂²×Dic₅₃

yes

characteristic

C₂³

#13133214

yes

characteristic

C₂×C₄

—

424

—

C₂×C₄

—

424

—

C₂×C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

C₂×C₄

C₂×C₄⋊Dic₅₃

yes

—

C₂×C₄

#13133217

yes

—

C₂×C₄

#13133217

yes

characteristic

C₂×C₄

C₂×D₄₂₄

yes

—

100

C₂×C₄

C₂×Dic₂₁₂

yes

—

101

C₂×C₄

#13133239

yes

—

102

C₈

—

103

C₈

#13133151

yes

—

104

C₂²

#16810493

yes

characteristic

105

C₂²

#16811000

yes

characteristic

106

C₂²

#16811003

yes

characteristic

107

C₂²

#16811007

yes

—

108

C₄

—

848

—

109

C₄

C₂×#13133151

yes

—

110

C₄

#16810661

yes

—

111

C₄

#16811025

yes

—

112

C₄

#16811028

yes

—

113

C₂

#20274321

yes

—

114

C₂

#20274325

yes

—

115

C₂

#20274875

yes

characteristic

116

C₂

#20274879

yes

—

	subgroup	G/H
center	C₂³	#13133214
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₈²×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	#127
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×C₈²	D₅₃
hypocenter	C₅₃	#8816
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂³

#13133214

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₂×C₈²×C₅₃

C₂

socle, prime radical

C₂³×C₅₃

#127

cosocle

C₂×C₄²×C₅₃

C₂³

hypercenter

C₂×C₈²

D₅₃

hypocenter

C₅₃

#8816

LCS

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
9. C₈²×C₅₃	C₄	semidirect
67. C₈²	Dic₅₃	semidirect
68. C₅₃	#8816	semidirect

H ≅ G/N

type

9. C₈²×C₅₃

C₄

semidirect

67. C₈²

Dic₅₃

semidirect

68. C₅₃

#8816

semidirect

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

subgroup	index
#20274875	2
#20274843	2
C₂×C₈²×C₅₃	2
#8816	53

subgroup

index

#20274875

#20274843

C₂×C₈²×C₅₃

#8816

Subgroups — #27277869

Conjugacy classes of subgroups

Complements

Maximal subgroups