Group #20270631 — subgroups

Subgroups — C₅₃×#1694 #20270631

Order 6784.

152 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 68 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	C₅₃×#500	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#462	C₂	1	2	yes	maximal
3	C₅₃×#446	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#334	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#332	C₂	1	2	yes	maximal
6	C₂×Q₁₆×C₅₃	—	2	2	no	—
7	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
8	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
9	#107×C₅₃	—	2	1	no	—
10	#107×C₅₃	—	2	1	no	—
11	C₄⋊C₈×C₅₃	—	2	1	no	—
12	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
14	#103×C₅₃	—	2	2	no	—
15	#97×C₅₃	—	2	1	no	—
16	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₅₃	—	2	2	no	—
18	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
19	Q₁₆×C₅₃	—	4	4	no	—
20	C₂×C₈×C₅₃	—	2	1	no	—
21	C₂×C₈×C₅₃	—	2	2	no	—
22	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
23	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₄⋊C₄×C₅₃	—	2	1	no	—
25	C₄⋊C₄×C₅₃	—	2	2	no	—
26	C₄⋊C₄×C₅₃	—	2	2	no	—
27	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
28	C₄²×C₅₃	—	2	1	no	—
29	C₄²×C₅₃	C₂³	1	1	yes	characteristic
30	Q₈×C₅₃	—	2	2	no	—
31	Q₈×C₅₃	—	4	2	no	—
32	Q₈×C₅₃	—	4	2	no	—
33	C₂×C₄×C₅₃	—	2	1	no	—
34	C₂×C₄×C₅₃	—	2	1	no	—
35	C₂×C₄×C₅₃	—	4	2	no	—
36	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
39	C₈×C₅₃	—	2	1	no	—
40	C₈×C₅₃	—	4	2	no	—
41	C₈×C₅₃	—	4	1	no	—
42	C₈×C₅₃	Q₁₆	1	2	yes	complemented
43	C₂²×C₅₃	#121	1	1	yes	characteristic
44	C₄×C₅₃	—	2	1	no	—
45	C₄×C₅₃	—	4	1	no	—
46	C₄×C₅₃	—	4	1	no	—
47	C₄×C₅₃	—	8	2	no	—
48	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
49	C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
50	C₄×C₅₃	#137	1	1	yes	characteristic
51	#1694	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂×C₅₃	#462	1	1	yes	characteristic
53	C₂×C₅₃	#469	1	1	yes	characteristic
54	C₂×C₅₃	#473	1	1	yes	characteristic
55	#500	C₂×C₅₃	1	1	yes	characteristic
56	#462	C₂×C₅₃	1	2	yes	—
57	#446	C₂×C₅₃	1	1	yes	characteristic
58	#334	C₂×C₅₃	1	1	yes	characteristic
59	#332	C₂×C₅₃	1	2	yes	—
60	C₅₃	#1694	1	1	yes	characteristic, direct product factor, Sylow
61	C₂×Q₁₆	—	2	2	no	—
62	C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
63	C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
64	#107	—	2	1	no	—
65	#107	—	2	1	no	—
66	C₄⋊C₈	—	2	1	no	—
67	C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
68	C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
69	#103	—	2	2	no	—
70	#97	—	2	1	no	—
71	C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
72	C₂×Q₈	—	2	2	no	—
73	C₂×Q₈	D₄×C₅₃	1	1	yes	characteristic
74	Q₁₆	—	4	4	no	—
75	C₂×C₈	—	2	1	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	D₄×C₅₃	1	1	yes	characteristic
78	C₂×C₈	D₄×C₅₃	1	1	yes	characteristic
79	C₄⋊C₄	—	2	1	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
83	C₄²	—	2	1	no	—
84	C₄²	C₂³×C₅₃	1	1	yes	characteristic
85	Q₈	—	2	2	no	—
86	Q₈	—	4	2	no	—
87	Q₈	—	4	2	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	4	2	no	—
91	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
92	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
93	C₂×C₄	#41×C₅₃	1	1	yes	characteristic
94	C₈	—	2	1	no	—
95	C₈	—	4	2	no	—
96	C₈	—	4	1	no	—
97	C₈	Q₁₆×C₅₃	1	2	yes	complemented
98	C₂²	#121×C₅₃	1	1	yes	characteristic
99	C₄	—	2	1	no	—
100	C₄	—	4	1	no	—
101	C₄	—	4	1	no	—
102	C₄	—	8	2	no	—
103	C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
104	C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
105	C₄	#137×C₅₃	1	1	yes	characteristic
106	C₂	C₅₃×#462	1	1	yes	characteristic
107	C₂	C₅₃×#469	1	1	yes	characteristic
108	C₂	C₅₃×#473	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#500

C₂

yes

maximal, characteristic

C₅₃×#462

C₂

yes

maximal

C₅₃×#446

C₂

yes

maximal, characteristic

C₅₃×#334

C₂

yes

maximal, characteristic

C₅₃×#332

C₂

yes

maximal

C₂×Q₁₆×C₅₃

—

C₄⋊Q₈×C₅₃

C₂²

yes

—

C₄×Q₈×C₅₃

C₂²

yes

characteristic

#107×C₅₃

—

#107×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

—

#103×C₅₃

—

#97×C₅₃

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

D₄

yes

characteristic

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

Q₁₆

yes

complemented

C₂²×C₅₃

#121

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×Q₁₆

yes

characteristic

C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₄×C₅₃

#137

yes

characteristic

#1694

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

characteristic

#462

C₂×C₅₃

yes

—

#446

C₂×C₅₃

yes

characteristic

#334

C₂×C₅₃

yes

characteristic

#332

C₂×C₅₃

yes

—

C₅₃

#1694

yes

characteristic, direct product factor, Sylow

C₂×Q₁₆

—

C₄⋊Q₈

C₂²×C₅₃

yes

—

C₄×Q₈

C₂²×C₅₃

yes

characteristic

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

C₂²×C₅₃

yes

characteristic

C₄⋊C₈

C₂²×C₅₃

yes

—

#103

—

#97

—

C₄×C₈

C₂²×C₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

D₄×C₅₃

yes

characteristic

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×C₅₃

yes

characteristic

C₂×C₈

D₄×C₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄×C₅₃

yes

characteristic

C₄²

—

C₄²

C₂³×C₅₃

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄×C₅₃

yes

characteristic

C₂×C₄

C₂×D₄×C₅₃

yes

characteristic

C₂×C₄

#41×C₅₃

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

Q₁₆×C₅₃

yes

complemented

C₂²

#121×C₅₃

yes

characteristic

C₄

—

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

C₂×Q₁₆×C₅₃

yes

characteristic

104

C₄

AGL(1,ℤ/8)×C₅₃

yes

—

105

C₄

#137×C₅₃

yes

characteristic

106

C₂

C₅₃×#462

yes

characteristic

107

C₂

C₅₃×#469

yes

characteristic

108

C₂

C₅₃×#473

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₅₃	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₅₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄²×C₅₃ ◁ G

subgroup

G/H

center, socle, prime radical

C₂²×C₅₃

#121

commutator, Frattini, cosocle

C₄²

C₂³×C₅₃

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₅₃ ◁ C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
42. C₈×C₅₃	Q₁₆	semidirect
51. #1694	C₅₃	direct (N × H)
60. C₅₃	#1694	direct (N × H)
97. C₈	Q₁₆×C₅₃	semidirect

H ≅ G/N

type

42. C₈×C₅₃

Q₁₆

semidirect

51. #1694

C₅₃

direct (N × H)

60. C₅₃

#1694

direct (N × H)

97. C₈

Q₁₆×C₅₃

semidirect

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

subgroup	index
C₅₃×#500	2
C₅₃×#462	2
C₅₃×#446	2
C₅₃×#334	2
C₅₃×#332	2
#1694	53

subgroup

index

C₅₃×#500

C₅₃×#462

C₅₃×#446

C₅₃×#334

C₅₃×#332

#1694

Subgroups — C₅₃×#1694 #20270631

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#1694 #20270631

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#1694 #20270631