Group #20270633 — subgroups

Subgroups — C₅₃×#1696 #20270633

Order 6784.

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

normal subgroups only

This group has 140 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×#477	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×#333	C₂	1	2	yes	maximal
6	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
7	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
8	#107×C₅₃	—	2	1	no	—
9	#107×C₅₃	—	2	1	no	—
10	#107×C₅₃	—	2	2	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	C₂×C₈×C₅₃	—	2	1	no	—
20	C₂×C₈×C₅₃	—	2	2	no	—
21	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
22	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
23	C₄⋊C₄×C₅₃	—	2	1	no	—
24	C₄⋊C₄×C₅₃	—	2	2	no	—
25	C₄⋊C₄×C₅₃	—	2	2	no	—
26	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
27	C₄²×C₅₃	—	2	1	no	—
28	C₄²×C₅₃	C₂³	1	1	yes	characteristic
29	Q₈×C₅₃	—	2	2	no	—
30	Q₈×C₅₃	—	8	2	no	—
31	C₂×C₄×C₅₃	—	2	1	no	—
32	C₂×C₄×C₅₃	—	2	1	no	—
33	C₂×C₄×C₅₃	—	4	2	no	—
34	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
37	C₈×C₅₃	—	2	1	no	—
38	C₈×C₅₃	—	4	1	no	—
39	C₈×C₅₃	—	4	2	no	—
40	C₈×C₅₃	Q₁₆	1	2	yes	—
41	C₂²×C₅₃	#121	1	1	yes	characteristic
42	C₄×C₅₃	—	2	1	no	—
43	C₄×C₅₃	—	4	1	no	—
44	C₄×C₅₃	—	4	1	no	—
45	C₄×C₅₃	—	8	2	no	—
46	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
47	C₄×C₅₃	#137	1	1	yes	characteristic
48	C₄×C₅₃	#137	1	2	yes	—
49	#1696	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂×C₅₃	#462	1	1	yes	characteristic
51	C₂×C₅₃	#470	1	1	yes	characteristic
52	C₂×C₅₃	#472	1	1	yes	characteristic
53	#500	C₂×C₅₃	1	1	yes	characteristic
54	#477	C₂×C₅₃	1	2	yes	—
55	#446	C₂×C₅₃	1	1	yes	characteristic
56	#334	C₂×C₅₃	1	1	yes	characteristic
57	#333	C₂×C₅₃	1	2	yes	—
58	C₅₃	#1696	1	1	yes	characteristic, direct product factor, Sylow
59	C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
60	C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
61	#107	—	2	1	no	—
62	#107	—	2	1	no	—
63	#107	—	2	2	no	—
64	C₄⋊C₈	—	2	1	no	—
65	C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
66	C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
67	#103	—	2	2	no	—
68	#97	—	2	1	no	—
69	C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
70	C₂×Q₈	—	2	2	no	—
71	C₂×Q₈	D₄×C₅₃	1	1	yes	characteristic
72	C₂×C₈	—	2	1	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	D₄×C₅₃	1	1	yes	characteristic
75	C₂×C₈	D₄×C₅₃	1	1	yes	characteristic
76	C₄⋊C₄	—	2	1	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄⋊C₄	—	2	2	no	—
79	C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
80	C₄²	—	2	1	no	—
81	C₄²	C₂³×C₅₃	1	1	yes	characteristic
82	Q₈	—	2	2	no	—
83	Q₈	—	8	2	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	4	2	no	—
87	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
88	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
89	C₂×C₄	#41×C₅₃	1	1	yes	characteristic
90	C₈	—	2	1	no	—
91	C₈	—	4	1	no	—
92	C₈	—	4	2	no	—
93	C₈	Q₁₆×C₅₃	1	2	yes	—
94	C₂²	#121×C₅₃	1	1	yes	characteristic
95	C₄	—	2	1	no	—
96	C₄	—	4	1	no	—
97	C₄	—	4	1	no	—
98	C₄	—	8	2	no	—
99	C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
100	C₄	#137×C₅₃	1	1	yes	characteristic
101	C₄	#137×C₅₃	1	2	yes	—
102	C₂	C₅₃×#462	1	1	yes	characteristic
103	C₂	C₅₃×#470	1	1	yes	characteristic
104	C₂	C₅₃×#472	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#500

C₂

yes

maximal, characteristic

C₅₃×#477

C₂

yes

maximal

C₅₃×#446

C₂

yes

maximal, characteristic

C₅₃×#334

C₂

yes

maximal, characteristic

C₅₃×#333

C₂

yes

maximal

C₄⋊Q₈×C₅₃

C₂²

yes

—

C₄×Q₈×C₅₃

C₂²

yes

characteristic

#107×C₅₃

—

#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

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

—

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

—

C₂²×C₅₃

#121

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×Q₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

#1696

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

characteristic

#477

C₂×C₅₃

yes

—

#446

C₂×C₅₃

yes

characteristic

#334

C₂×C₅₃

yes

characteristic

#333

C₂×C₅₃

yes

—

C₅₃

#1696

yes

characteristic, direct product factor, Sylow

C₄⋊Q₈

C₂²×C₅₃

yes

—

C₄×Q₈

C₂²×C₅₃

yes

—

—

—

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

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

—

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

—

C₂²

#121×C₅₃

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

C₂×Q₁₆×C₅₃

yes

characteristic

100

C₄

#137×C₅₃

yes

characteristic

101

C₄

#137×C₅₃

yes

—

102

C₂

C₅₃×#462

yes

characteristic

103

C₂

C₅₃×#470

yes

characteristic

104

C₂

C₅₃×#472

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
49. #1696	C₅₃	direct (N × H)
58. C₅₃	#1696	direct (N × H)

H ≅ G/N

type

49. #1696

C₅₃

direct (N × H)

58. C₅₃

#1696

direct (N × H)

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

subgroup	index
C₅₃×#500	2
C₅₃×#477	2
C₅₃×#446	2
C₅₃×#334	2
C₅₃×#333	2
#1696	53

subgroup

index

C₅₃×#500

C₅₃×#477

C₅₃×#446

C₅₃×#334

C₅₃×#333

#1696

Subgroups — C₅₃×#1696 #20270633

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#1696 #20270633

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#1696 #20270633