Group #20270498 — subgroups

Subgroups — C₅₃×#1561 #20270498

Order 6784.

210 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 96 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×#533	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#423	C₂	1	2	yes	maximal
3	C₅₃×#332	C₂	1	4	yes	maximal, complemented
4	C₂×M₁₆×C₅₃	—	2	2	no	—
5	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
6	#125×C₅₃	C₄	1	2	yes	—
7	C₂²⋊Q₈×C₅₃	—	2	2	no	—
8	C₄×Q₈×C₅₃	—	2	2	no	—
9	#117×C₅₃	C₄	1	2	yes	—
10	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₅₃	—	2	2	no	—
12	C₄⋊C₈×C₅₃	C₂²	1	4	yes	—
13	C₂×Q₈×C₅₃	—	2	2	no	—
14	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	D₄	1	2	yes	—
16	M₁₆×C₅₃	—	4	4	no	—
17	C₂×C₈×C₅₃	—	2	4	no	—
18	C₄⋊C₄×C₅₃	—	2	4	no	—
19	C₄⋊C₄×C₅₃	—	2	2	no	—
20	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
21	C₂²⋊C₄×C₅₃	—	2	2	no	—
22	C₄²×C₅₃	—	2	2	no	—
23	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₅₃	D₄	1	2	yes	—
25	C₄²×C₅₃	C₂³	1	1	yes	characteristic
26	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
27	Q₈×C₅₃	—	4	2	no	—
28	Q₈×C₅₃	—	4	2	no	—
29	C₂×C₄×C₅₃	—	2	2	no	—
30	C₂×C₄×C₅₃	—	2	2	no	—
31	C₂×C₄×C₅₃	—	2	2	no	—
32	C₂×C₄×C₅₃	—	2	4	no	—
33	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
34	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
37	C₈×C₅₃	—	4	4	no	—
38	C₂²×C₅₃	—	2	1	no	—
39	C₂²×C₅₃	—	2	2	no	—
40	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
41	C₄×C₅₃	—	2	1	no	—
42	C₄×C₅₃	—	2	2	no	—
43	C₄×C₅₃	—	4	4	no	—
44	C₄×C₅₃	#101	1	2	yes	—
45	C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
46	C₄×C₅₃	#137	1	2	yes	—
47	#1561	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂×C₅₃	—	4	1	no	—
49	C₂×C₅₃	C₂×#101	1	1	yes	characteristic
50	C₂×C₅₃	#417	1	2	yes	—
51	#533	C₂×C₅₃	1	1	yes	characteristic
52	#423	C₂×C₅₃	1	2	yes	—
53	#332	C₂×C₅₃	1	4	yes	complemented
54	C₅₃	#1561	1	1	yes	characteristic, direct product factor, Sylow
55	C₂×M₁₆	—	2	2	no	—
56	C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
57	#125	C₄×C₅₃	1	2	yes	—
58	C₂²⋊Q₈	—	2	2	no	—
59	C₄×Q₈	—	2	2	no	—
60	#117	C₄×C₅₃	1	2	yes	—
61	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
62	C₄⋊C₈	—	2	2	no	—
63	C₄⋊C₈	C₂²×C₅₃	1	4	yes	—
64	C₂×Q₈	—	2	2	no	—
65	C₂²×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
66	C₂²×C₄	D₄×C₅₃	1	2	yes	—
67	M₁₆	—	4	4	no	—
68	C₂×C₈	—	2	4	no	—
69	C₄⋊C₄	—	2	2	no	—
70	C₄⋊C₄	—	2	4	no	—
71	C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
72	C₂²⋊C₄	—	2	2	no	—
73	C₄²	—	2	2	no	—
74	C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
75	C₄²	D₄×C₅₃	1	2	yes	—
76	C₄²	C₂³×C₅₃	1	1	yes	characteristic
77	C₂³	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
78	Q₈	—	4	2	no	—
79	Q₈	—	4	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	4	no	—
84	C₂×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
85	C₂×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
86	C₂×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
87	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
88	C₈	—	4	4	no	—
89	C₂²	—	2	1	no	—
90	C₂²	—	2	2	no	—
91	C₂²	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
92	C₄	—	2	1	no	—
93	C₄	—	2	2	no	—
94	C₄	—	4	4	no	—
95	C₄	#101×C₅₃	1	2	yes	—
96	C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
97	C₄	#137×C₅₃	1	2	yes	—
98	C₂	—	4	1	no	—
99	C₂	C₂×#101×C₅₃	1	1	yes	characteristic
100	C₂	C₅₃×#417	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#533

C₂

yes

maximal, characteristic

C₅₃×#423

C₂

yes

maximal

C₅₃×#332

C₂

yes

maximal, complemented

C₂×M₁₆×C₅₃

—

C₄⋊Q₈×C₅₃

C₂²

yes

—

#125×C₅₃

C₄

yes

—

C₂²⋊Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

#117×C₅₃

C₄

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

C₂²

yes

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₈×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#101

yes

—

C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₄×C₅₃

#137

yes

—

#1561

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×#101

yes

characteristic

C₂×C₅₃

#417

yes

—

#533

C₂×C₅₃

yes

characteristic

#423

C₂×C₅₃

yes

—

#332

C₂×C₅₃

yes

complemented

C₅₃

#1561

yes

characteristic, direct product factor, Sylow

C₂×M₁₆

—

C₄⋊Q₈

C₂²×C₅₃

yes

—

#125

C₄×C₅₃

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₄×C₅₃

yes

—

C₂×C₄²

C₂²×C₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂²×C₅₃

yes

—

C₂×Q₈

—

C₂²×C₄

C₂×C₄×C₅₃

yes

characteristic

C₂²×C₄

D₄×C₅₃

yes

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₄×C₅₃

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×C₄×C₅₃

yes

characteristic

C₄²

D₄×C₅₃

yes

—

C₄²

C₂³×C₅₃

yes

characteristic

C₂³

C₂²⋊C₄×C₅₃

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊C₄×C₅₃

yes

—

C₂×C₄

C₂²⋊C₄×C₅₃

yes

characteristic

C₂×C₄

C₂²×C₄×C₅₃

yes

characteristic

C₂×C₄

C₂×D₄×C₅₃

yes

—

C₈

—

C₂²

—

C₂²

—

C₂²

C₂×C₂²⋊C₄×C₅₃

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#101×C₅₃

yes

—

C₄

AGL(1,ℤ/8)×C₅₃

yes

—

C₄

#137×C₅₃

yes

—

C₂

—

C₂

C₂×#101×C₅₃

yes

characteristic

100

C₂

C₅₃×#417

yes

—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₄²	C₂³×C₅₃
prime radical	C₂³×C₅₃	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

C₂×C₂²⋊C₄

commutator

C₂×C₄

C₂²×C₄×C₅₃

Frattini, cosocle

C₄²

C₂³×C₅₃

prime radical

C₂³×C₅₃

C₂²⋊C₄

LCS

G ▷ C₂×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
3. C₅₃×#332	C₂	semidirect
47. #1561	C₅₃	direct (N × H)
53. #332	C₂×C₅₃	semidirect
54. C₅₃	#1561	direct (N × H)

H ≅ G/N

type

3. C₅₃×#332

C₂

semidirect

47. #1561

C₅₃

direct (N × H)

53. #332

C₂×C₅₃

semidirect

54. C₅₃

#1561

direct (N × H)

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

subgroup	index
C₅₃×#533	2
C₅₃×#423	2
C₅₃×#332	2
#1561	53

subgroup

index

C₅₃×#533

C₅₃×#423

C₅₃×#332

#1561

Subgroups — C₅₃×#1561 #20270498

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#1561 #20270498

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#1561 #20270498