Group #20271026 — subgroups

Subgroups — C₅₃×#2089 #20271026

Order 6784.

176 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 84 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×#428	C₂	1	2	yes	maximal
2	C₂×#107×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#424	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#343	C₂	1	2	yes	maximal
5	C₅₃×#341	C₂	1	1	yes	maximal, characteristic
6	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
8	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	#117×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
11	#107×C₅₃	—	2	2	no	—
12	#107×C₅₃	—	2	2	no	—
13	#106×C₅₃	—	2	2	no	—
14	C₄⋊C₈×C₅₃	—	2	2	no	—
15	C₂²×C₄×C₅₃	—	2	2	no	—
16	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
17	M₁₆×C₅₃	—	2	2	no	—
18	M₁₆×C₅₃	—	2	2	no	—
19	M₁₆×C₅₃	Q₈	1	4	yes	—
20	C₂×C₈×C₅₃	—	2	2	no	—
21	C₂×C₈×C₅₃	—	2	2	no	—
22	C₂×C₈×C₅₃	—	2	1	no	—
23	C₂×C₈×C₅₃	D₄	1	2	yes	—
24	C₄⋊C₄×C₅₃	—	2	2	no	—
25	C₄⋊C₄×C₅₃	—	2	2	no	—
26	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
27	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
28	C₂²⋊C₄×C₅₃	—	2	1	no	—
29	C₄²×C₅₃	—	2	1	no	—
30	C₂³×C₅₃	#41	1	1	yes	characteristic
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	—	8	2	no	—
35	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₅₃	#41	1	2	yes	—
39	C₈×C₅₃	—	2	4	no	—
40	C₈×C₅₃	—	2	2	no	—
41	C₈×C₅₃	—	4	1	no	—
42	C₂²×C₅₃	—	2	1	no	—
43	C₂²×C₅₃	—	2	1	no	—
44	C₂²×C₅₃	#121	1	1	yes	characteristic
45	C₂²×C₅₃	#125	1	2	yes	—
46	C₄×C₅₃	—	4	1	no	—
47	C₄×C₅₃	—	4	1	no	—
48	C₄×C₅₃	—	8	2	no	—
49	C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
50	C₄×C₅₃	#123	1	1	yes	characteristic
51	C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
52	#2089	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂×C₅₃	—	2	2	no	—
54	C₂×C₅₃	#398	1	1	yes	characteristic
55	C₂×C₅₃	#472	1	1	yes	characteristic
56	C₂×C₅₃	#473	1	1	yes	characteristic
57	#428	C₂×C₅₃	1	2	yes	—
58	C₂×#107	C₂×C₅₃	1	1	yes	characteristic
59	#424	C₂×C₅₃	1	1	yes	characteristic
60	#343	C₂×C₅₃	1	2	yes	—
61	#341	C₂×C₅₃	1	1	yes	characteristic
62	C₅₃	#2089	1	1	yes	characteristic, direct product factor, Sylow
63	C₂×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
64	C₂×M₁₆	C₂²×C₅₃	1	2	yes	—
65	C₂²×C₈	C₂²×C₅₃	1	1	yes	characteristic
66	#117	C₂²×C₅₃	1	1	yes	characteristic
67	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
68	#107	—	2	2	no	—
69	#107	—	2	2	no	—
70	#106	—	2	2	no	—
71	C₄⋊C₈	—	2	2	no	—
72	C₂²×C₄	—	2	2	no	—
73	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
74	M₁₆	—	2	2	no	—
75	M₁₆	—	2	2	no	—
76	M₁₆	Q₈×C₅₃	1	4	yes	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	D₄×C₅₃	1	2	yes	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
84	C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
85	C₂²⋊C₄	—	2	1	no	—
86	C₄²	—	2	1	no	—
87	C₂³	#41×C₅₃	1	1	yes	characteristic
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	4	2	no	—
91	C₂×C₄	—	8	2	no	—
92	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
93	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
94	C₂×C₄	C₂×Q₈×C₅₃	1	2	yes	—
95	C₂×C₄	#41×C₅₃	1	2	yes	—
96	C₈	—	2	4	no	—
97	C₈	—	2	2	no	—
98	C₈	—	4	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	—	2	1	no	—
101	C₂²	#121×C₅₃	1	1	yes	characteristic
102	C₂²	#125×C₅₃	1	2	yes	—
103	C₄	—	4	1	no	—
104	C₄	—	4	1	no	—
105	C₄	—	8	2	no	—
106	C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
107	C₄	#123×C₅₃	1	1	yes	characteristic
108	C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
109	C₂	—	2	2	no	—
110	C₂	C₅₃×#398	1	1	yes	characteristic
111	C₂	C₅₃×#472	1	1	yes	characteristic
112	C₂	C₅₃×#473	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#428

C₂

yes

maximal

C₂×#107×C₅₃

C₂

yes

maximal, characteristic

C₅₃×#424

C₂

yes

maximal, characteristic

C₅₃×#343

C₂

yes

maximal

C₅₃×#341

C₂

yes

maximal, characteristic

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂×M₁₆×C₅₃

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

#117×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

#107×C₅₃

—

#107×C₅₃

—

#106×C₅₃

—

C₄⋊C₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

Q₈

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₂³×C₅₃

#41

yes

characteristic

C₂×C₄×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

C₂×Q₈

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#121

yes

characteristic

C₂²×C₅₃

#125

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊Q₈

yes

—

C₄×C₅₃

#123

yes

characteristic

C₄×C₅₃

C₄⋊Q₈

yes

characteristic

#2089

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×#107

C₂×C₅₃

yes

characteristic

#424

C₂×C₅₃

yes

characteristic

#343

C₂×C₅₃

yes

—

#341

C₂×C₅₃

yes

characteristic

C₅₃

#2089

yes

characteristic, direct product factor, Sylow

C₂×M₁₆

C₂²×C₅₃

yes

characteristic

C₂×M₁₆

C₂²×C₅₃

yes

—

C₂²×C₈

C₂²×C₅₃

yes

characteristic

#117

C₂²×C₅₃

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₅₃

yes

—

#107

—

#107

—

#106

—

C₄⋊C₈

—

C₂²×C₄

—

C₂²×C₄

C₂³×C₅₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

Q₈×C₅₃

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×C₅₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄×C₅₃

yes

characteristic

C₄⋊C₄

D₄×C₅₃

yes

characteristic

C₂²⋊C₄

—

C₄²

—

C₂³

#41×C₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄×C₅₃

yes

characteristic

C₂×C₄

C₂×D₄×C₅₃

yes

characteristic

C₂×C₄

C₂×Q₈×C₅₃

yes

—

C₂×C₄

#41×C₅₃

yes

—

C₈

—

C₈

—

C₈

—

C₂²

—

100

C₂²

—

101

C₂²

#121×C₅₃

yes

characteristic

102

C₂²

#125×C₅₃

yes

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

C₂²⋊Q₈×C₅₃

yes

—

107

C₄

#123×C₅₃

yes

characteristic

108

C₄

C₄⋊Q₈×C₅₃

yes

characteristic

109

C₂

—

110

C₂

C₅₃×#398

yes

characteristic

111

C₂

C₅₃×#472

yes

characteristic

112

C₂

C₅₃×#473

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₅₃
prime radical	C₂³×C₅₃	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

#121

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₅₃

C₂³×C₅₃

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ G

N	H ≅ G/N	type
52. #2089	C₅₃	direct (N × H)
62. C₅₃	#2089	direct (N × H)

H ≅ G/N

type

52. #2089

C₅₃

direct (N × H)

62. C₅₃

#2089

direct (N × H)

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

subgroup	index
C₅₃×#428	2
C₂×#107×C₅₃	2
C₅₃×#424	2
C₅₃×#343	2
C₅₃×#341	2
#2089	53

subgroup

index

C₅₃×#428

C₂×#107×C₅₃

C₅₃×#424

C₅₃×#343

C₅₃×#341

#2089

Subgroups — C₅₃×#2089 #20271026

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#2089 #20271026

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2089 #20271026