Group #20270966 — subgroups

Subgroups — C₅₃×#2029 #20270966

Order 6784.

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

normal subgroups only

This group has 466 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×#543	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#379	C₂	1	3	yes	maximal
3	C₅₃×#342	C₂	1	3	yes	maximal, complemented
4	C₂⁵×C₅₃	C₂²	1	1	yes	characteristic
5	C₂²⋊Q₈×C₅₃	—	2	6	no	—
6	C₂×C₂²⋊C₄×C₅₃	—	2	6	no	—
7	C₂×C₂²⋊C₄×C₅₃	C₂²	1	3	yes	—
8	C₂×C₂²⋊C₄×C₅₃	C₂²	1	3	yes	—
9	C₂⁴×C₅₃	—	2	3	no	—
10	C₂⁴×C₅₃	—	2	3	no	—
11	C₂⁴×C₅₃	—	4	4	no	—
12	C₂⁴×C₅₃	Q₈	1	2	yes	complemented
13	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
14	C₂×Q₈×C₅₃	—	2	2	no	—
15	C₂²×C₄×C₅₃	—	2	3	no	—
16	C₂²×C₄×C₅₃	D₄	1	6	yes	—
17	C₄⋊C₄×C₅₃	—	2	6	no	—
18	C₂²⋊C₄×C₅₃	—	2	6	no	—
19	C₂²⋊C₄×C₅₃	—	2	6	no	—
20	C₂²⋊C₄×C₅₃	—	4	12	no	—
21	C₂³×C₅₃	—	2	12	no	—
22	C₂³×C₅₃	—	2	6	no	—
23	C₂³×C₅₃	—	4	4	no	—
24	C₂³×C₅₃	—	4	2	no	—
25	C₂³×C₅₃	—	4	6	no	—
26	C₂³×C₅₃	—	4	12	no	—
27	C₂³×C₅₃	—	4	1	no	—
28	C₂³×C₅₃	—	4	3	no	—
29	C₂³×C₅₃	C₂×D₄	1	3	yes	—
30	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
31	C₂³×C₅₃	#41	1	3	yes	—
32	Q₈×C₅₃	—	8	2	no	—
33	C₂×C₄×C₅₃	—	2	6	no	—
34	C₂×C₄×C₅₃	—	4	6	no	—
35	C₂×C₄×C₅₃	—	4	6	no	—
36	C₂×C₄×C₅₃	—	4	3	no	—
37	C₂²×C₅₃	—	2	6	no	—
38	C₂²×C₅₃	—	2	12	no	—
39	C₂²×C₅₃	—	4	1	no	—
40	C₂²×C₅₃	—	4	3	no	—
41	C₂²×C₅₃	—	4	12	no	—
42	C₂²×C₅₃	—	4	6	no	—
43	C₂²×C₅₃	—	4	2	no	—
44	C₂²×C₅₃	—	4	4	no	—
45	C₂²×C₅₃	#120	1	1	yes	characteristic
46	C₂²×C₅₃	C₂²⋊Q₈	1	3	yes	—
47	C₂²×C₅₃	#124	1	3	yes	—
48	C₄×C₅₃	—	4	6	no	—
49	C₄×C₅₃	—	8	3	no	—
50	#2029	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂×C₅₃	—	2	3	no	—
52	C₂×C₅₃	—	2	3	no	—
53	C₂×C₅₃	—	4	4	no	—
54	C₂×C₅₃	#393	1	1	yes	characteristic
55	C₂×C₅₃	B(4,𝔽₂)	1	2	yes	—
56	#543	C₂×C₅₃	1	1	yes	characteristic, complemented
57	#379	C₂×C₅₃	1	3	yes	—
58	#342	C₂×C₅₃	1	3	yes	complemented
59	C₅₃	#2029	1	1	yes	characteristic, direct product factor, Sylow
60	C₂⁵	C₂²×C₅₃	1	1	yes	characteristic
61	C₂²⋊Q₈	—	2	6	no	—
62	C₂×C₂²⋊C₄	—	2	6	no	—
63	C₂×C₂²⋊C₄	C₂²×C₅₃	1	3	yes	—
64	C₂×C₂²⋊C₄	C₂²×C₅₃	1	3	yes	—
65	C₂⁴	—	2	3	no	—
66	C₂⁴	—	2	3	no	—
67	C₂⁴	—	4	4	no	—
68	C₂⁴	Q₈×C₅₃	1	2	yes	complemented
69	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
70	C₂×Q₈	—	2	2	no	—
71	C₂²×C₄	—	2	3	no	—
72	C₂²×C₄	D₄×C₅₃	1	6	yes	—
73	C₄⋊C₄	—	2	6	no	—
74	C₂²⋊C₄	—	2	6	no	—
75	C₂²⋊C₄	—	2	6	no	—
76	C₂²⋊C₄	—	4	12	no	—
77	C₂³	—	2	12	no	—
78	C₂³	—	2	6	no	—
79	C₂³	—	4	4	no	—
80	C₂³	—	4	2	no	—
81	C₂³	—	4	6	no	—
82	C₂³	—	4	12	no	—
83	C₂³	—	4	1	no	—
84	C₂³	—	4	3	no	—
85	C₂³	C₂×D₄×C₅₃	1	3	yes	—
86	C₂³	C₂×Q₈×C₅₃	1	1	yes	characteristic
87	C₂³	#41×C₅₃	1	3	yes	—
88	Q₈	—	8	2	no	—
89	C₂×C₄	—	2	6	no	—
90	C₂×C₄	—	4	6	no	—
91	C₂×C₄	—	4	6	no	—
92	C₂×C₄	—	4	3	no	—
93	C₂²	—	2	6	no	—
94	C₂²	—	2	12	no	—
95	C₂²	—	4	1	no	—
96	C₂²	—	4	3	no	—
97	C₂²	—	4	12	no	—
98	C₂²	—	4	6	no	—
99	C₂²	—	4	2	no	—
100	C₂²	—	4	4	no	—
101	C₂²	#120×C₅₃	1	1	yes	characteristic
102	C₂²	C₂²⋊Q₈×C₅₃	1	3	yes	—
103	C₂²	#124×C₅₃	1	3	yes	—
104	C₄	—	4	6	no	—
105	C₄	—	8	3	no	—
106	C₂	—	2	3	no	—
107	C₂	—	2	3	no	—
108	C₂	—	4	4	no	—
109	C₂	C₅₃×#393	1	1	yes	characteristic
110	C₂	C₅₃×B(4,𝔽₂)	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#543

C₂

yes

maximal, characteristic, complemented

C₅₃×#379

C₂

yes

maximal

C₅₃×#342

C₂

yes

maximal, complemented

C₂⁵×C₅₃

C₂²

yes

characteristic

C₂²⋊Q₈×C₅₃

—

C₂×C₂²⋊C₄×C₅₃

—

C₂×C₂²⋊C₄×C₅₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₅₃

C₂²

yes

—

C₂⁴×C₅₃

—

C₂⁴×C₅₃

—

C₂⁴×C₅₃

—

C₂⁴×C₅₃

Q₈

yes

complemented

C₂⁴×C₅₃

C₂³

yes

characteristic

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂×D₄

yes

—

C₂³×C₅₃

C₂×Q₈

yes

characteristic

C₂³×C₅₃

#41

yes

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#120

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

—

C₂²×C₅₃

#124

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

#2029

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#393

yes

characteristic

C₂×C₅₃

B(4,𝔽₂)

yes

—

#543

C₂×C₅₃

yes

characteristic, complemented

#379

C₂×C₅₃

yes

—

#342

C₂×C₅₃

yes

complemented

C₅₃

#2029

yes

characteristic, direct product factor, Sylow

C₂⁵

C₂²×C₅₃

yes

characteristic

C₂²⋊Q₈

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂²×C₅₃

yes

—

C₂×C₂²⋊C₄

C₂²×C₅₃

yes

—

C₂⁴

—

C₂⁴

—

C₂⁴

—

C₂⁴

Q₈×C₅₃

yes

complemented

C₂⁴

C₂³×C₅₃

yes

characteristic

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

D₄×C₅₃

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂×D₄×C₅₃

yes

—

C₂³

C₂×Q₈×C₅₃

yes

characteristic

C₂³

#41×C₅₃

yes

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

#120×C₅₃

yes

characteristic

102

C₂²

C₂²⋊Q₈×C₅₃

yes

—

103

C₂²

#124×C₅₃

yes

—

104

C₄

—

105

C₄

—

106

C₂

—

107

C₂

—

108

C₂

—

109

C₂

C₅₃×#393

yes

characteristic

110

C₂

C₅₃×B(4,𝔽₂)

yes

—

	subgroup	G/H
center, socle	C₂²×C₅₃	#120
commutator, Frattini, cosocle	C₂⁴	C₂³×C₅₃
prime radical	C₂⁵×C₅₃	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂⁴×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

#120

commutator, Frattini, cosocle

C₂⁴

C₂³×C₅₃

prime radical

C₂⁵×C₅₃

C₂²

LCS

G ▷ C₂⁴ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₅₃ ◁ C₂⁴×C₅₃ ◁ G

N	H ≅ G/N	type
1. C₅₃×#543	C₂	semidirect
3. C₅₃×#342	C₂	semidirect
12. C₂⁴×C₅₃	Q₈	semidirect
50. #2029	C₅₃	direct (N × H)
56. #543	C₂×C₅₃	semidirect
58. #342	C₂×C₅₃	semidirect
59. C₅₃	#2029	direct (N × H)
68. C₂⁴	Q₈×C₅₃	semidirect

H ≅ G/N

type

1. C₅₃×#543

C₂

semidirect

3. C₅₃×#342

C₂

semidirect

12. C₂⁴×C₅₃

Q₈

semidirect

50. #2029

C₅₃

direct (N × H)

56. #543

C₂×C₅₃

semidirect

58. #342

C₂×C₅₃

semidirect

59. C₅₃

#2029

direct (N × H)

68. C₂⁴

Q₈×C₅₃

semidirect

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

subgroup	index
C₅₃×#543	2
C₅₃×#379	2
C₅₃×#342	2
#2029	53

subgroup

index

C₅₃×#543

C₅₃×#379

C₅₃×#342

#2029

Subgroups — C₅₃×#2029 #20270966

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#2029 #20270966

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2029 #20270966