Group #20270473 — subgroups

Subgroups — C₅₃×#1536 #20270473

Order 6784.

238 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 88 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×#532	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#432	C₂	1	2	yes	maximal, complemented
3	C₅₃×#329	C₂	1	4	yes	maximal, complemented
4	C₂×#41×C₅₃	C₄	1	2	yes	—
5	C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
6	#127×C₅₃	C₄	1	1	yes	characteristic
7	#124×C₅₃	C₂²	1	2	yes	—
8	#121×C₅₃	—	2	2	no	—
9	C₄×D₄×C₅₃	—	2	2	no	—
10	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₅₃	—	2	2	no	—
12	C₂²⋊C₈×C₅₃	—	2	2	no	—
13	#97×C₅₃	C₂²	1	4	yes	—
14	#41×C₅₃	—	4	4	no	—
15	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
16	C₂×D₄×C₅₃	—	2	2	no	—
17	C₂×D₄×C₅₃	—	2	2	no	—
18	C₂×D₄×C₅₃	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₅₃	—	2	2	no	—
20	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	D₄	1	2	yes	—
22	C₂×C₈×C₅₃	—	2	4	no	—
23	C₄⋊C₄×C₅₃	—	2	2	no	—
24	C₂²⋊C₄×C₅₃	—	2	4	no	—
25	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₅₃	D₄	1	2	yes	—
27	C₄²×C₅₃	C₂³	1	1	yes	characteristic
28	C₂³×C₅₃	—	2	2	no	—
29	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
30	Q₈×C₅₃	—	4	2	no	—
31	D₄×C₅₃	—	4	2	no	—
32	D₄×C₅₃	—	4	4	no	—
33	D₄×C₅₃	—	4	4	no	—
34	C₂×C₄×C₅₃	—	2	2	no	—
35	C₂×C₄×C₅₃	—	2	2	no	—
36	C₂×C₄×C₅₃	—	4	2	no	—
37	C₂×C₄×C₅₃	—	4	4	no	—
38	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
42	C₈×C₅₃	—	4	4	no	—
43	C₂²×C₅₃	—	2	1	no	—
44	C₂²×C₅₃	—	2	2	no	—
45	C₂²×C₅₃	—	4	2	no	—
46	C₂²×C₅₃	—	4	4	no	—
47	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₄×C₅₃	—	2	1	no	—
49	C₄×C₅₃	—	2	2	no	—
50	C₄×C₅₃	—	2	2	no	—
51	C₄×C₅₃	—	4	2	no	—
52	C₄×C₅₃	#100	1	2	yes	—
53	#1536	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂×C₅₃	—	4	1	no	—
55	C₂×C₅₃	—	8	2	no	—
56	C₂×C₅₃	C₂×#100	1	1	yes	characteristic
57	C₂×C₅₃	#421	1	2	yes	—
58	#532	C₂×C₅₃	1	1	yes	characteristic
59	#432	C₂×C₅₃	1	2	yes	complemented
60	#329	C₂×C₅₃	1	4	yes	complemented
61	C₅₃	#1536	1	1	yes	characteristic, direct product factor, Sylow
62	C₂×#41	C₄×C₅₃	1	2	yes	—
63	C₄⋊Q₈	C₄×C₅₃	1	1	yes	characteristic
64	#127	C₄×C₅₃	1	1	yes	characteristic
65	#124	C₂²×C₅₃	1	2	yes	—
66	#121	—	2	2	no	—
67	C₄×D₄	—	2	2	no	—
68	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
69	C₄⋊C₈	—	2	2	no	—
70	C₂²⋊C₈	—	2	2	no	—
71	#97	C₂²×C₅₃	1	4	yes	—
72	#41	—	4	4	no	—
73	C₂×Q₈	C₂×C₄×C₅₃	1	2	yes	—
74	C₂×D₄	—	2	2	no	—
75	C₂×D₄	—	2	2	no	—
76	C₂×D₄	C₂×C₄×C₅₃	1	2	yes	—
77	C₂²×C₄	—	2	2	no	—
78	C₂²×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
79	C₂²×C₄	D₄×C₅₃	1	2	yes	—
80	C₂×C₈	—	2	4	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₂²⋊C₄	—	2	4	no	—
83	C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
84	C₄²	D₄×C₅₃	1	2	yes	—
85	C₄²	C₂³×C₅₃	1	1	yes	characteristic
86	C₂³	—	2	2	no	—
87	C₂³	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
88	Q₈	—	4	2	no	—
89	D₄	—	4	2	no	—
90	D₄	—	4	4	no	—
91	D₄	—	4	4	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	4	4	no	—
96	C₂×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
97	C₂×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
98	C₂×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
99	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
100	C₈	—	4	4	no	—
101	C₂²	—	2	1	no	—
102	C₂²	—	2	2	no	—
103	C₂²	—	4	2	no	—
104	C₂²	—	4	4	no	—
105	C₂²	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
106	C₄	—	2	1	no	—
107	C₄	—	2	2	no	—
108	C₄	—	2	2	no	—
109	C₄	—	4	2	no	—
110	C₄	#100×C₅₃	1	2	yes	—
111	C₂	—	4	1	no	—
112	C₂	—	8	2	no	—
113	C₂	C₂×#100×C₅₃	1	1	yes	characteristic
114	C₂	C₅₃×#421	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#532

C₂

yes

maximal, characteristic

C₅₃×#432

C₂

yes

maximal, complemented

C₅₃×#329

C₂

yes

maximal, complemented

C₂×#41×C₅₃

C₄

yes

—

C₄⋊Q₈×C₅₃

C₄

yes

characteristic

#127×C₅₃

C₄

yes

characteristic

#124×C₅₃

C₂²

yes

—

#121×C₅₃

—

C₄×D₄×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

#97×C₅₃

C₂²

yes

—

#41×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

C₂³×C₅₃

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×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

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#100

yes

—

#1536

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂×#100

yes

characteristic

C₂×C₅₃

#421

yes

—

#532

C₂×C₅₃

yes

characteristic

#432

C₂×C₅₃

yes

complemented

#329

C₂×C₅₃

yes

complemented

C₅₃

#1536

yes

characteristic, direct product factor, Sylow

C₂×#41

C₄×C₅₃

yes

—

C₄⋊Q₈

C₄×C₅₃

yes

characteristic

#127

C₄×C₅₃

yes

characteristic

#124

C₂²×C₅₃

yes

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂²×C₅₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

C₂²×C₅₃

yes

—

#41

—

C₂×Q₈

C₂×C₄×C₅₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×C₄×C₅₃

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×C₅₃

yes

characteristic

C₂²×C₄

D₄×C₅₃

yes

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×C₄×C₅₃

yes

characteristic

C₄²

D₄×C₅₃

yes

—

C₄²

C₂³×C₅₃

yes

characteristic

C₂³

—

C₂³

C₂²⋊C₄×C₅₃

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

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

—

100

C₈

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

C₂×C₂²⋊C₄×C₅₃

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

#100×C₅₃

yes

—

111

C₂

—

112

C₂

—

113

C₂

C₂×#100×C₅₃

yes

characteristic

114

C₂

C₅₃×#421

yes

—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₄²	C₂³×C₅₃
prime radical	C₅₃×#532	C₂
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

C₅₃×#532

C₂

LCS

G ▷ C₂×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
2. C₅₃×#432	C₂	semidirect
3. C₅₃×#329	C₂	semidirect
53. #1536	C₅₃	direct (N × H)
59. #432	C₂×C₅₃	semidirect
60. #329	C₂×C₅₃	semidirect
61. C₅₃	#1536	direct (N × H)

H ≅ G/N

type

2. C₅₃×#432

C₂

semidirect

3. C₅₃×#329

C₂

semidirect

53. #1536

C₅₃

direct (N × H)

59. #432

C₂×C₅₃

semidirect

60. #329

C₂×C₅₃

semidirect

61. C₅₃

#1536

direct (N × H)

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

subgroup	index
C₅₃×#532	2
C₅₃×#432	2
C₅₃×#329	2
#1536	53

subgroup

index

C₅₃×#532

C₅₃×#432

C₅₃×#329

#1536

Subgroups — C₅₃×#1536 #20270473

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#1536 #20270473

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#1536 #20270473