Group #50311659 — subgroups

Subgroups — C₅₃×#10624774 #50311659

Order 38637.

326 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 98 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	C₃²×3₋¹⁺²×C₅₃	C₃	1	1	yes	maximal, characteristic
2	C₅₃×#6895	C₃	1	3	yes	maximal, complemented
3	C₅₃×#6868	C₃	1	3	yes	maximal, complemented
4	C₅₃×#6867	C₃	1	3	yes	maximal, complemented
5	C₅₃×#6866	C₃	1	3	yes	maximal
6	C₃⁴×C₅₃	C₃²	1	1	yes	characteristic
7	C₃×3₋¹⁺²×C₅₃	—	3	3	no	—
8	C₃×3₋¹⁺²×C₅₃	—	3	3	no	—
9	C₃×3₋¹⁺²×C₅₃	—	3	3	no	—
10	C₃×3₋¹⁺²×C₅₃	—	3	3	no	—
11	C₃²×C₉×C₅₃	C₃²	1	1	yes	characteristic
12	C₃²×C₉×C₅₃	C₃²	1	1	yes	characteristic
13	C₃²×C₉×C₅₃	C₃²	1	1	yes	characteristic
14	C₅₃×C₉⋊C₉	—	3	9	no	—
15	C₅₃×C₃²⋊C₉	C₃²	1	9	yes	—
16	C₃³×C₅₃	—	3	1	no	—
17	C₃³×C₅₃	—	3	1	no	—
18	C₃³×C₅₃	—	3	1	no	—
19	C₃³×C₅₃	—	9	3	no	—
20	C₃³×C₅₃	3₋¹⁺²	1	3	yes	—
21	C₃³×C₅₃	C₃³	1	1	yes	characteristic
22	3₋¹⁺²×C₅₃	—	3	27	no	—
23	C₃×C₉×C₅₃	—	3	1	no	—
24	C₃×C₉×C₅₃	—	3	1	no	—
25	C₃×C₉×C₅₃	—	3	1	no	—
26	C₃×C₉×C₅₃	—	3	1	no	—
27	C₃×C₉×C₅₃	—	3	1	no	—
28	C₃×C₉×C₅₃	—	3	1	no	—
29	C₃×C₉×C₅₃	—	3	1	no	—
30	C₃×C₉×C₅₃	—	3	1	no	—
31	C₃×C₉×C₅₃	—	3	1	no	—
32	C₃×C₉×C₅₃	—	3	9	no	—
33	C₃×C₉×C₅₃	H(3,𝔽₃)	1	3	yes	—
34	C₃×C₉×C₅₃	3₋¹⁺²	1	3	yes	—
35	C₃×C₉×C₅₃	3₋¹⁺²	1	3	yes	—
36	#10624774	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₃²×C₅₃	—	3	3	no	—
38	C₃²×C₅₃	—	3	3	no	—
39	C₃²×C₅₃	—	9	3	no	—
40	C₃²×C₅₃	—	9	3	no	—
41	C₃²×C₅₃	—	9	3	no	—
42	C₃²×C₅₃	—	9	3	no	—
43	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
44	C₃²×C₅₃	C₃×3₋¹⁺²	1	1	yes	characteristic
45	C₃²×C₅₃	C₃×3₋¹⁺²	1	1	yes	characteristic
46	C₃²×C₅₃	C₃×3₋¹⁺²	1	1	yes	characteristic
47	C₉×C₅₃	—	3	3	no	—
48	C₉×C₅₃	—	3	3	no	—
49	C₉×C₅₃	—	3	3	no	—
50	C₉×C₅₃	—	9	9	no	—
51	C₃²×3₋¹⁺²	C₃×C₅₃	1	1	yes	characteristic
52	#6895	C₃×C₅₃	1	3	yes	complemented
53	#6868	C₃×C₅₃	1	3	yes	complemented
54	#6867	C₃×C₅₃	1	3	yes	complemented
55	#6866	C₃×C₅₃	1	3	yes	—
56	C₃×C₅₃	—	3	1	no	—
57	C₃×C₅₃	—	3	1	no	—
58	C₃×C₅₃	—	3	1	no	—
59	C₃×C₅₃	—	9	3	no	—
60	C₃×C₅₃	#6891	1	1	yes	characteristic
61	C₃×C₅₃	#6913	1	3	yes	—
62	C₃⁴	C₃²×C₅₃	1	1	yes	characteristic
63	C₃×3₋¹⁺²	—	3	3	no	—
64	C₃×3₋¹⁺²	—	3	3	no	—
65	C₃×3₋¹⁺²	—	3	3	no	—
66	C₃×3₋¹⁺²	—	3	3	no	—
67	C₃²×C₉	C₃²×C₅₃	1	1	yes	characteristic
68	C₃²×C₉	C₃²×C₅₃	1	1	yes	characteristic
69	C₃²×C₉	C₃²×C₅₃	1	1	yes	characteristic
70	C₉⋊C₉	—	3	9	no	—
71	C₃²⋊C₉	C₃²×C₅₃	1	9	yes	—
72	C₅₃	#10624774	1	1	yes	characteristic, direct product factor, Sylow
73	C₃³	—	3	1	no	—
74	C₃³	—	3	1	no	—
75	C₃³	—	3	1	no	—
76	C₃³	—	9	3	no	—
77	C₃³	3₋¹⁺²×C₅₃	1	3	yes	—
78	C₃³	C₃³×C₅₃	1	1	yes	characteristic
79	3₋¹⁺²	—	3	27	no	—
80	C₃×C₉	—	3	1	no	—
81	C₃×C₉	—	3	1	no	—
82	C₃×C₉	—	3	1	no	—
83	C₃×C₉	—	3	1	no	—
84	C₃×C₉	—	3	1	no	—
85	C₃×C₉	—	3	1	no	—
86	C₃×C₉	—	3	1	no	—
87	C₃×C₉	—	3	1	no	—
88	C₃×C₉	—	3	1	no	—
89	C₃×C₉	—	3	9	no	—
90	C₃×C₉	H(3,𝔽₃)×C₅₃	1	3	yes	—
91	C₃×C₉	3₋¹⁺²×C₅₃	1	3	yes	—
92	C₃×C₉	3₋¹⁺²×C₅₃	1	3	yes	—
93	C₃²	—	3	3	no	—
94	C₃²	—	3	3	no	—
95	C₃²	—	9	3	no	—
96	C₃²	—	9	3	no	—
97	C₃²	—	9	3	no	—
98	C₃²	—	9	3	no	—
99	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
100	C₃²	C₃×3₋¹⁺²×C₅₃	1	1	yes	characteristic
101	C₃²	C₃×3₋¹⁺²×C₅₃	1	1	yes	characteristic
102	C₃²	C₃×3₋¹⁺²×C₅₃	1	1	yes	characteristic
103	C₉	—	3	3	no	—
104	C₉	—	3	3	no	—
105	C₉	—	3	3	no	—
106	C₉	—	9	9	no	—
107	C₃	—	3	1	no	—
108	C₃	—	3	1	no	—
109	C₃	—	3	1	no	—
110	C₃	—	9	3	no	—
111	C₃	C₅₃×#6891	1	1	yes	characteristic
112	C₃	C₅₃×#6913	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×3₋¹⁺²×C₅₃

C₃

yes

maximal, characteristic

C₅₃×#6895

C₃

yes

maximal, complemented

C₅₃×#6868

C₃

yes

maximal, complemented

C₅₃×#6867

C₃

yes

maximal, complemented

C₅₃×#6866

C₃

yes

maximal

C₃⁴×C₅₃

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₅₃

—

C₃×3₋¹⁺²×C₅₃

—

C₃×3₋¹⁺²×C₅₃

—

C₃×3₋¹⁺²×C₅₃

—

C₃²×C₉×C₅₃

C₃²

yes

characteristic

C₃²×C₉×C₅₃

C₃²

yes

characteristic

C₃²×C₉×C₅₃

C₃²

yes

characteristic

C₅₃×C₉⋊C₉

—

C₅₃×C₃²⋊C₉

C₃²

yes

—

C₃³×C₅₃

—

C₃³×C₅₃

—

C₃³×C₅₃

—

C₃³×C₅₃

—

C₃³×C₅₃

3₋¹⁺²

yes

—

C₃³×C₅₃

C₃³

yes

characteristic

3₋¹⁺²×C₅₃

—

C₃×C₉×C₅₃

—

C₃×C₉×C₅₃

—

C₃×C₉×C₅₃

—

C₃×C₉×C₅₃

—

C₃×C₉×C₅₃

—

C₃×C₉×C₅₃

—

C₃×C₉×C₅₃

—

C₃×C₉×C₅₃

—

C₃×C₉×C₅₃

—

C₃×C₉×C₅₃

—

C₃×C₉×C₅₃

H(3,𝔽₃)

yes

—

C₃×C₉×C₅₃

3₋¹⁺²

yes

—

C₃×C₉×C₅₃

3₋¹⁺²

yes

—

#10624774

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₃²×C₅₃

—

C₃²×C₅₃

—

C₃²×C₅₃

—

C₃²×C₅₃

—

C₃²×C₅₃

—

C₃²×C₅₃

—

C₃²×C₅₃

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₅₃

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₅₃

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₅₃

C₃×3₋¹⁺²

yes

characteristic

C₉×C₅₃

—

C₉×C₅₃

—

C₉×C₅₃

—

C₉×C₅₃

—

C₃²×3₋¹⁺²

C₃×C₅₃

yes

characteristic

#6895

C₃×C₅₃

yes

complemented

#6868

C₃×C₅₃

yes

complemented

#6867

C₃×C₅₃

yes

complemented

#6866

C₃×C₅₃

yes

—

C₃×C₅₃

—

C₃×C₅₃

—

C₃×C₅₃

—

C₃×C₅₃

—

C₃×C₅₃

#6891

yes

characteristic

C₃×C₅₃

#6913

yes

—

C₃⁴

C₃²×C₅₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃²×C₉

C₃²×C₅₃

yes

characteristic

C₃²×C₉

C₃²×C₅₃

yes

characteristic

C₃²×C₉

C₃²×C₅₃

yes

characteristic

C₉⋊C₉

—

C₃²⋊C₉

C₃²×C₅₃

yes

—

C₅₃

#10624774

yes

characteristic, direct product factor, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

3₋¹⁺²×C₅₃

yes

—

C₃³

C₃³×C₅₃

yes

characteristic

3₋¹⁺²

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

H(3,𝔽₃)×C₅₃

yes

—

C₃×C₉

3₋¹⁺²×C₅₃

yes

—

C₃×C₉

3₋¹⁺²×C₅₃

yes

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃×H(3,𝔽₃)×C₅₃

yes

characteristic

100

C₃²

C₃×3₋¹⁺²×C₅₃

yes

characteristic

101

C₃²

C₃×3₋¹⁺²×C₅₃

yes

characteristic

102

C₃²

C₃×3₋¹⁺²×C₅₃

yes

characteristic

103

C₉

—

104

C₉

—

105

C₉

—

106

C₉

—

107

C₃

—

108

C₃

—

109

C₃

—

110

C₃

—

111

C₃

C₅₃×#6891

yes

characteristic

112

C₃

C₅₃×#6913

yes

—

	subgroup	G/H
center, socle	C₃²×C₅₃	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₅₃
prime radical	C₃⁴×C₅₃	C₃²
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₅₃ ◁ C₃²×C₉×C₅₃ ◁ G

subgroup

G/H

center, socle

C₃²×C₅₃

C₃×H(3,𝔽₃)

commutator, Frattini, cosocle

C₃³

C₃³×C₅₃

prime radical

C₃⁴×C₅₃

C₃²

LCS

G ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃²×C₅₃ ◁ C₃²×C₉×C₅₃ ◁ G

N	H ≅ G/N	type
2. C₅₃×#6895	C₃	semidirect
3. C₅₃×#6868	C₃	semidirect
4. C₅₃×#6867	C₃	semidirect
36. #10624774	C₅₃	direct (N × H)
52. #6895	C₃×C₅₃	semidirect
53. #6868	C₃×C₅₃	semidirect
54. #6867	C₃×C₅₃	semidirect
72. C₅₃	#10624774	direct (N × H)

H ≅ G/N

type

2. C₅₃×#6895

C₃

semidirect

3. C₅₃×#6868

C₃

semidirect

4. C₅₃×#6867

C₃

semidirect

36. #10624774

C₅₃

direct (N × H)

52. #6895

C₃×C₅₃

semidirect

53. #6868

C₃×C₅₃

semidirect

54. #6867

C₃×C₅₃

semidirect

72. C₅₃

#10624774

direct (N × H)

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

subgroup	index
C₃²×3₋¹⁺²×C₅₃	3
C₅₃×#6895	3
C₅₃×#6868	3
C₅₃×#6867	3
C₅₃×#6866	3
#10624774	53

subgroup

index

C₃²×3₋¹⁺²×C₅₃

C₅₃×#6895

C₅₃×#6868

C₅₃×#6867

C₅₃×#6866

#10624774

Subgroups — C₅₃×#10624774 #50311659

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#10624774 #50311659

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10624774 #50311659