Group #50311516 — subgroups

Subgroups — C₅₃×#10624631 #50311516

Order 38637.

344 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 80 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×#6891	C₃	1	1	yes	maximal, characteristic
3	C₅₃×#6891	C₃	1	2	yes	maximal
4	C₅₃×#6858	C₃	1	9	yes	maximal, complemented
5	C₃⁴×C₅₃	C₃²	1	1	yes	characteristic
6	C₃×3₋¹⁺²×C₅₃	—	3	1	no	—
7	C₃×3₋¹⁺²×C₅₃	—	3	1	no	—
8	C₃×3₋¹⁺²×C₅₃	—	3	2	no	—
9	C₃×3₋¹⁺²×C₅₃	—	3	1	no	—
10	C₃×3₋¹⁺²×C₅₃	—	3	9	no	—
11	C₃×3₋¹⁺²×C₅₃	C₃²	1	3	yes	—
12	C₃²×C₉×C₅₃	—	3	1	no	—
13	C₅₃×C₃²⋊C₉	—	3	1	no	—
14	C₅₃×C₃²⋊C₉	—	3	2	no	—
15	C₅₃×C₃²⋊C₉	—	3	1	no	—
16	C₅₃×C₃²⋊C₉	—	3	2	no	—
17	C₅₃×C₃²⋊C₉	C₃²	1	3	yes	—
18	C₅₃×C₃²⋊C₉	C₃²	1	6	yes	—
19	C₃³×C₅₃	—	3	3	no	—
20	C₃³×C₅₃	—	9	1	no	—
21	C₃³×C₅₃	—	9	2	no	—
22	C₃³×C₅₃	H(3,𝔽₃)	1	1	yes	characteristic
23	C₃³×C₅₃	H(3,𝔽₃)	1	1	yes	characteristic
24	C₃³×C₅₃	H(3,𝔽₃)	1	1	yes	characteristic
25	C₃³×C₅₃	C₃³	1	1	yes	characteristic
26	3₋¹⁺²×C₅₃	—	3	9	no	—
27	3₋¹⁺²×C₅₃	—	3	9	no	—
28	3₋¹⁺²×C₅₃	—	3	9	no	—
29	3₋¹⁺²×C₅₃	—	9	6	no	—
30	3₋¹⁺²×C₅₃	—	9	3	no	—
31	C₃×C₉×C₅₃	—	3	9	no	—
32	C₃×C₉×C₅₃	—	3	3	no	—
33	C₃×C₉×C₅₃	—	3	3	no	—
34	C₃×C₉×C₅₃	—	3	6	no	—
35	#10624631	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₃²×C₅₃	—	3	1	no	—
37	C₃²×C₅₃	—	3	2	no	—
38	C₃²×C₅₃	—	3	1	no	—
39	C₃²×C₅₃	—	3	1	no	—
40	C₃²×C₅₃	—	3	1	no	—
41	C₃²×C₅₃	—	3	1	no	—
42	C₃²×C₅₃	—	3	1	no	—
43	C₃²×C₅₃	—	3	1	no	—
44	C₃²×C₅₃	—	3	2	no	—
45	C₃²×C₅₃	—	3	2	no	—
46	C₃²×C₅₃	—	3	2	no	—
47	C₃²×C₅₃	—	9	9	no	—
48	C₃²×C₅₃	AΣL(1,𝔽₂₇)	1	3	yes	—
49	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
50	C₉×C₅₃	—	3	9	no	—
51	C₉×C₅₃	—	9	3	no	—
52	C₉×C₅₃	—	9	6	no	—
53	C₃²×3₋¹⁺²	C₃×C₅₃	1	1	yes	characteristic
54	#6891	C₃×C₅₃	1	1	yes	characteristic
55	#6891	C₃×C₅₃	1	2	yes	—
56	#6858	C₃×C₅₃	1	9	yes	complemented
57	C₃×C₅₃	—	3	3	no	—
58	C₃×C₅₃	—	9	1	no	—
59	C₃×C₅₃	—	9	1	no	—
60	C₃×C₅₃	—	9	1	no	—
61	C₃×C₅₃	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
62	C₃×C₅₃	#6910	1	2	yes	—
63	C₃×C₅₃	#6912	1	1	yes	characteristic
64	C₃⁴	C₃²×C₅₃	1	1	yes	characteristic
65	C₃×3₋¹⁺²	—	3	1	no	—
66	C₃×3₋¹⁺²	—	3	1	no	—
67	C₃×3₋¹⁺²	—	3	2	no	—
68	C₃×3₋¹⁺²	—	3	1	no	—
69	C₃×3₋¹⁺²	—	3	9	no	—
70	C₃×3₋¹⁺²	C₃²×C₅₃	1	3	yes	—
71	C₃²×C₉	—	3	1	no	—
72	C₃²⋊C₉	—	3	1	no	—
73	C₃²⋊C₉	—	3	2	no	—
74	C₃²⋊C₉	—	3	1	no	—
75	C₃²⋊C₉	—	3	2	no	—
76	C₃²⋊C₉	C₃²×C₅₃	1	3	yes	—
77	C₃²⋊C₉	C₃²×C₅₃	1	6	yes	—
78	C₅₃	#10624631	1	1	yes	characteristic, direct product factor, Sylow
79	C₃³	—	3	3	no	—
80	C₃³	—	9	2	no	—
81	C₃³	—	9	1	no	—
82	C₃³	H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
83	C₃³	H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
84	C₃³	H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
85	C₃³	C₃³×C₅₃	1	1	yes	characteristic
86	3₋¹⁺²	—	3	9	no	—
87	3₋¹⁺²	—	3	9	no	—
88	3₋¹⁺²	—	3	9	no	—
89	3₋¹⁺²	—	9	6	no	—
90	3₋¹⁺²	—	9	3	no	—
91	C₃×C₉	—	3	9	no	—
92	C₃×C₉	—	3	3	no	—
93	C₃×C₉	—	3	3	no	—
94	C₃×C₉	—	3	6	no	—
95	C₃²	—	3	2	no	—
96	C₃²	—	3	1	no	—
97	C₃²	—	3	1	no	—
98	C₃²	—	3	1	no	—
99	C₃²	—	3	1	no	—
100	C₃²	—	3	2	no	—
101	C₃²	—	3	2	no	—
102	C₃²	—	3	2	no	—
103	C₃²	—	3	1	no	—
104	C₃²	—	3	1	no	—
105	C₃²	—	3	1	no	—
106	C₃²	—	9	9	no	—
107	C₃²	C₅₃×AΣL(1,𝔽₂₇)	1	3	yes	—
108	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
109	C₉	—	3	9	no	—
110	C₉	—	9	3	no	—
111	C₉	—	9	6	no	—
112	C₃	—	3	3	no	—
113	C₃	—	9	1	no	—
114	C₃	—	9	1	no	—
115	C₃	—	9	1	no	—
116	C₃	C₃×C₅₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
117	C₃	C₅₃×#6910	1	2	yes	—
118	C₃	C₅₃×#6912	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×3₋¹⁺²×C₅₃

C₃

yes

maximal, characteristic

C₅₃×#6891

C₃

yes

maximal, characteristic

C₅₃×#6891

C₃

yes

maximal

C₅₃×#6858

C₃

yes

maximal, complemented

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

—

C₃×3₋¹⁺²×C₅₃

—

C₃×3₋¹⁺²×C₅₃

C₃²

yes

—

C₃²×C₉×C₅₃

—

C₅₃×C₃²⋊C₉

—

C₅₃×C₃²⋊C₉

—

C₅₃×C₃²⋊C₉

—

C₅₃×C₃²⋊C₉

—

C₅₃×C₃²⋊C₉

C₃²

yes

—

C₅₃×C₃²⋊C₉

C₃²

yes

—

C₃³×C₅₃

—

C₃³×C₅₃

—

C₃³×C₅₃

—

C₃³×C₅₃

yes

C₃³×C₅₃

yes

C₃³×C₅₃

yes

C₃³×C₅₃

C₃³

yes

characteristic

3₋¹⁺²×C₅₃

—

3₋¹⁺²×C₅₃

—

3₋¹⁺²×C₅₃

—

3₋¹⁺²×C₅₃

—

3₋¹⁺²×C₅₃

—

C₃×C₉×C₅₃

—

C₃×C₉×C₅₃

—

C₃×C₉×C₅₃

—

C₃×C₉×C₅₃

—

#10624631

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₃²×C₅₃

—

C₃²×C₅₃

—

C₃²×C₅₃

—

C₃²×C₅₃

—

C₃²×C₅₃

—

C₃²×C₅₃

—

C₃²×C₅₃

—

C₃²×C₅₃

—

C₃²×C₅₃

—

C₃²×C₅₃

—

C₃²×C₅₃

—

C₃²×C₅₃

—

C₃²×C₅₃

AΣL(1,𝔽₂₇)

yes

—

C₃²×C₅₃

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₅₃

—

C₉×C₅₃

—

C₉×C₅₃

—

C₃²×3₋¹⁺²

C₃×C₅₃

yes

characteristic

#6891

C₃×C₅₃

yes

characteristic

#6891

C₃×C₅₃

yes

—

#6858

C₃×C₅₃

yes

complemented

C₃×C₅₃

—

C₃×C₅₃

—

C₃×C₅₃

—

C₃×C₅₃

—

C₃×C₅₃

C₃×AΣL(1,𝔽₂₇)

yes

characteristic

C₃×C₅₃

#6910

yes

—

C₃×C₅₃

#6912

yes

characteristic

C₃⁴

C₃²×C₅₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃²×C₅₃

yes

—

C₃²×C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃²×C₅₃

yes

—

C₃²⋊C₉

C₃²×C₅₃

yes

—

C₅₃

#10624631

yes

characteristic, direct product factor, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

H(3,𝔽₃)×C₅₃

yes

characteristic

C₃³

H(3,𝔽₃)×C₅₃

yes

characteristic

C₃³

H(3,𝔽₃)×C₅₃

yes

characteristic

C₃³

C₃³×C₅₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

100

C₃²

—

101

C₃²

—

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

C₅₃×AΣL(1,𝔽₂₇)

yes

—

108

C₃²

C₃×H(3,𝔽₃)×C₅₃

yes

characteristic

109

C₉

—

110

C₉

—

111

C₉

—

112

C₃

—

113

C₃

—

114

C₃

—

115

C₃

—

116

C₃

C₃×C₅₃×AΣL(1,𝔽₂₇)

yes

characteristic

117

C₃

C₅₃×#6910

yes

—

118

C₃

C₅₃×#6912

yes

characteristic

	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 ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×C₅₃ ◁ C₃⁴×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 ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₃²×C₅₃ ◁ C₃⁴×C₅₃ ◁ G

N	H ≅ G/N	type
4. C₅₃×#6858	C₃	semidirect
35. #10624631	C₅₃	direct (N × H)
56. #6858	C₃×C₅₃	semidirect
78. C₅₃	#10624631	direct (N × H)

H ≅ G/N

type

4. C₅₃×#6858

C₃

semidirect

35. #10624631

C₅₃

direct (N × H)

56. #6858

C₃×C₅₃

semidirect

78. C₅₃

#10624631

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×#6891	3
C₅₃×#6858	3
#10624631	53

subgroup

index

C₃²×3₋¹⁺²×C₅₃

C₅₃×#6891

C₅₃×#6858

#10624631

Subgroups — C₅₃×#10624631 #50311516

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#10624631 #50311516

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10624631 #50311516