Group #50311643 — subgroups

Subgroups — C₅₃×#10624758 #50311643

Order 38637.

402 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 92 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	C₃×C₅₃×#737	C₃	1	1	yes	maximal, characteristic
2	C₅₃×#6912	C₃	1	6	yes	maximal, complemented
3	C₅₃×#6910	C₃	1	3	yes	maximal
4	C₃×C₅₃×#733	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅₃×#733	C₃	1	2	yes	maximal, complemented
6	C₅₃×#737	—	3	6	no	—
7	C₅₃×#737	—	3	3	no	—
8	C₃×3₋¹⁺²×C₅₃	—	3	2	no	—
9	C₃×3₋¹⁺²×C₅₃	C₃²	1	3	yes	—
10	C₃×3₋¹⁺²×C₅₃	C₃²	1	6	yes	—
11	C₃×3₋¹⁺²×C₅₃	C₃²	1	2	yes	—
12	C₃×H(3,𝔽₃)×C₅₃	C₃²	1	1	yes	characteristic
13	C₃²×C₉×C₅₃	—	3	1	no	—
14	C₃²×C₉×C₅₃	C₃²	1	1	yes	characteristic
15	C₅₃×#733	—	3	18	no	—
16	C₅₃×#733	—	3	1	no	—
17	C₅₃×#733	—	3	2	no	—
18	C₅₃×#733	—	3	2	no	—
19	C₅₃×#733	—	3	4	no	—
20	C₅₃×#731	—	3	9	no	—
21	C₃³×C₅₃	—	3	1	no	—
22	C₃³×C₅₃	C₃³	1	1	yes	characteristic
23	3₋¹⁺²×C₅₃	—	3	18	no	—
24	3₋¹⁺²×C₅₃	—	3	2	no	—
25	3₋¹⁺²×C₅₃	—	3	2	no	—
26	3₋¹⁺²×C₅₃	—	3	6	no	—
27	3₋¹⁺²×C₅₃	—	3	6	no	—
28	3₋¹⁺²×C₅₃	—	3	6	no	—
29	3₋¹⁺²×C₅₃	—	3	3	no	—
30	3₋¹⁺²×C₅₃	—	3	6	no	—
31	3₋¹⁺²×C₅₃	H(3,𝔽₃)	1	6	yes	—
32	H(3,𝔽₃)×C₅₃	—	3	2	no	—
33	H(3,𝔽₃)×C₅₃	H(3,𝔽₃)	1	3	yes	—
34	C₃×C₉×C₅₃	—	3	9	no	—
35	C₃×C₉×C₅₃	—	3	2	no	—
36	C₃×C₉×C₅₃	—	3	2	no	—
37	C₃×C₉×C₅₃	—	3	1	no	—
38	C₃×C₉×C₅₃	—	3	3	no	—
39	C₃×C₉×C₅₃	—	3	6	no	—
40	C₃×C₉×C₅₃	H(3,𝔽₃)	1	1	yes	characteristic
41	C₃×C₉×C₅₃	H(3,𝔽₃)	1	2	yes	—
42	#10624758	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₃²×C₅₃	—	3	3	no	—
44	C₃²×C₅₃	—	9	1	no	—
45	C₃²×C₅₃	—	9	3	no	—
46	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	2	yes	—
47	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
48	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
49	C₉×C₅₃	—	3	1	no	—
50	C₉×C₅₃	—	3	6	no	—
51	C₉×C₅₃	—	3	2	no	—
52	C₉×C₅₃	—	9	3	no	—
53	C₉×C₅₃	—	9	6	no	—
54	C₃×#737	C₃×C₅₃	1	1	yes	characteristic
55	#6912	C₃×C₅₃	1	6	yes	complemented
56	#6910	C₃×C₅₃	1	3	yes	—
57	C₃×#733	C₃×C₅₃	1	1	yes	characteristic, complemented
58	C₃×#733	C₃×C₅₃	1	2	yes	complemented
59	C₃×C₅₃	—	3	1	no	—
60	C₃×C₅₃	—	3	2	no	—
61	C₃×C₅₃	—	3	1	no	—
62	C₃×C₅₃	—	9	3	no	—
63	C₃×C₅₃	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
64	#737	—	3	6	no	—
65	#737	—	3	3	no	—
66	C₃×3₋¹⁺²	—	3	2	no	—
67	C₃×3₋¹⁺²	C₃²×C₅₃	1	3	yes	—
68	C₃×3₋¹⁺²	C₃²×C₅₃	1	6	yes	—
69	C₃×3₋¹⁺²	C₃²×C₅₃	1	2	yes	—
70	C₃×H(3,𝔽₃)	C₃²×C₅₃	1	1	yes	characteristic
71	C₃²×C₉	—	3	1	no	—
72	C₃²×C₉	C₃²×C₅₃	1	1	yes	characteristic
73	#733	—	3	18	no	—
74	#733	—	3	1	no	—
75	#733	—	3	2	no	—
76	#733	—	3	2	no	—
77	#733	—	3	4	no	—
78	#731	—	3	9	no	—
79	C₅₃	#10624758	1	1	yes	characteristic, direct product factor, Sylow
80	C₃³	—	3	1	no	—
81	C₃³	C₃³×C₅₃	1	1	yes	characteristic
82	3₋¹⁺²	—	3	18	no	—
83	3₋¹⁺²	—	3	2	no	—
84	3₋¹⁺²	—	3	2	no	—
85	3₋¹⁺²	—	3	6	no	—
86	3₋¹⁺²	—	3	6	no	—
87	3₋¹⁺²	—	3	6	no	—
88	3₋¹⁺²	—	3	3	no	—
89	3₋¹⁺²	—	3	6	no	—
90	3₋¹⁺²	H(3,𝔽₃)×C₅₃	1	6	yes	—
91	H(3,𝔽₃)	—	3	2	no	—
92	H(3,𝔽₃)	H(3,𝔽₃)×C₅₃	1	3	yes	—
93	C₃×C₉	—	3	9	no	—
94	C₃×C₉	—	3	2	no	—
95	C₃×C₉	—	3	2	no	—
96	C₃×C₉	—	3	1	no	—
97	C₃×C₉	—	3	3	no	—
98	C₃×C₉	—	3	6	no	—
99	C₃×C₉	H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
100	C₃×C₉	H(3,𝔽₃)×C₅₃	1	2	yes	—
101	C₃²	—	3	3	no	—
102	C₃²	—	9	1	no	—
103	C₃²	—	9	3	no	—
104	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	2	yes	—
105	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
106	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
107	C₉	—	3	1	no	—
108	C₉	—	3	6	no	—
109	C₉	—	3	2	no	—
110	C₉	—	9	3	no	—
111	C₉	—	9	6	no	—
112	C₃	—	3	1	no	—
113	C₃	—	3	2	no	—
114	C₃	—	3	1	no	—
115	C₃	—	9	3	no	—
116	C₃	C₅₃×C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₅₃×#737

C₃

yes

maximal, characteristic

C₅₃×#6912

C₃

yes

maximal, complemented

C₅₃×#6910

C₃

yes

maximal

C₃×C₅₃×#733

C₃

yes

maximal, characteristic, complemented

C₃×C₅₃×#733

C₃

yes

maximal, complemented

C₅₃×#737

—

C₅₃×#737

—

C₃×3₋¹⁺²×C₅₃

—

C₃×3₋¹⁺²×C₅₃

C₃²

yes

—

C₃×3₋¹⁺²×C₅₃

C₃²

yes

—

C₃×3₋¹⁺²×C₅₃

C₃²

yes

—

C₃×H(3,𝔽₃)×C₅₃

C₃²

yes

characteristic

C₃²×C₉×C₅₃

—

C₃²×C₉×C₅₃

C₃²

yes

characteristic

C₅₃×#733

—

C₅₃×#733

—

C₅₃×#733

—

C₅₃×#733

—

C₅₃×#733

—

C₅₃×#731

—

C₃³×C₅₃

—

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

—

3₋¹⁺²×C₅₃

—

3₋¹⁺²×C₅₃

—

3₋¹⁺²×C₅₃

—

3₋¹⁺²×C₅₃

H(3,𝔽₃)

yes

—

H(3,𝔽₃)×C₅₃

—

H(3,𝔽₃)×C₅₃

H(3,𝔽₃)

yes

—

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

characteristic

C₃×C₉×C₅₃

H(3,𝔽₃)

yes

—

#10624758

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₃²×C₅₃

—

C₃²×C₅₃

—

C₃²×C₅₃

—

C₃²×C₅₃

C₃×H(3,𝔽₃)

yes

—

C₃²×C₅₃

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₅₃

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₅₃

—

C₉×C₅₃

—

C₉×C₅₃

—

C₉×C₅₃

—

C₉×C₅₃

—

C₃×#737

C₃×C₅₃

yes

characteristic

#6912

C₃×C₅₃

yes

complemented

#6910

C₃×C₅₃

yes

—

C₃×#733

C₃×C₅₃

yes

characteristic, complemented

C₃×#733

C₃×C₅₃

yes

complemented

C₃×C₅₃

—

C₃×C₅₃

—

C₃×C₅₃

—

C₃×C₅₃

—

C₃×C₅₃

C₃²⋊H(3,𝔽₃)

yes

characteristic

#737

—

#737

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃²×C₅₃

yes

—

C₃×3₋¹⁺²

C₃²×C₅₃

yes

—

C₃×3₋¹⁺²

C₃²×C₅₃

yes

—

C₃×H(3,𝔽₃)

C₃²×C₅₃

yes

characteristic

C₃²×C₉

—

C₃²×C₉

C₃²×C₅₃

yes

—

—

—

—

—

—

C₅₃

#10624758

yes

characteristic, direct product factor, Sylow

C₃³

—

C₃³

C₃³×C₅₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

H(3,𝔽₃)×C₅₃

yes

—

H(3,𝔽₃)

—

H(3,𝔽₃)

H(3,𝔽₃)×C₅₃

yes

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

H(3,𝔽₃)×C₅₃

yes

characteristic

100

C₃×C₉

H(3,𝔽₃)×C₅₃

yes

—

101

C₃²

—

102

C₃²

—

103

C₃²

—

104

C₃²

C₃×H(3,𝔽₃)×C₅₃

yes

—

105

C₃²

C₃×H(3,𝔽₃)×C₅₃

yes

characteristic

106

C₃²

C₃×H(3,𝔽₃)×C₅₃

yes

characteristic

107

C₉

—

108

C₉

—

109

C₉

—

110

C₉

—

111

C₉

—

112

C₃

—

113

C₃

—

114

C₃

—

115

C₃

—

116

C₃

C₅₃×C₃²⋊H(3,𝔽₃)

yes

characteristic

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

subgroup

G/H

center, socle

C₃×C₅₃

C₃²⋊H(3,𝔽₃)

commutator, Frattini, cosocle

C₃³

C₃³×C₅₃

prime radical

C₃×H(3,𝔽₃)×C₅₃

C₃²

LCS

G ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃×C₅₃ ◁ C₃³×C₅₃ ◁ G

N	H ≅ G/N	type
2. C₅₃×#6912	C₃	semidirect
4. C₃×C₅₃×#733	C₃	semidirect
5. C₃×C₅₃×#733	C₃	semidirect
42. #10624758	C₅₃	direct (N × H)
55. #6912	C₃×C₅₃	semidirect
57. C₃×#733	C₃×C₅₃	semidirect
58. C₃×#733	C₃×C₅₃	semidirect
79. C₅₃	#10624758	direct (N × H)

H ≅ G/N

type

2. C₅₃×#6912

C₃

semidirect

4. C₃×C₅₃×#733

C₃

semidirect

5. C₃×C₅₃×#733

C₃

semidirect

42. #10624758

C₅₃

direct (N × H)

55. #6912

C₃×C₅₃

semidirect

57. C₃×#733

C₃×C₅₃

semidirect

58. C₃×#733

C₃×C₅₃

semidirect

79. C₅₃

#10624758

direct (N × H)

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

subgroup	index
C₃×C₅₃×#737	3
C₅₃×#6912	3
C₅₃×#6910	3
C₃×C₅₃×#733	3
#10624758	53

subgroup

index

C₃×C₅₃×#737

C₅₃×#6912

C₅₃×#6910

C₃×C₅₃×#733

#10624758

Subgroups — C₅₃×#10624758 #50311643

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#10624758 #50311643

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10624758 #50311643