Group #50311684 — subgroups

Subgroups — C₅₃×#10624799 #50311684

Order 38637.

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

normal subgroups only

This group has 222 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×#733	C₃	1	1	yes	maximal, characteristic
2	C₅₃×#6894	C₃	1	2	yes	maximal
3	C₉×3₋¹⁺²×C₅₃	C₃	1	1	yes	maximal, characteristic
4	C₃×C₅₃×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic
5	C₅₃×3₋¹⁺²⋊C₉	C₃	1	4	yes	maximal
6	C₅₃×3₋¹⁺²⋊C₉	C₃	1	2	yes	maximal
7	C₅₃×#6868	C₃	1	2	yes	maximal
8	C₃×3₋¹⁺²×C₅₃	C₃²	1	2	yes	—
9	C₃×3₋¹⁺²×C₅₃	C₃²	1	1	yes	characteristic
10	C₃²×C₉×C₅₃	C₃²	1	1	yes	characteristic
11	C₃²×C₉×C₅₃	C₃²	1	2	yes	—
12	C₃²×C₉×C₅₃	C₃²	1	1	yes	characteristic
13	C₅₃×#733	—	3	3	no	—
14	C₅₃×C₉⋊C₉	—	3	2	no	—
15	C₅₃×C₉⋊C₉	—	3	1	no	—
16	C₅₃×C₉⋊C₉	—	3	2	no	—
17	C₅₃×C₉⋊C₉	—	3	4	no	—
18	C₅₃×C₉⋊C₉	—	3	2	no	—
19	C₅₃×C₃²⋊C₉	C₃²	1	2	yes	—
20	C₅₃×C₃²⋊C₉	C₃²	1	4	yes	—
21	C₉²×C₅₃	—	3	1	no	—
22	C₃³×C₅₃	C₃³	1	1	yes	characteristic
23	3₋¹⁺²×C₅₃	—	3	2	no	—
24	3₋¹⁺²×C₅₃	—	3	2	no	—
25	3₋¹⁺²×C₅₃	—	3	4	no	—
26	3₋¹⁺²×C₅₃	3₋¹⁺²	1	3	yes	—
27	C₃×C₉×C₅₃	—	3	1	no	—
28	C₃×C₉×C₅₃	—	3	3	no	—
29	C₃×C₉×C₅₃	—	3	2	no	—
30	C₃×C₉×C₅₃	—	3	4	no	—
31	C₃×C₉×C₅₃	—	3	2	no	—
32	C₃×C₉×C₅₃	—	3	1	no	—
33	C₃×C₉×C₅₃	—	3	2	no	—
34	C₃×C₉×C₅₃	—	3	1	no	—
35	C₃×C₉×C₅₃	—	3	2	no	—
36	C₃×C₉×C₅₃	—	9	2	no	—
37	C₃×C₉×C₅₃	H(3,𝔽₃)	1	2	yes	—
38	C₃×C₉×C₅₃	H(3,𝔽₃)	1	1	yes	characteristic
39	#10624799	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₃²×C₅₃	—	3	2	no	—
41	C₃²×C₅₃	—	3	1	no	—
42	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
43	C₃²×C₅₃	C₃×3₋¹⁺²	1	1	yes	characteristic
44	C₃²×C₅₃	#737	1	2	yes	—
45	C₉×C₅₃	—	3	3	no	—
46	C₉×C₅₃	—	3	2	no	—
47	C₉×C₅₃	—	9	2	no	—
48	C₉×C₅₃	—	9	4	no	—
49	C₉×C₅₃	—	9	2	no	—
50	C₉×C₅₃	—	9	1	no	—
51	C₉×C₅₃	—	9	2	no	—
52	C₉×C₅₃	#733	1	3	yes	complemented
53	C₃×#733	C₃×C₅₃	1	1	yes	characteristic
54	#6894	C₃×C₅₃	1	2	yes	—
55	C₉×3₋¹⁺²	C₃×C₅₃	1	1	yes	characteristic
56	C₃×C₉⋊C₉	C₃×C₅₃	1	1	yes	characteristic
57	3₋¹⁺²⋊C₉	C₃×C₅₃	1	4	yes	—
58	3₋¹⁺²⋊C₉	C₃×C₅₃	1	2	yes	—
59	#6868	C₃×C₅₃	1	2	yes	—
60	C₃×C₅₃	—	3	1	no	—
61	C₃×C₅₃	—	3	2	no	—
62	C₃×C₅₃	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
63	C₃×C₅₃	C₃×#733	1	1	yes	characteristic
64	C₃×C₅₃	#6913	1	2	yes	—
65	C₃×3₋¹⁺²	C₃²×C₅₃	1	2	yes	—
66	C₃×3₋¹⁺²	C₃²×C₅₃	1	1	yes	characteristic
67	C₃²×C₉	C₃²×C₅₃	1	1	yes	characteristic
68	C₃²×C₉	C₃²×C₅₃	1	1	yes	characteristic
69	C₃²×C₉	C₃²×C₅₃	1	2	yes	—
70	#733	—	3	3	no	—
71	C₉⋊C₉	—	3	2	no	—
72	C₉⋊C₉	—	3	1	no	—
73	C₉⋊C₉	—	3	2	no	—
74	C₉⋊C₉	—	3	4	no	—
75	C₉⋊C₉	—	3	2	no	—
76	C₃²⋊C₉	C₃²×C₅₃	1	2	yes	—
77	C₃²⋊C₉	C₃²×C₅₃	1	4	yes	—
78	C₉²	—	3	1	no	—
79	C₅₃	#10624799	1	1	yes	characteristic, direct product factor, Sylow
80	C₃³	C₃³×C₅₃	1	1	yes	characteristic
81	3₋¹⁺²	—	3	2	no	—
82	3₋¹⁺²	—	3	2	no	—
83	3₋¹⁺²	—	3	4	no	—
84	3₋¹⁺²	3₋¹⁺²×C₅₃	1	3	yes	—
85	C₃×C₉	—	3	1	no	—
86	C₃×C₉	—	3	3	no	—
87	C₃×C₉	—	3	2	no	—
88	C₃×C₉	—	3	4	no	—
89	C₃×C₉	—	3	1	no	—
90	C₃×C₉	—	3	2	no	—
91	C₃×C₉	—	3	2	no	—
92	C₃×C₉	—	3	1	no	—
93	C₃×C₉	—	3	2	no	—
94	C₃×C₉	—	9	2	no	—
95	C₃×C₉	H(3,𝔽₃)×C₅₃	1	2	yes	—
96	C₃×C₉	H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
97	C₃²	—	3	1	no	—
98	C₃²	—	3	2	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	C₅₃×#737	1	2	yes	—
102	C₉	—	3	3	no	—
103	C₉	—	3	2	no	—
104	C₉	—	9	1	no	—
105	C₉	—	9	2	no	—
106	C₉	—	9	2	no	—
107	C₉	—	9	4	no	—
108	C₉	—	9	2	no	—
109	C₉	C₅₃×#733	1	3	yes	complemented
110	C₃	—	3	1	no	—
111	C₃	—	3	2	no	—
112	C₃	C₅₃×C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
113	C₃	C₃×C₅₃×#733	1	1	yes	characteristic
114	C₃	C₅₃×#6913	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₅₃×#733

C₃

yes

maximal, characteristic

C₅₃×#6894

C₃

yes

maximal

C₉×3₋¹⁺²×C₅₃

C₃

yes

maximal, characteristic

C₃×C₅₃×C₉⋊C₉

C₃

yes

maximal, characteristic

C₅₃×3₋¹⁺²⋊C₉

C₃

yes

maximal

C₅₃×3₋¹⁺²⋊C₉

C₃

yes

maximal

C₅₃×#6868

C₃

yes

maximal

C₃×3₋¹⁺²×C₅₃

C₃²

yes

—

C₃×3₋¹⁺²×C₅₃

C₃²

yes

characteristic

C₃²×C₉×C₅₃

C₃²

yes

characteristic

C₃²×C₉×C₅₃

C₃²

yes

—

C₃²×C₉×C₅₃

C₃²

yes

characteristic

C₅₃×#733

—

C₅₃×C₉⋊C₉

—

C₅₃×C₉⋊C₉

—

C₅₃×C₉⋊C₉

—

C₅₃×C₉⋊C₉

—

C₅₃×C₉⋊C₉

—

C₅₃×C₃²⋊C₉

C₃²

yes

—

C₅₃×C₃²⋊C₉

C₃²

yes

—

C₉²×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

yes

—

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

H(3,𝔽₃)

yes

characteristic

#10624799

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₃²×C₅₃

—

C₃²×C₅₃

—

C₃²×C₅₃

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₅₃

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₅₃

#737

yes

—

C₉×C₅₃

—

C₉×C₅₃

—

C₉×C₅₃

—

C₉×C₅₃

—

C₉×C₅₃

—

C₉×C₅₃

—

C₉×C₅₃

—

C₉×C₅₃

#733

yes

complemented

C₃×#733

C₃×C₅₃

yes

characteristic

#6894

C₃×C₅₃

yes

—

C₉×3₋¹⁺²

C₃×C₅₃

yes

characteristic

C₃×C₉⋊C₉

C₃×C₅₃

yes

characteristic

3₋¹⁺²⋊C₉

C₃×C₅₃

yes

—

3₋¹⁺²⋊C₉

C₃×C₅₃

yes

—

#6868

C₃×C₅₃

yes

—

C₃×C₅₃

—

C₃×C₅₃

—

C₃×C₅₃

C₉⋊H(3,𝔽₃)

yes

C₃×C₅₃

yes

C₃×C₅₃

yes

—

C₃×3₋¹⁺²

C₃²×C₅₃

yes

—

C₃×3₋¹⁺²

C₃²×C₅₃

yes

characteristic

C₃²×C₉

C₃²×C₅₃

yes

characteristic

C₃²×C₉

C₃²×C₅₃

yes

characteristic

C₃²×C₉

C₃²×C₅₃

yes

—

#733

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

C₃²×C₅₃

yes

—

C₃²⋊C₉

C₃²×C₅₃

yes

—

C₉²

—

C₅₃

#10624799

yes

characteristic, direct product factor, Sylow

C₃³

C₃³×C₅₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

3₋¹⁺²×C₅₃

yes

—

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

H(3,𝔽₃)×C₅₃

yes

characteristic

C₃²

—

C₃²

—

C₃²

C₃×H(3,𝔽₃)×C₅₃

yes

characteristic

100

C₃²

C₃×3₋¹⁺²×C₅₃

yes

characteristic

101

C₃²

C₅₃×#737

yes

—

102

C₉

—

103

C₉

—

104

C₉

—

105

C₉

—

106

C₉

—

107

C₉

—

108

C₉

—

109

C₉

C₅₃×#733

yes

complemented

110

C₃

—

111

C₃

—

112

C₃

C₅₃×C₉⋊H(3,𝔽₃)

yes

characteristic

113

C₃

C₃×C₅₃×#733

yes

characteristic

114

C₃

C₅₃×#6913

yes

—

	subgroup	G/H
center, socle	C₃²×C₅₃	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₅₃
prime radical	C₃³×C₅₃	C₃³
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

C₃³×C₅₃

C₃³

LCS

G ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃²×C₅₃ ◁ C₃²×C₉×C₅₃ ◁ G

N	H ≅ G/N	type
39. #10624799	C₅₃	direct (N × H)
52. C₉×C₅₃	#733	semidirect
79. C₅₃	#10624799	direct (N × H)
109. C₉	C₅₃×#733	semidirect

H ≅ G/N

type

39. #10624799

C₅₃

direct (N × H)

52. C₉×C₅₃

#733

semidirect

79. C₅₃

#10624799

direct (N × H)

109. C₉

C₅₃×#733

semidirect

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

subgroup	index
C₃×C₅₃×#733	3
C₅₃×#6894	3
C₉×3₋¹⁺²×C₅₃	3
C₃×C₅₃×C₉⋊C₉	3
C₅₃×3₋¹⁺²⋊C₉	3
C₅₃×#6868	3
#10624799	53

subgroup

index

C₃×C₅₃×#733

C₅₃×#6894

C₉×3₋¹⁺²×C₅₃

C₃×C₅₃×C₉⋊C₉

C₅₃×3₋¹⁺²⋊C₉

C₅₃×#6868

#10624799

Subgroups — C₅₃×#10624799 #50311684

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#10624799 #50311684

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10624799 #50311684