Group #23744730 — subgroups

Subgroups — C₃×C₁₃×#6857 #23744730

Order 9477.

680 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 116 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×C₁₃×3₋¹⁺²	C₃	1	2	yes	maximal, complemented
2	C₃²×C₁₃×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	C₃×C₁₃×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#6857	C₃	1	9	yes	maximal, direct product factor
5	C₃⁴×C₁₃	—	3	1	no	—
6	C₃⁴×C₁₃	C₃²	1	1	yes	characteristic
7	C₃×C₁₃×3₋¹⁺²	—	3	4	no	—
8	C₃×C₁₃×3₋¹⁺²	—	3	18	no	—
9	C₃×C₁₃×3₋¹⁺²	C₃²	1	6	yes	complemented
10	C₃×C₁₃×H(3,𝔽₃)	—	3	2	no	—
11	C₃×C₁₃×H(3,𝔽₃)	—	3	9	no	—
12	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	3	yes	—
13	C₃²×C₉×C₁₃	—	3	2	no	—
14	C₃²×C₉×C₁₃	—	3	1	no	—
15	C₁₃×C₃²⋊C₉	—	3	2	no	—
16	C₁₃×C₃²⋊C₉	C₃²	1	3	yes	complemented
17	C₃×#6857	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₃³×C₁₃	—	3	3	no	—
19	C₃³×C₁₃	—	3	9	no	—
20	C₃³×C₁₃	—	3	6	no	—
21	C₃³×C₁₃	—	3	3	no	—
22	C₃³×C₁₃	—	3	3	no	—
23	C₃³×C₁₃	—	9	6	no	—
24	C₃³×C₁₃	—	9	3	no	—
25	C₃³×C₁₃	H(3,𝔽₃)	1	2	yes	—
26	C₃³×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
27	C₃³×C₁₃	C₃³	1	1	yes	characteristic
28	C₁₃×3₋¹⁺²	—	3	36	no	—
29	C₁₃×3₋¹⁺²	—	3	18	no	—
30	C₁₃×H(3,𝔽₃)	—	3	9	no	—
31	C₁₃×H(3,𝔽₃)	—	3	18	no	—
32	C₃×C₉×C₁₃	—	3	18	no	—
33	C₃×C₉×C₁₃	—	3	6	no	—
34	C₃×C₉×C₁₃	—	3	3	no	—
35	C₃×C₉×C₁₃	—	9	3	no	—
36	C₃²×3₋¹⁺²	C₃×C₁₃	1	2	yes	complemented
37	C₃²×H(3,𝔽₃)	C₃×C₁₃	1	1	yes	characteristic
38	C₃×C₃²⋊C₉	C₃×C₁₃	1	1	yes	characteristic, complemented
39	#6857	C₃×C₁₃	1	9	yes	direct product factor
40	C₃²×C₁₃	—	3	2	no	—
41	C₃²×C₁₃	—	3	1	no	—
42	C₃²×C₁₃	—	3	9	no	—
43	C₃²×C₁₃	—	3	4	no	—
44	C₃²×C₁₃	—	3	2	no	—
45	C₃²×C₁₃	—	3	2	no	—
46	C₃²×C₁₃	—	3	1	no	—
47	C₃²×C₁₃	—	9	9	no	—
48	C₃²×C₁₃	—	9	27	no	—
49	C₃²×C₁₃	—	9	6	no	—
50	C₃²×C₁₃	—	9	3	no	—
51	C₃²×C₁₃	AΣL(1,𝔽₂₇)	1	6	yes	—
52	C₃²×C₁₃	AΣL(1,𝔽₂₇)	1	3	yes	—
53	C₃²×C₁₃	#731	1	3	yes	—
54	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
55	C₉×C₁₃	—	3	18	no	—
56	C₉×C₁₃	—	9	3	no	—
57	C₃⁴	—	3	1	no	—
58	C₃⁴	C₃²×C₁₃	1	1	yes	characteristic
59	C₃×3₋¹⁺²	—	3	4	no	—
60	C₃×3₋¹⁺²	—	3	18	no	—
61	C₃×3₋¹⁺²	C₃²×C₁₃	1	6	yes	complemented
62	C₃×H(3,𝔽₃)	—	3	9	no	—
63	C₃×H(3,𝔽₃)	—	3	2	no	—
64	C₃×H(3,𝔽₃)	C₃²×C₁₃	1	3	yes	—
65	C₃²×C₉	—	3	2	no	—
66	C₃²×C₉	—	3	1	no	—
67	C₃²⋊C₉	—	3	2	no	—
68	C₃²⋊C₉	C₃²×C₁₃	1	3	yes	complemented
69	C₃×C₁₃	—	9	2	no	—
70	C₃×C₁₃	—	9	1	no	—
71	C₃×C₁₃	—	9	9	no	—
72	C₃×C₁₃	#6857	1	9	yes	direct product factor
73	C₃×C₁₃	C₃×AΣL(1,𝔽₂₇)	1	2	yes	—
74	C₃×C₁₃	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
75	C₃×C₁₃	C₃×#731	1	1	yes	characteristic
76	C₃³	—	3	3	no	—
77	C₃³	—	3	9	no	—
78	C₃³	—	3	6	no	—
79	C₃³	—	3	3	no	—
80	C₃³	—	3	3	no	—
81	C₃³	—	9	6	no	—
82	C₃³	—	9	3	no	—
83	C₃³	C₁₃×H(3,𝔽₃)	1	2	yes	—
84	C₃³	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
85	C₃³	C₃³×C₁₃	1	1	yes	characteristic
86	3₋¹⁺²	—	3	36	no	—
87	3₋¹⁺²	—	3	18	no	—
88	H(3,𝔽₃)	—	3	9	no	—
89	H(3,𝔽₃)	—	3	18	no	—
90	C₃×C₉	—	3	18	no	—
91	C₃×C₉	—	3	6	no	—
92	C₃×C₉	—	3	3	no	—
93	C₃×C₉	—	9	3	no	—
94	C₁₃	C₃×#6857	1	1	yes	characteristic, direct product factor, Sylow
95	C₃²	—	3	2	no	—
96	C₃²	—	3	1	no	—
97	C₃²	—	3	9	no	—
98	C₃²	—	3	4	no	—
99	C₃²	—	3	2	no	—
100	C₃²	—	3	2	no	—
101	C₃²	—	3	1	no	—
102	C₃²	—	9	9	no	—
103	C₃²	—	9	27	no	—
104	C₃²	—	9	6	no	—
105	C₃²	—	9	3	no	—
106	C₃²	C₁₃×AΣL(1,𝔽₂₇)	1	6	yes	—
107	C₃²	C₁₃×AΣL(1,𝔽₂₇)	1	3	yes	—
108	C₃²	C₁₃×#731	1	3	yes	—
109	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
110	C₉	—	3	18	no	—
111	C₉	—	9	3	no	—
112	C₃	—	9	2	no	—
113	C₃	—	9	1	no	—
114	C₃	—	9	9	no	—
115	C₃	C₁₃×#6857	1	9	yes	direct product factor
116	C₃	C₃×C₁₃×AΣL(1,𝔽₂₇)	1	2	yes	—
117	C₃	C₃×C₁₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
118	C₃	C₃×C₁₃×#731	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₁₃×3₋¹⁺²

C₃

yes

maximal, complemented

C₃²×C₁₃×H(3,𝔽₃)

C₃

yes

maximal, characteristic

C₃×C₁₃×C₃²⋊C₉

C₃

yes

maximal, characteristic, complemented

C₁₃×#6857

C₃

yes

maximal, direct product factor

C₃⁴×C₁₃

—

C₃⁴×C₁₃

C₃²

yes

characteristic

C₃×C₁₃×3₋¹⁺²

—

C₃×C₁₃×3₋¹⁺²

—

C₃×C₁₃×3₋¹⁺²

C₃²

yes

complemented

C₃×C₁₃×H(3,𝔽₃)

—

C₃×C₁₃×H(3,𝔽₃)

—

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

—

C₃²×C₉×C₁₃

—

C₃²×C₉×C₁₃

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

C₃²

yes

complemented

C₃×#6857

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

H(3,𝔽₃)

yes

—

C₃³×C₁₃

H(3,𝔽₃)

yes

characteristic

C₃³×C₁₃

C₃³

yes

characteristic

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃²×3₋¹⁺²

C₃×C₁₃

yes

complemented

C₃²×H(3,𝔽₃)

C₃×C₁₃

yes

characteristic

C₃×C₃²⋊C₉

C₃×C₁₃

yes

characteristic, complemented

#6857

C₃×C₁₃

yes

direct product factor

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

AΣL(1,𝔽₂₇)

yes

—

C₃²×C₁₃

AΣL(1,𝔽₂₇)

yes

—

C₃²×C₁₃

#731

yes

—

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₁₃

—

C₉×C₁₃

—

C₃⁴

—

C₃⁴

C₃²×C₁₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃²×C₁₃

yes

complemented

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃²×C₁₃

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃²×C₁₃

yes

complemented

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

#6857

yes

direct product factor

C₃×C₁₃

C₃×AΣL(1,𝔽₂₇)

yes

—

C₃×C₁₃

C₃×AΣL(1,𝔽₂₇)

yes

characteristic

C₃×C₁₃

C₃×#731

yes

characteristic

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₁₃×H(3,𝔽₃)

yes

—

C₃³

C₁₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃³×C₁₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₁₃

C₃×#6857

yes

characteristic, direct product factor, Sylow

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

100

C₃²

—

101

C₃²

—

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

C₁₃×AΣL(1,𝔽₂₇)

yes

—

107

C₃²

C₁₃×AΣL(1,𝔽₂₇)

yes

—

108

C₃²

C₁₃×#731

yes

—

109

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

110

C₉

—

111

C₉

—

112

C₃

—

113

C₃

—

114

C₃

—

115

C₃

C₁₃×#6857

yes

direct product factor

116

C₃

C₃×C₁₃×AΣL(1,𝔽₂₇)

yes

—

117

C₃

C₃×C₁₃×AΣL(1,𝔽₂₇)

yes

characteristic

118

C₃

C₃×C₁₃×#731

yes

characteristic

	subgroup	G/H
center, socle	C₃³×C₁₃	H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₁₃
prime radical	C₃²×C₁₃×H(3,𝔽₃)	C₃
LCS	G ▷ C₃³ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃³×C₁₃ ◁ C₃⁴×C₁₃ ◁ G

subgroup

G/H

center, socle

C₃³×C₁₃

H(3,𝔽₃)

commutator, Frattini, cosocle

C₃³

C₃³×C₁₃

prime radical

C₃²×C₁₃×H(3,𝔽₃)

C₃

LCS

G ▷ C₃³ ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₃³×C₁₃ ◁ C₃⁴×C₁₃ ◁ G

N	H ≅ G/N	type
1. C₃²×C₁₃×3₋¹⁺²	C₃	semidirect
3. C₃×C₁₃×C₃²⋊C₉	C₃	semidirect
4. C₁₃×#6857	C₃	direct (N × H)
9. C₃×C₁₃×3₋¹⁺²	C₃²	semidirect
16. C₁₃×C₃²⋊C₉	C₃²	semidirect
17. C₃×#6857	C₁₃	direct (N × H)
36. C₃²×3₋¹⁺²	C₃×C₁₃	semidirect
38. C₃×C₃²⋊C₉	C₃×C₁₃	semidirect
39. #6857	C₃×C₁₃	direct (N × H)
61. C₃×3₋¹⁺²	C₃²×C₁₃	semidirect
68. C₃²⋊C₉	C₃²×C₁₃	semidirect
72. C₃×C₁₃	#6857	direct (N × H)
94. C₁₃	C₃×#6857	direct (N × H)
115. C₃	C₁₃×#6857	direct (N × H)

H ≅ G/N

type

1. C₃²×C₁₃×3₋¹⁺²

C₃

semidirect

3. C₃×C₁₃×C₃²⋊C₉

C₃

semidirect

4. C₁₃×#6857

C₃

direct (N × H)

9. C₃×C₁₃×3₋¹⁺²

C₃²

semidirect

16. C₁₃×C₃²⋊C₉

C₃²

semidirect

17. C₃×#6857

C₁₃

direct (N × H)

36. C₃²×3₋¹⁺²

C₃×C₁₃

semidirect

38. C₃×C₃²⋊C₉

C₃×C₁₃

semidirect

39. #6857

C₃×C₁₃

direct (N × H)

61. C₃×3₋¹⁺²

C₃²×C₁₃

semidirect

68. C₃²⋊C₉

C₃²×C₁₃

semidirect

72. C₃×C₁₃

#6857

direct (N × H)

94. C₁₃

C₃×#6857

direct (N × H)

115. C₃

C₁₃×#6857

direct (N × H)

Subgroups — C₃×C₁₃×#6857 #23744730

Conjugacy classes of subgroups

Complements

Subgroups — C3×C13×#6857 #23744730

Conjugacy classes of subgroups

Complements

Subgroups — C₃×C₁₃×#6857 #23744730