Group #23748523 — subgroups

Subgroups — C₃×#15691663 #23748523

Order 9477.

1304 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 1304 conjugacy classes of subgroups — too many to draw. Only its 274 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×#66848	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×H(3,𝔽₃)×C₁₃⋊C₃	C₃	1	3	yes	maximal, complemented
3	#15691663	C₃	1	27	yes	maximal, direct product factor
4	C₃×#11760441	C₃	1	2	yes	maximal, complemented
5	C₃×#11760440	C₃	1	6	yes	maximal, complemented
6	C₃×C₁₃×#732	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃³×C₁₃⋊C₃	—	3	3	no	—
8	C₃³×C₁₃⋊C₃	C₃²	1	1	yes	characteristic
9	C₃×#66848	—	3	3	no	—
10	C₃×#66848	—	3	6	no	—
11	C₃×#66848	C₃²	1	9	yes	complemented
12	C₃×C₃²⋊(C₁₃⋊C₃)	—	3	6	no	—
13	C₃×C₃²⋊(C₁₃⋊C₃)	C₃²	1	6	yes	complemented
14	H(3,𝔽₃)×C₁₃⋊C₃	—	3	18	no	—
15	H(3,𝔽₃)×C₁₃⋊C₃	C₃²	1	27	yes	complemented
16	C₃²×C₁₃⋊C₉	C₃²	1	2	yes	complemented
17	#11760441	C₃²	1	18	yes	complemented
18	#11760440	C₃²	1	54	yes	complemented
19	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	3	yes	complemented
20	C₃²×C₉×C₁₃	C₃²	1	1	yes	characteristic, complemented
21	C₁₃×#732	C₃²	1	9	yes	complemented
22	C₃×#6911	—	13	1	no	maximal, Sylow
23	C₃³×C₁₃	—	3	3	no	—
24	C₃³×C₁₃	C₃³	1	1	yes	characteristic
25	C₃²×C₁₃⋊C₃	—	3	6	no	—
26	C₃²×C₁₃⋊C₃	—	3	27	no	—
27	C₃²×C₁₃⋊C₃	—	9	3	no	—
28	C₃²×C₁₃⋊C₃	—	9	27	no	—
29	C₃²×C₁₃⋊C₃	H(3,𝔽₃)	1	3	yes	—
30	C₃²×C₁₃⋊C₃	H(3,𝔽₃)	1	6	yes	—
31	C₃²×C₁₃⋊C₃	C₃³	1	3	yes	—
32	C₁₃×H(3,𝔽₃)	—	3	6	no	—
33	C₁₃×H(3,𝔽₃)	C₃³	1	9	yes	complemented
34	C₃×C₉×C₁₃	—	3	2	no	—
35	C₃×C₉×C₁₃	—	3	1	no	—
36	C₃×C₉×C₁₃	C₃³	1	3	yes	complemented
37	C₃²⋊(C₁₃⋊C₃)	—	3	12	no	—
38	C₃²⋊(C₁₃⋊C₃)	—	3	54	no	—
39	C₃²⋊(C₁₃⋊C₃)	C₃³	1	18	yes	complemented
40	C₃×C₁₃⋊C₉	—	3	4	no	—
41	C₃×C₁₃⋊C₉	—	3	2	no	—
42	C₃×C₁₃⋊C₉	C₃³	1	6	yes	complemented
43	#66848	—	3	27	no	—
44	C₃²×3₋¹⁺²	—	13	1	no	—
45	C₃²×H(3,𝔽₃)	—	13	3	no	—
46	#6911	—	13	27	no	—
47	C₃×#732	—	13	6	no	—
48	C₃×#732	—	13	2	no	—
49	C₃×#732	C₁₃⋊C₃	1	1	yes	characteristic, complemented
50	C₃²×C₁₃	—	3	3	no	—
51	C₃²×C₁₃	—	3	9	no	—
52	C₃²×C₁₃	—	9	9	no	—
53	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	2	yes	—
54	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
55	C₃²×C₁₃	C₃⁴	1	1	yes	characteristic
56	C₃×C₁₃⋊C₃	—	3	9	no	—
57	C₃×C₁₃⋊C₃	—	3	18	no	—
58	C₃×C₁₃⋊C₃	—	9	6	no	—
59	C₃×C₁₃⋊C₃	—	9	3	no	—
60	C₃×C₁₃⋊C₃	—	9	81	no	—
61	C₃×C₁₃⋊C₃	—	9	18	no	—
62	C₃×C₁₃⋊C₃	C₃×H(3,𝔽₃)	1	9	yes	—
63	C₉×C₁₃	—	3	3	no	—
64	C₁₃⋊C₉	—	3	6	no	—
65	C₃⁴	—	13	1	no	—
66	C₃⁴	—	39	3	no	—
67	C₃×3₋¹⁺²	—	13	9	no	—
68	C₃×3₋¹⁺²	—	39	6	no	—
69	C₃×3₋¹⁺²	—	39	3	no	—
70	C₃×H(3,𝔽₃)	—	13	6	no	—
71	C₃×H(3,𝔽₃)	—	13	27	no	—
72	C₃×H(3,𝔽₃)	—	39	18	no	—
73	C₃×H(3,𝔽₃)	—	39	6	no	—
74	C₃×H(3,𝔽₃)	C₃×C₁₃⋊C₃	1	3	yes	complemented
75	C₃²×C₉	—	13	2	no	—
76	C₃²×C₉	C₃×C₁₃⋊C₃	1	1	yes	characteristic, complemented
77	#732	—	13	54	no	—
78	#732	—	13	18	no	—
79	#732	C₃×C₁₃⋊C₃	1	9	yes	complemented
80	C₃×C₁₃	—	3	2	no	—
81	C₃×C₁₃	—	3	1	no	—
82	C₃×C₁₃	—	9	9	no	—
83	C₃×C₁₃	#6911	1	3	yes	complemented
84	C₃×C₁₃	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
85	C₁₃⋊C₃	—	3	9	no	—
86	C₁₃⋊C₃	—	9	18	no	—
87	C₃³	—	3	3	no	—
88	C₃³	—	13	3	no	—
89	C₃³	—	13	6	no	—
90	C₃³	—	13	3	no	—
91	C₃³	—	39	27	no	—
92	C₃³	—	39	6	no	—
93	C₃³	—	117	3	no	—
94	C₃³	—	117	27	no	—
95	C₃³	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
96	3₋¹⁺²	—	39	27	no	—
97	H(3,𝔽₃)	—	3	6	no	—
98	H(3,𝔽₃)	—	13	18	no	—
99	H(3,𝔽₃)	—	39	54	no	—
100	H(3,𝔽₃)	—	39	12	no	—
101	H(3,𝔽₃)	C₃²×C₁₃⋊C₃	1	9	yes	complemented
102	C₃×C₉	—	3	2	no	—
103	C₃×C₉	—	3	1	no	—
104	C₃×C₉	—	13	6	no	—
105	C₃×C₉	—	39	4	no	—
106	C₃×C₉	—	39	2	no	—
107	C₃×C₉	C₃²×C₁₃⋊C₃	1	3	yes	complemented
108	C₁₃	C₃×#6911	1	1	yes	characteristic, complemented, Sylow
109	C₃²	—	3	3	no	—
110	C₃²	—	3	9	no	—
111	C₃²	—	9	9	no	—
112	C₃²	—	13	9	no	—
113	C₃²	—	39	9	no	—
114	C₃²	—	39	18	no	—
115	C₃²	—	117	6	no	—
116	C₃²	—	117	3	no	—
117	C₃²	—	117	81	no	—
118	C₃²	—	117	18	no	—
119	C₃²	H(3,𝔽₃)×C₁₃⋊C₃	1	2	yes	—
120	C₃²	H(3,𝔽₃)×C₁₃⋊C₃	1	1	yes	characteristic
121	C₃²	C₃³×C₁₃⋊C₃	1	1	yes	characteristic
122	C₉	—	3	3	no	—
123	C₉	—	39	6	no	—
124	C₃	—	3	2	no	—
125	C₃	—	3	1	no	—
126	C₃	—	9	9	no	—
127	C₃	—	39	9	no	—
128	C₃	—	117	18	no	—
129	C₃	#15691663	1	3	yes	direct product factor
130	C₃	C₃×H(3,𝔽₃)×C₁₃⋊C₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#66848

C₃

yes

maximal, characteristic, complemented

C₃×H(3,𝔽₃)×C₁₃⋊C₃

C₃

yes

maximal, complemented

#15691663

C₃

yes

maximal, direct product factor

C₃×#11760441

C₃

yes

maximal, complemented

C₃×#11760440

C₃

yes

maximal, complemented

C₃×C₁₃×#732

C₃

yes

maximal, characteristic, complemented

C₃³×C₁₃⋊C₃

—

C₃³×C₁₃⋊C₃

C₃²

yes

—

—

C₃²

yes

complemented

C₃×C₃²⋊(C₁₃⋊C₃)

—

C₃×C₃²⋊(C₁₃⋊C₃)

C₃²

yes

complemented

H(3,𝔽₃)×C₁₃⋊C₃

—

H(3,𝔽₃)×C₁₃⋊C₃

C₃²

yes

complemented

C₃²×C₁₃⋊C₉

C₃²

yes

complemented

#11760441

C₃²

yes

complemented

#11760440

C₃²

yes

complemented

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

complemented

C₃²×C₉×C₁₃

C₃²

yes

characteristic, complemented

C₁₃×#732

C₃²

yes

complemented

C₃×#6911

—

maximal, Sylow

C₃³×C₁₃

—

C₃³×C₁₃

C₃³

yes

characteristic

C₃²×C₁₃⋊C₃

—

C₃²×C₁₃⋊C₃

—

C₃²×C₁₃⋊C₃

—

C₃²×C₁₃⋊C₃

—

C₃²×C₁₃⋊C₃

H(3,𝔽₃)

yes

—

C₃²×C₁₃⋊C₃

H(3,𝔽₃)

yes

—

C₃²×C₁₃⋊C₃

C₃³

yes

—

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

C₃³

yes

complemented

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

C₃³

yes

complemented

C₃²⋊(C₁₃⋊C₃)

—

C₃²⋊(C₁₃⋊C₃)

—

C₃²⋊(C₁₃⋊C₃)

C₃³

yes

complemented

C₃×C₁₃⋊C₉

—

C₃×C₁₃⋊C₉

—

C₃×C₁₃⋊C₉

C₃³

yes

complemented

#66848

—

C₃²×3₋¹⁺²

—

C₃²×H(3,𝔽₃)

—

—

—

—

C₁₃⋊C₃

yes

characteristic, complemented

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

—

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₁₃

C₃⁴

yes

characteristic

C₃×C₁₃⋊C₃

—

C₃×C₁₃⋊C₃

—

C₃×C₁₃⋊C₃

—

C₃×C₁₃⋊C₃

—

C₃×C₁₃⋊C₃

—

C₃×C₁₃⋊C₃

—

C₃×C₁₃⋊C₃

C₃×H(3,𝔽₃)

yes

—

C₉×C₁₃

—

C₁₃⋊C₉

—

C₃⁴

—

C₃⁴

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃×C₁₃⋊C₃

yes

complemented

C₃²×C₉

—

C₃²×C₉

C₃×C₁₃⋊C₃

yes

characteristic, complemented

#732

—

#732

—

#732

C₃×C₁₃⋊C₃

yes

complemented

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

#6911

yes

complemented

C₃×C₁₃

C₃²×H(3,𝔽₃)

yes

characteristic

C₁₃⋊C₃

—

C₁₃⋊C₃

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

117

—

C₃³

—

117

—

C₃³

C₃²×C₁₃⋊C₃

yes

characteristic

3₋¹⁺²

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

100

H(3,𝔽₃)

—

101

H(3,𝔽₃)

C₃²×C₁₃⋊C₃

yes

complemented

102

C₃×C₉

—

103

C₃×C₉

—

104

C₃×C₉

—

105

C₃×C₉

—

106

C₃×C₉

—

107

C₃×C₉

C₃²×C₁₃⋊C₃

yes

complemented

108

C₁₃

C₃×#6911

yes

characteristic, complemented, Sylow

109

C₃²

—

110

C₃²

—

111

C₃²

—

112

C₃²

—

113

C₃²

—

114

C₃²

—

115

C₃²

—

117

—

116

C₃²

—

117

—

117

C₃²

—

117

—

118

C₃²

—

117

—

119

C₃²

H(3,𝔽₃)×C₁₃⋊C₃

yes

—

120

C₃²

H(3,𝔽₃)×C₁₃⋊C₃

yes

characteristic

121

C₃²

C₃³×C₁₃⋊C₃

yes

characteristic

122

C₉

—

123

C₉

—

124

C₃

—

125

C₃

—

126

C₃

—

127

C₃

—

128

C₃

—

117

—

129

C₃

#15691663

yes

direct product factor

130

C₃

C₃×H(3,𝔽₃)×C₁₃⋊C₃

yes

characteristic

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₁₃⋊C₃
commutator, cosocle	C₃²×C₁₃	C₃⁴
Frattini	C₃²	C₃³×C₁₃⋊C₃
Fitting	C₃×C₁₃×#732	C₃
socle	C₃²×C₁₃	C₃×H(3,𝔽₃)
hypercenter	C₃×#732	C₁₃⋊C₃
hypocenter	C₁₃	C₃×#6911
LCS	G ▷ C₃²×C₁₃ ▷ C₃×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×#732

subgroup

G/H

center

C₃²

H(3,𝔽₃)×C₁₃⋊C₃

commutator, cosocle

C₃²×C₁₃

C₃⁴

Frattini

C₃²

C₃³×C₁₃⋊C₃

Fitting

C₃×C₁₃×#732

C₃

socle

C₃²×C₁₃

C₃×H(3,𝔽₃)

C₁₃⋊C₃

C₁₃

G ▷ C₃²×C₁₃ ▷ C₃×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ C₃×#732

N	H ≅ G/N	type
1. C₃²×#66848	C₃	semidirect
2. C₃×H(3,𝔽₃)×C₁₃⋊C₃	C₃	semidirect
3. #15691663	C₃	direct (N × H)
4. C₃×#11760441	C₃	semidirect
5. C₃×#11760440	C₃	semidirect
6. C₃×C₁₃×#732	C₃	semidirect
11. C₃×#66848	C₃²	semidirect
13. C₃×C₃²⋊(C₁₃⋊C₃)	C₃²	semidirect
15. H(3,𝔽₃)×C₁₃⋊C₃	C₃²	semidirect
16. C₃²×C₁₃⋊C₉	C₃²	semidirect
17. #11760441	C₃²	semidirect
18. #11760440	C₃²	semidirect
19. C₃×C₁₃×H(3,𝔽₃)	C₃²	semidirect
20. C₃²×C₉×C₁₃	C₃²	semidirect
21. C₁₃×#732	C₃²	semidirect
33. C₁₃×H(3,𝔽₃)	C₃³	semidirect
36. C₃×C₉×C₁₃	C₃³	semidirect
39. C₃²⋊(C₁₃⋊C₃)	C₃³	semidirect
42. C₃×C₁₃⋊C₉	C₃³	semidirect
49. C₃×#732	C₁₃⋊C₃	semidirect
74. C₃×H(3,𝔽₃)	C₃×C₁₃⋊C₃	semidirect
76. C₃²×C₉	C₃×C₁₃⋊C₃	semidirect
79. #732	C₃×C₁₃⋊C₃	semidirect
83. C₃×C₁₃	#6911	semidirect
101. H(3,𝔽₃)	C₃²×C₁₃⋊C₃	semidirect
107. C₃×C₉	C₃²×C₁₃⋊C₃	semidirect
108. C₁₃	C₃×#6911	semidirect
129. C₃	#15691663	direct (N × H)

H ≅ G/N

type

1. C₃²×#66848

C₃

semidirect

2. C₃×H(3,𝔽₃)×C₁₃⋊C₃

C₃

semidirect

3. #15691663

C₃

direct (N × H)

4. C₃×#11760441

C₃

semidirect

5. C₃×#11760440

C₃

semidirect

6. C₃×C₁₃×#732

C₃

semidirect

11. C₃×#66848

C₃²

semidirect

13. C₃×C₃²⋊(C₁₃⋊C₃)

C₃²

semidirect

15. H(3,𝔽₃)×C₁₃⋊C₃

C₃²

semidirect

16. C₃²×C₁₃⋊C₉

C₃²

semidirect

17. #11760441

C₃²

semidirect

18. #11760440

C₃²

semidirect

19. C₃×C₁₃×H(3,𝔽₃)

C₃²

semidirect

20. C₃²×C₉×C₁₃

C₃²

semidirect

21. C₁₃×#732

C₃²

semidirect

33. C₁₃×H(3,𝔽₃)

C₃³

semidirect

36. C₃×C₉×C₁₃

C₃³

semidirect

39. C₃²⋊(C₁₃⋊C₃)

C₃³

semidirect

42. C₃×C₁₃⋊C₉

C₃³

semidirect

49. C₃×#732

C₁₃⋊C₃

semidirect

74. C₃×H(3,𝔽₃)

C₃×C₁₃⋊C₃

semidirect

76. C₃²×C₉

C₃×C₁₃⋊C₃

semidirect

79. #732

C₃×C₁₃⋊C₃

semidirect

83. C₃×C₁₃

#6911

semidirect

101. H(3,𝔽₃)

C₃²×C₁₃⋊C₃

semidirect

107. C₃×C₉

C₃²×C₁₃⋊C₃

semidirect

108. C₁₃

C₃×#6911

semidirect

129. C₃

#15691663

direct (N × H)

Subgroups — C₃×#15691663 #23748523

Conjugacy classes of subgroups

Complements

Subgroups — C3×#15691663 #23748523

Conjugacy classes of subgroups

Complements

Subgroups — C₃×#15691663 #23748523