Group #23748519 — subgroups

Subgroups — C₃×#15691660 #23748519

Order 9477.

1088 conjugacy classes of subgroups; only the 292 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 292 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×#66851	C₃	1	2	yes	maximal, complemented
2	C₃²×#66851	C₃	1	1	yes	maximal, characteristic, complemented
3	#15691660	C₃	1	27	yes	maximal, direct product factor
4	C₃×#11760437	C₃	1	9	yes	maximal, complemented
5	C₃²×C₁₃×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
6	C₃×#66851	—	3	2	no	—
7	C₃×#66851	—	3	2	no	—
8	C₃×#66851	—	3	12	no	—
9	C₃×#66851	—	3	2	no	—
10	C₃×#66851	—	3	1	no	—
11	C₃×#66851	—	3	6	no	—
12	C₃×#66851	C₃²	1	6	yes	complemented
13	C₃×#66851	C₃²	1	18	yes	complemented
14	C₃×#66851	C₃²	1	9	yes	complemented
15	C₃²×C₁₃⋊C₉	—	3	2	no	—
16	C₃²×C₁₃⋊C₉	C₃²	1	3	yes	complemented
17	#11760437	C₃²	1	81	yes	complemented
18	C₃⁴×C₁₃	—	3	1	no	—
19	C₃⁴×C₁₃	C₃²	1	1	yes	characteristic
20	C₃×C₁₃×H(3,𝔽₃)	—	3	2	no	—
21	C₃×C₁₃×H(3,𝔽₃)	—	3	6	no	—
22	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	3	yes	—
23	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	9	yes	—
24	C₃×#6910	—	13	1	no	maximal, Sylow
25	C₃³×C₁₃	—	3	9	no	—
26	C₃³×C₁₃	—	3	3	no	—
27	C₃³×C₁₃	—	9	3	no	—
28	C₃³×C₁₃	—	9	9	no	—
29	C₃³×C₁₃	H(3,𝔽₃)	1	2	yes	—
30	C₃³×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
31	C₃³×C₁₃	H(3,𝔽₃)	1	6	yes	—
32	C₃³×C₁₃	C₃³	1	1	yes	characteristic
33	C₃³×C₁₃	C₃³	1	3	yes	—
34	C₁₃×H(3,𝔽₃)	—	3	6	no	—
35	C₁₃×H(3,𝔽₃)	—	3	18	no	—
36	C₁₃×H(3,𝔽₃)	C₃³	1	9	yes	—
37	#66851	—	3	12	no	—
38	#66851	—	3	18	no	—
39	#66851	—	3	18	no	—
40	#66851	—	3	18	no	—
41	#66851	—	3	9	no	—
42	#66851	C₃³	1	18	yes	complemented
43	C₃×C₁₃⋊C₉	—	3	18	no	—
44	C₃×C₁₃⋊C₉	—	3	6	no	—
45	C₃×C₁₃⋊C₉	—	3	6	no	—
46	C₃×C₁₃⋊C₉	—	3	3	no	—
47	C₃×C₁₃⋊C₉	C₃³	1	9	yes	complemented
48	C₃²×3₋¹⁺²	—	13	1	no	—
49	C₃²×3₋¹⁺²	—	13	2	no	—
50	C₃²×H(3,𝔽₃)	C₁₃⋊C₃	1	1	yes	characteristic
51	#6910	—	13	27	no	—
52	C₃×#731	—	13	9	no	—
53	C₃²×C₁₃	—	3	3	no	—
54	C₃²×C₁₃	—	3	6	no	—
55	C₃²×C₁₃	—	3	3	no	—
56	C₃²×C₁₃	—	3	9	no	—
57	C₃²×C₁₃	—	9	6	no	—
58	C₃²×C₁₃	—	9	3	no	—
59	C₃²×C₁₃	—	9	27	no	—
60	C₃²×C₁₃	—	9	9	no	—
61	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	6	yes	—
62	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	2	yes	—
63	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	3	yes	—
64	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
65	C₃²×C₁₃	C₃⁴	1	1	yes	characteristic
66	C₁₃⋊C₉	—	3	18	no	—
67	C₁₃⋊C₉	—	3	9	no	—
68	C₃⁴	—	3	1	no	—
69	C₃⁴	C₃×C₁₃⋊C₃	1	1	yes	characteristic
70	C₃×3₋¹⁺²	—	13	9	no	—
71	C₃×3₋¹⁺²	—	13	6	no	—
72	C₃×3₋¹⁺²	—	13	18	no	—
73	C₃×3₋¹⁺²	—	39	6	no	—
74	C₃×3₋¹⁺²	—	39	12	no	—
75	C₃×3₋¹⁺²	—	39	2	no	—
76	C₃×3₋¹⁺²	—	39	1	no	—
77	C₃×3₋¹⁺²	—	39	2	no	—
78	C₃×3₋¹⁺²	—	39	2	no	—
79	C₃×H(3,𝔽₃)	—	3	6	no	—
80	C₃×H(3,𝔽₃)	—	3	2	no	—
81	C₃×H(3,𝔽₃)	C₃×C₁₃⋊C₃	1	3	yes	—
82	C₃×H(3,𝔽₃)	C₃×C₁₃⋊C₃	1	9	yes	—
83	C₃²×C₉	—	13	3	no	—
84	C₃²×C₉	—	39	2	no	—
85	#731	—	13	81	no	—
86	C₃×C₁₃	—	3	3	no	—
87	C₃×C₁₃	—	3	6	no	—
88	C₃×C₁₃	—	3	2	no	—
89	C₃×C₁₃	—	3	1	no	—
90	C₃×C₁₃	—	9	9	no	—
91	C₃×C₁₃	#6910	1	3	yes	complemented
92	C₃×C₁₃	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
93	C₃³	—	3	9	no	—
94	C₃³	—	3	3	no	—
95	C₃³	—	9	3	no	—
96	C₃³	—	9	9	no	—
97	C₃³	C₃²⋊(C₁₃⋊C₃)	1	2	yes	—
98	C₃³	C₃²⋊(C₁₃⋊C₃)	1	1	yes	characteristic
99	C₃³	C₃²⋊(C₁₃⋊C₃)	1	6	yes	—
100	C₃³	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
101	C₃³	C₃²×C₁₃⋊C₃	1	3	yes	—
102	3₋¹⁺²	—	13	18	no	—
103	3₋¹⁺²	—	39	18	no	—
104	3₋¹⁺²	—	39	9	no	—
105	3₋¹⁺²	—	39	12	no	—
106	3₋¹⁺²	—	39	18	no	—
107	3₋¹⁺²	—	39	18	no	—
108	H(3,𝔽₃)	—	3	6	no	—
109	H(3,𝔽₃)	—	3	18	no	—
110	H(3,𝔽₃)	C₃²×C₁₃⋊C₃	1	9	yes	—
111	C₃×C₉	—	13	9	no	—
112	C₃×C₉	—	39	6	no	—
113	C₃×C₉	—	39	18	no	—
114	C₃×C₉	—	39	3	no	—
115	C₃×C₉	—	39	6	no	—
116	C₁₃	C₃×#6910	1	1	yes	characteristic, complemented, Sylow
117	C₃²	—	3	3	no	—
118	C₃²	—	3	6	no	—
119	C₃²	—	3	3	no	—
120	C₃²	—	3	9	no	—
121	C₃²	—	9	6	no	—
122	C₃²	—	9	3	no	—
123	C₃²	—	9	27	no	—
124	C₃²	—	9	9	no	—
125	C₃²	C₃×C₃²⋊(C₁₃⋊C₃)	1	2	yes	—
126	C₃²	C₃×C₃²⋊(C₁₃⋊C₃)	1	6	yes	—
127	C₃²	C₃×C₃²⋊(C₁₃⋊C₃)	1	3	yes	—
128	C₃²	C₃×C₃²⋊(C₁₃⋊C₃)	1	1	yes	characteristic
129	C₃²	C₃³×C₁₃⋊C₃	1	1	yes	characteristic
130	C₉	—	39	9	no	—
131	C₉	—	39	18	no	—
132	C₃	—	3	3	no	—
133	C₃	—	3	6	no	—
134	C₃	—	3	2	no	—
135	C₃	—	3	1	no	—
136	C₃	—	9	9	no	—
137	C₃	#15691660	1	3	yes	direct product factor
138	C₃	C₃²×C₃²⋊(C₁₃⋊C₃)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#66851

C₃

yes

maximal, complemented

C₃²×#66851

C₃

yes

maximal, characteristic, complemented

#15691660

C₃

yes

maximal, direct product factor

C₃×#11760437

C₃

yes

maximal, complemented

C₃²×C₁₃×H(3,𝔽₃)

C₃

yes

maximal, characteristic

—

—

—

—

—

—

C₃²

yes

complemented

C₃×#66851

C₃²

yes

complemented

C₃×#66851

C₃²

yes

complemented

C₃²×C₁₃⋊C₉

—

C₃²×C₁₃⋊C₉

C₃²

yes

complemented

#11760437

C₃²

yes

complemented

C₃⁴×C₁₃

—

C₃⁴×C₁₃

C₃²

yes

characteristic

C₃×C₁₃×H(3,𝔽₃)

—

C₃×C₁₃×H(3,𝔽₃)

—

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

—

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

—

C₃×#6910

—

maximal, Sylow

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

H(3,𝔽₃)

yes

—

C₃³×C₁₃

H(3,𝔽₃)

yes

characteristic

C₃³×C₁₃

H(3,𝔽₃)

yes

—

C₃³×C₁₃

C₃³

yes

characteristic

C₃³×C₁₃

C₃³

yes

—

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

C₃³

yes

—

—

—

—

—

—

C₃³

yes

complemented

C₃×C₁₃⋊C₉

—

C₃×C₁₃⋊C₉

—

C₃×C₁₃⋊C₉

—

C₃×C₁₃⋊C₉

—

C₃×C₁₃⋊C₉

C₃³

yes

complemented

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

—

C₃²×H(3,𝔽₃)

C₁₃⋊C₃

yes

characteristic

#6910

—

C₃×#731

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

—

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

—

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

—

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₁₃

C₃⁴

yes

characteristic

C₁₃⋊C₉

—

C₁₃⋊C₉

—

C₃⁴

—

C₃⁴

C₃×C₁₃⋊C₃

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

—

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×C₁₃⋊C₃

yes

—

C₃×H(3,𝔽₃)

C₃×C₁₃⋊C₃

yes

—

C₃²×C₉

—

C₃²×C₉

—

#731

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

#6910

yes

complemented

C₃×C₁₃

C₃²×H(3,𝔽₃)

yes

characteristic

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²⋊(C₁₃⋊C₃)

yes

—

C₃³

C₃²⋊(C₁₃⋊C₃)

yes

characteristic

C₃³

C₃²⋊(C₁₃⋊C₃)

yes

—

100

C₃³

C₃²×C₁₃⋊C₃

yes

characteristic

101

C₃³

C₃²×C₁₃⋊C₃

yes

—

102

3₋¹⁺²

—

103

3₋¹⁺²

—

104

3₋¹⁺²

—

105

3₋¹⁺²

—

106

3₋¹⁺²

—

107

3₋¹⁺²

—

108

H(3,𝔽₃)

—

109

H(3,𝔽₃)

—

110

H(3,𝔽₃)

C₃²×C₁₃⋊C₃

yes

—

111

C₃×C₉

—

112

C₃×C₉

—

113

C₃×C₉

—

114

C₃×C₉

—

115

C₃×C₉

—

116

C₁₃

C₃×#6910

yes

characteristic, complemented, Sylow

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

—

122

C₃²

—

123

C₃²

—

124

C₃²

—

125

C₃²

C₃×C₃²⋊(C₁₃⋊C₃)

yes

—

126

C₃²

C₃×C₃²⋊(C₁₃⋊C₃)

yes

—

127

C₃²

C₃×C₃²⋊(C₁₃⋊C₃)

yes

—

128

C₃²

C₃×C₃²⋊(C₁₃⋊C₃)

yes

characteristic

129

C₃²

C₃³×C₁₃⋊C₃

yes

characteristic

130

C₉

—

131

C₉

—

132

C₃

—

133

C₃

—

134

C₃

—

135

C₃

—

136

C₃

—

137

C₃

#15691660

yes

direct product factor

138

C₃

C₃²×C₃²⋊(C₁₃⋊C₃)

yes

characteristic

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₁₃⋊C₃)
commutator, cosocle	C₃²×C₁₃	C₃⁴
Frattini	C₃²	C₃³×C₁₃⋊C₃
Fitting, prime radical	C₃²×C₁₃×H(3,𝔽₃)	C₃
socle	C₃²×C₁₃	C₃×H(3,𝔽₃)
hypercenter	C₃²×H(3,𝔽₃)	C₁₃⋊C₃
hypocenter	C₁₃	C₃×#6910
LCS	G ▷ C₃²×C₁₃ ▷ C₃×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃²×H(3,𝔽₃)

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₁₃⋊C₃)

commutator, cosocle

C₃²×C₁₃

C₃⁴

Frattini

C₃²

C₃³×C₁₃⋊C₃

Fitting, prime radical

C₃²×C₁₃×H(3,𝔽₃)

C₃

socle

C₃²×C₁₃

C₃×H(3,𝔽₃)

hypercenter

C₃²×H(3,𝔽₃)

C₁₃⋊C₃

hypocenter

C₁₃

C₃×#6910

LCS

G ▷ C₃²×C₁₃ ▷ C₃×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃²×H(3,𝔽₃)

N	H ≅ G/N	type
1. C₃²×#66851	C₃	semidirect
2. C₃²×#66851	C₃	semidirect
3. #15691660	C₃	direct (N × H)
4. C₃×#11760437	C₃	semidirect
12. C₃×#66851	C₃²	semidirect
13. C₃×#66851	C₃²	semidirect
14. C₃×#66851	C₃²	semidirect
16. C₃²×C₁₃⋊C₉	C₃²	semidirect
17. #11760437	C₃²	semidirect
42. #66851	C₃³	semidirect
47. C₃×C₁₃⋊C₉	C₃³	semidirect
91. C₃×C₁₃	#6910	semidirect
116. C₁₃	C₃×#6910	semidirect
137. C₃	#15691660	direct (N × H)

H ≅ G/N

type

1. C₃²×#66851

C₃

semidirect

2. C₃²×#66851

C₃

semidirect

3. #15691660

C₃

direct (N × H)

4. C₃×#11760437

C₃

semidirect

12. C₃×#66851

C₃²

semidirect

13. C₃×#66851

C₃²

semidirect

14. C₃×#66851

C₃²

semidirect

16. C₃²×C₁₃⋊C₉

C₃²

semidirect

17. #11760437

C₃²

semidirect

42. #66851

C₃³

semidirect

47. C₃×C₁₃⋊C₉

C₃³

semidirect

91. C₃×C₁₃

#6910

semidirect

116. C₁₃

C₃×#6910

semidirect

137. C₃

#15691660

direct (N × H)

Subgroups — C₃×#15691660 #23748519

Conjugacy classes of subgroups

Complements

Subgroups — C3×#15691660 #23748519

Conjugacy classes of subgroups

Complements

Subgroups — C₃×#15691660 #23748519