Group #23748529 — subgroups

Subgroups — C₃×#15691668 #23748529

Order 9477.

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

normal subgroups only

This group has 980 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×#66849	C₃	1	1	yes	maximal, characteristic
2	C₃×3₋¹⁺²×C₁₃⋊C₃	C₃	1	3	yes	maximal
3	#15691668	C₃	1	27	yes	maximal, direct product factor
4	C₃×#11760443	C₃	1	2	yes	maximal, complemented
5	C₃×#11760442	C₃	1	6	yes	maximal, complemented
6	C₃×C₁₃×#733	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃³×C₁₃⋊C₃	C₃²	1	1	yes	characteristic
8	C₃×#66851	C₃²	1	6	yes	—
9	C₃×#66848	—	3	3	no	—
10	C₃×#66849	—	3	3	no	—
11	C₃×#66849	—	3	3	no	—
12	C₃×#66849	—	3	6	no	—
13	C₃×#66849	C₃²	1	9	yes	—
14	3₋¹⁺²×C₁₃⋊C₃	—	3	18	no	—
15	3₋¹⁺²×C₁₃⋊C₃	C₃²	1	27	yes	—
16	C₃²×C₁₃⋊C₉	C₃²	1	2	yes	—
17	C₃×C₉×C₁₃⋊C₃	—	3	3	no	—
18	#11760443	C₃²	1	18	yes	complemented
19	#11760442	C₃²	1	54	yes	complemented
20	C₃×C₁₃×3₋¹⁺²	C₃²	1	3	yes	—
21	C₃²×C₉×C₁₃	C₃²	1	1	yes	characteristic
22	C₁₃×#733	C₃²	1	9	yes	complemented
23	C₃×#6912	—	13	1	no	maximal, Sylow
24	C₃³×C₁₃	C₃³	1	1	yes	characteristic
25	C₃²×C₁₃⋊C₃	—	9	3	no	—
26	C₃²×C₁₃⋊C₃	H(3,𝔽₃)	1	3	yes	—
27	C₃²×C₁₃⋊C₃	H(3,𝔽₃)	1	6	yes	—
28	C₃²×C₁₃⋊C₃	C₃³	1	3	yes	—
29	C₁₃×3₋¹⁺²	—	3	6	no	—
30	C₁₃×3₋¹⁺²	C₃³	1	9	yes	—
31	C₃×C₉×C₁₃	—	3	2	no	—
32	C₃×C₉×C₁₃	—	3	1	no	—
33	C₃×C₉×C₁₃	—	3	3	no	—
34	C₃×C₉×C₁₃	C₃³	1	3	yes	—
35	#66851	—	3	12	no	—
36	#66851	C₃³	1	18	yes	—
37	C₃×C₁₃⋊C₉	—	3	6	no	—
38	C₃×C₁₃⋊C₉	—	3	4	no	—
39	C₃×C₁₃⋊C₉	—	3	2	no	—
40	C₃×C₁₃⋊C₉	C₃³	1	6	yes	—
41	#66849	—	3	27	no	—
42	#66849	—	3	27	no	—
43	#66848	—	3	27	no	—
44	C₉×C₁₃⋊C₃	—	3	27	no	—
45	C₃²×3₋¹⁺²	—	13	1	no	—
46	C₃²×3₋¹⁺²	—	13	3	no	—
47	#6912	—	13	27	no	—
48	C₃×#733	—	13	6	no	—
49	C₃×#733	—	13	2	no	—
50	C₃×#733	C₁₃⋊C₃	1	1	yes	characteristic, complemented
51	C₃²×C₁₃	—	3	3	no	—
52	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	2	yes	—
53	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
54	C₃²×C₁₃	C₃⁴	1	1	yes	characteristic
55	C₃×C₁₃⋊C₃	—	3	9	no	—
56	C₃×C₁₃⋊C₃	—	9	6	no	—
57	C₃×C₁₃⋊C₃	—	9	3	no	—
58	C₃×C₁₃⋊C₃	C₃×H(3,𝔽₃)	1	9	yes	—
59	C₉×C₁₃	—	3	9	no	—
60	C₉×C₁₃	—	3	3	no	—
61	C₁₃⋊C₉	—	3	18	no	—
62	C₁₃⋊C₉	—	3	6	no	—
63	C₃⁴	—	13	1	no	—
64	C₃×3₋¹⁺²	—	13	9	no	—
65	C₃×3₋¹⁺²	—	13	6	no	—
66	C₃×3₋¹⁺²	—	13	27	no	—
67	C₃×3₋¹⁺²	—	39	6	no	—
68	C₃×3₋¹⁺²	—	39	3	no	—
69	C₃×3₋¹⁺²	—	39	3	no	—
70	C₃×3₋¹⁺²	—	39	3	no	—
71	C₃×3₋¹⁺²	—	39	18	no	—
72	C₃×3₋¹⁺²	C₃×C₁₃⋊C₃	1	3	yes	—
73	C₃²×C₉	—	13	2	no	—
74	C₃²×C₉	—	39	3	no	—
75	C₃²×C₉	C₃×C₁₃⋊C₃	1	1	yes	characteristic
76	#733	—	13	54	no	—
77	#733	—	13	18	no	—
78	#733	C₃×C₁₃⋊C₃	1	9	yes	complemented
79	C₃×C₁₃	—	3	2	no	—
80	C₃×C₁₃	—	3	1	no	—
81	C₃×C₁₃	#6912	1	3	yes	complemented
82	C₃×C₁₃	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
83	C₁₃⋊C₃	—	3	9	no	—
84	C₃³	—	13	3	no	—
85	C₃³	—	13	6	no	—
86	C₃³	—	13	3	no	—
87	C₃³	—	117	3	no	—
88	C₃³	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
89	3₋¹⁺²	—	3	6	no	—
90	3₋¹⁺²	—	13	18	no	—
91	3₋¹⁺²	—	39	27	no	—
92	3₋¹⁺²	—	39	27	no	—
93	3₋¹⁺²	—	39	27	no	—
94	3₋¹⁺²	—	39	12	no	—
95	3₋¹⁺²	C₃²×C₁₃⋊C₃	1	9	yes	—
96	C₃×C₉	—	3	2	no	—
97	C₃×C₉	—	3	3	no	—
98	C₃×C₉	—	3	1	no	—
99	C₃×C₉	—	13	6	no	—
100	C₃×C₉	—	39	27	no	—
101	C₃×C₉	—	39	4	no	—
102	C₃×C₉	—	39	2	no	—
103	C₃×C₉	—	39	6	no	—
104	C₃×C₉	C₃²×C₁₃⋊C₃	1	3	yes	—
105	C₁₃	C₃×#6912	1	1	yes	characteristic, complemented, Sylow
106	C₃²	—	3	3	no	—
107	C₃²	—	13	9	no	—
108	C₃²	—	39	9	no	—
109	C₃²	—	117	6	no	—
110	C₃²	—	117	3	no	—
111	C₃²	H(3,𝔽₃)×C₁₃⋊C₃	1	2	yes	—
112	C₃²	H(3,𝔽₃)×C₁₃⋊C₃	1	1	yes	characteristic
113	C₃²	C₃³×C₁₃⋊C₃	1	1	yes	characteristic
114	C₉	—	3	3	no	—
115	C₉	—	3	9	no	—
116	C₉	—	39	6	no	—
117	C₉	—	39	18	no	—
118	C₃	—	3	2	no	—
119	C₃	—	3	1	no	—
120	C₃	—	39	9	no	—
121	C₃	#15691668	1	3	yes	direct product factor
122	C₃	C₃×H(3,𝔽₃)×C₁₃⋊C₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#66849

C₃

yes

maximal, characteristic

C₃×3₋¹⁺²×C₁₃⋊C₃

C₃

yes

maximal

#15691668

C₃

yes

maximal, direct product factor

C₃×#11760443

C₃

yes

maximal, complemented

C₃×#11760442

C₃

yes

maximal, complemented

C₃×C₁₃×#733

C₃

yes

maximal, characteristic, complemented

C₃³×C₁₃⋊C₃

C₃²

yes

characteristic

C₃×#66851

C₃²

yes

—

—

—

—

—

C₃²

yes

—

3₋¹⁺²×C₁₃⋊C₃

—

3₋¹⁺²×C₁₃⋊C₃

C₃²

yes

—

C₃²×C₁₃⋊C₉

C₃²

yes

—

C₃×C₉×C₁₃⋊C₃

—

#11760443

C₃²

yes

complemented

#11760442

C₃²

yes

complemented

C₃×C₁₃×3₋¹⁺²

C₃²

yes

—

C₃²×C₉×C₁₃

C₃²

yes

characteristic

C₁₃×#733

C₃²

yes

complemented

C₃×#6912

—

maximal, Sylow

C₃³×C₁₃

C₃³

yes

characteristic

C₃²×C₁₃⋊C₃

—

C₃²×C₁₃⋊C₃

H(3,𝔽₃)

yes

—

C₃²×C₁₃⋊C₃

H(3,𝔽₃)

yes

—

C₃²×C₁₃⋊C₃

C₃³

yes

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

C₃³

yes

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

C₃³

yes

—

#66851

—

#66851

C₃³

yes

—

C₃×C₁₃⋊C₉

—

C₃×C₁₃⋊C₉

—

C₃×C₁₃⋊C₉

—

C₃×C₁₃⋊C₉

C₃³

yes

—

#66849

—

#66849

—

#66848

—

C₉×C₁₃⋊C₃

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

—

—

—

—

C₁₃⋊C₃

yes

characteristic, complemented

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×H(3,𝔽₃)

yes

—

C₉×C₁₃

—

C₉×C₁₃

—

C₁₃⋊C₉

—

C₁₃⋊C₉

—

C₃⁴

—

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×C₁₃⋊C₃

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₁₃⋊C₃

yes

—

—

C₃×C₁₃⋊C₃

yes

complemented

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

#6912

yes

complemented

C₃×C₁₃

C₃²×H(3,𝔽₃)

yes

characteristic

C₁₃⋊C₃

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

117

—

C₃³

C₃²×C₁₃⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

C₃²×C₁₃⋊C₃

yes

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

100

C₃×C₉

—

101

C₃×C₉

—

102

C₃×C₉

—

103

C₃×C₉

—

104

C₃×C₉

C₃²×C₁₃⋊C₃

yes

—

105

C₁₃

C₃×#6912

yes

characteristic, complemented, Sylow

106

C₃²

—

107

C₃²

—

108

C₃²

—

109

C₃²

—

117

—

110

C₃²

—

117

—

111

C₃²

H(3,𝔽₃)×C₁₃⋊C₃

yes

—

112

C₃²

H(3,𝔽₃)×C₁₃⋊C₃

yes

characteristic

113

C₃²

C₃³×C₁₃⋊C₃

yes

characteristic

114

C₉

—

115

C₉

—

116

C₉

—

117

C₉

—

118

C₃

—

119

C₃

—

120

C₃

—

121

C₃

#15691668

yes

direct product factor

122

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×#733	C₃
socle	C₃²×C₁₃	C₃×H(3,𝔽₃)
prime radical	C₃³×C₁₃⋊C₃	C₃²
hypercenter	C₃×#733	C₁₃⋊C₃
hypocenter	C₁₃	C₃×#6912
LCS	G ▷ C₃²×C₁₃ ▷ C₃×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×#733

subgroup

G/H

center

C₃²

H(3,𝔽₃)×C₁₃⋊C₃

commutator, cosocle

C₃²×C₁₃

C₃⁴

Frattini

C₃²

C₃³×C₁₃⋊C₃

Fitting

C₃×C₁₃×#733

C₃

socle

C₃²×C₁₃

C₃×H(3,𝔽₃)

prime radical

C₃³×C₁₃⋊C₃

C₃²

C₁₃⋊C₃

C₁₃

G ▷ C₃²×C₁₃ ▷ C₃×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ C₃×#733

N	H ≅ G/N	type
3. #15691668	C₃	direct (N × H)
4. C₃×#11760443	C₃	semidirect
5. C₃×#11760442	C₃	semidirect
6. C₃×C₁₃×#733	C₃	semidirect
18. #11760443	C₃²	semidirect
19. #11760442	C₃²	semidirect
22. C₁₃×#733	C₃²	semidirect
50. C₃×#733	C₁₃⋊C₃	semidirect
78. #733	C₃×C₁₃⋊C₃	semidirect
81. C₃×C₁₃	#6912	semidirect
105. C₁₃	C₃×#6912	semidirect
121. C₃	#15691668	direct (N × H)

H ≅ G/N

type

3. #15691668

C₃

direct (N × H)

4. C₃×#11760443

C₃

semidirect

5. C₃×#11760442

C₃

semidirect

6. C₃×C₁₃×#733

C₃

semidirect

18. #11760443

C₃²

semidirect

19. #11760442

C₃²

semidirect

22. C₁₃×#733

C₃²

semidirect

50. C₃×#733

C₁₃⋊C₃

semidirect

78. #733

C₃×C₁₃⋊C₃

semidirect

81. C₃×C₁₃

#6912

semidirect

105. C₁₃

C₃×#6912

semidirect

121. C₃

#15691668

direct (N × H)

Subgroups — C₃×#15691668 #23748529

Conjugacy classes of subgroups

Complements

Subgroups — C3×#15691668 #23748529

Conjugacy classes of subgroups

Complements

Subgroups — C₃×#15691668 #23748529