Group #23748537 — subgroups

Subgroups — C₃×#15691675 #23748537

Order 9477.

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

normal subgroups only

This group has 992 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	C₃×#11760452	C₃	1	3	yes	maximal, complemented
2	C₃²×#66848	C₃	1	1	yes	maximal, characteristic, complemented
3	#15691675	C₃	1	27	yes	maximal, direct product factor
4	C₃×#11760443	C₃	1	2	yes	maximal, complemented
5	C₃×#11760437	C₃	1	3	yes	maximal, complemented
6	C₃×#11760434	C₃	1	3	yes	maximal, complemented
7	C₃×C₁₃×#732	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃³×C₁₃⋊C₃	C₃²	1	1	yes	characteristic
9	#11760452	—	3	18	no	—
10	#11760452	C₃²	1	27	yes	complemented
11	C₃×#66851	—	3	6	no	—
12	C₃×#66851	C₃²	1	6	yes	—
13	C₃×#66848	—	3	3	no	—
14	C₃×#66848	—	3	6	no	—
15	C₃×#66848	C₃²	1	9	yes	complemented
16	C₃²×C₁₃⋊C₉	—	3	3	no	—
17	C₃²×C₁₃⋊C₉	C₃²	1	1	yes	characteristic
18	C₃²×C₁₃⋊C₉	C₃²	1	1	yes	characteristic
19	#11760443	C₃²	1	18	yes	complemented
20	#11760437	C₃²	1	27	yes	complemented
21	#11760434	C₃²	1	27	yes	complemented
22	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	3	yes	—
23	C₃²×C₉×C₁₃	C₃²	1	1	yes	characteristic
24	C₁₃×#732	C₃²	1	9	yes	complemented
25	C₃×#6913	—	13	1	no	maximal, Sylow
26	C₃³×C₁₃	—	3	3	no	—
27	C₃³×C₁₃	C₃³	1	1	yes	characteristic
28	C₃²×C₁₃⋊C₃	—	3	2	no	—
29	C₃²×C₁₃⋊C₃	—	3	1	no	—
30	C₃²×C₁₃⋊C₃	—	9	3	no	—
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	—
34	C₃×C₉×C₁₃	—	3	2	no	—
35	C₃×C₉×C₁₃	—	3	1	no	—
36	C₃×C₉×C₁₃	C₃³	1	3	yes	—
37	#66851	—	3	12	no	—
38	#66851	—	3	54	no	—
39	#66851	C₃³	1	18	yes	—
40	C₃×C₁₃⋊C₉	—	3	6	no	—
41	C₃×C₁₃⋊C₉	—	3	2	no	—
42	C₃×C₁₃⋊C₉	—	3	1	no	—
43	C₃×C₁₃⋊C₉	—	3	27	no	—
44	C₃×C₁₃⋊C₉	H(3,𝔽₃)	1	3	yes	—
45	C₃×C₁₃⋊C₉	H(3,𝔽₃)	1	6	yes	—
46	C₃×C₁₃⋊C₉	C₃³	1	3	yes	—
47	C₃×C₁₃⋊C₉	C₃³	1	3	yes	—
48	#66848	—	3	27	no	—
49	C₃×#737	—	13	3	no	—
50	C₃²×3₋¹⁺²	—	13	1	no	—
51	#6913	—	13	27	no	—
52	C₃×#733	—	13	2	no	—
53	C₃×#732	C₁₃⋊C₃	1	1	yes	characteristic, complemented
54	C₃×#731	—	13	3	no	—
55	C₃×AΣL(1,𝔽₂₇)	—	13	3	no	—
56	C₃²×C₁₃	—	3	3	no	—
57	C₃²×C₁₃	—	3	9	no	—
58	C₃²×C₁₃	—	9	9	no	—
59	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	2	yes	—
60	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
61	C₃²×C₁₃	C₃⁴	1	1	yes	characteristic
62	C₃×C₁₃⋊C₃	—	3	3	no	—
63	C₃×C₁₃⋊C₃	—	9	6	no	—
64	C₃×C₁₃⋊C₃	—	9	3	no	—
65	C₃×C₁₃⋊C₃	—	9	3	no	—
66	C₉×C₁₃	—	3	3	no	—
67	C₁₃⋊C₉	—	3	3	no	—
68	C₁₃⋊C₉	—	3	18	no	—
69	C₁₃⋊C₉	C₃×H(3,𝔽₃)	1	9	yes	—
70	C₃⁴	—	13	1	no	—
71	#737	—	13	27	no	—
72	#737	—	39	18	no	—
73	C₃×3₋¹⁺²	—	13	9	no	—
74	C₃×3₋¹⁺²	—	13	6	no	—
75	C₃×3₋¹⁺²	—	39	6	no	—
76	C₃×3₋¹⁺²	—	39	3	no	—
77	C₃×3₋¹⁺²	—	39	6	no	—
78	C₃×H(3,𝔽₃)	C₃×C₁₃⋊C₃	1	3	yes	—
79	C₃²×C₉	—	13	1	no	—
80	C₃²×C₉	—	13	1	no	—
81	C₃²×C₉	—	39	3	no	—
82	C₃²×C₉	C₃×C₁₃⋊C₃	1	1	yes	characteristic
83	#733	—	13	18	no	—
84	#732	C₃×C₁₃⋊C₃	1	9	yes	complemented
85	#731	—	13	27	no	—
86	AΣL(1,𝔽₂₇)	—	13	27	no	—
87	C₃×C₁₃	—	3	2	no	—
88	C₃×C₁₃	—	3	1	no	—
89	C₃×C₁₃	—	9	9	no	—
90	C₃×C₁₃	#6913	1	3	yes	complemented
91	C₃×C₁₃	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
92	C₁₃⋊C₃	—	9	3	no	—
93	C₃³	—	3	3	no	—
94	C₃³	—	13	3	no	—
95	C₃³	—	39	2	no	—
96	C₃³	—	39	1	no	—
97	C₃³	—	117	3	no	—
98	C₃³	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
99	3₋¹⁺²	—	13	18	no	—
100	3₋¹⁺²	—	39	27	no	—
101	3₋¹⁺²	—	39	54	no	—
102	3₋¹⁺²	—	39	12	no	—
103	H(3,𝔽₃)	—	3	6	no	—
104	H(3,𝔽₃)	C₃²×C₁₃⋊C₃	1	9	yes	—
105	C₃×C₉	—	3	2	no	—
106	C₃×C₉	—	3	1	no	—
107	C₃×C₉	—	13	6	no	—
108	C₃×C₉	—	13	3	no	—
109	C₃×C₉	—	13	3	no	—
110	C₃×C₉	—	13	3	no	—
111	C₃×C₉	—	39	27	no	—
112	C₃×C₉	—	39	2	no	—
113	C₃×C₉	—	39	1	no	—
114	C₃×C₉	—	39	6	no	—
115	C₃×C₉	C₃²×C₁₃⋊C₃	1	3	yes	—
116	C₁₃	C₃×#6913	1	1	yes	characteristic, complemented, Sylow
117	C₃²	—	3	3	no	—
118	C₃²	—	3	9	no	—
119	C₃²	—	9	9	no	—
120	C₃²	—	39	3	no	—
121	C₃²	—	117	3	no	—
122	C₃²	—	117	6	no	—
123	C₃²	—	117	3	no	—
124	C₃²	H(3,𝔽₃)×C₁₃⋊C₃	1	2	yes	—
125	C₃²	H(3,𝔽₃)×C₁₃⋊C₃	1	1	yes	characteristic
126	C₃²	C₃³×C₁₃⋊C₃	1	1	yes	characteristic
127	C₉	—	3	3	no	—
128	C₉	—	13	9	no	—
129	C₉	—	39	3	no	—
130	C₉	—	39	18	no	—
131	C₃	—	3	2	no	—
132	C₃	—	3	1	no	—
133	C₃	—	9	9	no	—
134	C₃	—	117	3	no	—
135	C₃	#15691675	1	3	yes	direct product factor
136	C₃	C₃×H(3,𝔽₃)×C₁₃⋊C₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#11760452

C₃

yes

maximal, complemented

C₃²×#66848

C₃

yes

maximal, characteristic, complemented

#15691675

C₃

yes

maximal, direct product factor

C₃×#11760443

C₃

yes

maximal, complemented

C₃×#11760437

C₃

yes

maximal, complemented

C₃×#11760434

C₃

yes

maximal, complemented

C₃×C₁₃×#732

C₃

yes

maximal, characteristic, complemented

C₃³×C₁₃⋊C₃

C₃²

yes

characteristic

#11760452

—

#11760452

C₃²

yes

complemented

C₃×#66851

—

C₃×#66851

C₃²

yes

—

C₃×#66848

—

C₃×#66848

—

C₃×#66848

C₃²

yes

complemented

C₃²×C₁₃⋊C₉

—

C₃²×C₁₃⋊C₉

C₃²

yes

characteristic

C₃²×C₁₃⋊C₉

C₃²

yes

characteristic

#11760443

C₃²

yes

complemented

#11760437

C₃²

yes

complemented

#11760434

C₃²

yes

complemented

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

—

C₃²×C₉×C₁₃

C₃²

yes

characteristic

C₁₃×#732

C₃²

yes

complemented

C₃×#6913

—

maximal, Sylow

C₃³×C₁₃

—

C₃³×C₁₃

C₃³

yes

characteristic

C₃²×C₁₃⋊C₃

—

C₃²×C₁₃⋊C₃

—

C₃²×C₁₃⋊C₃

—

C₃²×C₁₃⋊C₃

C₃³

yes

—

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

C₃³

yes

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

C₃³

yes

—

#66851

—

#66851

—

#66851

C₃³

yes

—

C₃×C₁₃⋊C₉

—

C₃×C₁₃⋊C₉

—

C₃×C₁₃⋊C₉

—

C₃×C₁₃⋊C₉

—

C₃×C₁₃⋊C₉

H(3,𝔽₃)

yes

—

C₃×C₁₃⋊C₉

H(3,𝔽₃)

yes

—

C₃×C₁₃⋊C₉

C₃³

yes

—

C₃×C₁₃⋊C₉

C₃³

yes

—

#66848

—

C₃×#737

—

C₃²×3₋¹⁺²

—

#6913

—

C₃×#733

—

C₃×#732

C₁₃⋊C₃

yes

characteristic, complemented

C₃×#731

—

C₃×AΣL(1,𝔽₂₇)

—

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

—

C₉×C₁₃

—

C₁₃⋊C₉

—

C₁₃⋊C₉

—

C₁₃⋊C₉

C₃×H(3,𝔽₃)

yes

—

C₃⁴

—

#737

—

#737

—

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×C₁₃⋊C₃

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₁₃⋊C₃

yes

characteristic

#733

—

#732

C₃×C₁₃⋊C₃

yes

complemented

#731

—

AΣL(1,𝔽₂₇)

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

#6913

yes

complemented

C₃×C₁₃

C₃²×H(3,𝔽₃)

yes

characteristic

C₁₃⋊C₃

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

117

—

C₃³

C₃²×C₁₃⋊C₃

yes

characteristic

3₋¹⁺²

—

100

3₋¹⁺²

—

101

3₋¹⁺²

—

102

3₋¹⁺²

—

103

H(3,𝔽₃)

—

104

H(3,𝔽₃)

C₃²×C₁₃⋊C₃

yes

—

105

C₃×C₉

—

106

C₃×C₉

—

107

C₃×C₉

—

108

C₃×C₉

—

109

C₃×C₉

—

110

C₃×C₉

—

111

C₃×C₉

—

112

C₃×C₉

—

113

C₃×C₉

—

114

C₃×C₉

—

115

C₃×C₉

C₃²×C₁₃⋊C₃

yes

—

116

C₁₃

C₃×#6913

yes

characteristic, complemented, Sylow

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

—

117

—

122

C₃²

—

117

—

123

C₃²

—

117

—

124

C₃²

H(3,𝔽₃)×C₁₃⋊C₃

yes

—

125

C₃²

H(3,𝔽₃)×C₁₃⋊C₃

yes

characteristic

126

C₃²

C₃³×C₁₃⋊C₃

yes

characteristic

127

C₉

—

128

C₉

—

129

C₉

—

130

C₉

—

131

C₃

—

132

C₃

—

133

C₃

—

134

C₃

—

117

—

135

C₃

#15691675

yes

direct product factor

136

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×#6913
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. C₃×#11760452	C₃	semidirect
2. C₃²×#66848	C₃	semidirect
3. #15691675	C₃	direct (N × H)
4. C₃×#11760443	C₃	semidirect
5. C₃×#11760437	C₃	semidirect
6. C₃×#11760434	C₃	semidirect
7. C₃×C₁₃×#732	C₃	semidirect
10. #11760452	C₃²	semidirect
15. C₃×#66848	C₃²	semidirect
19. #11760443	C₃²	semidirect
20. #11760437	C₃²	semidirect
21. #11760434	C₃²	semidirect
24. C₁₃×#732	C₃²	semidirect
53. C₃×#732	C₁₃⋊C₃	semidirect
84. #732	C₃×C₁₃⋊C₃	semidirect
90. C₃×C₁₃	#6913	semidirect
116. C₁₃	C₃×#6913	semidirect
135. C₃	#15691675	direct (N × H)

H ≅ G/N

type

1. C₃×#11760452

C₃

semidirect

2. C₃²×#66848

C₃

semidirect

3. #15691675

C₃

direct (N × H)

4. C₃×#11760443

C₃

semidirect

5. C₃×#11760437

C₃

semidirect

6. C₃×#11760434

C₃

semidirect

7. C₃×C₁₃×#732

C₃

semidirect

10. #11760452

C₃²

semidirect

15. C₃×#66848

C₃²

semidirect

19. #11760443

C₃²

semidirect

20. #11760437

C₃²

semidirect

21. #11760434

C₃²

semidirect

24. C₁₃×#732

C₃²

semidirect

53. C₃×#732

C₁₃⋊C₃

semidirect

84. #732

C₃×C₁₃⋊C₃

semidirect

90. C₃×C₁₃

#6913

semidirect

116. C₁₃

C₃×#6913

semidirect

135. C₃

#15691675

direct (N × H)

Subgroups — C₃×#15691675 #23748537

Conjugacy classes of subgroups

Complements

Subgroups — C3×#15691675 #23748537

Conjugacy classes of subgroups

Complements

Subgroups — C₃×#15691675 #23748537