Group #23748542 — subgroups

Subgroups — C₃×#15691680 #23748542

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×#11760454	C₃	1	3	yes	maximal, complemented
2	C₃²×#66848	C₃	1	1	yes	maximal, characteristic, complemented
3	#15691680	C₃	1	27	yes	maximal, direct product factor
4	C₃×#11760443	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃×#11760441	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃×#11760439	C₃	1	3	yes	maximal, complemented
7	C₃×#11760436	C₃	1	3	yes	maximal, complemented
8	C₃×C₁₃×#733	C₃	1	1	yes	maximal, characteristic, complemented
9	C₃³×C₁₃⋊C₃	C₃²	1	1	yes	characteristic
10	#11760454	—	3	18	no	—
11	#11760454	C₃²	1	27	yes	complemented
12	C₃×#66851	C₃²	1	3	yes	—
13	C₃×#66848	—	3	3	no	—
14	C₃×#66848	—	3	6	no	—
15	C₃×#66848	—	3	3	no	—
16	C₃×#66848	C₃²	1	9	yes	complemented
17	C₃×#66849	—	3	3	no	—
18	C₃×C₃²⋊(C₁₃⋊C₃)	C₃²	1	3	yes	—
19	C₃²×C₁₃⋊C₉	C₃²	1	1	yes	characteristic
20	C₃²×C₁₃⋊C₉	C₃²	1	1	yes	characteristic
21	C₃×C₉×C₁₃⋊C₃	—	3	3	no	—
22	#11760443	C₃²	1	9	yes	complemented
23	#11760441	C₃²	1	9	yes	complemented
24	#11760439	C₃²	1	27	yes	complemented
25	#11760436	C₃²	1	27	yes	complemented
26	C₃×C₁₃×3₋¹⁺²	C₃²	1	3	yes	—
27	C₃²×C₉×C₁₃	C₃²	1	1	yes	characteristic
28	C₁₃×#733	C₃²	1	9	yes	complemented
29	C₃×#6913	—	13	1	no	maximal, Sylow
30	C₃³×C₁₃	C₃³	1	1	yes	characteristic
31	C₃²×C₁₃⋊C₃	—	3	3	no	—
32	C₃²×C₁₃⋊C₃	—	3	2	no	—
33	C₃²×C₁₃⋊C₃	—	3	1	no	—
34	C₃²×C₁₃⋊C₃	—	9	3	no	—
35	C₃²×C₁₃⋊C₃	C₃³	1	3	yes	—
36	C₁₃×3₋¹⁺²	—	3	6	no	—
37	C₁₃×3₋¹⁺²	C₃³	1	9	yes	—
38	C₃×C₉×C₁₃	—	3	2	no	—
39	C₃×C₉×C₁₃	—	3	1	no	—
40	C₃×C₉×C₁₃	—	3	3	no	—
41	C₃×C₉×C₁₃	C₃³	1	3	yes	—
42	C₃²⋊(C₁₃⋊C₃)	—	3	6	no	—
43	C₃²⋊(C₁₃⋊C₃)	C₃³	1	9	yes	—
44	#66851	—	3	6	no	—
45	#66851	C₃³	1	9	yes	—
46	C₃×C₁₃⋊C₉	—	3	3	no	—
47	C₃×C₁₃⋊C₉	—	3	2	no	—
48	C₃×C₁₃⋊C₉	—	3	1	no	—
49	C₃×C₁₃⋊C₉	H(3,𝔽₃)	1	3	yes	—
50	C₃×C₁₃⋊C₉	H(3,𝔽₃)	1	6	yes	—
51	C₃×C₁₃⋊C₉	C₃³	1	3	yes	—
52	C₃×C₁₃⋊C₉	C₃³	1	3	yes	—
53	#66849	—	3	27	no	—
54	#66848	—	3	27	no	—
55	#66848	—	3	27	no	—
56	C₉×C₁₃⋊C₃	—	3	27	no	—
57	C₃×#737	—	13	3	no	—
58	C₃²×3₋¹⁺²	—	13	1	no	—
59	#6913	—	13	27	no	—
60	C₃×#733	—	13	1	no	—
61	C₃×#733	C₁₃⋊C₃	1	1	yes	characteristic, complemented
62	C₃×#732	—	13	1	no	—
63	C₃×#731	—	13	3	no	—
64	C₃×AΣL(1,𝔽₂₇)	—	13	3	no	—
65	C₃²×C₁₃	—	3	3	no	—
66	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	2	yes	—
67	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
68	C₃²×C₁₃	C₃⁴	1	1	yes	characteristic
69	C₃×C₁₃⋊C₃	—	3	9	no	—
70	C₃×C₁₃⋊C₃	—	3	3	no	—
71	C₃×C₁₃⋊C₃	—	9	6	no	—
72	C₃×C₁₃⋊C₃	—	9	9	no	—
73	C₃×C₁₃⋊C₃	—	9	3	no	—
74	C₃×C₁₃⋊C₃	—	9	3	no	—
75	C₉×C₁₃	—	3	9	no	—
76	C₉×C₁₃	—	3	3	no	—
77	C₁₃⋊C₉	—	3	9	no	—
78	C₁₃⋊C₉	—	3	3	no	—
79	C₁₃⋊C₉	C₃×H(3,𝔽₃)	1	9	yes	—
80	C₃⁴	—	13	1	no	—
81	#737	—	13	27	no	—
82	#737	—	39	18	no	—
83	C₃×3₋¹⁺²	—	13	9	no	—
84	C₃×3₋¹⁺²	—	13	3	no	—
85	C₃×3₋¹⁺²	—	39	6	no	—
86	C₃×3₋¹⁺²	—	39	3	no	—
87	C₃×3₋¹⁺²	—	39	3	no	—
88	C₃×3₋¹⁺²	—	39	3	no	—
89	C₃×3₋¹⁺²	C₃×C₁₃⋊C₃	1	3	yes	—
90	C₃×H(3,𝔽₃)	—	13	3	no	—
91	C₃²×C₉	—	13	1	no	—
92	C₃²×C₉	—	13	1	no	—
93	C₃²×C₉	—	39	3	no	—
94	C₃²×C₉	C₃×C₁₃⋊C₃	1	1	yes	characteristic
95	#733	—	13	9	no	—
96	#733	C₃×C₁₃⋊C₃	1	9	yes	complemented
97	#732	—	13	9	no	—
98	#731	—	13	27	no	—
99	AΣL(1,𝔽₂₇)	—	13	27	no	—
100	C₃×C₁₃	—	3	2	no	—
101	C₃×C₁₃	—	3	1	no	—
102	C₃×C₁₃	#6913	1	3	yes	complemented
103	C₃×C₁₃	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
104	C₁₃⋊C₃	—	9	9	no	—
105	C₁₃⋊C₃	—	9	3	no	—
106	C₃³	—	13	3	no	—
107	C₃³	—	39	2	no	—
108	C₃³	—	39	3	no	—
109	C₃³	—	39	1	no	—
110	C₃³	—	117	3	no	—
111	C₃³	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
112	3₋¹⁺²	—	3	6	no	—
113	3₋¹⁺²	—	13	9	no	—
114	3₋¹⁺²	—	39	27	no	—
115	3₋¹⁺²	—	39	27	no	—
116	3₋¹⁺²	—	39	27	no	—
117	3₋¹⁺²	—	39	6	no	—
118	3₋¹⁺²	C₃²×C₁₃⋊C₃	1	9	yes	—
119	H(3,𝔽₃)	—	13	9	no	—
120	H(3,𝔽₃)	—	39	6	no	—
121	C₃×C₉	—	3	2	no	—
122	C₃×C₉	—	3	3	no	—
123	C₃×C₉	—	3	1	no	—
124	C₃×C₉	—	13	6	no	—
125	C₃×C₉	—	13	3	no	—
126	C₃×C₉	—	13	3	no	—
127	C₃×C₉	—	13	3	no	—
128	C₃×C₉	—	39	27	no	—
129	C₃×C₉	—	39	2	no	—
130	C₃×C₉	—	39	1	no	—
131	C₃×C₉	—	39	3	no	—
132	C₃×C₉	C₃²×C₁₃⋊C₃	1	3	yes	—
133	C₁₃	C₃×#6913	1	1	yes	characteristic, complemented, Sylow
134	C₃²	—	3	3	no	—
135	C₃²	—	39	9	no	—
136	C₃²	—	39	3	no	—
137	C₃²	—	117	3	no	—
138	C₃²	—	117	6	no	—
139	C₃²	—	117	3	no	—
140	C₃²	—	117	9	no	—
141	C₃²	H(3,𝔽₃)×C₁₃⋊C₃	1	2	yes	—
142	C₃²	H(3,𝔽₃)×C₁₃⋊C₃	1	1	yes	characteristic
143	C₃²	C₃³×C₁₃⋊C₃	1	1	yes	characteristic
144	C₉	—	3	3	no	—
145	C₉	—	3	9	no	—
146	C₉	—	13	9	no	—
147	C₉	—	39	3	no	—
148	C₉	—	39	9	no	—
149	C₃	—	3	2	no	—
150	C₃	—	3	1	no	—
151	C₃	—	117	3	no	—
152	C₃	—	117	9	no	—
153	C₃	#15691680	1	3	yes	direct product factor
154	C₃	C₃×H(3,𝔽₃)×C₁₃⋊C₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#11760454

C₃

yes

maximal, complemented

C₃²×#66848

C₃

yes

maximal, characteristic, complemented

#15691680

C₃

yes

maximal, direct product factor

C₃×#11760443

C₃

yes

maximal, characteristic, complemented

C₃×#11760441

C₃

yes

maximal, characteristic, complemented

C₃×#11760439

C₃

yes

maximal, complemented

C₃×#11760436

C₃

yes

maximal, complemented

C₃×C₁₃×#733

C₃

yes

maximal, characteristic, complemented

C₃³×C₁₃⋊C₃

C₃²

yes

characteristic

#11760454

—

#11760454

C₃²

yes

complemented

C₃×#66851

C₃²

yes

—

—

—

—

C₃²

yes

complemented

C₃×#66849

—

C₃×C₃²⋊(C₁₃⋊C₃)

C₃²

yes

—

C₃²×C₁₃⋊C₉

C₃²

yes

characteristic

C₃²×C₁₃⋊C₉

C₃²

yes

characteristic

C₃×C₉×C₁₃⋊C₃

—

#11760443

C₃²

yes

complemented

#11760441

C₃²

yes

complemented

#11760439

C₃²

yes

complemented

#11760436

C₃²

yes

complemented

C₃×C₁₃×3₋¹⁺²

C₃²

yes

—

C₃²×C₉×C₁₃

C₃²

yes

characteristic

C₁₃×#733

C₃²

yes

complemented

C₃×#6913

—

maximal, Sylow

C₃³×C₁₃

C₃³

yes

characteristic

C₃²×C₁₃⋊C₃

—

C₃²×C₁₃⋊C₃

—

C₃²×C₁₃⋊C₃

—

C₃²×C₁₃⋊C₃

—

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

—

C₃²⋊(C₁₃⋊C₃)

—

C₃²⋊(C₁₃⋊C₃)

C₃³

yes

—

#66851

—

#66851

C₃³

yes

—

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

—

#66849

—

#66848

—

#66848

—

C₉×C₁₃⋊C₃

—

C₃×#737

—

C₃²×3₋¹⁺²

—

#6913

—

C₃×#733

—

C₃×#733

C₁₃⋊C₃

yes

characteristic, complemented

C₃×#732

—

C₃×#731

—

C₃×AΣL(1,𝔽₂₇)

—

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

—

C₉×C₁₃

—

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

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₁₃⋊C₃

yes

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₁₃⋊C₃

yes

characteristic

#733

—

#733

C₃×C₁₃⋊C₃

yes

complemented

#732

—

#731

—

AΣL(1,𝔽₂₇)

—

100

C₃×C₁₃

—

101

C₃×C₁₃

—

102

C₃×C₁₃

#6913

yes

complemented

103

C₃×C₁₃

C₃²×H(3,𝔽₃)

yes

characteristic

104

C₁₃⋊C₃

—

105

C₁₃⋊C₃

—

106

C₃³

—

107

C₃³

—

108

C₃³

—

109

C₃³

—

110

C₃³

—

117

—

111

C₃³

C₃²×C₁₃⋊C₃

yes

characteristic

112

3₋¹⁺²

—

113

3₋¹⁺²

—

114

3₋¹⁺²

—

115

3₋¹⁺²

—

116

3₋¹⁺²

—

117

3₋¹⁺²

—

118

3₋¹⁺²

C₃²×C₁₃⋊C₃

yes

—

119

H(3,𝔽₃)

—

120

H(3,𝔽₃)

—

121

C₃×C₉

—

122

C₃×C₉

—

123

C₃×C₉

—

124

C₃×C₉

—

125

C₃×C₉

—

126

C₃×C₉

—

127

C₃×C₉

—

128

C₃×C₉

—

129

C₃×C₉

—

130

C₃×C₉

—

131

C₃×C₉

—

132

C₃×C₉

C₃²×C₁₃⋊C₃

yes

—

133

C₁₃

C₃×#6913

yes

characteristic, complemented, Sylow

134

C₃²

—

135

C₃²

—

136

C₃²

—

137

C₃²

—

117

—

138

C₃²

—

117

—

139

C₃²

—

117

—

140

C₃²

—

117

—

141

C₃²

H(3,𝔽₃)×C₁₃⋊C₃

yes

—

142

C₃²

H(3,𝔽₃)×C₁₃⋊C₃

yes

characteristic

143

C₃²

C₃³×C₁₃⋊C₃

yes

characteristic

144

C₉

—

145

C₉

—

146

C₉

—

147

C₉

—

148

C₉

—

149

C₃

—

150

C₃

—

151

C₃

—

117

—

152

C₃

—

117

—

153

C₃

#15691680

yes

direct product factor

154

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)
hypercenter	C₃×#733	C₁₃⋊C₃
hypocenter	C₁₃	C₃×#6913
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)

C₁₃⋊C₃

C₁₃

G ▷ C₃²×C₁₃ ▷ C₃×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ C₃×#733

N	H ≅ G/N	type
1. C₃×#11760454	C₃	semidirect
2. C₃²×#66848	C₃	semidirect
3. #15691680	C₃	direct (N × H)
4. C₃×#11760443	C₃	semidirect
5. C₃×#11760441	C₃	semidirect
6. C₃×#11760439	C₃	semidirect
7. C₃×#11760436	C₃	semidirect
8. C₃×C₁₃×#733	C₃	semidirect
11. #11760454	C₃²	semidirect
16. C₃×#66848	C₃²	semidirect
22. #11760443	C₃²	semidirect
23. #11760441	C₃²	semidirect
24. #11760439	C₃²	semidirect
25. #11760436	C₃²	semidirect
28. C₁₃×#733	C₃²	semidirect
61. C₃×#733	C₁₃⋊C₃	semidirect
96. #733	C₃×C₁₃⋊C₃	semidirect
102. C₃×C₁₃	#6913	semidirect
133. C₁₃	C₃×#6913	semidirect
153. C₃	#15691680	direct (N × H)

H ≅ G/N

type

1. C₃×#11760454

C₃

semidirect

2. C₃²×#66848

C₃

semidirect

3. #15691680

C₃

direct (N × H)

4. C₃×#11760443

C₃

semidirect

5. C₃×#11760441

C₃

semidirect

6. C₃×#11760439

C₃

semidirect

7. C₃×#11760436

C₃

semidirect

8. C₃×C₁₃×#733

C₃

semidirect

11. #11760454

C₃²

semidirect

16. C₃×#66848

C₃²

semidirect

22. #11760443

C₃²

semidirect

23. #11760441

C₃²

semidirect

24. #11760439

C₃²

semidirect

25. #11760436

C₃²

semidirect

28. C₁₃×#733

C₃²

semidirect

61. C₃×#733

C₁₃⋊C₃

semidirect

96. #733

C₃×C₁₃⋊C₃

semidirect

102. C₃×C₁₃

#6913

semidirect

133. C₁₃

C₃×#6913

semidirect

153. C₃

#15691680

direct (N × H)

Subgroups — C₃×#15691680 #23748542

Conjugacy classes of subgroups

Complements

Subgroups — C3×#15691680 #23748542

Conjugacy classes of subgroups

Complements

Subgroups — C₃×#15691680 #23748542