Group #23748566 — subgroups

Subgroups — #23748566

Order 9477.

1062 conjugacy classes of subgroups; only the 268 normal subgroups are shown.

normal subgroups only

This group has 1062 conjugacy classes of subgroups — too many to draw. Only its 268 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	C₁₃⋊H(5,𝔽₃)	C₃	1	3	yes	maximal, complemented
2	C₃×#11760454	C₃	1	1	yes	maximal, characteristic, complemented
3	#15691666	C₃	1	2	yes	maximal, complemented
4	#15691665	C₃	1	3	yes	maximal, complemented
5	#15691664	C₃	1	9	yes	maximal, complemented
6	#15691663	C₃	1	9	yes	maximal, complemented
7	C₃×#11760440	C₃	1	3	yes	maximal, complemented
8	C₁₃⋊C₃×#732	C₃	1	9	yes	maximal, complemented
9	C₁₃×#6911	C₃	1	1	yes	maximal, characteristic, complemented
10	#11760454	—	3	9	no	—
11	C₃×#66851	C₃²	1	1	yes	characteristic, complemented
12	C₃×#66848	C₃²	1	3	yes	complemented
13	C₃×#66849	C₃²	1	3	yes	complemented
14	C₃×C₃²⋊(C₁₃⋊C₃)	—	3	9	no	—
15	C₃×C₃²⋊(C₁₃⋊C₃)	C₃²	1	1	yes	characteristic
16	C₃×C₃²⋊(C₁₃⋊C₃)	C₃²	1	6	yes	complemented
17	H(3,𝔽₃)×C₁₃⋊C₃	—	3	18	no	—
18	H(3,𝔽₃)×C₁₃⋊C₃	C₃²	1	27	yes	complemented
19	C₃²×C₁₃⋊C₉	C₃²	1	1	yes	characteristic, complemented
20	C₃×C₉×C₁₃⋊C₃	C₃²	1	3	yes	complemented
21	#11760441	C₃²	1	18	yes	complemented
22	#11760440	C₃²	1	9	yes	complemented
23	#11760440	C₃²	1	27	yes	complemented
24	#11760440	C₃²	1	9	yes	complemented
25	#11760440	C₃²	1	9	yes	complemented
26	C₃×C₁₃×3₋¹⁺²	C₃²	1	1	yes	characteristic, complemented
27	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	3	yes	complemented
28	C₁₃×#732	C₃²	1	9	yes	complemented
29	#10624932	—	13	1	no	maximal, Sylow
30	C₃³×C₁₃	—	3	3	no	—
31	C₃³×C₁₃	C₃³	1	1	yes	characteristic
32	C₃²×C₁₃⋊C₃	—	3	6	no	—
33	C₃²×C₁₃⋊C₃	—	3	27	no	—
34	C₃²×C₁₃⋊C₃	C₃³	1	3	yes	—
35	C₁₃×3₋¹⁺²	—	3	3	no	—
36	C₁₃×H(3,𝔽₃)	—	3	6	no	—
37	C₁₃×H(3,𝔽₃)	C₃³	1	9	yes	—
38	C₃×C₉×C₁₃	C₃³	1	3	yes	—
39	C₃²⋊(C₁₃⋊C₃)	—	3	6	no	—
40	C₃²⋊(C₁₃⋊C₃)	—	3	6	no	—
41	C₃²⋊(C₁₃⋊C₃)	—	3	54	no	—
42	C₃²⋊(C₁₃⋊C₃)	H(3,𝔽₃)	1	9	yes	—
43	C₃²⋊(C₁₃⋊C₃)	C₃³	1	18	yes	—
44	#66851	—	3	3	no	—
45	C₃×C₁₃⋊C₉	—	3	3	no	—
46	C₃×C₁₃⋊C₉	C₃³	1	1	yes	characteristic
47	C₃×C₁₃⋊C₉	C₃³	1	3	yes	—
48	C₃×C₁₃⋊C₉	C₃³	1	1	yes	characteristic
49	C₃×C₁₃⋊C₉	C₃³	1	1	yes	characteristic
50	#66849	—	3	9	no	—
51	#66848	—	3	9	no	—
52	C₉×C₁₃⋊C₃	—	3	9	no	—
53	H(5,𝔽₃)	—	13	3	no	—
54	C₃×#737	—	13	1	no	—
55	#6911	—	13	2	no	—
56	#6911	—	13	3	no	—
57	#6911	—	13	9	no	—
58	#6911	—	13	9	no	—
59	#6911	C₁₃⋊C₃	1	1	yes	characteristic, complemented
60	C₃×#732	—	13	3	no	—
61	C₃×#732	—	13	9	no	—
62	C₃²×C₁₃	—	3	9	no	—
63	C₃²×C₁₃	—	9	1	no	—
64	C₃²×C₁₃	—	27	3	no	—
65	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	3	yes	—
66	C₃²×C₁₃	C₃⁴	1	1	yes	characteristic
67	C₃×C₁₃⋊C₃	—	3	18	no	—
68	C₃×C₁₃⋊C₃	—	9	3	no	—
69	C₃×C₁₃⋊C₃	—	27	27	no	—
70	C₃×C₁₃⋊C₃	—	27	6	no	—
71	C₃×C₁₃⋊C₃	C₃×H(3,𝔽₃)	1	9	yes	—
72	C₉×C₁₃	—	3	3	no	—
73	C₁₃⋊C₉	—	3	1	no	—
74	C₁₃⋊C₉	—	3	3	no	—
75	C₁₃⋊C₉	—	3	1	no	—
76	C₁₃⋊C₉	—	3	1	no	—
77	#737	—	39	9	no	—
78	C₃×3₋¹⁺²	—	13	1	no	—
79	C₃×3₋¹⁺²	—	13	3	no	—
80	C₃×3₋¹⁺²	—	13	3	no	—
81	C₃×3₋¹⁺²	C₃×C₁₃⋊C₃	1	1	yes	characteristic, complemented
82	C₃×H(3,𝔽₃)	—	13	1	no	—
83	C₃×H(3,𝔽₃)	—	13	6	no	—
84	C₃×H(3,𝔽₃)	—	13	27	no	—
85	C₃×H(3,𝔽₃)	—	39	18	no	—
86	C₃×H(3,𝔽₃)	—	39	9	no	—
87	C₃×H(3,𝔽₃)	C₃×C₁₃⋊C₃	1	3	yes	complemented
88	C₃²×C₉	—	13	1	no	—
89	C₃²×C₉	—	13	3	no	—
90	#732	—	13	9	no	—
91	#732	—	13	27	no	—
92	#732	—	13	9	no	—
93	#732	—	13	9	no	—
94	#732	—	13	18	no	—
95	#732	C₃×C₁₃⋊C₃	1	9	yes	complemented
96	C₃×C₁₃	—	3	1	no	—
97	C₃×C₁₃	—	3	3	no	—
98	C₃×C₁₃	—	9	9	no	—
99	C₃×C₁₃	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
100	C₁₃⋊C₃	—	3	9	no	—
101	C₁₃⋊C₃	—	9	18	no	—
102	C₃³	—	3	3	no	—
103	C₃³	—	13	3	no	—
104	C₃³	—	39	27	no	—
105	C₃³	—	39	6	no	—
106	C₃³	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
107	3₋¹⁺²	—	3	3	no	—
108	3₋¹⁺²	—	39	9	no	—
109	3₋¹⁺²	—	39	9	no	—
110	3₋¹⁺²	—	39	3	no	—
111	H(3,𝔽₃)	—	3	6	no	—
112	H(3,𝔽₃)	—	13	9	no	—
113	H(3,𝔽₃)	—	13	18	no	—
114	H(3,𝔽₃)	—	39	54	no	—
115	H(3,𝔽₃)	—	39	6	no	—
116	H(3,𝔽₃)	—	39	6	no	—
117	H(3,𝔽₃)	C₃²×C₁₃⋊C₃	1	9	yes	—
118	C₃×C₉	—	13	1	no	—
119	C₃×C₉	—	13	3	no	—
120	C₃×C₉	—	13	1	no	—
121	C₃×C₉	—	13	1	no	—
122	C₃×C₉	—	39	9	no	—
123	C₃×C₉	—	39	3	no	—
124	C₃×C₉	C₃²×C₁₃⋊C₃	1	3	yes	—
125	C₁₃	#10624932	1	1	yes	characteristic, complemented, Sylow
126	C₃²	—	3	9	no	—
127	C₃²	—	9	1	no	—
128	C₃²	—	13	9	no	—
129	C₃²	—	27	3	no	—
130	C₃²	—	39	18	no	—
131	C₃²	—	351	6	no	—
132	C₃²	—	351	27	no	—
133	C₃²	—	117	3	no	—
134	C₃²	H(3,𝔽₃)×C₁₃⋊C₃	1	3	yes	—
135	C₃²	C₃³×C₁₃⋊C₃	1	1	yes	characteristic
136	C₉	—	3	3	no	—
137	C₉	—	39	1	no	—
138	C₉	—	39	3	no	—
139	C₉	—	39	1	no	—
140	C₉	—	39	1	no	—
141	C₃	—	3	3	no	—
142	C₃	—	3	1	no	—
143	C₃	—	9	9	no	—
144	C₃	—	39	9	no	—
145	C₃	—	117	18	no	—
146	C₃	C₃×H(3,𝔽₃)×C₁₃⋊C₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃⋊H(5,𝔽₃)

C₃

yes

maximal, complemented

C₃×#11760454

C₃

yes

maximal, characteristic, complemented

#15691666

C₃

yes

maximal, complemented

#15691665

C₃

yes

maximal, complemented

#15691664

C₃

yes

maximal, complemented

#15691663

C₃

yes

maximal, complemented

C₃×#11760440

C₃

yes

maximal, complemented

C₁₃⋊C₃×#732

C₃

yes

maximal, complemented

C₁₃×#6911

C₃

yes

maximal, characteristic, complemented

#11760454

—

C₃×#66851

C₃²

yes

characteristic, complemented

C₃×#66848

C₃²

yes

complemented

C₃×#66849

C₃²

yes

complemented

C₃×C₃²⋊(C₁₃⋊C₃)

—

C₃×C₃²⋊(C₁₃⋊C₃)

C₃²

yes

characteristic

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

characteristic, complemented

C₃×C₉×C₁₃⋊C₃

C₃²

yes

complemented

#11760441

C₃²

yes

complemented

#11760440

C₃²

yes

complemented

#11760440

C₃²

yes

complemented

#11760440

C₃²

yes

complemented

#11760440

C₃²

yes

complemented

C₃×C₁₃×3₋¹⁺²

C₃²

yes

characteristic, complemented

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

complemented

C₁₃×#732

C₃²

yes

complemented

#10624932

—

maximal, Sylow

C₃³×C₁₃

—

C₃³×C₁₃

C₃³

yes

characteristic

C₃²×C₁₃⋊C₃

—

C₃²×C₁₃⋊C₃

—

C₃²×C₁₃⋊C₃

C₃³

yes

—

C₁₃×3₋¹⁺²

—

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

C₃³

yes

—

C₃×C₉×C₁₃

C₃³

yes

—

C₃²⋊(C₁₃⋊C₃)

—

C₃²⋊(C₁₃⋊C₃)

—

C₃²⋊(C₁₃⋊C₃)

—

C₃²⋊(C₁₃⋊C₃)

H(3,𝔽₃)

yes

—

C₃²⋊(C₁₃⋊C₃)

C₃³

yes

—

#66851

—

C₃×C₁₃⋊C₉

—

C₃×C₁₃⋊C₉

C₃³

yes

characteristic

C₃×C₁₃⋊C₉

C₃³

yes

—

C₃×C₁₃⋊C₉

C₃³

yes

characteristic

C₃×C₁₃⋊C₉

C₃³

yes

characteristic

#66849

—

#66848

—

C₉×C₁₃⋊C₃

—

—

—

—

—

—

—

C₁₃⋊C₃

yes

characteristic, complemented

C₃×#732

—

C₃×#732

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

—

C₃²×C₁₃

C₃⁴

yes

characteristic

C₃×C₁₃⋊C₃

—

C₃×C₁₃⋊C₃

—

C₃×C₁₃⋊C₃

—

C₃×C₁₃⋊C₃

—

C₃×C₁₃⋊C₃

C₃×H(3,𝔽₃)

yes

—

C₉×C₁₃

—

C₁₃⋊C₉

—

C₁₃⋊C₉

—

C₁₃⋊C₉

—

C₁₃⋊C₉

—

#737

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₁₃⋊C₃

yes

characteristic, complemented

C₃×H(3,𝔽₃)

—

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

complemented

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

C₃²×H(3,𝔽₃)

yes

characteristic

100

C₁₃⋊C₃

—

101

C₁₃⋊C₃

—

102

C₃³

—

103

C₃³

—

104

C₃³

—

105

C₃³

—

106

C₃³

C₃²×C₁₃⋊C₃

yes

characteristic

107

3₋¹⁺²

—

108

3₋¹⁺²

—

109

3₋¹⁺²

—

110

3₋¹⁺²

—

111

H(3,𝔽₃)

—

112

H(3,𝔽₃)

—

113

H(3,𝔽₃)

—

114

H(3,𝔽₃)

—

115

H(3,𝔽₃)

—

116

H(3,𝔽₃)

—

117

H(3,𝔽₃)

C₃²×C₁₃⋊C₃

yes

—

118

C₃×C₉

—

119

C₃×C₉

—

120

C₃×C₉

—

121

C₃×C₉

—

122

C₃×C₉

—

123

C₃×C₉

—

124

C₃×C₉

C₃²×C₁₃⋊C₃

yes

—

125

C₁₃

#10624932

yes

characteristic, complemented, Sylow

126

C₃²

—

127

C₃²

—

128

C₃²

—

129

C₃²

—

130

C₃²

—

131

C₃²

—

351

—

132

C₃²

—

351

—

133

C₃²

—

117

—

134

C₃²

H(3,𝔽₃)×C₁₃⋊C₃

yes

—

135

C₃²

C₃³×C₁₃⋊C₃

yes

characteristic

136

C₉

—

137

C₉

—

138

C₉

—

139

C₉

—

140

C₉

—

141

C₃

—

142

C₃

—

143

C₃

—

144

C₃

—

145

C₃

—

117

—

146

C₃

C₃×H(3,𝔽₃)×C₁₃⋊C₃

yes

characteristic

	subgroup	G/H
center	C₃	C₃×H(3,𝔽₃)×C₁₃⋊C₃
commutator, cosocle	C₃²×C₁₃	C₃⁴
Frattini	C₃²	C₃³×C₁₃⋊C₃
Fitting	C₁₃×#6911	C₃
socle	C₃×C₁₃	C₃²×H(3,𝔽₃)
hypercenter	#6911	C₁₃⋊C₃
hypocenter	C₁₃	#10624932
LCS	G ▷ C₃²×C₁₃ ▷ C₃×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ #6911

subgroup

G/H

center

C₃

C₃×H(3,𝔽₃)×C₁₃⋊C₃

commutator, cosocle

C₃²×C₁₃

C₃⁴

Frattini

C₃²

C₃³×C₁₃⋊C₃

Fitting

C₁₃×#6911

C₃

socle

C₃×C₁₃

C₃²×H(3,𝔽₃)

C₁₃⋊C₃

C₁₃

G ▷ C₃²×C₁₃ ▷ C₃×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₃ ◁ C₃³ ◁ #6911

N	H ≅ G/N	type
1. C₁₃⋊H(5,𝔽₃)	C₃	semidirect
2. C₃×#11760454	C₃	semidirect
3. #15691666	C₃	semidirect
4. #15691665	C₃	semidirect
5. #15691664	C₃	semidirect
6. #15691663	C₃	semidirect
7. C₃×#11760440	C₃	semidirect
8. C₁₃⋊C₃×#732	C₃	semidirect
9. C₁₃×#6911	C₃	semidirect
11. C₃×#66851	C₃²	semidirect
12. C₃×#66848	C₃²	semidirect
13. C₃×#66849	C₃²	semidirect
16. C₃×C₃²⋊(C₁₃⋊C₃)	C₃²	semidirect
18. H(3,𝔽₃)×C₁₃⋊C₃	C₃²	semidirect
19. C₃²×C₁₃⋊C₉	C₃²	semidirect
20. C₃×C₉×C₁₃⋊C₃	C₃²	semidirect
21. #11760441	C₃²	semidirect
22. #11760440	C₃²	semidirect
23. #11760440	C₃²	semidirect
24. #11760440	C₃²	semidirect
25. #11760440	C₃²	semidirect
26. C₃×C₁₃×3₋¹⁺²	C₃²	semidirect
27. C₃×C₁₃×H(3,𝔽₃)	C₃²	semidirect
28. C₁₃×#732	C₃²	semidirect
59. #6911	C₁₃⋊C₃	semidirect
81. C₃×3₋¹⁺²	C₃×C₁₃⋊C₃	semidirect
87. C₃×H(3,𝔽₃)	C₃×C₁₃⋊C₃	semidirect
95. #732	C₃×C₁₃⋊C₃	semidirect
125. C₁₃	#10624932	semidirect

H ≅ G/N

type

1. C₁₃⋊H(5,𝔽₃)

C₃

semidirect

2. C₃×#11760454

C₃

semidirect

3. #15691666

C₃

semidirect

4. #15691665

C₃

semidirect

5. #15691664

C₃

semidirect

6. #15691663

C₃

semidirect

7. C₃×#11760440

C₃

semidirect

8. C₁₃⋊C₃×#732

C₃

semidirect

9. C₁₃×#6911

C₃

semidirect

11. C₃×#66851

C₃²

semidirect

12. C₃×#66848

C₃²

semidirect

13. C₃×#66849

C₃²

semidirect

16. C₃×C₃²⋊(C₁₃⋊C₃)

C₃²

semidirect

18. H(3,𝔽₃)×C₁₃⋊C₃

C₃²

semidirect

19. C₃²×C₁₃⋊C₉

C₃²

semidirect

20. C₃×C₉×C₁₃⋊C₃

C₃²

semidirect

21. #11760441

C₃²

semidirect

22. #11760440

C₃²

semidirect

23. #11760440

C₃²

semidirect

24. #11760440

C₃²

semidirect

25. #11760440

C₃²

semidirect

26. C₃×C₁₃×3₋¹⁺²

C₃²

semidirect

27. C₃×C₁₃×H(3,𝔽₃)

C₃²

semidirect

28. C₁₃×#732

C₃²

semidirect

59. #6911

C₁₃⋊C₃

semidirect

81. C₃×3₋¹⁺²

C₃×C₁₃⋊C₃

semidirect

87. C₃×H(3,𝔽₃)

C₃×C₁₃⋊C₃

semidirect

95. #732

C₃×C₁₃⋊C₃

semidirect

125. C₁₃

#10624932

semidirect

Subgroups — #23748566

Conjugacy classes of subgroups

Complements