Group #23748552 — subgroups

Subgroups — #23748552

Order 9477.

960 conjugacy classes of subgroups; only the 286 normal subgroups are shown.

normal subgroups only

This group has 960 conjugacy classes of subgroups — too many to draw. Only its 286 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×#11760453	C₃	1	3	yes	maximal, complemented
2	#15691679	C₃	1	18	yes	maximal, complemented
3	#15691670	C₃	1	2	yes	maximal, complemented
4	#15691666	C₃	1	1	yes	maximal, characteristic, complemented
5	#15691661	C₃	1	3	yes	maximal, complemented
6	#15691656	C₃	1	3	yes	maximal, complemented
7	#15691651	C₃	1	9	yes	maximal, complemented
8	C₃×C₁₃×#737	C₃	1	1	yes	maximal, characteristic, complemented
9	#11760453	—	3	6	no	—
10	#11760453	—	3	6	no	—
11	#11760453	—	3	6	no	—
12	#11760453	C₃²	1	9	yes	complemented
13	#11760453	C₃²	1	18	yes	complemented
14	C₃×#66851	C₃²	1	6	yes	—
15	C₃×#66848	—	3	3	no	—
16	C₃×#66849	—	3	3	no	—
17	C₃×C₃²⋊(C₁₃⋊C₃)	C₃²	1	3	yes	—
18	C₃×C₉×C₁₃⋊C₃	—	3	3	no	—
19	#11760443	C₃²	1	18	yes	complemented
20	#11760441	C₃²	1	9	yes	—
21	#11760438	C₃²	1	27	yes	complemented
22	#11760435	C₃²	1	27	yes	complemented
23	C₁₃×#737	—	3	3	no	—
24	C₁₃×#737	—	3	3	no	—
25	C₁₃×#737	—	3	3	no	—
26	C₃×C₁₃×3₋¹⁺²	C₃²	1	1	yes	characteristic
27	C₃×C₁₃×3₋¹⁺²	C₃²	1	1	yes	characteristic
28	C₃×C₁₃×3₋¹⁺²	C₃²	1	6	yes	complemented
29	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
30	C₃²×C₉×C₁₃	C₃²	1	1	yes	characteristic
31	C₃²×C₉×C₁₃	C₃²	1	3	yes	complemented
32	#10624929	—	13	1	no	maximal, Sylow
33	C₃³×C₁₃	C₃³	1	3	yes	—
34	C₃³×C₁₃	C₃³	1	1	yes	characteristic
35	C₃²×C₁₃⋊C₃	—	3	3	no	—
36	C₁₃×3₋¹⁺²	—	3	3	no	—
37	C₁₃×3₋¹⁺²	—	3	3	no	—
38	C₁₃×3₋¹⁺²	—	3	18	no	—
39	C₁₃×H(3,𝔽₃)	—	3	3	no	—
40	C₃×C₉×C₁₃	—	3	3	no	—
41	C₃×C₉×C₁₃	—	3	3	no	—
42	C₃×C₉×C₁₃	—	3	3	no	—
43	C₃×C₉×C₁₃	H(3,𝔽₃)	1	3	yes	—
44	C₃×C₉×C₁₃	H(3,𝔽₃)	1	3	yes	—
45	C₃×C₉×C₁₃	H(3,𝔽₃)	1	3	yes	—
46	C₃×C₉×C₁₃	C₃³	1	3	yes	—
47	C₃×C₉×C₁₃	C₃³	1	3	yes	—
48	C₃×C₉×C₁₃	C₃³	1	1	yes	characteristic
49	C₃×C₉×C₁₃	C₃³	1	2	yes	—
50	C₃²⋊(C₁₃⋊C₃)	—	3	6	no	—
51	C₃²⋊(C₁₃⋊C₃)	C₃³	1	9	yes	—
52	#66851	—	3	12	no	—
53	#66851	C₃³	1	18	yes	—
54	C₃×C₁₃⋊C₉	—	3	6	no	—
55	#66849	—	3	18	no	—
56	#66849	—	3	9	no	—
57	#66848	—	3	18	no	—
58	#66848	—	3	9	no	—
59	C₉×C₁₃⋊C₃	—	3	18	no	—
60	C₉×C₁₃⋊C₃	—	3	9	no	—
61	C₃×#737	—	13	3	no	—
62	C₃×#737	C₁₃⋊C₃	1	1	yes	characteristic, complemented
63	#6913	—	13	18	no	—
64	#6912	—	13	2	no	—
65	#6911	—	13	1	no	—
66	#6910	—	13	3	no	—
67	#6909	—	13	3	no	—
68	#6908	—	13	9	no	—
69	C₃²×C₁₃	—	3	3	no	—
70	C₃²×C₁₃	—	9	1	no	—
71	C₃²×C₁₃	—	9	3	no	—
72	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	3	yes	—
73	C₃²×C₁₃	C₃⁴	1	1	yes	characteristic
74	C₃×C₁₃⋊C₃	—	3	9	no	—
75	C₃×C₁₃⋊C₃	—	9	9	no	—
76	C₉×C₁₃	—	3	3	no	—
77	C₉×C₁₃	—	3	3	no	—
78	C₉×C₁₃	C₃×H(3,𝔽₃)	1	6	yes	—
79	C₉×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
80	C₉×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
81	C₉×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
82	C₁₃⋊C₉	—	3	18	no	—
83	#737	—	3	3	no	—
84	#737	—	3	3	no	—
85	#737	—	3	3	no	—
86	#737	—	13	18	no	—
87	#737	—	13	9	no	—
88	#737	—	39	6	no	—
89	#737	—	39	6	no	—
90	#737	—	39	6	no	—
91	C₃×3₋¹⁺²	—	13	6	no	—
92	C₃×3₋¹⁺²	—	39	3	no	—
93	C₃×3₋¹⁺²	—	39	3	no	—
94	C₃×3₋¹⁺²	C₃×C₁₃⋊C₃	1	1	yes	characteristic
95	C₃×3₋¹⁺²	C₃×C₁₃⋊C₃	1	1	yes	characteristic
96	C₃×3₋¹⁺²	C₃×C₁₃⋊C₃	1	6	yes	complemented
97	C₃×H(3,𝔽₃)	—	13	3	no	—
98	C₃×H(3,𝔽₃)	C₃×C₁₃⋊C₃	1	1	yes	characteristic
99	C₃²×C₉	—	39	3	no	—
100	C₃²×C₉	C₃×C₁₃⋊C₃	1	1	yes	characteristic
101	C₃²×C₉	C₃×C₁₃⋊C₃	1	3	yes	complemented
102	#733	—	13	18	no	—
103	#732	—	13	9	no	—
104	#731	—	13	27	no	—
105	AΣL(1,𝔽₂₇)	—	13	27	no	—
106	C₃×C₁₃	—	3	3	no	—
107	C₃×C₁₃	—	3	1	no	—
108	C₃×C₁₃	—	9	3	no	—
109	C₃×C₁₃	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
110	C₁₃⋊C₃	—	9	9	no	—
111	C₃³	—	39	3	no	—
112	C₃³	C₃²×C₁₃⋊C₃	1	3	yes	—
113	C₃³	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
114	3₋¹⁺²	—	3	3	no	—
115	3₋¹⁺²	—	3	3	no	—
116	3₋¹⁺²	—	3	18	no	—
117	3₋¹⁺²	—	13	18	no	—
118	3₋¹⁺²	—	39	12	no	—
119	3₋¹⁺²	—	39	9	no	—
120	3₋¹⁺²	—	39	18	no	—
121	3₋¹⁺²	—	39	9	no	—
122	3₋¹⁺²	—	39	18	no	—
123	H(3,𝔽₃)	—	3	3	no	—
124	H(3,𝔽₃)	—	13	9	no	—
125	H(3,𝔽₃)	—	39	6	no	—
126	C₃×C₉	—	3	3	no	—
127	C₃×C₉	—	3	3	no	—
128	C₃×C₉	—	3	3	no	—
129	C₃×C₉	—	39	6	no	—
130	C₃×C₉	—	39	18	no	—
131	C₃×C₉	—	39	9	no	—
132	C₃×C₉	C₃²⋊(C₁₃⋊C₃)	1	3	yes	—
133	C₃×C₉	C₃²⋊(C₁₃⋊C₃)	1	3	yes	—
134	C₃×C₉	C₃²⋊(C₁₃⋊C₃)	1	3	yes	—
135	C₃×C₉	C₃²×C₁₃⋊C₃	1	3	yes	—
136	C₃×C₉	C₃²×C₁₃⋊C₃	1	3	yes	—
137	C₃×C₉	C₃²×C₁₃⋊C₃	1	2	yes	—
138	C₃×C₉	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
139	C₁₃	#10624929	1	1	yes	characteristic, complemented, Sylow
140	C₃²	—	3	3	no	—
141	C₃²	—	9	1	no	—
142	C₃²	—	9	3	no	—
143	C₃²	—	39	9	no	—
144	C₃²	—	117	9	no	—
145	C₃²	C₃×C₃²⋊(C₁₃⋊C₃)	1	3	yes	—
146	C₃²	C₃³×C₁₃⋊C₃	1	1	yes	characteristic
147	C₉	—	3	3	no	—
148	C₉	—	3	3	no	—
149	C₉	—	39	18	no	—
150	C₉	C₃×C₃²⋊(C₁₃⋊C₃)	1	1	yes	characteristic
151	C₉	C₃×C₃²⋊(C₁₃⋊C₃)	1	1	yes	characteristic
152	C₉	C₃×C₃²⋊(C₁₃⋊C₃)	1	6	yes	—
153	C₉	C₃×C₃²⋊(C₁₃⋊C₃)	1	1	yes	characteristic
154	C₃	—	3	1	no	—
155	C₃	—	3	3	no	—
156	C₃	—	9	3	no	—
157	C₃	—	117	9	no	—
158	C₃	C₃²×C₃²⋊(C₁₃⋊C₃)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#11760453

C₃

yes

maximal, complemented

#15691679

C₃

yes

maximal, complemented

#15691670

C₃

yes

maximal, complemented

#15691666

C₃

yes

maximal, characteristic, complemented

#15691661

C₃

yes

maximal, complemented

#15691656

C₃

yes

maximal, complemented

#15691651

C₃

yes

maximal, complemented

C₃×C₁₃×#737

C₃

yes

maximal, characteristic, complemented

—

—

—

C₃²

yes

complemented

#11760453

C₃²

yes

complemented

C₃×#66851

C₃²

yes

—

C₃×#66848

—

C₃×#66849

—

C₃×C₃²⋊(C₁₃⋊C₃)

C₃²

yes

—

C₃×C₉×C₁₃⋊C₃

—

#11760443

C₃²

yes

complemented

#11760441

C₃²

yes

—

#11760438

C₃²

yes

complemented

#11760435

C₃²

yes

complemented

C₁₃×#737

—

C₁₃×#737

—

C₁₃×#737

—

C₃×C₁₃×3₋¹⁺²

C₃²

yes

characteristic

C₃×C₁₃×3₋¹⁺²

C₃²

yes

characteristic

C₃×C₁₃×3₋¹⁺²

C₃²

yes

complemented

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

characteristic

C₃²×C₉×C₁₃

C₃²

yes

characteristic

C₃²×C₉×C₁₃

C₃²

yes

complemented

#10624929

—

maximal, Sylow

C₃³×C₁₃

C₃³

yes

—

C₃³×C₁₃

C₃³

yes

characteristic

C₃²×C₁₃⋊C₃

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×H(3,𝔽₃)

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

H(3,𝔽₃)

yes

—

C₃×C₉×C₁₃

H(3,𝔽₃)

yes

—

C₃×C₉×C₁₃

H(3,𝔽₃)

yes

—

C₃×C₉×C₁₃

C₃³

yes

—

C₃×C₉×C₁₃

C₃³

yes

—

C₃×C₉×C₁₃

C₃³

yes

characteristic

C₃×C₉×C₁₃

C₃³

yes

—

C₃²⋊(C₁₃⋊C₃)

—

C₃²⋊(C₁₃⋊C₃)

C₃³

yes

—

#66851

—

#66851

C₃³

yes

—

C₃×C₁₃⋊C₉

—

—

—

—

—

C₉×C₁₃⋊C₃

—

C₉×C₁₃⋊C₃

—

C₃×#737

—

C₃×#737

C₁₃⋊C₃

yes

characteristic, complemented

—

—

—

—

—

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

—

C₃²×C₁₃

C₃⁴

yes

characteristic

C₃×C₁₃⋊C₃

—

C₃×C₁₃⋊C₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

C₃×H(3,𝔽₃)

yes

—

C₉×C₁₃

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₁₃

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₁₃

C₃×H(3,𝔽₃)

yes

characteristic

C₁₃⋊C₉

—

—

—

—

—

—

—

—

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₁₃⋊C₃

yes

characteristic

C₃×3₋¹⁺²

C₃×C₁₃⋊C₃

yes

characteristic

C₃×3₋¹⁺²

C₃×C₁₃⋊C₃

yes

complemented

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃×C₁₃⋊C₃

yes

characteristic

C₃²×C₉

—

100

C₃²×C₉

C₃×C₁₃⋊C₃

yes

characteristic

101

C₃²×C₉

C₃×C₁₃⋊C₃

yes

complemented

102

#733

—

103

#732

—

104

#731

—

105

AΣL(1,𝔽₂₇)

—

106

C₃×C₁₃

—

107

C₃×C₁₃

—

108

C₃×C₁₃

—

109

C₃×C₁₃

C₃²×H(3,𝔽₃)

yes

characteristic

110

C₁₃⋊C₃

—

111

C₃³

—

112

C₃³

C₃²×C₁₃⋊C₃

yes

—

113

C₃³

C₃²×C₁₃⋊C₃

yes

characteristic

114

3₋¹⁺²

—

115

3₋¹⁺²

—

116

3₋¹⁺²

—

117

3₋¹⁺²

—

118

3₋¹⁺²

—

119

3₋¹⁺²

—

120

3₋¹⁺²

—

121

3₋¹⁺²

—

122

3₋¹⁺²

—

123

H(3,𝔽₃)

—

124

H(3,𝔽₃)

—

125

H(3,𝔽₃)

—

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

C₃×C₉

C₃²⋊(C₁₃⋊C₃)

yes

—

134

C₃×C₉

C₃²⋊(C₁₃⋊C₃)

yes

—

135

C₃×C₉

C₃²×C₁₃⋊C₃

yes

—

136

C₃×C₉

C₃²×C₁₃⋊C₃

yes

—

137

C₃×C₉

C₃²×C₁₃⋊C₃

yes

—

138

C₃×C₉

C₃²×C₁₃⋊C₃

yes

characteristic

139

C₁₃

#10624929

yes

characteristic, complemented, Sylow

140

C₃²

—

141

C₃²

—

142

C₃²

—

143

C₃²

—

144

C₃²

—

117

—

145

C₃²

C₃×C₃²⋊(C₁₃⋊C₃)

yes

—

146

C₃²

C₃³×C₁₃⋊C₃

yes

characteristic

147

C₉

—

148

C₉

—

149

C₉

—

150

C₉

C₃×C₃²⋊(C₁₃⋊C₃)

yes

characteristic

151

C₉

C₃×C₃²⋊(C₁₃⋊C₃)

yes

characteristic

152

C₉

C₃×C₃²⋊(C₁₃⋊C₃)

yes

—

153

C₉

C₃×C₃²⋊(C₁₃⋊C₃)

yes

characteristic

154

C₃

—

155

C₃

—

156

C₃

—

157

C₃

—

117

—

158

C₃

C₃²×C₃²⋊(C₁₃⋊C₃)

yes

characteristic

	subgroup	G/H
center	C₉	C₃×C₃²⋊(C₁₃⋊C₃)
commutator, cosocle	C₃²×C₁₃	C₃⁴
Frattini	C₃²	C₃³×C₁₃⋊C₃
Fitting	C₃×C₁₃×#737	C₃
socle	C₃×C₁₃	C₃²×H(3,𝔽₃)
hypercenter	C₃×#737	C₁₃⋊C₃
hypocenter	C₁₃	#10624929
LCS	G ▷ C₃²×C₁₃ ▷ C₃×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₉ ◁ C₃²×C₉ ◁ C₃×#737

subgroup

G/H

center

C₉

C₃×C₃²⋊(C₁₃⋊C₃)

commutator, cosocle

C₃²×C₁₃

C₃⁴

Frattini

C₃²

C₃³×C₁₃⋊C₃

Fitting

C₃×C₁₃×#737

C₃

socle

C₃×C₁₃

C₃²×H(3,𝔽₃)

C₁₃⋊C₃

C₁₃

G ▷ C₃²×C₁₃ ▷ C₃×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₉ ◁ C₃²×C₉ ◁ C₃×#737

N	H ≅ G/N	type
1. C₃×#11760453	C₃	semidirect
2. #15691679	C₃	semidirect
3. #15691670	C₃	semidirect
4. #15691666	C₃	semidirect
5. #15691661	C₃	semidirect
6. #15691656	C₃	semidirect
7. #15691651	C₃	semidirect
8. C₃×C₁₃×#737	C₃	semidirect
12. #11760453	C₃²	semidirect
13. #11760453	C₃²	semidirect
19. #11760443	C₃²	semidirect
21. #11760438	C₃²	semidirect
22. #11760435	C₃²	semidirect
28. C₃×C₁₃×3₋¹⁺²	C₃²	semidirect
31. C₃²×C₉×C₁₃	C₃²	semidirect
62. C₃×#737	C₁₃⋊C₃	semidirect
96. C₃×3₋¹⁺²	C₃×C₁₃⋊C₃	semidirect
101. C₃²×C₉	C₃×C₁₃⋊C₃	semidirect
139. C₁₃	#10624929	semidirect

H ≅ G/N

type

1. C₃×#11760453

C₃

semidirect

2. #15691679

C₃

semidirect

3. #15691670

C₃

semidirect

4. #15691666

C₃

semidirect

5. #15691661

C₃

semidirect

6. #15691656

C₃

semidirect

7. #15691651

C₃

semidirect

8. C₃×C₁₃×#737

C₃

semidirect

12. #11760453

C₃²

semidirect

13. #11760453

C₃²

semidirect

19. #11760443

C₃²

semidirect

21. #11760438

C₃²

semidirect

22. #11760435

C₃²

semidirect

28. C₃×C₁₃×3₋¹⁺²

C₃²

semidirect

31. C₃²×C₉×C₁₃

C₃²

semidirect

62. C₃×#737

C₁₃⋊C₃

semidirect

96. C₃×3₋¹⁺²

C₃×C₁₃⋊C₃

semidirect

101. C₃²×C₉

C₃×C₁₃⋊C₃

semidirect

139. C₁₃

#10624929

semidirect

Subgroups — #23748552

Conjugacy classes of subgroups

Complements