Group #23748550 — subgroups

Subgroups — #23748550

Order 9477.

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

normal subgroups only

This group has 960 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	C₃×#11760454	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×#11760452	C₃	1	3	yes	maximal, complemented
3	#15691676	C₃	1	9	yes	maximal, complemented
4	#15691675	C₃	1	9	yes	maximal, complemented
5	#15691670	C₃	1	2	yes	maximal, complemented
6	#15691660	C₃	1	3	yes	maximal, complemented
7	#15691655	C₃	1	3	yes	maximal, complemented
8	#15691649	C₃	1	9	yes	maximal, complemented
9	C₁₃×#6911	C₃	1	1	yes	maximal, characteristic, complemented
10	#11760454	—	3	3	no	—
11	#11760454	—	3	6	no	—
12	#11760452	—	3	6	no	—
13	#11760452	—	3	6	no	—
14	#11760452	—	3	6	no	—
15	#11760452	C₃²	1	9	yes	complemented
16	#11760452	C₃²	1	9	yes	complemented
17	#11760452	C₃²	1	9	yes	complemented
18	C₃×#66851	—	3	6	no	—
19	C₃×#66851	C₃²	1	6	yes	—
20	C₃×#66851	C₃²	1	1	yes	characteristic
21	C₃×#66848	C₃²	1	3	yes	complemented
22	C₃×#66849	C₃²	1	3	yes	complemented
23	C₃×C₃²⋊(C₁₃⋊C₃)	C₃²	1	1	yes	characteristic
24	C₃²×C₁₃⋊C₉	—	3	3	no	—
25	C₃²×C₁₃⋊C₉	C₃²	1	1	yes	characteristic
26	C₃×C₉×C₁₃⋊C₃	C₃²	1	3	yes	complemented
27	#11760443	C₃²	1	18	yes	complemented
28	#11760437	C₃²	1	27	yes	complemented
29	#11760434	C₃²	1	27	yes	complemented
30	C₃×C₁₃×3₋¹⁺²	C₃²	1	1	yes	characteristic
31	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	3	yes	—
32	C₁₃×#732	C₃²	1	9	yes	—
33	#10624929	—	13	1	no	maximal, Sylow
34	C₃³×C₁₃	—	3	3	no	—
35	C₃³×C₁₃	C₃³	1	1	yes	characteristic
36	C₃²×C₁₃⋊C₃	C₃³	1	3	yes	—
37	C₁₃×3₋¹⁺²	—	3	3	no	—
38	C₁₃×H(3,𝔽₃)	—	3	6	no	—
39	C₁₃×H(3,𝔽₃)	C₃³	1	9	yes	—
40	C₃×C₉×C₁₃	C₃³	1	3	yes	—
41	C₃²⋊(C₁₃⋊C₃)	—	3	3	no	—
42	#66851	—	3	12	no	—
43	#66851	—	3	3	no	—
44	#66851	—	3	18	no	—
45	#66851	—	3	18	no	—
46	#66851	—	3	18	no	—
47	#66851	C₃³	1	18	yes	—
48	C₃×C₁₃⋊C₉	—	3	6	no	—
49	C₃×C₁₃⋊C₉	—	3	9	no	—
50	C₃×C₁₃⋊C₉	—	3	9	no	—
51	C₃×C₁₃⋊C₉	—	3	9	no	—
52	C₃×C₁₃⋊C₉	H(3,𝔽₃)	1	3	yes	—
53	C₃×C₁₃⋊C₉	H(3,𝔽₃)	1	6	yes	—
54	C₃×C₁₃⋊C₉	C₃³	1	3	yes	—
55	C₃×C₁₃⋊C₉	C₃³	1	1	yes	characteristic
56	C₃×C₁₃⋊C₉	C₃³	1	1	yes	characteristic
57	C₃×C₁₃⋊C₉	C₃³	1	1	yes	characteristic
58	#66849	—	3	9	no	—
59	#66848	—	3	9	no	—
60	C₉×C₁₃⋊C₃	—	3	3	no	—
61	C₉×C₁₃⋊C₃	—	3	6	no	—
62	C₃×#737	—	13	1	no	—
63	C₃×#737	—	13	3	no	—
64	#6913	—	13	9	no	—
65	#6913	—	13	9	no	—
66	#6912	—	13	2	no	—
67	#6911	C₁₃⋊C₃	1	1	yes	characteristic, complemented
68	#6910	—	13	3	no	—
69	#6909	—	13	3	no	—
70	#6908	—	13	9	no	—
71	C₃²×C₁₃	—	3	9	no	—
72	C₃²×C₁₃	—	9	1	no	—
73	C₃²×C₁₃	—	9	9	no	—
74	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	3	yes	—
75	C₃²×C₁₃	C₃⁴	1	1	yes	characteristic
76	C₃×C₁₃⋊C₃	—	3	3	no	—
77	C₃×C₁₃⋊C₃	—	9	3	no	—
78	C₉×C₁₃	—	3	3	no	—
79	C₁₃⋊C₉	—	3	3	no	—
80	C₁₃⋊C₉	—	3	18	no	—
81	C₁₃⋊C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
82	C₁₃⋊C₉	C₃×H(3,𝔽₃)	1	2	yes	—
83	C₁₃⋊C₉	C₃×H(3,𝔽₃)	1	3	yes	—
84	C₁₃⋊C₉	C₃×H(3,𝔽₃)	1	3	yes	—
85	#737	—	13	9	no	—
86	#737	—	13	9	no	—
87	#737	—	13	9	no	—
88	#737	—	39	6	no	—
89	#737	—	39	6	no	—
90	#737	—	39	6	no	—
91	#737	—	39	3	no	—
92	#737	—	39	6	no	—
93	C₃×3₋¹⁺²	—	13	1	no	—
94	C₃×3₋¹⁺²	—	13	3	no	—
95	C₃×3₋¹⁺²	—	13	3	no	—
96	C₃×3₋¹⁺²	—	13	6	no	—
97	C₃×3₋¹⁺²	—	39	6	no	—
98	C₃×3₋¹⁺²	C₃×C₁₃⋊C₃	1	1	yes	characteristic
99	C₃×H(3,𝔽₃)	—	13	1	no	—
100	C₃×H(3,𝔽₃)	C₃×C₁₃⋊C₃	1	3	yes	—
101	C₃²×C₉	—	13	1	no	—
102	C₃²×C₉	—	13	3	no	—
103	C₃²×C₉	—	39	3	no	—
104	#733	—	13	18	no	—
105	#732	C₃×C₁₃⋊C₃	1	9	yes	—
106	#731	—	13	27	no	—
107	AΣL(1,𝔽₂₇)	—	13	27	no	—
108	C₃×C₁₃	—	3	1	no	—
109	C₃×C₁₃	—	3	3	no	—
110	C₃×C₁₃	—	9	9	no	—
111	C₃×C₁₃	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
112	C₁₃⋊C₃	—	9	3	no	—
113	C₃³	—	3	3	no	—
114	C₃³	—	13	3	no	—
115	C₃³	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
116	3₋¹⁺²	—	3	3	no	—
117	3₋¹⁺²	—	13	18	no	—
118	3₋¹⁺²	—	39	9	no	—
119	3₋¹⁺²	—	39	9	no	—
120	3₋¹⁺²	—	39	18	no	—
121	3₋¹⁺²	—	39	18	no	—
122	3₋¹⁺²	—	39	18	no	—
123	3₋¹⁺²	—	39	3	no	—
124	3₋¹⁺²	—	39	12	no	—
125	H(3,𝔽₃)	—	3	6	no	—
126	H(3,𝔽₃)	—	39	3	no	—
127	H(3,𝔽₃)	C₃²×C₁₃⋊C₃	1	9	yes	—
128	C₃×C₉	—	13	3	no	—
129	C₃×C₉	—	13	6	no	—
130	C₃×C₉	—	13	1	no	—
131	C₃×C₉	—	13	1	no	—
132	C₃×C₉	—	13	1	no	—
133	C₃×C₉	—	13	3	no	—
134	C₃×C₉	—	39	3	no	—
135	C₃×C₉	—	39	6	no	—
136	C₃×C₉	—	39	9	no	—
137	C₃×C₉	—	39	9	no	—
138	C₃×C₉	—	39	9	no	—
139	C₃×C₉	—	39	6	no	—
140	C₃×C₉	C₃²×C₁₃⋊C₃	1	3	yes	—
141	C₁₃	#10624929	1	1	yes	characteristic, complemented, Sylow
142	C₃²	—	3	9	no	—
143	C₃²	—	9	1	no	—
144	C₃²	—	9	9	no	—
145	C₃²	—	39	3	no	—
146	C₃²	—	117	3	no	—
147	C₃²	H(3,𝔽₃)×C₁₃⋊C₃	1	3	yes	—
148	C₃²	C₃³×C₁₃⋊C₃	1	1	yes	characteristic
149	C₉	—	3	3	no	—
150	C₉	—	13	1	no	—
151	C₉	—	13	3	no	—
152	C₉	—	13	3	no	—
153	C₉	—	13	2	no	—
154	C₉	—	39	3	no	—
155	C₉	—	39	18	no	—
156	C₃	—	3	3	no	—
157	C₃	—	3	1	no	—
158	C₃	—	9	9	no	—
159	C₃	—	117	3	no	—
160	C₃	C₃×H(3,𝔽₃)×C₁₃⋊C₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#11760454

C₃

yes

maximal, characteristic, complemented

C₃×#11760452

C₃

yes

maximal, complemented

#15691676

C₃

yes

maximal, complemented

#15691675

C₃

yes

maximal, complemented

#15691670

C₃

yes

maximal, complemented

#15691660

C₃

yes

maximal, complemented

#15691655

C₃

yes

maximal, complemented

#15691649

C₃

yes

maximal, complemented

C₁₃×#6911

C₃

yes

maximal, characteristic, complemented

—

—

—

—

—

C₃²

yes

complemented

#11760452

C₃²

yes

complemented

#11760452

C₃²

yes

complemented

C₃×#66851

—

C₃×#66851

C₃²

yes

—

C₃×#66851

C₃²

yes

characteristic

C₃×#66848

C₃²

yes

complemented

C₃×#66849

C₃²

yes

complemented

C₃×C₃²⋊(C₁₃⋊C₃)

C₃²

yes

characteristic

C₃²×C₁₃⋊C₉

—

C₃²×C₁₃⋊C₉

C₃²

yes

characteristic

C₃×C₉×C₁₃⋊C₃

C₃²

yes

complemented

#11760443

C₃²

yes

complemented

#11760437

C₃²

yes

complemented

#11760434

C₃²

yes

complemented

C₃×C₁₃×3₋¹⁺²

C₃²

yes

characteristic

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

—

C₁₃×#732

C₃²

yes

—

#10624929

—

maximal, Sylow

C₃³×C₁₃

—

C₃³×C₁₃

C₃³

yes

characteristic

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)

—

—

—

—

—

—

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

characteristic

C₃×C₁₃⋊C₉

C₃³

yes

characteristic

C₃×C₁₃⋊C₉

C₃³

yes

characteristic

#66849

—

#66848

—

C₉×C₁₃⋊C₃

—

C₉×C₁₃⋊C₃

—

—

—

—

—

—

C₁₃⋊C₃

yes

characteristic, complemented

#6910

—

#6909

—

#6908

—

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

—

C₁₃⋊C₉

—

C₁₃⋊C₉

—

C₁₃⋊C₉

C₃×H(3,𝔽₃)

yes

characteristic

C₁₃⋊C₉

C₃×H(3,𝔽₃)

yes

—

C₁₃⋊C₉

C₃×H(3,𝔽₃)

yes

—

C₁₃⋊C₉

C₃×H(3,𝔽₃)

yes

—

—

—

—

—

—

—

—

—

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

characteristic

C₃×H(3,𝔽₃)

—

100

C₃×H(3,𝔽₃)

C₃×C₁₃⋊C₃

yes

—

101

C₃²×C₉

—

102

C₃²×C₉

—

103

C₃²×C₉

—

104

#733

—

105

#732

C₃×C₁₃⋊C₃

yes

—

106

#731

—

107

AΣL(1,𝔽₂₇)

—

108

C₃×C₁₃

—

109

C₃×C₁₃

—

110

C₃×C₁₃

—

111

C₃×C₁₃

C₃²×H(3,𝔽₃)

yes

characteristic

112

C₁₃⋊C₃

—

113

C₃³

—

114

C₃³

—

115

C₃³

C₃²×C₁₃⋊C₃

yes

characteristic

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

3₋¹⁺²

—

124

3₋¹⁺²

—

125

H(3,𝔽₃)

—

126

H(3,𝔽₃)

—

127

H(3,𝔽₃)

C₃²×C₁₃⋊C₃

yes

—

128

C₃×C₉

—

129

C₃×C₉

—

130

C₃×C₉

—

131

C₃×C₉

—

132

C₃×C₉

—

133

C₃×C₉

—

134

C₃×C₉

—

135

C₃×C₉

—

136

C₃×C₉

—

137

C₃×C₉

—

138

C₃×C₉

—

139

C₃×C₉

—

140

C₃×C₉

C₃²×C₁₃⋊C₃

yes

—

141

C₁₃

#10624929

yes

characteristic, complemented, Sylow

142

C₃²

—

143

C₃²

—

144

C₃²

—

145

C₃²

—

146

C₃²

—

117

—

147

C₃²

H(3,𝔽₃)×C₁₃⋊C₃

yes

—

148

C₃²

C₃³×C₁₃⋊C₃

yes

characteristic

149

C₉

—

150

C₉

—

151

C₉

—

152

C₉

—

153

C₉

—

154

C₉

—

155

C₉

—

156

C₃

—

157

C₃

—

158

C₃

—

159

C₃

—

117

—

160

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	#10624929
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. C₃×#11760454	C₃	semidirect
2. C₃×#11760452	C₃	semidirect
3. #15691676	C₃	semidirect
4. #15691675	C₃	semidirect
5. #15691670	C₃	semidirect
6. #15691660	C₃	semidirect
7. #15691655	C₃	semidirect
8. #15691649	C₃	semidirect
9. C₁₃×#6911	C₃	semidirect
15. #11760452	C₃²	semidirect
16. #11760452	C₃²	semidirect
17. #11760452	C₃²	semidirect
21. C₃×#66848	C₃²	semidirect
22. C₃×#66849	C₃²	semidirect
26. C₃×C₉×C₁₃⋊C₃	C₃²	semidirect
27. #11760443	C₃²	semidirect
28. #11760437	C₃²	semidirect
29. #11760434	C₃²	semidirect
67. #6911	C₁₃⋊C₃	semidirect
141. C₁₃	#10624929	semidirect

H ≅ G/N

type

1. C₃×#11760454

C₃

semidirect

2. C₃×#11760452

C₃

semidirect

3. #15691676

C₃

semidirect

4. #15691675

C₃

semidirect

5. #15691670

C₃

semidirect

6. #15691660

C₃

semidirect

7. #15691655

C₃

semidirect

8. #15691649

C₃

semidirect

9. C₁₃×#6911

C₃

semidirect

15. #11760452

C₃²

semidirect

16. #11760452

C₃²

semidirect

17. #11760452

C₃²

semidirect

21. C₃×#66848

C₃²

semidirect

22. C₃×#66849

C₃²

semidirect

26. C₃×C₉×C₁₃⋊C₃

C₃²

semidirect

27. #11760443

C₃²

semidirect

28. #11760437

C₃²

semidirect

29. #11760434

C₃²

semidirect

67. #6911

C₁₃⋊C₃

semidirect

141. C₁₃

#10624929

semidirect

Subgroups — #23748550

Conjugacy classes of subgroups

Complements