Group #23748576 — subgroups

Subgroups — #23748576

Order 9477.

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

normal subgroups only

This group has 980 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	#15691698	C₃	1	3	yes	maximal, complemented
2	C₃×3₋¹⁺²×C₁₃⋊C₃	C₃	1	1	yes	maximal, characteristic, 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	#15691649	C₃	1	9	yes	maximal, complemented
8	C₃×#11760434	C₃	1	3	yes	maximal, complemented
9	C₁₃×#6911	C₃	1	1	yes	maximal, characteristic, complemented
10	C₃³×C₁₃⋊C₃	C₃²	1	1	yes	characteristic
11	#11760452	—	3	18	no	—
12	#11760452	C₃²	1	27	yes	complemented
13	C₃×#66851	—	3	9	no	—
14	C₃×#66851	C₃²	1	6	yes	—
15	C₃×#66851	C₃²	1	1	yes	characteristic
16	C₃×#66851	C₃²	1	1	yes	characteristic
17	C₃×#66848	C₃²	1	3	yes	complemented
18	C₃×#66849	C₃²	1	3	yes	complemented
19	3₋¹⁺²×C₁₃⋊C₃	—	3	9	no	—
20	C₃×C₉×C₁₃⋊C₃	C₃²	1	3	yes	complemented
21	#11760443	C₃²	1	18	yes	complemented
22	#11760437	C₃²	1	27	yes	complemented
23	#11760434	C₃²	1	9	yes	complemented
24	#11760434	C₃²	1	9	yes	complemented
25	#11760434	C₃²	1	9	yes	complemented
26	C₃×C₁₃×3₋¹⁺²	C₃²	1	1	yes	characteristic
27	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	3	yes	—
28	C₁₃×#732	C₃²	1	9	yes	complemented
29	#10624934	—	13	1	no	maximal, Sylow
30	C₃³×C₁₃	—	3	3	no	—
31	C₃³×C₁₃	C₃³	1	1	yes	characteristic
32	C₃²×C₁₃⋊C₃	—	3	3	no	—
33	C₃²×C₁₃⋊C₃	—	9	3	no	—
34	C₃²×C₁₃⋊C₃	C₃³	1	1	yes	characteristic
35	C₃²×C₁₃⋊C₃	C₃³	1	1	yes	characteristic
36	C₃²×C₁₃⋊C₃	C₃³	1	1	yes	characteristic
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	#66851	—	3	3	no	—
42	#66851	—	3	6	no	—
43	#66851	—	3	6	no	—
44	#66851	—	3	54	no	—
45	#66851	H(3,𝔽₃)	1	9	yes	—
46	#66851	C₃³	1	18	yes	—
47	C₃×C₁₃⋊C₉	—	3	6	no	—
48	C₃×C₁₃⋊C₉	—	3	27	no	—
49	C₃×C₁₃⋊C₉	C₃³	1	3	yes	—
50	C₃×C₁₃⋊C₉	C₃³	1	3	yes	—
51	#66849	—	3	9	no	—
52	#66848	—	3	9	no	—
53	C₉×C₁₃⋊C₃	—	3	9	no	—
54	3₋¹⁺⁴	—	13	3	no	—
55	C₃²×3₋¹⁺²	—	13	1	no	—
56	#6913	—	13	9	no	—
57	#6913	—	13	9	no	—
58	#6912	—	13	2	no	—
59	#6911	C₁₃⋊C₃	1	1	yes	characteristic, complemented
60	#6910	—	13	3	no	—
61	#6908	—	13	9	no	—
62	C₃×AΣL(1,𝔽₂₇)	—	13	3	no	—
63	C₃²×C₁₃	—	3	9	no	—
64	C₃²×C₁₃	—	9	1	no	—
65	C₃²×C₁₃	—	27	3	no	—
66	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	3	yes	—
67	C₃²×C₁₃	C₃⁴	1	1	yes	characteristic
68	C₃×C₁₃⋊C₃	—	3	1	no	—
69	C₃×C₁₃⋊C₃	—	3	3	no	—
70	C₃×C₁₃⋊C₃	—	3	1	no	—
71	C₃×C₁₃⋊C₃	—	3	1	no	—
72	C₃×C₁₃⋊C₃	—	9	9	no	—
73	C₃×C₁₃⋊C₃	—	9	1	no	—
74	C₃×C₁₃⋊C₃	—	9	1	no	—
75	C₉×C₁₃	—	3	3	no	—
76	C₁₃⋊C₉	—	3	3	no	—
77	C₁₃⋊C₉	—	3	18	no	—
78	C₁₃⋊C₉	C₃×H(3,𝔽₃)	1	9	yes	—
79	C₃⁴	—	13	1	no	—
80	#737	—	13	27	no	—
81	#737	—	39	18	no	—
82	C₃×3₋¹⁺²	—	13	1	no	—
83	C₃×3₋¹⁺²	—	13	1	no	—
84	C₃×3₋¹⁺²	—	13	3	no	—
85	C₃×3₋¹⁺²	—	13	3	no	—
86	C₃×3₋¹⁺²	—	13	6	no	—
87	C₃×3₋¹⁺²	—	39	9	no	—
88	C₃×3₋¹⁺²	—	39	9	no	—
89	C₃×3₋¹⁺²	C₃×C₁₃⋊C₃	1	1	yes	characteristic
90	C₃×H(3,𝔽₃)	C₃×C₁₃⋊C₃	1	3	yes	—
91	C₃²×C₉	—	13	3	no	—
92	#733	—	13	18	no	—
93	#732	C₃×C₁₃⋊C₃	1	9	yes	complemented
94	#731	—	13	27	no	—
95	AΣL(1,𝔽₂₇)	—	13	9	no	—
96	AΣL(1,𝔽₂₇)	—	13	9	no	—
97	AΣL(1,𝔽₂₇)	—	13	9	no	—
98	C₃×C₁₃	—	3	1	no	—
99	C₃×C₁₃	—	3	3	no	—
100	C₃×C₁₃	—	9	9	no	—
101	C₃×C₁₃	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
102	C₁₃⋊C₃	—	3	3	no	—
103	C₁₃⋊C₃	—	9	1	no	—
104	C₁₃⋊C₃	—	9	1	no	—
105	C₃³	—	3	3	no	—
106	C₃³	—	13	1	no	—
107	C₃³	—	13	1	no	—
108	C₃³	—	13	1	no	—
109	C₃³	—	39	3	no	—
110	C₃³	—	117	3	no	—
111	C₃³	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
112	3₋¹⁺²	—	3	3	no	—
113	3₋¹⁺²	—	13	9	no	—
114	3₋¹⁺²	—	13	18	no	—
115	3₋¹⁺²	—	39	9	no	—
116	3₋¹⁺²	—	39	9	no	—
117	3₋¹⁺²	—	39	54	no	—
118	3₋¹⁺²	—	39	3	no	—
119	3₋¹⁺²	—	39	6	no	—
120	3₋¹⁺²	—	39	6	no	—
121	H(3,𝔽₃)	—	3	6	no	—
122	H(3,𝔽₃)	C₃²×C₁₃⋊C₃	1	9	yes	—
123	C₃×C₉	—	13	3	no	—
124	C₃×C₉	—	13	3	no	—
125	C₃×C₉	—	39	9	no	—
126	C₃×C₉	—	39	27	no	—
127	C₃×C₉	—	39	6	no	—
128	C₃×C₉	C₃²×C₁₃⋊C₃	1	3	yes	—
129	C₁₃	#10624934	1	1	yes	characteristic, complemented, Sylow
130	C₃²	—	3	9	no	—
131	C₃²	—	9	1	no	—
132	C₃²	—	27	3	no	—
133	C₃²	—	39	3	no	—
134	C₃²	—	39	1	no	—
135	C₃²	—	39	1	no	—
136	C₃²	—	39	1	no	—
137	C₃²	—	117	9	no	—
138	C₃²	—	117	1	no	—
139	C₃²	—	117	1	no	—
140	C₃²	H(3,𝔽₃)×C₁₃⋊C₃	1	3	yes	—
141	C₃²	C₃³×C₁₃⋊C₃	1	1	yes	characteristic
142	C₉	—	3	3	no	—
143	C₉	—	13	9	no	—
144	C₉	—	39	3	no	—
145	C₉	—	39	18	no	—
146	C₃	—	3	3	no	—
147	C₃	—	3	1	no	—
148	C₃	—	9	9	no	—
149	C₃	—	39	3	no	—
150	C₃	—	117	1	no	—
151	C₃	—	117	1	no	—
152	C₃	C₃×H(3,𝔽₃)×C₁₃⋊C₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15691698

C₃

yes

maximal, complemented

C₃×3₋¹⁺²×C₁₃⋊C₃

C₃

yes

maximal, characteristic, complemented

#15691676

C₃

yes

maximal, complemented

#15691675

C₃

yes

maximal, complemented

#15691670

C₃

yes

maximal, complemented

#15691660

C₃

yes

maximal, complemented

#15691649

C₃

yes

maximal, complemented

C₃×#11760434

C₃

yes

maximal, complemented

C₁₃×#6911

C₃

yes

maximal, characteristic, complemented

C₃³×C₁₃⋊C₃

C₃²

yes

characteristic

#11760452

—

#11760452

C₃²

yes

complemented

C₃×#66851

—

C₃×#66851

C₃²

yes

—

C₃×#66851

C₃²

yes

characteristic

C₃×#66851

C₃²

yes

characteristic

C₃×#66848

C₃²

yes

complemented

C₃×#66849

C₃²

yes

complemented

3₋¹⁺²×C₁₃⋊C₃

—

C₃×C₉×C₁₃⋊C₃

C₃²

yes

complemented

#11760443

C₃²

yes

complemented

#11760437

C₃²

yes

complemented

#11760434

C₃²

yes

complemented

#11760434

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

complemented

#10624934

—

maximal, Sylow

C₃³×C₁₃

—

C₃³×C₁₃

C₃³

yes

characteristic

C₃²×C₁₃⋊C₃

—

C₃²×C₁₃⋊C₃

—

C₃²×C₁₃⋊C₃

C₃³

yes

characteristic

C₃²×C₁₃⋊C₃

C₃³

yes

characteristic

C₃²×C₁₃⋊C₃

C₃³

yes

characteristic

C₁₃×3₋¹⁺²

—

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

C₃³

yes

—

C₃×C₉×C₁₃

C₃³

yes

—

—

—

—

—

yes

—

#66851

C₃³

yes

—

C₃×C₁₃⋊C₉

—

C₃×C₁₃⋊C₉

—

C₃×C₁₃⋊C₉

C₃³

yes

—

C₃×C₁₃⋊C₉

C₃³

yes

—

#66849

—

#66848

—

C₉×C₁₃⋊C₃

—

3₋¹⁺⁴

—

C₃²×3₋¹⁺²

—

—

—

—

C₁₃⋊C₃

yes

characteristic, complemented

#6910

—

#6908

—

C₃×AΣL(1,𝔽₂₇)

—

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×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

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₁₃⋊C₃

yes

characteristic

C₃×H(3,𝔽₃)

C₃×C₁₃⋊C₃

yes

—

C₃²×C₉

—

#733

—

#732

C₃×C₁₃⋊C₃

yes

complemented

#731

—

AΣL(1,𝔽₂₇)

—

AΣL(1,𝔽₂₇)

—

AΣL(1,𝔽₂₇)

—

C₃×C₁₃

—

C₃×C₁₃

—

100

C₃×C₁₃

—

101

C₃×C₁₃

C₃²×H(3,𝔽₃)

yes

characteristic

102

C₁₃⋊C₃

—

103

C₁₃⋊C₃

—

104

C₁₃⋊C₃

—

105

C₃³

—

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

—

119

3₋¹⁺²

—

120

3₋¹⁺²

—

121

H(3,𝔽₃)

—

122

H(3,𝔽₃)

C₃²×C₁₃⋊C₃

yes

—

123

C₃×C₉

—

124

C₃×C₉

—

125

C₃×C₉

—

126

C₃×C₉

—

127

C₃×C₉

—

128

C₃×C₉

C₃²×C₁₃⋊C₃

yes

—

129

C₁₃

#10624934

yes

characteristic, complemented, Sylow

130

C₃²

—

131

C₃²

—

132

C₃²

—

133

C₃²

—

134

C₃²

—

135

C₃²

—

136

C₃²

—

137

C₃²

—

117

—

138

C₃²

—

117

—

139

C₃²

—

117

—

140

C₃²

H(3,𝔽₃)×C₁₃⋊C₃

yes

—

141

C₃²

C₃³×C₁₃⋊C₃

yes

characteristic

142

C₉

—

143

C₉

—

144

C₉

—

145

C₉

—

146

C₃

—

147

C₃

—

148

C₃

—

149

C₃

—

150

C₃

—

117

—

151

C₃

—

117

—

152

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	#10624934
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. #15691698	C₃	semidirect
2. C₃×3₋¹⁺²×C₁₃⋊C₃	C₃	semidirect
3. #15691676	C₃	semidirect
4. #15691675	C₃	semidirect
5. #15691670	C₃	semidirect
6. #15691660	C₃	semidirect
7. #15691649	C₃	semidirect
8. C₃×#11760434	C₃	semidirect
9. C₁₃×#6911	C₃	semidirect
12. #11760452	C₃²	semidirect
17. C₃×#66848	C₃²	semidirect
18. C₃×#66849	C₃²	semidirect
20. C₃×C₉×C₁₃⋊C₃	C₃²	semidirect
21. #11760443	C₃²	semidirect
22. #11760437	C₃²	semidirect
23. #11760434	C₃²	semidirect
24. #11760434	C₃²	semidirect
25. #11760434	C₃²	semidirect
28. C₁₃×#732	C₃²	semidirect
59. #6911	C₁₃⋊C₃	semidirect
93. #732	C₃×C₁₃⋊C₃	semidirect
129. C₁₃	#10624934	semidirect

H ≅ G/N

type

1. #15691698

C₃

semidirect

2. C₃×3₋¹⁺²×C₁₃⋊C₃

C₃

semidirect

3. #15691676

C₃

semidirect

4. #15691675

C₃

semidirect

5. #15691670

C₃

semidirect

6. #15691660

C₃

semidirect

7. #15691649

C₃

semidirect

8. C₃×#11760434

C₃

semidirect

9. C₁₃×#6911

C₃

semidirect

12. #11760452

C₃²

semidirect

17. C₃×#66848

C₃²

semidirect

18. C₃×#66849

C₃²

semidirect

20. C₃×C₉×C₁₃⋊C₃

C₃²

semidirect

21. #11760443

C₃²

semidirect

22. #11760437

C₃²

semidirect

23. #11760434

C₃²

semidirect

24. #11760434

C₃²

semidirect

25. #11760434

C₃²

semidirect

28. C₁₃×#732

C₃²

semidirect

59. #6911

C₁₃⋊C₃

semidirect

93. #732

C₃×C₁₃⋊C₃

semidirect

129. C₁₃

#10624934

semidirect

Subgroups — #23748576

Conjugacy classes of subgroups

Complements