Group #23748570 — subgroups

Subgroups — #23748570

Order 9477.

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

normal subgroups only

This group has 936 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	#15691697	C₃	1	3	yes	maximal
2	C₃×#11760454	C₃	1	1	yes	maximal, characteristic
3	#15691670	C₃	1	2	yes	maximal, complemented
4	#15691669	C₃	1	3	yes	maximal, complemented
5	#15691668	C₃	1	9	yes	maximal, complemented
6	#15691667	C₃	1	9	yes	maximal, complemented
7	C₃×#11760442	C₃	1	3	yes	maximal, complemented
8	C₁₃⋊C₃×#733	C₃	1	9	yes	maximal, complemented
9	C₁₃×#6912	C₃	1	1	yes	maximal, characteristic, complemented
10	#11760454	—	3	9	no	—
11	#11760454	—	3	18	no	—
12	#11760453	—	3	9	no	—
13	C₃×#66851	C₃²	1	6	yes	—
14	C₃×#66851	C₃²	1	1	yes	characteristic
15	C₃×#66848	C₃²	1	3	yes	—
16	C₃×#66849	C₃²	1	3	yes	—
17	3₋¹⁺²×C₁₃⋊C₃	C₃²	1	27	yes	—
18	C₃×C₃²⋊(C₁₃⋊C₃)	C₃²	1	1	yes	characteristic
19	C₃²×C₁₃⋊C₉	C₃²	1	1	yes	characteristic
20	C₃×C₉×C₁₃⋊C₃	C₃²	1	3	yes	—
21	#11760443	C₃²	1	18	yes	—
22	#11760442	C₃²	1	9	yes	—
23	#11760442	C₃²	1	27	yes	—
24	#11760442	C₃²	1	9	yes	—
25	#11760442	C₃²	1	9	yes	—
26	C₃×C₁₃×3₋¹⁺²	C₃²	1	3	yes	—
27	C₃×C₁₃×3₋¹⁺²	C₃²	1	1	yes	characteristic
28	C₁₃×#733	C₃²	1	9	yes	—
29	#10624933	—	13	1	no	maximal, Sylow
30	C₃³×C₁₃	C₃³	1	1	yes	characteristic
31	C₃²×C₁₃⋊C₃	C₃³	1	3	yes	—
32	C₁₃×3₋¹⁺²	—	3	6	no	—
33	C₁₃×3₋¹⁺²	—	3	3	no	—
34	C₁₃×3₋¹⁺²	C₃³	1	9	yes	—
35	C₃×C₉×C₁₃	—	3	3	no	—
36	C₃×C₉×C₁₃	C₃³	1	3	yes	—
37	C₃²⋊(C₁₃⋊C₃)	H(3,𝔽₃)	1	9	yes	—
38	#66851	—	3	6	no	—
39	#66851	—	3	6	no	—
40	#66851	—	3	3	no	—
41	#66851	C₃³	1	18	yes	—
42	C₃×C₁₃⋊C₉	—	3	3	no	—
43	C₃×C₁₃⋊C₉	—	3	6	no	—
44	C₃×C₁₃⋊C₉	C₃³	1	1	yes	characteristic
45	C₃×C₁₃⋊C₉	C₃³	1	3	yes	—
46	C₃×C₁₃⋊C₉	C₃³	1	1	yes	characteristic
47	C₃×C₁₃⋊C₉	C₃³	1	1	yes	characteristic
48	#66849	—	3	27	no	—
49	#66849	—	3	9	no	—
50	#66848	—	3	27	no	—
51	#66848	—	3	9	no	—
52	C₉×C₁₃⋊C₃	—	3	9	no	—
53	C₉×C₁₃⋊C₃	—	3	27	no	—
54	3₋¹⁺⁴	—	13	3	no	—
55	C₃×#737	—	13	1	no	—
56	#6912	—	13	3	no	—
57	#6912	—	13	2	no	—
58	#6912	—	13	9	no	—
59	#6912	—	13	9	no	—
60	#6912	C₁₃⋊C₃	1	1	yes	characteristic, complemented
61	C₃×#733	—	13	3	no	—
62	C₃×#733	—	13	9	no	—
63	C₃²×C₁₃	—	9	1	no	—
64	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	3	yes	—
65	C₃²×C₁₃	C₃⁴	1	1	yes	characteristic
66	C₃×C₁₃⋊C₃	—	9	3	no	—
67	C₃×C₁₃⋊C₃	C₃×H(3,𝔽₃)	1	9	yes	—
68	C₉×C₁₃	—	3	9	no	—
69	C₉×C₁₃	—	3	3	no	—
70	C₁₃⋊C₉	—	3	1	no	—
71	C₁₃⋊C₉	—	3	18	no	—
72	C₁₃⋊C₉	—	3	3	no	—
73	C₁₃⋊C₉	—	3	1	no	—
74	C₁₃⋊C₉	—	3	1	no	—
75	#737	—	39	9	no	—
76	#737	—	39	9	no	—
77	#737	—	39	18	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	—	13	6	no	—
82	C₃×3₋¹⁺²	—	13	27	no	—
83	C₃×3₋¹⁺²	C₃×C₁₃⋊C₃	1	3	yes	—
84	C₃×3₋¹⁺²	C₃×C₁₃⋊C₃	1	1	yes	characteristic
85	C₃×H(3,𝔽₃)	—	13	1	no	—
86	C₃²×C₉	—	13	1	no	—
87	C₃²×C₉	—	13	3	no	—
88	#733	—	13	27	no	—
89	#733	—	13	9	no	—
90	#733	—	13	9	no	—
91	#733	—	13	9	no	—
92	#733	—	13	18	no	—
93	#733	C₃×C₁₃⋊C₃	1	9	yes	—
94	C₃×C₁₃	—	3	1	no	—
95	C₃×C₁₃	—	3	3	no	—
96	C₃×C₁₃	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
97	C₁₃⋊C₃	—	3	9	no	—
98	C₃³	—	13	3	no	—
99	C₃³	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
100	3₋¹⁺²	—	3	6	no	—
101	3₋¹⁺²	—	3	3	no	—
102	3₋¹⁺²	—	13	18	no	—
103	3₋¹⁺²	—	39	27	no	—
104	3₋¹⁺²	—	39	9	no	—
105	3₋¹⁺²	—	39	27	no	—
106	3₋¹⁺²	—	39	9	no	—
107	3₋¹⁺²	—	39	3	no	—
108	3₋¹⁺²	—	39	6	no	—
109	3₋¹⁺²	—	39	6	no	—
110	3₋¹⁺²	C₃²×C₁₃⋊C₃	1	9	yes	—
111	H(3,𝔽₃)	—	13	9	no	—
112	C₃×C₉	—	3	3	no	—
113	C₃×C₉	—	13	1	no	—
114	C₃×C₉	—	13	3	no	—
115	C₃×C₉	—	13	1	no	—
116	C₃×C₉	—	13	1	no	—
117	C₃×C₉	—	39	9	no	—
118	C₃×C₉	—	39	27	no	—
119	C₃×C₉	—	39	3	no	—
120	C₃×C₉	—	39	6	no	—
121	C₃×C₉	C₃²×C₁₃⋊C₃	1	3	yes	—
122	C₁₃	#10624933	1	1	yes	characteristic, complemented, Sylow
123	C₃²	—	9	1	no	—
124	C₃²	—	13	9	no	—
125	C₃²	—	117	3	no	—
126	C₃²	H(3,𝔽₃)×C₁₃⋊C₃	1	3	yes	—
127	C₃²	C₃³×C₁₃⋊C₃	1	1	yes	characteristic
128	C₉	—	3	9	no	—
129	C₉	—	3	3	no	—
130	C₉	—	39	1	no	—
131	C₉	—	39	3	no	—
132	C₉	—	39	1	no	—
133	C₉	—	39	1	no	—
134	C₉	—	39	18	no	—
135	C₃	—	3	1	no	—
136	C₃	—	3	3	no	—
137	C₃	—	39	9	no	—
138	C₃	C₃×H(3,𝔽₃)×C₁₃⋊C₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15691697

C₃

yes

maximal

C₃×#11760454

C₃

yes

maximal, characteristic

#15691670

C₃

yes

maximal, complemented

#15691669

C₃

yes

maximal, complemented

#15691668

C₃

yes

maximal, complemented

#15691667

C₃

yes

maximal, complemented

C₃×#11760442

C₃

yes

maximal, complemented

C₁₃⋊C₃×#733

C₃

yes

maximal, complemented

C₁₃×#6912

C₃

yes

maximal, characteristic, complemented

—

—

—

C₃²

yes

—

C₃×#66851

C₃²

yes

characteristic

C₃×#66848

C₃²

yes

—

C₃×#66849

C₃²

yes

—

3₋¹⁺²×C₁₃⋊C₃

C₃²

yes

—

C₃×C₃²⋊(C₁₃⋊C₃)

C₃²

yes

characteristic

C₃²×C₁₃⋊C₉

C₃²

yes

characteristic

C₃×C₉×C₁₃⋊C₃

C₃²

yes

—

#11760443

C₃²

yes

—

#11760442

C₃²

yes

—

#11760442

C₃²

yes

—

#11760442

C₃²

yes

—

#11760442

C₃²

yes

—

C₃×C₁₃×3₋¹⁺²

C₃²

yes

—

C₃×C₁₃×3₋¹⁺²

C₃²

yes

characteristic

C₁₃×#733

C₃²

yes

—

#10624933

—

maximal, Sylow

C₃³×C₁₃

C₃³

yes

characteristic

C₃²×C₁₃⋊C₃

C₃³

yes

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

C₃³

yes

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

C₃³

yes

—

C₃²⋊(C₁₃⋊C₃)

H(3,𝔽₃)

yes

—

—

—

—

C₃³

yes

—

C₃×C₁₃⋊C₉

—

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

—

—

—

—

C₉×C₁₃⋊C₃

—

C₉×C₁₃⋊C₃

—

3₋¹⁺⁴

—

—

—

—

—

—

C₁₃⋊C₃

yes

characteristic, complemented

C₃×#733

—

C₃×#733

—

C₃²×C₁₃

—

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

—

C₃²×C₁₃

C₃⁴

yes

characteristic

C₃×C₁₃⋊C₃

—

C₃×C₁₃⋊C₃

C₃×H(3,𝔽₃)

yes

—

C₉×C₁₃

—

C₉×C₁₃

—

C₁₃⋊C₉

—

C₁₃⋊C₉

—

C₁₃⋊C₉

—

C₁₃⋊C₉

—

C₁₃⋊C₉

—

#737

—

#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×C₁₃⋊C₃

yes

—

C₃×3₋¹⁺²

C₃×C₁₃⋊C₃

yes

characteristic

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

—

—

—

—

—

—

C₃×C₁₃⋊C₃

yes

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

C₃²×H(3,𝔽₃)

yes

characteristic

C₁₃⋊C₃

—

C₃³

—

C₃³

C₃²×C₁₃⋊C₃

yes

characteristic

100

3₋¹⁺²

—

101

3₋¹⁺²

—

102

3₋¹⁺²

—

103

3₋¹⁺²

—

104

3₋¹⁺²

—

105

3₋¹⁺²

—

106

3₋¹⁺²

—

107

3₋¹⁺²

—

108

3₋¹⁺²

—

109

3₋¹⁺²

—

110

3₋¹⁺²

C₃²×C₁₃⋊C₃

yes

—

111

H(3,𝔽₃)

—

112

C₃×C₉

—

113

C₃×C₉

—

114

C₃×C₉

—

115

C₃×C₉

—

116

C₃×C₉

—

117

C₃×C₉

—

118

C₃×C₉

—

119

C₃×C₉

—

120

C₃×C₉

—

121

C₃×C₉

C₃²×C₁₃⋊C₃

yes

—

122

C₁₃

#10624933

yes

characteristic, complemented, Sylow

123

C₃²

—

124

C₃²

—

125

C₃²

—

117

—

126

C₃²

H(3,𝔽₃)×C₁₃⋊C₃

yes

—

127

C₃²

C₃³×C₁₃⋊C₃

yes

characteristic

128

C₉

—

129

C₉

—

130

C₉

—

131

C₉

—

132

C₉

—

133

C₉

—

134

C₉

—

135

C₃

—

136

C₃

—

137

C₃

—

138

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×#6912	C₃
socle	C₃×C₁₃	C₃²×H(3,𝔽₃)
prime radical	C₃×C₃²⋊(C₁₃⋊C₃)	C₃²
hypercenter	#6912	C₁₃⋊C₃
hypocenter	C₁₃	#10624933
LCS	G ▷ C₃²×C₁₃ ▷ C₃×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ #6912

subgroup

G/H

center

C₃

C₃×H(3,𝔽₃)×C₁₃⋊C₃

commutator, cosocle

C₃²×C₁₃

C₃⁴

Frattini

C₃²

C₃³×C₁₃⋊C₃

Fitting

C₁₃×#6912

C₃

socle

C₃×C₁₃

C₃²×H(3,𝔽₃)

prime radical

C₃×C₃²⋊(C₁₃⋊C₃)

C₃²

C₁₃⋊C₃

C₁₃

G ▷ C₃²×C₁₃ ▷ C₃×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₃ ◁ C₃³ ◁ #6912

N	H ≅ G/N	type
3. #15691670	C₃	semidirect
4. #15691669	C₃	semidirect
5. #15691668	C₃	semidirect
6. #15691667	C₃	semidirect
7. C₃×#11760442	C₃	semidirect
8. C₁₃⋊C₃×#733	C₃	semidirect
9. C₁₃×#6912	C₃	semidirect
60. #6912	C₁₃⋊C₃	semidirect
122. C₁₃	#10624933	semidirect

H ≅ G/N

type

3. #15691670

C₃

semidirect

4. #15691669

C₃

semidirect

5. #15691668

C₃

semidirect

6. #15691667

C₃

semidirect

7. C₃×#11760442

C₃

semidirect

8. C₁₃⋊C₃×#733

C₃

semidirect

9. C₁₃×#6912

C₃

semidirect

60. #6912

C₁₃⋊C₃

semidirect

122. C₁₃

#10624933

semidirect

Subgroups — #23748570

Conjugacy classes of subgroups

Complements