Group #23748590 — subgroups

Subgroups — #23748590

Order 9477.

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

normal subgroups only

This group has 948 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	#15691697	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	#15691656	C₃	1	3	yes	maximal, complemented
6	#15691651	C₃	1	9	yes	maximal, complemented
7	C₃×#11760438	C₃	1	3	yes	maximal, complemented
8	C₃×C₁₃×#737	C₃	1	1	yes	maximal, characteristic, complemented
9	#11760454	—	3	18	no	—
10	#11760453	C₃²	1	27	yes	complemented
11	C₃×#66851	C₃²	1	6	yes	—
12	3₋¹⁺²×C₁₃⋊C₃	—	3	9	no	—
13	C₃×C₃²⋊(C₁₃⋊C₃)	C₃²	1	3	yes	—
14	#11760443	C₃²	1	18	yes	complemented
15	#11760441	C₃²	1	9	yes	—
16	#11760438	C₃²	1	9	yes	complemented
17	#11760438	C₃²	1	18	yes	complemented
18	#11760435	C₃²	1	27	yes	complemented
19	C₁₃×#737	—	3	9	no	—
20	C₃×C₁₃×3₋¹⁺²	C₃²	1	1	yes	characteristic
21	C₃×C₁₃×3₋¹⁺²	C₃²	1	1	yes	characteristic
22	C₃×C₁₃×3₋¹⁺²	C₃²	1	6	yes	complemented
23	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
24	C₃²×C₉×C₁₃	C₃²	1	1	yes	characteristic
25	C₃²×C₉×C₁₃	C₃²	1	3	yes	complemented
26	#10624935	—	13	1	no	maximal, Sylow
27	C₃³×C₁₃	C₃³	1	3	yes	—
28	C₃³×C₁₃	C₃³	1	1	yes	characteristic
29	C₃²×C₁₃⋊C₃	—	3	3	no	—
30	C₁₃×3₋¹⁺²	—	3	3	no	—
31	C₁₃×3₋¹⁺²	—	3	18	no	—
32	C₁₃×3₋¹⁺²	H(3,𝔽₃)	1	9	yes	—
33	C₁₃×H(3,𝔽₃)	—	3	3	no	—
34	C₃×C₉×C₁₃	—	3	3	no	—
35	C₃×C₉×C₁₃	—	3	9	no	—
36	C₃×C₉×C₁₃	C₃³	1	1	yes	characteristic
37	C₃×C₉×C₁₃	C₃³	1	2	yes	—
38	C₃×C₉×C₁₃	C₃³	1	3	yes	—
39	C₃×C₉×C₁₃	C₃³	1	3	yes	—
40	C₃²⋊(C₁₃⋊C₃)	—	3	6	no	—
41	C₃²⋊(C₁₃⋊C₃)	C₃³	1	9	yes	—
42	#66851	—	3	6	no	—
43	#66851	—	3	6	no	—
44	#66851	C₃³	1	18	yes	—
45	C₃×C₁₃⋊C₉	—	3	6	no	—
46	#66849	—	3	27	no	—
47	#66848	—	3	27	no	—
48	C₉×C₁₃⋊C₃	—	3	27	no	—
49	3₋¹⁺⁴	—	13	3	no	—
50	C₃×#737	C₁₃⋊C₃	1	1	yes	characteristic, complemented
51	#6913	—	13	18	no	—
52	#6912	—	13	2	no	—
53	#6911	—	13	1	no	—
54	#6909	—	13	3	no	—
55	#6908	—	13	9	no	—
56	C₃×#731	—	13	3	no	—
57	C₃²×C₁₃	—	3	3	no	—
58	C₃²×C₁₃	—	9	1	no	—
59	C₃²×C₁₃	—	9	3	no	—
60	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	3	yes	—
61	C₃²×C₁₃	C₃⁴	1	1	yes	characteristic
62	C₃×C₁₃⋊C₃	—	3	9	no	—
63	C₃×C₁₃⋊C₃	—	27	3	no	—
64	C₉×C₁₃	—	3	1	no	—
65	C₉×C₁₃	—	3	2	no	—
66	C₉×C₁₃	—	3	3	no	—
67	C₉×C₁₃	C₃×H(3,𝔽₃)	1	9	yes	—
68	C₁₃⋊C₉	—	3	18	no	—
69	#737	—	3	9	no	—
70	#737	—	13	27	no	—
71	#737	—	39	18	no	—
72	C₃×3₋¹⁺²	—	13	6	no	—
73	C₃×3₋¹⁺²	—	39	9	no	—
74	C₃×3₋¹⁺²	C₃×C₁₃⋊C₃	1	1	yes	characteristic
75	C₃×3₋¹⁺²	C₃×C₁₃⋊C₃	1	1	yes	characteristic
76	C₃×3₋¹⁺²	C₃×C₁₃⋊C₃	1	6	yes	complemented
77	C₃×H(3,𝔽₃)	—	13	3	no	—
78	C₃×H(3,𝔽₃)	C₃×C₁₃⋊C₃	1	1	yes	characteristic
79	C₃²×C₉	C₃×C₁₃⋊C₃	1	1	yes	characteristic
80	C₃²×C₉	C₃×C₁₃⋊C₃	1	3	yes	complemented
81	#733	—	13	18	no	—
82	#732	—	13	9	no	—
83	#731	—	13	18	no	—
84	#731	—	13	9	no	—
85	AΣL(1,𝔽₂₇)	—	13	27	no	—
86	C₃×C₁₃	—	3	3	no	—
87	C₃×C₁₃	—	3	1	no	—
88	C₃×C₁₃	—	9	3	no	—
89	C₃×C₁₃	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
90	C₁₃⋊C₃	—	9	9	no	—
91	C₃³	—	39	3	no	—
92	C₃³	C₃²×C₁₃⋊C₃	1	3	yes	—
93	C₃³	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
94	3₋¹⁺²	—	3	3	no	—
95	3₋¹⁺²	—	3	18	no	—
96	3₋¹⁺²	—	13	18	no	—
97	3₋¹⁺²	—	39	6	no	—
98	3₋¹⁺²	—	39	6	no	—
99	3₋¹⁺²	—	39	27	no	—
100	3₋¹⁺²	—	39	27	no	—
101	3₋¹⁺²	C₃²⋊(C₁₃⋊C₃)	1	9	yes	—
102	H(3,𝔽₃)	—	3	3	no	—
103	H(3,𝔽₃)	—	13	9	no	—
104	H(3,𝔽₃)	—	39	6	no	—
105	C₃×C₉	—	3	3	no	—
106	C₃×C₉	—	3	9	no	—
107	C₃×C₉	—	39	6	no	—
108	C₃×C₉	—	39	27	no	—
109	C₃×C₉	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
110	C₃×C₉	C₃²×C₁₃⋊C₃	1	2	yes	—
111	C₃×C₉	C₃²×C₁₃⋊C₃	1	3	yes	—
112	C₃×C₉	C₃²×C₁₃⋊C₃	1	3	yes	—
113	C₁₃	#10624935	1	1	yes	characteristic, complemented, Sylow
114	C₃²	—	3	3	no	—
115	C₃²	—	9	1	no	—
116	C₃²	—	9	3	no	—
117	C₃²	—	39	9	no	—
118	C₃²	—	351	3	no	—
119	C₃²	C₃×C₃²⋊(C₁₃⋊C₃)	1	3	yes	—
120	C₃²	C₃³×C₁₃⋊C₃	1	1	yes	characteristic
121	C₉	—	3	1	no	—
122	C₉	—	3	2	no	—
123	C₉	—	3	3	no	—
124	C₉	—	39	18	no	—
125	C₉	C₃×C₃²⋊(C₁₃⋊C₃)	1	9	yes	—
126	C₃	—	3	1	no	—
127	C₃	—	3	3	no	—
128	C₃	—	9	3	no	—
129	C₃	—	117	9	no	—
130	C₃	C₃²×C₃²⋊(C₁₃⋊C₃)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15691697

C₃

yes

maximal, complemented

#15691679

C₃

yes

maximal, complemented

#15691670

C₃

yes

maximal, complemented

#15691666

C₃

yes

maximal, characteristic, complemented

#15691656

C₃

yes

maximal, complemented

#15691651

C₃

yes

maximal, complemented

C₃×#11760438

C₃

yes

maximal, complemented

C₃×C₁₃×#737

C₃

yes

maximal, characteristic, complemented

#11760454

—

#11760453

C₃²

yes

complemented

C₃×#66851

C₃²

yes

—

3₋¹⁺²×C₁₃⋊C₃

—

C₃×C₃²⋊(C₁₃⋊C₃)

C₃²

yes

—

#11760443

C₃²

yes

complemented

#11760441

C₃²

yes

—

#11760438

C₃²

yes

complemented

#11760438

C₃²

yes

complemented

#11760435

C₃²

yes

complemented

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

#10624935

—

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

H(3,𝔽₃)

yes

—

C₁₃×H(3,𝔽₃)

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

C₃³

yes

characteristic

C₃×C₉×C₁₃

C₃³

yes

—

C₃×C₉×C₁₃

C₃³

yes

—

C₃×C₉×C₁₃

C₃³

yes

—

C₃²⋊(C₁₃⋊C₃)

—

C₃²⋊(C₁₃⋊C₃)

C₃³

yes

—

#66851

—

#66851

—

#66851

C₃³

yes

—

C₃×C₁₃⋊C₉

—

#66849

—

#66848

—

C₉×C₁₃⋊C₃

—

3₋¹⁺⁴

—

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

—

C₉×C₁₃

C₃×H(3,𝔽₃)

yes

—

C₁₃⋊C₉

—

#737

—

#737

—

#737

—

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

C₃×C₁₃⋊C₃

yes

characteristic

C₃²×C₉

C₃×C₁₃⋊C₃

yes

—

—

—

—

AΣL(1,𝔽₂₇)

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

C₃²×H(3,𝔽₃)

yes

characteristic

C₁₃⋊C₃

—

C₃³

—

C₃³

C₃²×C₁₃⋊C₃

yes

—

C₃³

C₃²×C₁₃⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

100

3₋¹⁺²

—

101

3₋¹⁺²

C₃²⋊(C₁₃⋊C₃)

yes

—

102

H(3,𝔽₃)

—

103

H(3,𝔽₃)

—

104

H(3,𝔽₃)

—

105

C₃×C₉

—

106

C₃×C₉

—

107

C₃×C₉

—

108

C₃×C₉

—

109

C₃×C₉

C₃²×C₁₃⋊C₃

yes

characteristic

110

C₃×C₉

C₃²×C₁₃⋊C₃

yes

—

111

C₃×C₉

C₃²×C₁₃⋊C₃

yes

—

112

C₃×C₉

C₃²×C₁₃⋊C₃

yes

—

113

C₁₃

#10624935

yes

characteristic, complemented, Sylow

114

C₃²

—

115

C₃²

—

116

C₃²

—

117

C₃²

—

118

C₃²

—

351

—

119

C₃²

C₃×C₃²⋊(C₁₃⋊C₃)

yes

—

120

C₃²

C₃³×C₁₃⋊C₃

yes

characteristic

121

C₉

—

122

C₉

—

123

C₉

—

124

C₉

—

125

C₉

C₃×C₃²⋊(C₁₃⋊C₃)

yes

—

126

C₃

—

127

C₃

—

128

C₃

—

129

C₃

—

117

—

130

C₃

C₃²×C₃²⋊(C₁₃⋊C₃)

yes

characteristic

	subgroup	G/H
center	C₃	C₃²×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	#10624935
LCS	G ▷ C₃²×C₁₃ ▷ C₃×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₃ ◁ C₃×3₋¹⁺² ◁ C₃×#737

subgroup

G/H

center

C₃

C₃²×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 ◁ C₃ ◁ C₃×3₋¹⁺² ◁ C₃×#737

N	H ≅ G/N	type
1. #15691697	C₃	semidirect
2. #15691679	C₃	semidirect
3. #15691670	C₃	semidirect
4. #15691666	C₃	semidirect
5. #15691656	C₃	semidirect
6. #15691651	C₃	semidirect
7. C₃×#11760438	C₃	semidirect
8. C₃×C₁₃×#737	C₃	semidirect
10. #11760453	C₃²	semidirect
14. #11760443	C₃²	semidirect
16. #11760438	C₃²	semidirect
17. #11760438	C₃²	semidirect
18. #11760435	C₃²	semidirect
22. C₃×C₁₃×3₋¹⁺²	C₃²	semidirect
25. C₃²×C₉×C₁₃	C₃²	semidirect
50. C₃×#737	C₁₃⋊C₃	semidirect
76. C₃×3₋¹⁺²	C₃×C₁₃⋊C₃	semidirect
80. C₃²×C₉	C₃×C₁₃⋊C₃	semidirect
113. C₁₃	#10624935	semidirect

H ≅ G/N

type

1. #15691697

C₃

semidirect

2. #15691679

C₃

semidirect

3. #15691670

C₃

semidirect

4. #15691666

C₃

semidirect

5. #15691656

C₃

semidirect

6. #15691651

C₃

semidirect

7. C₃×#11760438

C₃

semidirect

8. C₃×C₁₃×#737

C₃

semidirect

10. #11760453

C₃²

semidirect

14. #11760443

C₃²

semidirect

16. #11760438

C₃²

semidirect

17. #11760438

C₃²

semidirect

18. #11760435

C₃²

semidirect

22. C₃×C₁₃×3₋¹⁺²

C₃²

semidirect

25. C₃²×C₉×C₁₃

C₃²

semidirect

50. C₃×#737

C₁₃⋊C₃

semidirect

76. C₃×3₋¹⁺²

C₃×C₁₃⋊C₃

semidirect

80. C₃²×C₉

C₃×C₁₃⋊C₃

semidirect

113. C₁₃

#10624935

semidirect

Subgroups — #23748590

Conjugacy classes of subgroups

Complements