Group #23748622 — subgroups

Subgroups — #23748622

Order 9477.

5490 conjugacy classes of subgroups; only the 2878 normal subgroups are shown.

normal subgroups only

This group has 5490 conjugacy classes of subgroups — too many to draw. Only its 2878 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	2	yes	maximal, complemented
2	#15691697	C₃	1	24	yes	maximal, complemented
3	#15691696	C₃	1	27	yes	maximal, complemented
4	#15691695	C₃	1	27	yes	maximal, complemented
5	C₁₃⋊H(5,𝔽₃)	C₃	1	1	yes	maximal, characteristic
6	C₃×#11760453	C₃	1	12	yes	maximal, complemented
7	C₁₃⋊C₃×#737	C₃	1	27	yes	maximal, complemented
8	C₃×C₁₃×#737	C₃	1	1	yes	maximal, characteristic, complemented
9	#11760454	C₃²	1	432	yes	complemented
10	#11760453	C₃²	1	72	yes	complemented
11	#11760453	C₃²	1	9	yes	complemented
12	#11760453	C₃²	1	72	yes	complemented
13	#11760453	C₃²	1	72	yes	complemented
14	#11760453	C₃²	1	72	yes	complemented
15	#11760452	C₃²	1	54	yes	complemented
16	C₃×#66851	C₃²	1	24	yes	complemented
17	C₃×#66848	C₃²	1	36	yes	complemented
18	C₃×#66849	C₃²	1	36	yes	complemented
19	3₋¹⁺²×C₁₃⋊C₃	C₃²	1	216	yes	complemented
20	C₃×C₃²⋊(C₁₃⋊C₃)	C₃²	1	12	yes	—
21	H(3,𝔽₃)×C₁₃⋊C₃	C₃²	1	27	yes	—
22	C₃×C₉×C₁₃⋊C₃	C₃²	1	36	yes	complemented
23	C₁₃×#737	C₃²	1	9	yes	complemented
24	C₁₃×#737	C₃²	1	9	yes	complemented
25	C₁₃×#737	C₃²	1	9	yes	complemented
26	C₃×C₁₃×3₋¹⁺²	C₃²	1	8	yes	complemented
27	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
28	C₃²×C₉×C₁₃	C₃²	1	4	yes	complemented
29	#10624943	—	13	1	no	maximal, Sylow
30	C₃³×C₁₃	C₃³	1	4	yes	—
31	C₃²×C₁₃⋊C₃	C₃³	1	36	yes	—
32	C₁₃×3₋¹⁺²	C₃³	1	72	yes	—
33	C₁₃×H(3,𝔽₃)	C₃³	1	9	yes	—
34	C₃×C₉×C₁₃	C₃³	1	1	yes	characteristic
35	C₃×C₉×C₁₃	C₃³	1	8	yes	—
36	C₃×C₉×C₁₃	C₃³	1	12	yes	—
37	C₃×C₉×C₁₃	C₃³	1	12	yes	—
38	C₃×C₉×C₁₃	C₃³	1	12	yes	—
39	C₃²⋊(C₁₃⋊C₃)	C₃³	1	72	yes	—
40	C₃²⋊(C₁₃⋊C₃)	C₃³	1	9	yes	—
41	#66851	C₃³	1	144	yes	—
42	#66851	C₃³	1	18	yes	—
43	C₃×C₁₃⋊C₉	C₃³	1	72	yes	—
44	#66849	C₃³	1	216	yes	—
45	#66849	C₃³	1	27	yes	—
46	#66848	C₃³	1	216	yes	—
47	#66848	C₃³	1	27	yes	—
48	C₉×C₁₃⋊C₃	C₃³	1	27	yes	—
49	C₉×C₁₃⋊C₃	C₃³	1	216	yes	—
50	3₋¹⁺⁴	—	13	27	no	—
51	3₋¹⁺⁴	—	13	27	no	—
52	3₋¹⁺⁴	—	13	24	no	—
53	3₋¹⁺⁴	—	13	2	no	—
54	H(5,𝔽₃)	—	13	1	no	—
55	C₃×#737	—	13	27	no	—
56	C₃×#737	—	13	12	no	—
57	C₃×#737	C₁₃⋊C₃	1	1	yes	characteristic, complemented
58	C₃²×C₁₃	—	9	4	no	—
59	C₃²×C₁₃	C₃⁴	1	12	yes	—
60	C₃²×C₁₃	C₃⁴	1	1	yes	characteristic
61	C₃×C₁₃⋊C₃	—	9	36	no	—
62	C₃×C₁₃⋊C₃	C₃⁴	1	27	yes	—
63	C₉×C₁₃	C₃⁴	1	24	yes	—
64	C₉×C₁₃	C₃⁴	1	1	yes	characteristic
65	C₉×C₁₃	C₃⁴	1	1	yes	characteristic
66	C₉×C₁₃	C₃⁴	1	1	yes	characteristic
67	C₁₃⋊C₉	C₃⁴	1	54	yes	—
68	#737	—	13	432	no	—
69	#737	—	13	54	no	—
70	#737	—	13	72	no	—
71	#737	—	13	9	no	—
72	#737	—	13	72	no	—
73	#737	—	13	72	no	—
74	#737	—	13	72	no	—
75	#737	C₃×C₁₃⋊C₃	1	9	yes	complemented
76	#737	C₃×C₁₃⋊C₃	1	9	yes	complemented
77	#737	C₃×C₁₃⋊C₃	1	9	yes	complemented
78	C₃×3₋¹⁺²	—	13	24	no	—
79	C₃×3₋¹⁺²	—	13	216	no	—
80	C₃×3₋¹⁺²	—	13	36	no	—
81	C₃×3₋¹⁺²	—	13	36	no	—
82	C₃×3₋¹⁺²	C₃×C₁₃⋊C₃	1	8	yes	complemented
83	C₃×H(3,𝔽₃)	—	13	12	no	—
84	C₃×H(3,𝔽₃)	—	13	27	no	—
85	C₃×H(3,𝔽₃)	C₃×C₁₃⋊C₃	1	1	yes	characteristic
86	C₃²×C₉	—	13	36	no	—
87	C₃²×C₉	C₃×C₁₃⋊C₃	1	4	yes	complemented
88	C₃×C₁₃	—	3	12	no	—
89	C₃×C₁₃	—	3	1	no	—
90	C₃×C₁₃	C₃⁵	1	1	yes	characteristic
91	C₁₃⋊C₃	—	3	27	no	—
92	C₃³	—	13	36	no	—
93	C₃³	C₃²×C₁₃⋊C₃	1	4	yes	—
94	3₋¹⁺²	—	13	144	no	—
95	3₋¹⁺²	—	13	18	no	—
96	3₋¹⁺²	—	13	216	no	—
97	3₋¹⁺²	—	13	27	no	—
98	3₋¹⁺²	—	13	216	no	—
99	3₋¹⁺²	—	13	27	no	—
100	3₋¹⁺²	C₃²×C₁₃⋊C₃	1	72	yes	—
101	H(3,𝔽₃)	—	13	72	no	—
102	H(3,𝔽₃)	—	13	9	no	—
103	H(3,𝔽₃)	C₃²×C₁₃⋊C₃	1	9	yes	—
104	C₃×C₉	—	13	72	no	—
105	C₃×C₉	—	13	216	no	—
106	C₃×C₉	—	13	27	no	—
107	C₃×C₉	C₃²×C₁₃⋊C₃	1	12	yes	—
108	C₃×C₉	C₃²×C₁₃⋊C₃	1	12	yes	—
109	C₃×C₉	C₃²×C₁₃⋊C₃	1	8	yes	—
110	C₃×C₉	C₃²×C₁₃⋊C₃	1	12	yes	—
111	C₃×C₉	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
112	C₁₃	#10624943	1	1	yes	characteristic, complemented, Sylow
113	C₃²	—	9	4	no	—
114	C₃²	—	13	27	no	—
115	C₃²	—	117	36	no	—
116	C₃²	C₃³×C₁₃⋊C₃	1	12	yes	—
117	C₃²	C₃³×C₁₃⋊C₃	1	1	yes	characteristic
118	C₉	—	13	54	no	—
119	C₉	C₃³×C₁₃⋊C₃	1	24	yes	—
120	C₉	C₃³×C₁₃⋊C₃	1	1	yes	characteristic
121	C₉	C₃³×C₁₃⋊C₃	1	1	yes	characteristic
122	C₉	C₃³×C₁₃⋊C₃	1	1	yes	characteristic
123	C₃	—	3	1	no	—
124	C₃	—	3	12	no	—
125	C₃	—	39	27	no	—
126	C₃	C₃⁴×C₁₃⋊C₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15691698

C₃

yes

maximal, complemented

#15691697

C₃

yes

maximal, complemented

#15691696

C₃

yes

maximal, complemented

#15691695

C₃

yes

maximal, complemented

C₁₃⋊H(5,𝔽₃)

C₃

yes

maximal, characteristic

C₃×#11760453

C₃

yes

maximal, complemented

C₁₃⋊C₃×#737

C₃

yes

maximal, complemented

C₃×C₁₃×#737

C₃

yes

maximal, characteristic, complemented

#11760454

C₃²

432

yes

complemented

#11760453

C₃²

yes

complemented

#11760453

C₃²

yes

complemented

#11760453

C₃²

yes

complemented

#11760453

C₃²

yes

complemented

#11760453

C₃²

yes

complemented

#11760452

C₃²

yes

complemented

C₃×#66851

C₃²

yes

complemented

C₃×#66848

C₃²

yes

complemented

C₃×#66849

C₃²

yes

complemented

3₋¹⁺²×C₁₃⋊C₃

C₃²

216

yes

complemented

C₃×C₃²⋊(C₁₃⋊C₃)

C₃²

yes

—

H(3,𝔽₃)×C₁₃⋊C₃

C₃²

yes

—

C₃×C₉×C₁₃⋊C₃

C₃²

yes

complemented

C₁₃×#737

C₃²

yes

complemented

C₁₃×#737

C₃²

yes

complemented

C₁₃×#737

C₃²

yes

complemented

C₃×C₁₃×3₋¹⁺²

C₃²

yes

complemented

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

characteristic

C₃²×C₉×C₁₃

C₃²

yes

complemented

#10624943

—

maximal, Sylow

C₃³×C₁₃

C₃³

yes

—

C₃²×C₁₃⋊C₃

C₃³

yes

—

C₁₃×3₋¹⁺²

C₃³

yes

—

C₁₃×H(3,𝔽₃)

C₃³

yes

—

C₃×C₉×C₁₃

C₃³

yes

characteristic

C₃×C₉×C₁₃

C₃³

yes

—

C₃×C₉×C₁₃

C₃³

yes

—

C₃×C₉×C₁₃

C₃³

yes

—

C₃×C₉×C₁₃

C₃³

yes

—

C₃²⋊(C₁₃⋊C₃)

C₃³

yes

—

C₃²⋊(C₁₃⋊C₃)

C₃³

yes

—

#66851

C₃³

144

yes

—

#66851

C₃³

yes

—

C₃×C₁₃⋊C₉

C₃³

yes

—

#66849

C₃³

216

yes

—

#66849

C₃³

yes

—

#66848

C₃³

216

yes

—

#66848

C₃³

yes

—

C₉×C₁₃⋊C₃

C₃³

yes

—

C₉×C₁₃⋊C₃

C₃³

216

yes

—

3₋¹⁺⁴

—

3₋¹⁺⁴

—

3₋¹⁺⁴

—

3₋¹⁺⁴

—

—

—

—

C₁₃⋊C₃

yes

characteristic, complemented

C₃²×C₁₃

—

C₃²×C₁₃

C₃⁴

yes

—

C₃²×C₁₃

C₃⁴

yes

characteristic

C₃×C₁₃⋊C₃

—

C₃×C₁₃⋊C₃

C₃⁴

yes

—

C₉×C₁₃

C₃⁴

yes

—

C₉×C₁₃

C₃⁴

yes

characteristic

C₉×C₁₃

C₃⁴

yes

characteristic

C₉×C₁₃

C₃⁴

yes

characteristic

C₁₃⋊C₉

C₃⁴

yes

—

#737

—

432

—

—

—

—

—

—

—

C₃×C₁₃⋊C₃

yes

complemented

#737

C₃×C₁₃⋊C₃

yes

complemented

#737

C₃×C₁₃⋊C₃

yes

complemented

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

216

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₁₃⋊C₃

yes

complemented

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃×C₁₃⋊C₃

yes

characteristic

C₃²×C₉

—

C₃²×C₉

C₃×C₁₃⋊C₃

yes

complemented

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

C₃⁵

yes

characteristic

C₁₃⋊C₃

—

C₃³

—

C₃³

C₃²×C₁₃⋊C₃

yes

—

3₋¹⁺²

—

144

—

3₋¹⁺²

—

3₋¹⁺²

—

216

—

3₋¹⁺²

—

3₋¹⁺²

—

216

—

3₋¹⁺²

—

100

3₋¹⁺²

C₃²×C₁₃⋊C₃

yes

—

101

H(3,𝔽₃)

—

102

H(3,𝔽₃)

—

103

H(3,𝔽₃)

C₃²×C₁₃⋊C₃

yes

—

104

C₃×C₉

—

105

C₃×C₉

—

216

—

106

C₃×C₉

—

107

C₃×C₉

C₃²×C₁₃⋊C₃

yes

—

108

C₃×C₉

C₃²×C₁₃⋊C₃

yes

—

109

C₃×C₉

C₃²×C₁₃⋊C₃

yes

—

110

C₃×C₉

C₃²×C₁₃⋊C₃

yes

—

111

C₃×C₉

C₃²×C₁₃⋊C₃

yes

characteristic

112

C₁₃

#10624943

yes

characteristic, complemented, Sylow

113

C₃²

—

114

C₃²

—

115

C₃²

—

117

—

116

C₃²

C₃³×C₁₃⋊C₃

yes

—

117

C₃²

C₃³×C₁₃⋊C₃

yes

characteristic

118

C₉

—

119

C₉

C₃³×C₁₃⋊C₃

yes

—

120

C₉

C₃³×C₁₃⋊C₃

yes

characteristic

121

C₉

C₃³×C₁₃⋊C₃

yes

characteristic

122

C₉

C₃³×C₁₃⋊C₃

yes

characteristic

123

C₃

—

124

C₃

—

125

C₃

—

126

C₃

C₃⁴×C₁₃⋊C₃

yes

characteristic

	subgroup	G/H
center	C₉	C₃³×C₁₃⋊C₃
commutator, socle, cosocle	C₃×C₁₃	C₃⁵
Frattini	C₃	C₃⁴×C₁₃⋊C₃
Fitting	C₃×C₁₃×#737	C₃
prime radical	C₁₃⋊H(5,𝔽₃)	C₃
hypercenter	C₃×#737	C₁₃⋊C₃
hypocenter	C₁₃	#10624943
LCS	G ▷ C₃×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₉ ◁ C₃×#737

subgroup

G/H

center

C₉

C₃³×C₁₃⋊C₃

commutator, socle, cosocle

C₃×C₁₃

C₃⁵

Frattini

C₃

C₃⁴×C₁₃⋊C₃

Fitting

C₃×C₁₃×#737

C₃

prime radical

C₁₃⋊H(5,𝔽₃)

C₃

C₁₃⋊C₃

C₁₃

G ▷ C₃×C₁₃ ▷ C₁₃

C₁ ◁ C₉ ◁ C₃×#737

N	H ≅ G/N	type
1. #15691698	C₃	semidirect
2. #15691697	C₃	semidirect
3. #15691696	C₃	semidirect
4. #15691695	C₃	semidirect
6. C₃×#11760453	C₃	semidirect
7. C₁₃⋊C₃×#737	C₃	semidirect
8. C₃×C₁₃×#737	C₃	semidirect
9. #11760454	C₃²	semidirect
10. #11760453	C₃²	semidirect
11. #11760453	C₃²	semidirect
12. #11760453	C₃²	semidirect
13. #11760453	C₃²	semidirect
14. #11760453	C₃²	semidirect
15. #11760452	C₃²	semidirect
16. C₃×#66851	C₃²	semidirect
17. C₃×#66848	C₃²	semidirect
18. C₃×#66849	C₃²	semidirect
19. 3₋¹⁺²×C₁₃⋊C₃	C₃²	semidirect
22. C₃×C₉×C₁₃⋊C₃	C₃²	semidirect
23. C₁₃×#737	C₃²	semidirect
24. C₁₃×#737	C₃²	semidirect
25. C₁₃×#737	C₃²	semidirect
26. C₃×C₁₃×3₋¹⁺²	C₃²	semidirect
28. C₃²×C₉×C₁₃	C₃²	semidirect
57. C₃×#737	C₁₃⋊C₃	semidirect
75. #737	C₃×C₁₃⋊C₃	semidirect
76. #737	C₃×C₁₃⋊C₃	semidirect
77. #737	C₃×C₁₃⋊C₃	semidirect
82. C₃×3₋¹⁺²	C₃×C₁₃⋊C₃	semidirect
87. C₃²×C₉	C₃×C₁₃⋊C₃	semidirect
112. C₁₃	#10624943	semidirect

H ≅ G/N

type

1. #15691698

C₃

semidirect

2. #15691697

C₃

semidirect

3. #15691696

C₃

semidirect

4. #15691695

C₃

semidirect

6. C₃×#11760453

C₃

semidirect

7. C₁₃⋊C₃×#737

C₃

semidirect

8. C₃×C₁₃×#737

C₃

semidirect

9. #11760454

C₃²

semidirect

10. #11760453

C₃²

semidirect

11. #11760453

C₃²

semidirect

12. #11760453

C₃²

semidirect

13. #11760453

C₃²

semidirect

14. #11760453

C₃²

semidirect

15. #11760452

C₃²

semidirect

16. C₃×#66851

C₃²

semidirect

17. C₃×#66848

C₃²

semidirect

18. C₃×#66849

C₃²

semidirect

19. 3₋¹⁺²×C₁₃⋊C₃

C₃²

semidirect

22. C₃×C₉×C₁₃⋊C₃

C₃²

semidirect

23. C₁₃×#737

C₃²

semidirect

24. C₁₃×#737

C₃²

semidirect

25. C₁₃×#737

C₃²

semidirect

26. C₃×C₁₃×3₋¹⁺²

C₃²

semidirect

28. C₃²×C₉×C₁₃

C₃²

semidirect

57. C₃×#737

C₁₃⋊C₃

semidirect

75. #737

C₃×C₁₃⋊C₃

semidirect

76. #737

C₃×C₁₃⋊C₃

semidirect

77. #737

C₃×C₁₃⋊C₃

semidirect

82. C₃×3₋¹⁺²

C₃×C₁₃⋊C₃

semidirect

87. C₃²×C₉

C₃×C₁₃⋊C₃

semidirect

112. C₁₃

#10624943

semidirect

Subgroups — #23748622

Conjugacy classes of subgroups

Complements