Group #23744836 — subgroups

Subgroups — C₃×C₁₃×#6866 #23744836

Order 9477.

1988 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 1988 conjugacy classes of subgroups — too many to draw. Only its 188 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	C₃⁵×C₁₃	C₃	1	1	yes	maximal, characteristic
2	C₃×C₁₃×C₃²⋊C₉	C₃	1	3	yes	maximal, complemented
3	C₁₃×#6866	C₃	1	9	yes	maximal, direct product factor
4	C₃⁴×C₁₃	—	3	2	no	—
5	C₃⁴×C₁₃	—	3	1	no	—
6	C₃⁴×C₁₃	—	3	6	no	—
7	C₃⁴×C₁₃	—	3	27	no	—
8	C₃⁴×C₁₃	C₉	1	9	yes	complemented
9	C₃⁴×C₁₃	C₃²	1	3	yes	—
10	C₃⁴×C₁₃	C₃²	1	1	yes	characteristic
11	C₃²×C₉×C₁₃	—	3	3	no	—
12	C₁₃×C₃²⋊C₉	—	3	6	no	—
13	C₁₃×C₃²⋊C₉	C₃²	1	9	yes	complemented
14	C₃×#6866	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₃³×C₁₃	—	3	81	no	—
16	C₃³×C₁₃	—	3	27	no	—
17	C₃³×C₁₃	—	3	162	no	—
18	C₃³×C₁₃	—	3	54	no	—
19	C₃³×C₁₃	—	3	18	no	—
20	C₃³×C₁₃	—	3	9	no	—
21	C₃³×C₁₃	—	3	27	no	—
22	C₃³×C₁₃	—	3	9	no	—
23	C₃³×C₁₃	—	3	3	no	—
24	C₃³×C₁₃	—	3	6	no	—
25	C₃³×C₁₃	C₃×C₉	1	3	yes	—
26	C₃³×C₁₃	C₃×C₉	1	9	yes	complemented
27	C₃³×C₁₃	H(3,𝔽₃)	1	2	yes	—
28	C₃³×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
29	C₃³×C₁₃	3₋¹⁺²	1	6	yes	—
30	C₃³×C₁₃	C₃³	1	1	yes	characteristic
31	C₃×C₉×C₁₃	—	3	9	no	—
32	C₃×C₉×C₁₃	—	9	9	no	—
33	C₃⁵	C₃×C₁₃	1	1	yes	characteristic
34	C₃×C₃²⋊C₉	C₃×C₁₃	1	3	yes	complemented
35	#6866	C₃×C₁₃	1	9	yes	direct product factor
36	C₃²×C₁₃	—	3	27	no	—
37	C₃²×C₁₃	—	3	162	no	—
38	C₃²×C₁₃	—	3	54	no	—
39	C₃²×C₁₃	—	3	81	no	—
40	C₃²×C₁₃	—	3	9	no	—
41	C₃²×C₁₃	—	3	18	no	—
42	C₃²×C₁₃	—	3	9	no	—
43	C₃²×C₁₃	—	3	27	no	—
44	C₃²×C₁₃	—	3	6	no	—
45	C₃²×C₁₃	—	3	3	no	—
46	C₃²×C₁₃	C₃²⋊C₉	1	6	yes	—
47	C₃²×C₁₃	C₃²⋊C₉	1	3	yes	—
48	C₃²×C₁₃	AΣL(1,𝔽₂₇)	1	9	yes	—
49	C₃²×C₁₃	C₃²×C₉	1	1	yes	characteristic
50	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
51	C₃²×C₁₃	C₃×3₋¹⁺²	1	2	yes	—
52	C₉×C₁₃	—	9	9	no	—
53	C₃⁴	—	3	27	no	—
54	C₃⁴	—	3	2	no	—
55	C₃⁴	—	3	1	no	—
56	C₃⁴	—	3	6	no	—
57	C₃⁴	C₉×C₁₃	1	9	yes	complemented
58	C₃⁴	C₃²×C₁₃	1	3	yes	—
59	C₃⁴	C₃²×C₁₃	1	1	yes	characteristic
60	C₃²×C₉	—	3	3	no	—
61	C₃²⋊C₉	—	3	6	no	—
62	C₃²⋊C₉	C₃²×C₁₃	1	9	yes	complemented
63	C₃×C₁₃	—	3	27	no	—
64	C₃×C₁₃	—	3	6	no	—
65	C₃×C₁₃	—	3	1	no	—
66	C₃×C₁₃	—	3	2	no	—
67	C₃×C₁₃	#6866	1	9	yes	direct product factor
68	C₃×C₁₃	C₃×C₃²⋊C₉	1	1	yes	characteristic
69	C₃×C₁₃	C₃×AΣL(1,𝔽₂₇)	1	3	yes	—
70	C₃³	—	3	81	no	—
71	C₃³	—	3	27	no	—
72	C₃³	—	3	162	no	—
73	C₃³	—	3	54	no	—
74	C₃³	—	3	18	no	—
75	C₃³	—	3	9	no	—
76	C₃³	—	3	27	no	—
77	C₃³	—	3	9	no	—
78	C₃³	—	3	3	no	—
79	C₃³	—	3	6	no	—
80	C₃³	C₃×C₉×C₁₃	1	3	yes	—
81	C₃³	C₃×C₉×C₁₃	1	9	yes	complemented
82	C₃³	C₁₃×H(3,𝔽₃)	1	2	yes	—
83	C₃³	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
84	C₃³	C₁₃×3₋¹⁺²	1	6	yes	—
85	C₃³	C₃³×C₁₃	1	1	yes	characteristic
86	C₃×C₉	—	3	9	no	—
87	C₃×C₉	—	9	9	no	—
88	C₁₃	C₃×#6866	1	1	yes	characteristic, direct product factor, Sylow
89	C₃²	—	3	27	no	—
90	C₃²	—	3	162	no	—
91	C₃²	—	3	54	no	—
92	C₃²	—	3	9	no	—
93	C₃²	—	3	18	no	—
94	C₃²	—	3	81	no	—
95	C₃²	—	3	9	no	—
96	C₃²	—	3	6	no	—
97	C₃²	—	3	3	no	—
98	C₃²	—	3	27	no	—
99	C₃²	C₁₃×C₃²⋊C₉	1	6	yes	—
100	C₃²	C₁₃×C₃²⋊C₉	1	3	yes	—
101	C₃²	C₁₃×AΣL(1,𝔽₂₇)	1	9	yes	—
102	C₃²	C₃²×C₉×C₁₃	1	1	yes	characteristic
103	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
104	C₃²	C₃×C₁₃×3₋¹⁺²	1	2	yes	—
105	C₉	—	9	9	no	—
106	C₃	—	3	6	no	—
107	C₃	—	3	1	no	—
108	C₃	—	3	2	no	—
109	C₃	—	3	27	no	—
110	C₃	C₁₃×#6866	1	9	yes	direct product factor
111	C₃	C₃×C₁₃×C₃²⋊C₉	1	1	yes	characteristic
112	C₃	C₃×C₁₃×AΣL(1,𝔽₂₇)	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃⁵×C₁₃

C₃

yes

maximal, characteristic

C₃×C₁₃×C₃²⋊C₉

C₃

yes

maximal, complemented

C₁₃×#6866

C₃

yes

maximal, direct product factor

C₃⁴×C₁₃

—

C₃⁴×C₁₃

—

C₃⁴×C₁₃

—

C₃⁴×C₁₃

—

C₃⁴×C₁₃

C₉

yes

complemented

C₃⁴×C₁₃

C₃²

yes

—

C₃⁴×C₁₃

C₃²

yes

characteristic

C₃²×C₉×C₁₃

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

C₃²

yes

complemented

C₃×#6866

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

162

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

C₃×C₉

yes

—

C₃³×C₁₃

C₃×C₉

yes

complemented

C₃³×C₁₃

H(3,𝔽₃)

yes

—

C₃³×C₁₃

H(3,𝔽₃)

yes

characteristic

C₃³×C₁₃

3₋¹⁺²

yes

—

C₃³×C₁₃

C₃³

yes

characteristic

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃⁵

C₃×C₁₃

yes

characteristic

C₃×C₃²⋊C₉

C₃×C₁₃

yes

complemented

#6866

C₃×C₁₃

yes

direct product factor

C₃²×C₁₃

—

C₃²×C₁₃

—

162

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

C₃²⋊C₉

yes

—

C₃²×C₁₃

C₃²⋊C₉

yes

—

C₃²×C₁₃

AΣL(1,𝔽₂₇)

yes

—

C₃²×C₁₃

C₃²×C₉

yes

characteristic

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₁₃

C₃×3₋¹⁺²

yes

—

C₉×C₁₃

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

C₉×C₁₃

yes

complemented

C₃⁴

C₃²×C₁₃

yes

—

C₃⁴

C₃²×C₁₃

yes

characteristic

C₃²×C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃²×C₁₃

yes

complemented

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

#6866

yes

direct product factor

C₃×C₁₃

C₃×C₃²⋊C₉

yes

characteristic

C₃×C₁₃

C₃×AΣL(1,𝔽₂₇)

yes

—

C₃³

—

C₃³

—

C₃³

—

162

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×C₉×C₁₃

yes

—

C₃³

C₃×C₉×C₁₃

yes

complemented

C₃³

C₁₃×H(3,𝔽₃)

yes

—

C₃³

C₁₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₁₃×3₋¹⁺²

yes

—

C₃³

C₃³×C₁₃

yes

characteristic

C₃×C₉

—

C₃×C₉

—

C₁₃

C₃×#6866

yes

characteristic, direct product factor, Sylow

C₃²

—

C₃²

—

162

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₁₃×C₃²⋊C₉

yes

—

100

C₃²

C₁₃×C₃²⋊C₉

yes

—

101

C₃²

C₁₃×AΣL(1,𝔽₂₇)

yes

—

102

C₃²

C₃²×C₉×C₁₃

yes

characteristic

103

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

104

C₃²

C₃×C₁₃×3₋¹⁺²

yes

—

105

C₉

—

106

C₃

—

107

C₃

—

108

C₃

—

109

C₃

—

110

C₃

C₁₃×#6866

yes

direct product factor

111

C₃

C₃×C₁₃×C₃²⋊C₉

yes

characteristic

112

C₃

C₃×C₁₃×AΣL(1,𝔽₂₇)

yes

—

	subgroup	G/H
center, socle	C₃³×C₁₃	H(3,𝔽₃)
commutator	C₃²	C₃²×C₉×C₁₃
Frattini, cosocle	C₃³	C₃³×C₁₃
prime radical	C₃⁵×C₁₃	C₃
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃³×C₁₃ ◁ C₃⁴×C₁₃ ◁ G

subgroup

G/H

center, socle

C₃³×C₁₃

H(3,𝔽₃)

commutator

C₃²

C₃²×C₉×C₁₃

Frattini, cosocle

C₃³

C₃³×C₁₃

prime radical

C₃⁵×C₁₃

C₃

LCS

G ▷ C₃² ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃³×C₁₃ ◁ C₃⁴×C₁₃ ◁ G

N	H ≅ G/N	type
2. C₃×C₁₃×C₃²⋊C₉	C₃	semidirect
3. C₁₃×#6866	C₃	direct (N × H)
8. C₃⁴×C₁₃	C₉	semidirect
13. C₁₃×C₃²⋊C₉	C₃²	semidirect
14. C₃×#6866	C₁₃	direct (N × H)
26. C₃³×C₁₃	C₃×C₉	semidirect
34. C₃×C₃²⋊C₉	C₃×C₁₃	semidirect
35. #6866	C₃×C₁₃	direct (N × H)
57. C₃⁴	C₉×C₁₃	semidirect
62. C₃²⋊C₉	C₃²×C₁₃	semidirect
67. C₃×C₁₃	#6866	direct (N × H)
81. C₃³	C₃×C₉×C₁₃	semidirect
88. C₁₃	C₃×#6866	direct (N × H)
110. C₃	C₁₃×#6866	direct (N × H)

H ≅ G/N

type

2. C₃×C₁₃×C₃²⋊C₉

C₃

semidirect

3. C₁₃×#6866

C₃

direct (N × H)

8. C₃⁴×C₁₃

C₉

semidirect

13. C₁₃×C₃²⋊C₉

C₃²

semidirect

14. C₃×#6866

C₁₃

direct (N × H)

26. C₃³×C₁₃

C₃×C₉

semidirect

34. C₃×C₃²⋊C₉

C₃×C₁₃

semidirect

35. #6866

C₃×C₁₃

direct (N × H)

57. C₃⁴

C₉×C₁₃

semidirect

62. C₃²⋊C₉

C₃²×C₁₃

semidirect

67. C₃×C₁₃

#6866

direct (N × H)

81. C₃³

C₃×C₉×C₁₃

semidirect

88. C₁₃

C₃×#6866

direct (N × H)

110. C₃

C₁₃×#6866

direct (N × H)

Subgroups — C₃×C₁₃×#6866 #23744836

Conjugacy classes of subgroups

Complements

Subgroups — C3×C13×#6866 #23744836

Conjugacy classes of subgroups

Complements

Subgroups — C₃×C₁₃×#6866 #23744836