Group #23744732 — subgroups

Subgroups — C₃×C₁₃×#6859 #23744732

Order 9477.

488 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 116 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	C₃²×C₁₃×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
2	C₃×C₁₃×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₁₃×C₃²⋊C₉	C₃	1	2	yes	maximal, complemented
4	C₁₃×#6859	C₃	1	9	yes	maximal, direct product factor
5	C₃⁴×C₁₃	—	3	1	no	—
6	C₃⁴×C₁₃	C₃²	1	1	yes	characteristic
7	C₃×C₁₃×H(3,𝔽₃)	—	3	2	no	—
8	C₃×C₁₃×H(3,𝔽₃)	—	3	9	no	—
9	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	3	yes	—
10	C₃²×C₉×C₁₃	—	3	1	no	—
11	C₃²×C₉×C₁₃	—	3	2	no	—
12	C₁₃×C₃²⋊C₉	—	3	2	no	—
13	C₁₃×C₃²⋊C₉	—	3	4	no	—
14	C₁₃×C₃²⋊C₉	C₃²	1	3	yes	complemented
15	C₁₃×C₃²⋊C₉	C₃²	1	6	yes	complemented
16	C₃×#6859	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₃³×C₁₃	—	3	3	no	—
18	C₃³×C₁₃	—	3	9	no	—
19	C₃³×C₁₃	—	3	6	no	—
20	C₃³×C₁₃	—	3	3	no	—
21	C₃³×C₁₃	—	3	3	no	—
22	C₃³×C₁₃	—	9	6	no	—
23	C₃³×C₁₃	—	9	3	no	—
24	C₃³×C₁₃	H(3,𝔽₃)	1	2	yes	—
25	C₃³×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
26	C₃³×C₁₃	C₃³	1	1	yes	characteristic
27	C₁₃×H(3,𝔽₃)	—	3	9	no	—
28	C₁₃×H(3,𝔽₃)	—	3	18	no	—
29	C₃×C₉×C₁₃	—	3	3	no	—
30	C₃×C₉×C₁₃	—	3	6	no	—
31	C₃×C₉×C₁₃	—	9	3	no	—
32	C₃×C₉×C₁₃	—	9	6	no	—
33	C₃²×H(3,𝔽₃)	C₃×C₁₃	1	1	yes	characteristic
34	C₃×C₃²⋊C₉	C₃×C₁₃	1	1	yes	characteristic, complemented
35	C₃×C₃²⋊C₉	C₃×C₁₃	1	2	yes	complemented
36	#6859	C₃×C₁₃	1	9	yes	direct product factor
37	C₃²×C₁₃	—	3	2	no	—
38	C₃²×C₁₃	—	3	1	no	—
39	C₃²×C₁₃	—	3	9	no	—
40	C₃²×C₁₃	—	3	4	no	—
41	C₃²×C₁₃	—	3	2	no	—
42	C₃²×C₁₃	—	3	2	no	—
43	C₃²×C₁₃	—	3	1	no	—
44	C₃²×C₁₃	—	9	9	no	—
45	C₃²×C₁₃	—	9	27	no	—
46	C₃²×C₁₃	—	9	6	no	—
47	C₃²×C₁₃	—	9	3	no	—
48	C₃²×C₁₃	AΣL(1,𝔽₂₇)	1	3	yes	—
49	C₃²×C₁₃	#731	1	6	yes	—
50	C₃²×C₁₃	#732	1	3	yes	—
51	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
52	C₉×C₁₃	—	9	3	no	—
53	C₉×C₁₃	—	9	6	no	—
54	C₃⁴	—	3	1	no	—
55	C₃⁴	C₃²×C₁₃	1	1	yes	characteristic
56	C₃×H(3,𝔽₃)	—	3	9	no	—
57	C₃×H(3,𝔽₃)	—	3	2	no	—
58	C₃×H(3,𝔽₃)	C₃²×C₁₃	1	3	yes	—
59	C₃²×C₉	—	3	1	no	—
60	C₃²×C₉	—	3	2	no	—
61	C₃²⋊C₉	—	3	2	no	—
62	C₃²⋊C₉	—	3	4	no	—
63	C₃²⋊C₉	C₃²×C₁₃	1	3	yes	complemented
64	C₃²⋊C₉	C₃²×C₁₃	1	6	yes	complemented
65	C₃×C₁₃	—	9	2	no	—
66	C₃×C₁₃	—	9	1	no	—
67	C₃×C₁₃	—	9	9	no	—
68	C₃×C₁₃	#6859	1	9	yes	direct product factor
69	C₃×C₁₃	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
70	C₃×C₁₃	C₃×#731	1	2	yes	—
71	C₃×C₁₃	C₃×#732	1	1	yes	characteristic
72	C₃³	—	3	3	no	—
73	C₃³	—	3	9	no	—
74	C₃³	—	3	6	no	—
75	C₃³	—	3	3	no	—
76	C₃³	—	3	3	no	—
77	C₃³	—	9	6	no	—
78	C₃³	—	9	3	no	—
79	C₃³	C₁₃×H(3,𝔽₃)	1	2	yes	—
80	C₃³	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
81	C₃³	C₃³×C₁₃	1	1	yes	characteristic
82	H(3,𝔽₃)	—	3	9	no	—
83	H(3,𝔽₃)	—	3	18	no	—
84	C₃×C₉	—	3	3	no	—
85	C₃×C₉	—	3	6	no	—
86	C₃×C₉	—	9	3	no	—
87	C₃×C₉	—	9	6	no	—
88	C₁₃	C₃×#6859	1	1	yes	characteristic, direct product factor, Sylow
89	C₃²	—	3	2	no	—
90	C₃²	—	3	1	no	—
91	C₃²	—	3	9	no	—
92	C₃²	—	3	4	no	—
93	C₃²	—	3	2	no	—
94	C₃²	—	3	2	no	—
95	C₃²	—	3	1	no	—
96	C₃²	—	9	9	no	—
97	C₃²	—	9	27	no	—
98	C₃²	—	9	6	no	—
99	C₃²	—	9	3	no	—
100	C₃²	C₁₃×AΣL(1,𝔽₂₇)	1	3	yes	—
101	C₃²	C₁₃×#731	1	6	yes	—
102	C₃²	C₁₃×#732	1	3	yes	—
103	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
104	C₉	—	9	3	no	—
105	C₉	—	9	6	no	—
106	C₃	—	9	2	no	—
107	C₃	—	9	1	no	—
108	C₃	—	9	9	no	—
109	C₃	C₁₃×#6859	1	9	yes	direct product factor
110	C₃	C₃×C₁₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
111	C₃	C₃×C₁₃×#731	1	2	yes	—
112	C₃	C₃×C₁₃×#732	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₁₃×H(3,𝔽₃)

C₃

yes

maximal, characteristic

C₃×C₁₃×C₃²⋊C₉

C₃

yes

maximal, characteristic, complemented

C₃×C₁₃×C₃²⋊C₉

C₃

yes

maximal, complemented

C₁₃×#6859

C₃

yes

maximal, direct product factor

C₃⁴×C₁₃

—

C₃⁴×C₁₃

C₃²

yes

characteristic

C₃×C₁₃×H(3,𝔽₃)

—

C₃×C₁₃×H(3,𝔽₃)

—

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

—

C₃²×C₉×C₁₃

—

C₃²×C₉×C₁₃

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

C₃²

yes

complemented

C₁₃×C₃²⋊C₉

C₃²

yes

complemented

C₃×#6859

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

H(3,𝔽₃)

yes

—

C₃³×C₁₃

H(3,𝔽₃)

yes

characteristic

C₃³×C₁₃

C₃³

yes

characteristic

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃²×H(3,𝔽₃)

C₃×C₁₃

yes

characteristic

C₃×C₃²⋊C₉

C₃×C₁₃

yes

characteristic, complemented

C₃×C₃²⋊C₉

C₃×C₁₃

yes

complemented

#6859

C₃×C₁₃

yes

direct product factor

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

AΣL(1,𝔽₂₇)

yes

—

C₃²×C₁₃

#731

yes

—

C₃²×C₁₃

#732

yes

—

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₁₃

—

C₉×C₁₃

—

C₃⁴

—

C₃⁴

C₃²×C₁₃

yes

characteristic

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃²×C₁₃

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃²×C₁₃

yes

complemented

C₃²⋊C₉

C₃²×C₁₃

yes

complemented

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

#6859

yes

direct product factor

C₃×C₁₃

C₃×AΣL(1,𝔽₂₇)

yes

characteristic

C₃×C₁₃

C₃×#731

yes

—

C₃×C₁₃

C₃×#732

yes

characteristic

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₁₃×H(3,𝔽₃)

yes

—

C₃³

C₁₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃³×C₁₃

yes

characteristic

H(3,𝔽₃)

—

H(3,𝔽₃)

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₁₃

C₃×#6859

yes

characteristic, direct product factor, Sylow

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

100

C₃²

C₁₃×AΣL(1,𝔽₂₇)

yes

—

101

C₃²

C₁₃×#731

yes

—

102

C₃²

C₁₃×#732

yes

—

103

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

104

C₉

—

105

C₉

—

106

C₃

—

107

C₃

—

108

C₃

—

109

C₃

C₁₃×#6859

yes

direct product factor

110

C₃

C₃×C₁₃×AΣL(1,𝔽₂₇)

yes

characteristic

111

C₃

C₃×C₁₃×#731

yes

—

112

C₃

C₃×C₁₃×#732

yes

characteristic

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

subgroup

G/H

center, socle

C₃³×C₁₃

H(3,𝔽₃)

commutator, Frattini, cosocle

C₃³

C₃³×C₁₃

prime radical

C₃²×C₁₃×H(3,𝔽₃)

C₃

LCS

G ▷ C₃³ ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₃³×C₁₃ ◁ C₃⁴×C₁₃ ◁ G

N	H ≅ G/N	type
2. C₃×C₁₃×C₃²⋊C₉	C₃	semidirect
3. C₃×C₁₃×C₃²⋊C₉	C₃	semidirect
4. C₁₃×#6859	C₃	direct (N × H)
14. C₁₃×C₃²⋊C₉	C₃²	semidirect
15. C₁₃×C₃²⋊C₉	C₃²	semidirect
16. C₃×#6859	C₁₃	direct (N × H)
34. C₃×C₃²⋊C₉	C₃×C₁₃	semidirect
35. C₃×C₃²⋊C₉	C₃×C₁₃	semidirect
36. #6859	C₃×C₁₃	direct (N × H)
63. C₃²⋊C₉	C₃²×C₁₃	semidirect
64. C₃²⋊C₉	C₃²×C₁₃	semidirect
68. C₃×C₁₃	#6859	direct (N × H)
88. C₁₃	C₃×#6859	direct (N × H)
109. C₃	C₁₃×#6859	direct (N × H)

H ≅ G/N

type

2. C₃×C₁₃×C₃²⋊C₉

C₃

semidirect

3. C₃×C₁₃×C₃²⋊C₉

C₃

semidirect

4. C₁₃×#6859

C₃

direct (N × H)

14. C₁₃×C₃²⋊C₉

C₃²

semidirect

15. C₁₃×C₃²⋊C₉

C₃²

semidirect

16. C₃×#6859

C₁₃

direct (N × H)

34. C₃×C₃²⋊C₉

C₃×C₁₃

semidirect

35. C₃×C₃²⋊C₉

C₃×C₁₃

semidirect

36. #6859

C₃×C₁₃

direct (N × H)

63. C₃²⋊C₉

C₃²×C₁₃

semidirect

64. C₃²⋊C₉

C₃²×C₁₃

semidirect

68. C₃×C₁₃

#6859

direct (N × H)

88. C₁₃

C₃×#6859

direct (N × H)

109. C₃

C₁₃×#6859

direct (N × H)

Subgroups — C₃×C₁₃×#6859 #23744732

Conjugacy classes of subgroups

Complements

Subgroups — C3×C13×#6859 #23744732

Conjugacy classes of subgroups

Complements

Subgroups — C₃×C₁₃×#6859 #23744732