Group #23744731 — subgroups

Subgroups — C₃×C₁₃×#6858 #23744731

Order 9477.

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

normal subgroups only

This group has 380 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×C₁₃×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
2	C₃×C₁₃×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
3	C₃×C₁₃×C₃²⋊C₉	C₃	1	2	yes	maximal
4	C₁₃×#6858	C₃	1	9	yes	maximal, direct product factor
5	C₃⁴×C₁₃	C₃²	1	1	yes	characteristic
6	C₃×C₁₃×3₋¹⁺²	—	3	2	no	—
7	C₃×C₁₃×3₋¹⁺²	—	3	9	no	—
8	C₃×C₁₃×3₋¹⁺²	C₃²	1	3	yes	—
9	C₃²×C₉×C₁₃	—	3	1	no	—
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	—
15	C₁₃×C₃²⋊C₉	C₃²	1	6	yes	—
16	C₃×#6858	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₃³×C₁₃	—	3	3	no	—
18	C₃³×C₁₃	—	3	6	no	—
19	C₃³×C₁₃	—	3	3	no	—
20	C₃³×C₁₃	H(3,𝔽₃)	1	2	yes	—
21	C₃³×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
22	C₃³×C₁₃	C₃³	1	1	yes	characteristic
23	C₁₃×3₋¹⁺²	—	3	18	no	—
24	C₁₃×3₋¹⁺²	—	3	9	no	—
25	C₃×C₉×C₁₃	—	3	9	no	—
26	C₃×C₉×C₁₃	—	3	3	no	—
27	C₃×C₉×C₁₃	—	3	3	no	—
28	C₃×C₉×C₁₃	—	3	6	no	—
29	C₃×C₉×C₁₃	—	9	3	no	—
30	C₃×C₉×C₁₃	—	9	6	no	—
31	C₃²×3₋¹⁺²	C₃×C₁₃	1	1	yes	characteristic
32	C₃×C₃²⋊C₉	C₃×C₁₃	1	1	yes	characteristic
33	C₃×C₃²⋊C₉	C₃×C₁₃	1	2	yes	—
34	#6858	C₃×C₁₃	1	9	yes	direct product factor
35	C₃²×C₁₃	—	3	2	no	—
36	C₃²×C₁₃	—	3	1	no	—
37	C₃²×C₁₃	—	3	4	no	—
38	C₃²×C₁₃	—	3	2	no	—
39	C₃²×C₁₃	—	3	2	no	—
40	C₃²×C₁₃	—	3	1	no	—
41	C₃²×C₁₃	—	9	9	no	—
42	C₃²×C₁₃	AΣL(1,𝔽₂₇)	1	3	yes	—
43	C₃²×C₁₃	#731	1	6	yes	—
44	C₃²×C₁₃	#733	1	3	yes	—
45	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
46	C₉×C₁₃	—	3	9	no	—
47	C₉×C₁₃	—	9	3	no	—
48	C₉×C₁₃	—	9	6	no	—
49	C₃⁴	C₃²×C₁₃	1	1	yes	characteristic
50	C₃×3₋¹⁺²	—	3	2	no	—
51	C₃×3₋¹⁺²	—	3	9	no	—
52	C₃×3₋¹⁺²	C₃²×C₁₃	1	3	yes	—
53	C₃²×C₉	—	3	1	no	—
54	C₃²×C₉	—	3	1	no	—
55	C₃²×C₉	—	3	2	no	—
56	C₃²⋊C₉	—	3	2	no	—
57	C₃²⋊C₉	—	3	4	no	—
58	C₃²⋊C₉	C₃²×C₁₃	1	3	yes	—
59	C₃²⋊C₉	C₃²×C₁₃	1	6	yes	—
60	C₃×C₁₃	—	9	2	no	—
61	C₃×C₁₃	—	9	1	no	—
62	C₃×C₁₃	#6858	1	9	yes	direct product factor
63	C₃×C₁₃	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
64	C₃×C₁₃	C₃×#731	1	2	yes	—
65	C₃×C₁₃	C₃×#733	1	1	yes	characteristic
66	C₃³	—	3	3	no	—
67	C₃³	—	3	6	no	—
68	C₃³	—	3	3	no	—
69	C₃³	C₁₃×H(3,𝔽₃)	1	2	yes	—
70	C₃³	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
71	C₃³	C₃³×C₁₃	1	1	yes	characteristic
72	3₋¹⁺²	—	3	18	no	—
73	3₋¹⁺²	—	3	9	no	—
74	C₃×C₉	—	3	9	no	—
75	C₃×C₉	—	3	3	no	—
76	C₃×C₉	—	3	3	no	—
77	C₃×C₉	—	3	6	no	—
78	C₃×C₉	—	9	3	no	—
79	C₃×C₉	—	9	6	no	—
80	C₁₃	C₃×#6858	1	1	yes	characteristic, direct product factor, Sylow
81	C₃²	—	3	2	no	—
82	C₃²	—	3	1	no	—
83	C₃²	—	3	4	no	—
84	C₃²	—	3	2	no	—
85	C₃²	—	3	2	no	—
86	C₃²	—	3	1	no	—
87	C₃²	—	9	9	no	—
88	C₃²	C₁₃×AΣL(1,𝔽₂₇)	1	3	yes	—
89	C₃²	C₁₃×#731	1	6	yes	—
90	C₃²	C₁₃×#733	1	3	yes	—
91	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
92	C₉	—	3	9	no	—
93	C₉	—	9	3	no	—
94	C₉	—	9	6	no	—
95	C₃	—	9	2	no	—
96	C₃	—	9	1	no	—
97	C₃	C₁₃×#6858	1	9	yes	direct product factor
98	C₃	C₃×C₁₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
99	C₃	C₃×C₁₃×#731	1	2	yes	—
100	C₃	C₃×C₁₃×#733	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₁₃×3₋¹⁺²

C₃

yes

maximal, characteristic

C₃×C₁₃×C₃²⋊C₉

C₃

yes

maximal, characteristic

C₃×C₁₃×C₃²⋊C₉

C₃

yes

maximal

C₁₃×#6858

C₃

yes

maximal, direct product factor

C₃⁴×C₁₃

C₃²

yes

characteristic

C₃×C₁₃×3₋¹⁺²

—

C₃×C₁₃×3₋¹⁺²

—

C₃×C₁₃×3₋¹⁺²

C₃²

yes

—

C₃²×C₉×C₁₃

—

C₃²×C₉×C₁₃

—

C₃²×C₉×C₁₃

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

C₃²

yes

—

C₁₃×C₃²⋊C₉

C₃²

yes

—

C₃×#6858

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

H(3,𝔽₃)

yes

—

C₃³×C₁₃

H(3,𝔽₃)

yes

characteristic

C₃³×C₁₃

C₃³

yes

characteristic

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃²×3₋¹⁺²

C₃×C₁₃

yes

characteristic

C₃×C₃²⋊C₉

C₃×C₁₃

yes

characteristic

C₃×C₃²⋊C₉

C₃×C₁₃

yes

—

#6858

C₃×C₁₃

yes

direct product factor

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

#733

yes

—

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₃⁴

C₃²×C₁₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃²×C₁₃

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃²×C₁₃

yes

—

C₃²⋊C₉

C₃²×C₁₃

yes

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

#6858

yes

direct product factor

C₃×C₁₃

C₃×AΣL(1,𝔽₂₇)

yes

characteristic

C₃×C₁₃

C₃×#731

yes

—

C₃×C₁₃

C₃×#733

yes

characteristic

C₃³

—

C₃³

—

C₃³

—

C₃³

C₁₃×H(3,𝔽₃)

yes

—

C₃³

C₁₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃³×C₁₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₁₃

C₃×#6858

yes

characteristic, direct product factor, Sylow

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₁₃×AΣL(1,𝔽₂₇)

yes

—

C₃²

C₁₃×#731

yes

—

C₃²

C₁₃×#733

yes

—

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

C₉

—

C₉

—

C₉

—

C₃

—

C₃

—

C₃

C₁₃×#6858

yes

direct product factor

C₃

C₃×C₁₃×AΣL(1,𝔽₂₇)

yes

characteristic

C₃

C₃×C₁₃×#731

yes

—

100

C₃

C₃×C₁₃×#733

yes

characteristic

	subgroup	G/H
center, socle	C₃³×C₁₃	H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₁₃
prime radical	C₃⁴×C₁₃	C₃²
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

C₃⁴×C₁₃

C₃²

LCS

G ▷ C₃³ ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₃³×C₁₃ ◁ C₃⁴×C₁₃ ◁ G

N	H ≅ G/N	type
4. C₁₃×#6858	C₃	direct (N × H)
16. C₃×#6858	C₁₃	direct (N × H)
34. #6858	C₃×C₁₃	direct (N × H)
62. C₃×C₁₃	#6858	direct (N × H)
80. C₁₃	C₃×#6858	direct (N × H)
97. C₃	C₁₃×#6858	direct (N × H)

H ≅ G/N

type

4. C₁₃×#6858

C₃

direct (N × H)

16. C₃×#6858

C₁₃

direct (N × H)

34. #6858

C₃×C₁₃

direct (N × H)

62. C₃×C₁₃

#6858

direct (N × H)

80. C₁₃

C₃×#6858

direct (N × H)

97. C₃

C₁₃×#6858

direct (N × H)

Subgroups — C₃×C₁₃×#6858 #23744731

Conjugacy classes of subgroups

Complements

Subgroups — C3×C13×#6858 #23744731

Conjugacy classes of subgroups

Complements

Subgroups — C₃×C₁₃×#6858 #23744731