Group #23745049 — subgroups

Subgroups — C₁₃×3₋¹⁺²² #23745049

Order 9477.

836 conjugacy classes of subgroups; only the 518 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 518 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	2	yes	maximal
2	C₁₃×#6900	C₃	1	9	yes	maximal, complemented
3	C₁₃×#6896	C₃	1	9	yes	maximal, complemented
4	C₁₃×#6894	C₃	1	12	yes	maximal, complemented
5	C₁₃×#6891	C₃	1	2	yes	maximal
6	C₉×C₁₃×3₋¹⁺²	C₃	1	6	yes	maximal, complemented
7	C₃⁴×C₁₃	C₃²	1	1	yes	characteristic
8	C₃×C₁₃×3₋¹⁺²	—	3	18	no	—
9	C₃×C₁₃×3₋¹⁺²	C₃²	1	4	yes	—
10	C₃×C₁₃×3₋¹⁺²	C₃²	1	6	yes	—
11	C₃×C₁₃×3₋¹⁺²	C₃²	1	6	yes	—
12	C₃×C₁₃×3₋¹⁺²	C₃²	1	6	yes	—
13	C₃×C₁₃×3₋¹⁺²	C₃²	1	2	yes	—
14	C₃²×C₉×C₁₃	C₃²	1	6	yes	—
15	C₁₃×C₉⋊C₉	C₃²	1	36	yes	complemented
16	C₁₃×C₉⋊C₉	C₃²	1	36	yes	complemented
17	C₁₃×C₃²⋊C₉	C₃²	1	12	yes	—
18	C₁₃×C₃²⋊C₉	C₃²	1	6	yes	—
19	C₉²×C₁₃	C₃²	1	9	yes	complemented
20	3₋¹⁺²²	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₃³×C₁₃	—	3	6	no	—
22	C₃³×C₁₃	—	9	2	no	—
23	C₃³×C₁₃	C₃³	1	1	yes	characteristic
24	C₃³×C₁₃	C₃³	1	1	yes	characteristic
25	C₃³×C₁₃	C₃³	1	2	yes	—
26	C₁₃×3₋¹⁺²	—	3	12	no	—
27	C₁₃×3₋¹⁺²	—	3	18	no	—
28	C₁₃×3₋¹⁺²	—	3	18	no	—
29	C₁₃×3₋¹⁺²	—	9	6	no	—
30	C₁₃×3₋¹⁺²	3₋¹⁺²	1	18	yes	direct product factor
31	C₃×C₉×C₁₃	—	3	18	no	—
32	C₃×C₉×C₁₃	C₃³	1	18	yes	—
33	C₃×C₉×C₁₃	C₃³	1	12	yes	—
34	C₃×C₉×C₁₃	C₃³	1	6	yes	—
35	C₃²×3₋¹⁺²	C₃×C₁₃	1	2	yes	—
36	#6900	C₃×C₁₃	1	9	yes	complemented
37	#6896	C₃×C₁₃	1	9	yes	complemented
38	#6894	C₃×C₁₃	1	12	yes	complemented
39	#6891	C₃×C₁₃	1	2	yes	—
40	C₉×3₋¹⁺²	C₃×C₁₃	1	6	yes	complemented
41	C₃²×C₁₃	—	3	2	no	—
42	C₃²×C₁₃	—	3	4	no	—
43	C₃²×C₁₃	—	3	2	no	—
44	C₃²×C₁₃	—	3	2	no	—
45	C₃²×C₁₃	—	3	2	no	—
46	C₃²×C₁₃	—	3	2	no	—
47	C₃²×C₁₃	—	9	9	no	—
48	C₃²×C₁₃	C₃×3₋¹⁺²	1	6	yes	—
49	C₃²×C₁₃	C₃⁴	1	1	yes	characteristic
50	C₉×C₁₃	—	3	12	no	—
51	C₉×C₁₃	—	3	18	no	—
52	C₉×C₁₃	C₃×3₋¹⁺²	1	18	yes	complemented
53	C₃⁴	C₃²×C₁₃	1	1	yes	characteristic
54	C₃×3₋¹⁺²	—	3	18	no	—
55	C₃×3₋¹⁺²	C₃²×C₁₃	1	4	yes	—
56	C₃×3₋¹⁺²	C₃²×C₁₃	1	6	yes	—
57	C₃×3₋¹⁺²	C₃²×C₁₃	1	6	yes	—
58	C₃×3₋¹⁺²	C₃²×C₁₃	1	6	yes	—
59	C₃×3₋¹⁺²	C₃²×C₁₃	1	2	yes	—
60	C₃²×C₉	C₃²×C₁₃	1	6	yes	—
61	C₉⋊C₉	C₃²×C₁₃	1	36	yes	complemented
62	C₉⋊C₉	C₃²×C₁₃	1	36	yes	complemented
63	C₃²⋊C₉	C₃²×C₁₃	1	12	yes	—
64	C₃²⋊C₉	C₃²×C₁₃	1	6	yes	—
65	C₉²	C₃²×C₁₃	1	9	yes	complemented
66	C₃×C₁₃	—	3	6	no	—
67	C₃×C₁₃	—	9	1	no	—
68	C₃×C₁₃	—	9	1	no	—
69	C₃×C₁₃	C₃²×3₋¹⁺²	1	2	yes	—
70	C₃×C₁₃	3₋¹⁺⁴	1	2	yes	—
71	C₃³	—	3	6	no	—
72	C₃³	—	9	2	no	—
73	C₃³	C₃³×C₁₃	1	1	yes	characteristic
74	C₃³	C₃³×C₁₃	1	1	yes	characteristic
75	C₃³	C₃³×C₁₃	1	2	yes	—
76	3₋¹⁺²	—	3	12	no	—
77	3₋¹⁺²	—	3	18	no	—
78	3₋¹⁺²	—	3	18	no	—
79	3₋¹⁺²	—	9	6	no	—
80	3₋¹⁺²	C₁₃×3₋¹⁺²	1	18	yes	direct product factor
81	C₃×C₉	—	3	18	no	—
82	C₃×C₉	C₃³×C₁₃	1	18	yes	—
83	C₃×C₉	C₃³×C₁₃	1	12	yes	—
84	C₃×C₉	C₃³×C₁₃	1	6	yes	—
85	C₁₃	3₋¹⁺²²	1	1	yes	characteristic, direct product factor, Sylow
86	C₃²	—	3	2	no	—
87	C₃²	—	3	4	no	—
88	C₃²	—	3	2	no	—
89	C₃²	—	3	2	no	—
90	C₃²	—	3	2	no	—
91	C₃²	—	3	2	no	—
92	C₃²	—	9	9	no	—
93	C₃²	C₃×C₁₃×3₋¹⁺²	1	6	yes	—
94	C₃²	C₃⁴×C₁₃	1	1	yes	characteristic
95	C₉	—	3	12	no	—
96	C₉	—	3	18	no	—
97	C₉	C₃×C₁₃×3₋¹⁺²	1	18	yes	complemented
98	C₃	—	3	6	no	—
99	C₃	—	9	1	no	—
100	C₃	—	9	1	no	—
101	C₃	C₃²×C₁₃×3₋¹⁺²	1	2	yes	—
102	C₃	C₁₃×3₋¹⁺⁴	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₁₃×3₋¹⁺²

C₃

yes

maximal

C₁₃×#6900

C₃

yes

maximal, complemented

C₁₃×#6896

C₃

yes

maximal, complemented

C₁₃×#6894

C₃

yes

maximal, complemented

C₁₃×#6891

C₃

yes

maximal

C₉×C₁₃×3₋¹⁺²

C₃

yes

maximal, complemented

C₃⁴×C₁₃

C₃²

yes

characteristic

C₃×C₁₃×3₋¹⁺²

—

C₃×C₁₃×3₋¹⁺²

C₃²

yes

—

C₃×C₁₃×3₋¹⁺²

C₃²

yes

—

C₃×C₁₃×3₋¹⁺²

C₃²

yes

—

C₃×C₁₃×3₋¹⁺²

C₃²

yes

—

C₃×C₁₃×3₋¹⁺²

C₃²

yes

—

C₃²×C₉×C₁₃

C₃²

yes

—

C₁₃×C₉⋊C₉

C₃²

yes

complemented

C₁₃×C₉⋊C₉

C₃²

yes

complemented

C₁₃×C₃²⋊C₉

C₃²

yes

—

C₁₃×C₃²⋊C₉

C₃²

yes

—

C₉²×C₁₃

C₃²

yes

complemented

3₋¹⁺²²

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

C₃³

yes

characteristic

C₃³×C₁₃

C₃³

yes

characteristic

C₃³×C₁₃

C₃³

yes

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

3₋¹⁺²

yes

direct product factor

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

C₃³

yes

—

C₃×C₉×C₁₃

C₃³

yes

—

C₃×C₉×C₁₃

C₃³

yes

—

C₃²×3₋¹⁺²

C₃×C₁₃

yes

—

#6900

C₃×C₁₃

yes

complemented

#6896

C₃×C₁₃

yes

complemented

#6894

C₃×C₁₃

yes

complemented

#6891

C₃×C₁₃

yes

—

C₉×3₋¹⁺²

C₃×C₁₃

yes

complemented

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

C₃×3₋¹⁺²

yes

—

C₃²×C₁₃

C₃⁴

yes

characteristic

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

C₃×3₋¹⁺²

yes

complemented

C₃⁴

C₃²×C₁₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃²×C₁₃

yes

—

C₃×3₋¹⁺²

C₃²×C₁₃

yes

—

C₃×3₋¹⁺²

C₃²×C₁₃

yes

—

C₃×3₋¹⁺²

C₃²×C₁₃

yes

—

C₃×3₋¹⁺²

C₃²×C₁₃

yes

—

C₃²×C₉

C₃²×C₁₃

yes

—

C₉⋊C₉

C₃²×C₁₃

yes

complemented

C₉⋊C₉

C₃²×C₁₃

yes

complemented

C₃²⋊C₉

C₃²×C₁₃

yes

—

C₃²⋊C₉

C₃²×C₁₃

yes

—

C₉²

C₃²×C₁₃

yes

complemented

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

C₃²×3₋¹⁺²

yes

—

C₃×C₁₃

3₋¹⁺⁴

yes

—

C₃³

—

C₃³

—

C₃³

C₃³×C₁₃

yes

characteristic

C₃³

C₃³×C₁₃

yes

characteristic

C₃³

C₃³×C₁₃

yes

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

C₁₃×3₋¹⁺²

yes

direct product factor

C₃×C₉

—

C₃×C₉

C₃³×C₁₃

yes

—

C₃×C₉

C₃³×C₁₃

yes

—

C₃×C₉

C₃³×C₁₃

yes

—

C₁₃

3₋¹⁺²²

yes

characteristic, direct product factor, Sylow

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃×C₁₃×3₋¹⁺²

yes

—

C₃²

C₃⁴×C₁₃

yes

characteristic

C₉

—

C₉

—

C₉

C₃×C₁₃×3₋¹⁺²

yes

complemented

C₃

—

C₃

—

100

C₃

—

101

C₃

C₃²×C₁₃×3₋¹⁺²

yes

—

102

C₃

C₁₃×3₋¹⁺⁴

yes

—

	subgroup	G/H
center, socle	C₃²×C₁₃	C₃⁴
commutator, Frattini, cosocle	C₃²	C₃⁴×C₁₃
prime radical	C₃⁴×C₁₃	C₃²
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×C₁₃ ◁ G

subgroup

G/H

center, socle

C₃²×C₁₃

C₃⁴

commutator, Frattini, cosocle

C₃²

C₃⁴×C₁₃

prime radical

C₃⁴×C₁₃

C₃²

LCS

G ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₃²×C₁₃ ◁ G

N	H ≅ G/N	type
2. C₁₃×#6900	C₃	semidirect
3. C₁₃×#6896	C₃	semidirect
4. C₁₃×#6894	C₃	semidirect
6. C₉×C₁₃×3₋¹⁺²	C₃	semidirect
15. C₁₃×C₉⋊C₉	C₃²	semidirect
16. C₁₃×C₉⋊C₉	C₃²	semidirect
19. C₉²×C₁₃	C₃²	semidirect
20. 3₋¹⁺²²	C₁₃	direct (N × H)
30. C₁₃×3₋¹⁺²	3₋¹⁺²	direct (N × H)
36. #6900	C₃×C₁₃	semidirect
37. #6896	C₃×C₁₃	semidirect
38. #6894	C₃×C₁₃	semidirect
40. C₉×3₋¹⁺²	C₃×C₁₃	semidirect
52. C₉×C₁₃	C₃×3₋¹⁺²	semidirect
61. C₉⋊C₉	C₃²×C₁₃	semidirect
62. C₉⋊C₉	C₃²×C₁₃	semidirect
65. C₉²	C₃²×C₁₃	semidirect
80. 3₋¹⁺²	C₁₃×3₋¹⁺²	direct (N × H)
85. C₁₃	3₋¹⁺²²	direct (N × H)
97. C₉	C₃×C₁₃×3₋¹⁺²	semidirect

H ≅ G/N

type

2. C₁₃×#6900

C₃

semidirect

3. C₁₃×#6896

C₃

semidirect

4. C₁₃×#6894

C₃

semidirect

6. C₉×C₁₃×3₋¹⁺²

C₃

semidirect

15. C₁₃×C₉⋊C₉

C₃²

semidirect

16. C₁₃×C₉⋊C₉

C₃²

semidirect

19. C₉²×C₁₃

C₃²

semidirect

20. 3₋¹⁺²²

C₁₃

direct (N × H)

30. C₁₃×3₋¹⁺²

3₋¹⁺²

direct (N × H)

36. #6900

C₃×C₁₃

semidirect

37. #6896

C₃×C₁₃

semidirect

38. #6894

C₃×C₁₃

semidirect

40. C₉×3₋¹⁺²

C₃×C₁₃

semidirect

52. C₉×C₁₃

C₃×3₋¹⁺²

semidirect

61. C₉⋊C₉

C₃²×C₁₃

semidirect

62. C₉⋊C₉

C₃²×C₁₃

semidirect

65. C₉²

C₃²×C₁₃

semidirect

80. 3₋¹⁺²

C₁₃×3₋¹⁺²

direct (N × H)

85. C₁₃

3₋¹⁺²²

direct (N × H)

97. C₉

C₃×C₁₃×3₋¹⁺²

semidirect

Subgroups — C₁₃×3₋¹⁺²² #23745049

Conjugacy classes of subgroups

Complements

Subgroups — C13×3−1+22 #23745049

Conjugacy classes of subgroups

Complements

Subgroups — C₁₃×3₋¹⁺²² #23745049