Group #41007534 — subgroups

Subgroups — C₃×#22582551 #41007534

Order 26973.

2636 conjugacy classes of subgroups; only the 382 normal subgroups are shown.

normal subgroups only

This group has 2636 conjugacy classes of subgroups — too many to draw. Only its 382 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	C₃⁴×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
2	C₃²×#11758608	C₃	1	2	yes	maximal, complemented
3	#22582551	C₃	1	27	yes	maximal, direct product factor
4	C₃×#15688502	C₃	1	9	yes	maximal, complemented
5	C₃²×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃³×C₃₇⋊C₃	—	3	9	no	—
7	C₃³×C₃₇⋊C₃	—	3	27	no	—
8	C₃³×C₃₇⋊C₃	C₃²	1	9	yes	—
9	C₃³×C₃₇⋊C₃	C₃²	1	3	yes	—
10	C₃×#11758608	—	3	18	no	—
11	C₃×#11758608	C₃²	1	18	yes	complemented
12	C₃²×#66524	C₃²	1	6	yes	complemented
13	#15688502	C₃²	1	81	yes	complemented
14	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
15	C₃×3₋¹⁺²×C₃₇	—	3	9	no	—
16	C₃×3₋¹⁺²×C₃₇	C₃²	1	9	yes	complemented
17	C₃²×C₉×C₃₇	C₃²	1	3	yes	complemented
18	C₃³×C₃₇	—	3	9	no	—
19	C₃³×C₃₇	H(3,𝔽₃)	1	9	yes	—
20	C₃³×C₃₇	C₃³	1	3	yes	—
21	C₃³×C₃₇	C₃³	1	1	yes	characteristic
22	C₃²×C₃₇⋊C₃	—	3	243	no	—
23	C₃²×C₃₇⋊C₃	—	3	27	no	—
24	C₃²×C₃₇⋊C₃	—	3	9	no	—
25	C₃²×C₃₇⋊C₃	—	3	18	no	—
26	C₃²×C₃₇⋊C₃	—	3	81	no	—
27	C₃²×C₃₇⋊C₃	3₋¹⁺²	1	27	yes	complemented
28	C₃²×C₃₇⋊C₃	C₃³	1	9	yes	—
29	3₋¹⁺²×C₃₇	—	3	27	no	—
30	C₃×C₉×C₃₇	—	3	9	no	—
31	C₃×C₉×C₃₇	C₃³	1	9	yes	complemented
32	#11758608	—	3	54	no	—
33	C₃×#66524	—	3	18	no	—
34	C₃×#66524	C₃³	1	18	yes	complemented
35	C₃×#6891	—	37	1	no	maximal, Sylow
36	C₃²×C₃₇	—	3	9	no	—
37	C₃²×C₃₇	—	3	27	no	—
38	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	9	yes	—
39	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	3	yes	—
40	C₃²×C₃₇	C₃×3₋¹⁺²	1	9	yes	—
41	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
42	C₃×C₃₇⋊C₃	—	3	243	no	—
43	C₃×C₃₇⋊C₃	—	3	9	no	—
44	C₃×C₃₇⋊C₃	—	3	18	no	—
45	C₃×C₃₇⋊C₃	—	3	81	no	—
46	C₃×C₃₇⋊C₃	C₃×3₋¹⁺²	1	27	yes	complemented
47	C₉×C₃₇	—	3	9	no	—
48	#66524	—	3	18	no	—
49	C₃⁵	—	37	1	no	—
50	C₃²×3₋¹⁺²	—	37	2	no	—
51	C₃²×3₋¹⁺²	C₃₇⋊C₃	1	1	yes	characteristic, complemented
52	#6891	—	37	27	no	—
53	C₃×C₃²⋊C₉	—	37	9	no	—
54	C₃×C₃₇	—	3	9	no	—
55	C₃×C₃₇	#6891	1	9	yes	complemented
56	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
57	C₃×C₃₇	C₃²×3₋¹⁺²	1	3	yes	—
58	C₃₇⋊C₃	—	3	27	no	—
59	C₃⁴	—	37	3	no	—
60	C₃⁴	—	37	9	no	—
61	C₃⁴	—	111	27	no	—
62	C₃⁴	—	111	9	no	—
63	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
64	C₃×3₋¹⁺²	—	3	9	no	—
65	C₃×3₋¹⁺²	—	37	18	no	—
66	C₃×3₋¹⁺²	—	111	18	no	—
67	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	9	yes	complemented
68	C₃²×C₉	—	37	6	no	—
69	C₃²×C₉	C₃×C₃₇⋊C₃	1	3	yes	complemented
70	C₃²⋊C₉	—	37	81	no	—
71	C₃₇	C₃×#6891	1	1	yes	characteristic, complemented, Sylow
72	C₃³	—	3	9	no	—
73	C₃³	—	37	27	no	—
74	C₃³	—	37	9	no	—
75	C₃³	—	111	81	no	—
76	C₃³	—	111	243	no	—
77	C₃³	—	111	27	no	—
78	C₃³	—	111	9	no	—
79	C₃³	—	111	18	no	—
80	C₃³	C₃²⋊(C₃₇⋊C₃)	1	9	yes	—
81	C₃³	C₃²×C₃₇⋊C₃	1	3	yes	—
82	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
83	3₋¹⁺²	—	3	27	no	—
84	3₋¹⁺²	—	111	54	no	—
85	C₃×C₉	—	3	9	no	—
86	C₃×C₉	—	37	18	no	—
87	C₃×C₉	—	111	18	no	—
88	C₃×C₉	C₃²×C₃₇⋊C₃	1	9	yes	complemented
89	C₃²	—	3	27	no	—
90	C₃²	—	3	9	no	—
91	C₃²	—	37	27	no	—
92	C₃²	—	111	243	no	—
93	C₃²	—	111	81	no	—
94	C₃²	—	111	9	no	—
95	C₃²	—	111	18	no	—
96	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	9	yes	—
97	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
98	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	9	yes	—
99	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
100	C₉	—	3	9	no	—
101	C₉	—	111	18	no	—
102	C₃	—	3	9	no	—
103	C₃	—	111	27	no	—
104	C₃	#22582551	1	9	yes	direct product factor
105	C₃	C₃²×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
106	C₃	C₃×3₋¹⁺²×C₃₇⋊C₃	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃⁴×C₃₇⋊C₃

C₃

yes

maximal, characteristic

C₃²×#11758608

C₃

yes

maximal, complemented

#22582551

C₃

yes

maximal, direct product factor

C₃×#15688502

C₃

yes

maximal, complemented

C₃²×3₋¹⁺²×C₃₇

C₃

yes

maximal, characteristic, complemented

C₃³×C₃₇⋊C₃

—

C₃³×C₃₇⋊C₃

—

C₃³×C₃₇⋊C₃

C₃²

yes

—

C₃³×C₃₇⋊C₃

C₃²

yes

—

C₃×#11758608

—

C₃×#11758608

C₃²

yes

complemented

C₃²×#66524

C₃²

yes

complemented

#15688502

C₃²

yes

complemented

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₃₇

—

C₃×3₋¹⁺²×C₃₇

C₃²

yes

complemented

C₃²×C₉×C₃₇

C₃²

yes

complemented

C₃³×C₃₇

—

C₃³×C₃₇

H(3,𝔽₃)

yes

—

C₃³×C₃₇

C₃³

yes

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

243

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

3₋¹⁺²

yes

complemented

C₃²×C₃₇⋊C₃

C₃³

yes

—

3₋¹⁺²×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

C₃³

yes

—

—

C₃³

yes

complemented

C₃×#6891

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

—

C₃²×C₃₇

C₃×3₋¹⁺²

yes

—

C₃²×C₃₇

C₃⁴

yes

characteristic

C₃×C₃₇⋊C₃

—

243

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

C₃×3₋¹⁺²

yes

complemented

C₉×C₃₇

—

#66524

—

C₃⁵

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

C₃₇⋊C₃

yes

characteristic, complemented

#6891

—

C₃×C₃²⋊C₉

—

C₃×C₃₇

—

C₃×C₃₇

#6891

yes

complemented

C₃×C₃₇

C₃²×H(3,𝔽₃)

yes

characteristic

C₃×C₃₇

C₃²×3₋¹⁺²

yes

—

C₃₇⋊C₃

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

111

—

C₃⁴

—

111

—

C₃⁴

C₃×C₃₇⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

complemented

C₃²×C₉

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

complemented

C₃²⋊C₉

—

C₃₇

C₃×#6891

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

—

111

—

C₃³

—

111

243

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

—

C₃³

C₃²×C₃₇⋊C₃

yes

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

111

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

111

—

C₃×C₉

C₃²×C₃₇⋊C₃

yes

complemented

C₃²

—

C₃²

—

C₃²

—

C₃²

—

111

243

—

C₃²

—

111

—

C₃²

—

111

—

C₃²

—

111

—

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

—

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

—

C₃²

3₋¹⁺²×C₃₇⋊C₃

yes

—

C₃²

C₃³×C₃₇⋊C₃

yes

characteristic

100

C₉

—

101

C₉

—

111

—

102

C₃

—

103

C₃

—

111

—

104

C₃

#22582551

yes

direct product factor

105

C₃

C₃²×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

106

C₃

C₃×3₋¹⁺²×C₃₇⋊C₃

yes

—

	subgroup	G/H
center	C₃³	C₃²×C₃₇⋊C₃
commutator, cosocle	C₃²×C₃₇	C₃⁴
Frattini	C₃²	C₃³×C₃₇⋊C₃
Fitting	C₃²×3₋¹⁺²×C₃₇	C₃
socle	C₃³×C₃₇	C₃³
prime radical	C₃⁴×C₃₇⋊C₃	C₃
hypercenter	C₃²×3₋¹⁺²	C₃₇⋊C₃
hypocenter	C₃₇	C₃×#6891
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃³ ◁ C₃²×3₋¹⁺²

subgroup

G/H

center

C₃³

C₃²×C₃₇⋊C₃

commutator, cosocle

C₃²×C₃₇

C₃⁴

Frattini

C₃²

C₃³×C₃₇⋊C₃

Fitting

C₃²×3₋¹⁺²×C₃₇

C₃

socle

C₃³×C₃₇

C₃³

prime radical

C₃⁴×C₃₇⋊C₃

C₃

hypercenter

C₃²×3₋¹⁺²

C₃₇⋊C₃

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃₇

C₁ ◁ C₃³ ◁ C₃²×3₋¹⁺²

N	H ≅ G/N	type
2. C₃²×#11758608	C₃	semidirect
3. #22582551	C₃	direct (N × H)
4. C₃×#15688502	C₃	semidirect
5. C₃²×3₋¹⁺²×C₃₇	C₃	semidirect
11. C₃×#11758608	C₃²	semidirect
12. C₃²×#66524	C₃²	semidirect
13. #15688502	C₃²	semidirect
16. C₃×3₋¹⁺²×C₃₇	C₃²	semidirect
17. C₃²×C₉×C₃₇	C₃²	semidirect
27. C₃²×C₃₇⋊C₃	3₋¹⁺²	semidirect
31. C₃×C₉×C₃₇	C₃³	semidirect
34. C₃×#66524	C₃³	semidirect
46. C₃×C₃₇⋊C₃	C₃×3₋¹⁺²	semidirect
51. C₃²×3₋¹⁺²	C₃₇⋊C₃	semidirect
55. C₃×C₃₇	#6891	semidirect
67. C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	semidirect
69. C₃²×C₉	C₃×C₃₇⋊C₃	semidirect
71. C₃₇	C₃×#6891	semidirect
88. C₃×C₉	C₃²×C₃₇⋊C₃	semidirect
104. C₃	#22582551	direct (N × H)

H ≅ G/N

type

2. C₃²×#11758608

C₃

semidirect

3. #22582551

C₃

direct (N × H)

4. C₃×#15688502

C₃

semidirect

5. C₃²×3₋¹⁺²×C₃₇

C₃

semidirect

11. C₃×#11758608

C₃²

semidirect

12. C₃²×#66524

C₃²

semidirect

13. #15688502

C₃²

semidirect

16. C₃×3₋¹⁺²×C₃₇

C₃²

semidirect

17. C₃²×C₉×C₃₇

C₃²

semidirect

27. C₃²×C₃₇⋊C₃

3₋¹⁺²

semidirect

31. C₃×C₉×C₃₇

C₃³

semidirect

34. C₃×#66524

C₃³

semidirect

46. C₃×C₃₇⋊C₃

C₃×3₋¹⁺²

semidirect

51. C₃²×3₋¹⁺²

C₃₇⋊C₃

semidirect

55. C₃×C₃₇

#6891

semidirect

67. C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

semidirect

69. C₃²×C₉

C₃×C₃₇⋊C₃

semidirect

71. C₃₇

C₃×#6891

semidirect

88. C₃×C₉

C₃²×C₃₇⋊C₃

semidirect

104. C₃

#22582551

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃⁴×C₃₇⋊C₃	3
C₃²×#11758608	3
#22582551	3
C₃×#15688502	3
C₃²×3₋¹⁺²×C₃₇	3
C₃×#6891	37

subgroup

index

C₃⁴×C₃₇⋊C₃

C₃²×3₋¹⁺²×C₃₇

Subgroups — C₃×#22582551 #41007534

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#22582551 #41007534

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#22582551 #41007534