Group #41007625 — subgroups

Subgroups — C₃×#22582604 #41007625

Order 26973.

1076 conjugacy classes of subgroups; only the 382 normal subgroups are shown.

normal subgroups only

This group has 1076 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	C₃²×#11758606	C₃	1	3	yes	maximal
2	C₃²×#11758606	C₃	1	1	yes	maximal, characteristic
3	#22582604	C₃	1	27	yes	maximal, direct product factor
4	C₃×#15688505	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃×#15688504	C₃	1	6	yes	maximal, complemented
6	C₃×C₉×#66524	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃×C₃₇×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
9	C₃×#11758606	—	3	9	no	—
10	C₃×#11758606	—	3	27	no	—
11	C₃×#11758606	C₃²	1	9	yes	—
12	C₃×#11758606	C₃²	1	27	yes	—
13	C₃²×#66524	C₃²	1	2	yes	—
14	C₃²×#66524	C₃²	1	3	yes	—
15	C₃²×#66524	C₃²	1	3	yes	—
16	#15688505	C₃²	1	9	yes	complemented
17	#15688504	C₃²	1	54	yes	complemented
18	C₉×#66524	C₃²	1	9	yes	complemented
19	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
20	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
21	C₃₇×C₉⋊C₉	C₃²	1	9	yes	complemented
22	C₃³×C₃₇	C₃³	1	1	yes	characteristic
23	C₃²×C₃₇⋊C₃	—	3	9	no	—
24	C₃²×C₃₇⋊C₃	—	3	3	no	—
25	C₃²×C₃₇⋊C₃	C₃³	1	3	yes	—
26	C₃×C₉×C₃₇	—	3	9	no	—
27	C₃×C₉×C₃₇	3₋¹⁺²	1	9	yes	complemented
28	C₃×C₉×C₃₇	C₃³	1	9	yes	—
29	C₃×C₉×C₃₇	C₃³	1	3	yes	—
30	C₃×#66524	—	3	9	no	—
31	C₃×#66524	—	3	6	no	—
32	C₃×#66524	3₋¹⁺²	1	27	yes	complemented
33	C₃×#66524	C₃³	1	6	yes	—
34	C₃×#66524	C₃³	1	9	yes	—
35	C₃×#66524	C₃³	1	9	yes	—
36	#11758606	—	3	27	no	—
37	#11758606	—	3	81	no	—
38	C₃×#6900	—	37	1	no	maximal, Sylow
39	C₃²×C₃₇	C₃×3₋¹⁺²	1	9	yes	—
40	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
41	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
42	C₃×C₃₇⋊C₃	—	3	9	no	—
43	C₃×C₃₇⋊C₃	—	3	3	no	—
44	C₃×C₃₇⋊C₃	—	9	9	no	—
45	C₉×C₃₇	—	3	9	no	—
46	C₉×C₃₇	C₃×3₋¹⁺²	1	9	yes	complemented
47	#66524	—	3	9	no	—
48	#66524	—	3	6	no	—
49	#66524	C₃×3₋¹⁺²	1	27	yes	complemented
50	C₃²×3₋¹⁺²	—	37	3	no	—
51	C₃²×3₋¹⁺²	—	37	1	no	—
52	#6900	—	37	27	no	—
53	C₃×C₉⋊C₉	—	37	6	no	—
54	C₃×C₉⋊C₉	—	37	1	no	—
55	C₃×C₉⋊C₉	C₃₇⋊C₃	1	1	yes	characteristic, complemented
56	C₃×C₉²	—	37	1	no	—
57	C₃×C₃₇	#6900	1	9	yes	complemented
58	C₃×C₃₇	C₃²×3₋¹⁺²	1	3	yes	—
59	C₃×C₃₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
60	C₃₇⋊C₃	—	9	3	no	—
61	C₃⁴	—	37	1	no	—
62	C₃×3₋¹⁺²	—	37	27	no	—
63	C₃×3₋¹⁺²	—	37	9	no	—
64	C₃×3₋¹⁺²	—	111	27	no	—
65	C₃×3₋¹⁺²	—	111	9	no	—
66	C₃²×C₉	—	37	3	no	—
67	C₃²×C₉	—	37	3	no	—
68	C₃²×C₉	—	37	2	no	—
69	C₃²×C₉	C₃×C₃₇⋊C₃	1	3	yes	—
70	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
71	C₉⋊C₉	—	37	54	no	—
72	C₉⋊C₉	—	37	9	no	—
73	C₉⋊C₉	C₃×C₃₇⋊C₃	1	9	yes	complemented
74	C₉²	—	37	9	no	—
75	C₃₇	C₃×#6900	1	1	yes	characteristic, complemented, Sylow
76	C₃³	—	37	3	no	—
77	C₃³	—	111	9	no	—
78	C₃³	—	111	3	no	—
79	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
80	3₋¹⁺²	—	111	81	no	—
81	3₋¹⁺²	—	111	27	no	—
82	C₃×C₉	—	3	9	no	—
83	C₃×C₉	—	37	27	no	—
84	C₃×C₉	—	37	9	no	—
85	C₃×C₉	—	37	9	no	—
86	C₃×C₉	—	37	6	no	—
87	C₃×C₉	—	111	9	no	—
88	C₃×C₉	—	111	6	no	—
89	C₃×C₉	#11758606	1	9	yes	complemented
90	C₃×C₉	C₃²×C₃₇⋊C₃	1	9	yes	—
91	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
92	C₃²	—	333	9	no	—
93	C₃²	—	111	9	no	—
94	C₃²	—	111	3	no	—
95	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	9	yes	—
96	C₃²	C₃×#11758606	1	3	yes	—
97	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
98	C₉	—	3	9	no	—
99	C₉	—	37	27	no	—
100	C₉	—	111	9	no	—
101	C₉	—	111	6	no	—
102	C₉	C₃×#11758606	1	9	yes	complemented
103	C₃	—	333	3	no	—
104	C₃	#22582604	1	9	yes	direct product factor
105	C₃	C₃×3₋¹⁺²×C₃₇⋊C₃	1	3	yes	—
106	C₃	C₃²×#11758606	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#11758606

C₃

yes

maximal

C₃²×#11758606

C₃

yes

maximal, characteristic

#22582604

C₃

yes

maximal, direct product factor

C₃×#15688505

C₃

yes

maximal, characteristic, complemented

C₃×#15688504

C₃

yes

maximal, complemented

C₃×C₉×#66524

C₃

yes

maximal, characteristic, complemented

C₃×C₃₇×C₉⋊C₉

C₃

yes

maximal, characteristic, complemented

C₃³×C₃₇⋊C₃

C₃²

yes

—

—

C₃²

yes

—

C₃×#11758606

C₃²

yes

—

C₃²×#66524

C₃²

yes

—

C₃²×#66524

C₃²

yes

—

C₃²×#66524

C₃²

yes

—

#15688505

C₃²

yes

complemented

#15688504

C₃²

yes

complemented

C₉×#66524

C₃²

yes

complemented

C₃²×C₉×C₃₇

C₃²

yes

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×C₉⋊C₉

C₃²

yes

complemented

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

C₃³

yes

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

3₋¹⁺²

yes

complemented

C₃×C₉×C₃₇

C₃³

yes

—

C₃×C₉×C₃₇

C₃³

yes

—

C₃×#66524

—

C₃×#66524

—

C₃×#66524

3₋¹⁺²

yes

complemented

C₃×#66524

C₃³

yes

—

C₃×#66524

C₃³

yes

—

C₃×#66524

C₃³

yes

—

—

—

—

C₃²×C₃₇

C₃×3₋¹⁺²

yes

—

C₃²×C₃₇

C₃×3₋¹⁺²

yes

—

C₃²×C₃₇

C₃⁴

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

C₉×C₃₇

C₃×3₋¹⁺²

yes

—

—

C₃×3₋¹⁺²

yes

complemented

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

—

#6900

—

C₃×C₉⋊C₉

—

C₃×C₉⋊C₉

—

C₃×C₉⋊C₉

C₃₇⋊C₃

yes

characteristic, complemented

C₃×C₉²

—

C₃×C₃₇

#6900

yes

complemented

C₃×C₃₇

C₃²×3₋¹⁺²

yes

—

C₃×C₃₇

C₃²×3₋¹⁺²

yes

characteristic

C₃₇⋊C₃

—

C₃⁴

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

C₃×C₃₇⋊C₃

yes

complemented

C₉²

—

C₃₇

C₃×#6900

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

#11758606

yes

complemented

C₃×C₉

C₃²×C₃₇⋊C₃

yes

—

C₃×C₉

C₃²×C₃₇⋊C₃

yes

—

C₃²

—

333

—

C₃²

—

111

—

C₃²

—

111

—

C₃²

3₋¹⁺²×C₃₇⋊C₃

yes

—

C₃²

C₃×#11758606

yes

—

C₃²

C₃³×C₃₇⋊C₃

yes

characteristic

C₉

—

C₉

—

100

C₉

—

111

—

101

C₉

—

111

—

102

C₉

C₃×#11758606

yes

complemented

103

C₃

—

333

—

104

C₃

#22582604

yes

direct product factor

105

C₃

C₃×3₋¹⁺²×C₃₇⋊C₃

yes

—

106

C₃

C₃²×#11758606

yes

characteristic

	subgroup	G/H
center	C₃³	C₃²×C₃₇⋊C₃
commutator, cosocle	C₃²×C₃₇	C₃⁴
Frattini	C₃²	C₃³×C₃₇⋊C₃
Fitting	C₃×C₃₇×C₉⋊C₉	C₃
socle	C₃³×C₃₇	C₃³
prime radical	C₃³×C₃₇⋊C₃	C₃²
hypercenter	C₃×C₉⋊C₉	C₃₇⋊C₃
hypocenter	C₃₇	C₃×#6900
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃³ ◁ C₃×C₉⋊C₉

subgroup

G/H

center

C₃³

C₃²×C₃₇⋊C₃

commutator, cosocle

C₃²×C₃₇

C₃⁴

Frattini

C₃²

C₃³×C₃₇⋊C₃

Fitting

C₃×C₃₇×C₉⋊C₉

C₃

socle

C₃³×C₃₇

C₃³

prime radical

C₃³×C₃₇⋊C₃

C₃²

hypercenter

C₃×C₉⋊C₉

C₃₇⋊C₃

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃₇

C₁ ◁ C₃³ ◁ C₃×C₉⋊C₉

N	H ≅ G/N	type
3. #22582604	C₃	direct (N × H)
4. C₃×#15688505	C₃	semidirect
5. C₃×#15688504	C₃	semidirect
6. C₃×C₉×#66524	C₃	semidirect
7. C₃×C₃₇×C₉⋊C₉	C₃	semidirect
16. #15688505	C₃²	semidirect
17. #15688504	C₃²	semidirect
18. C₉×#66524	C₃²	semidirect
21. C₃₇×C₉⋊C₉	C₃²	semidirect
27. C₃×C₉×C₃₇	3₋¹⁺²	semidirect
32. C₃×#66524	3₋¹⁺²	semidirect
46. C₉×C₃₇	C₃×3₋¹⁺²	semidirect
49. #66524	C₃×3₋¹⁺²	semidirect
55. C₃×C₉⋊C₉	C₃₇⋊C₃	semidirect
57. C₃×C₃₇	#6900	semidirect
73. C₉⋊C₉	C₃×C₃₇⋊C₃	semidirect
75. C₃₇	C₃×#6900	semidirect
89. C₃×C₉	#11758606	semidirect
102. C₉	C₃×#11758606	semidirect
104. C₃	#22582604	direct (N × H)

H ≅ G/N

type

3. #22582604

C₃

direct (N × H)

4. C₃×#15688505

C₃

semidirect

5. C₃×#15688504

C₃

semidirect

6. C₃×C₉×#66524

C₃

semidirect

7. C₃×C₃₇×C₉⋊C₉

C₃

semidirect

16. #15688505

C₃²

semidirect

17. #15688504

C₃²

semidirect

18. C₉×#66524

C₃²

semidirect

21. C₃₇×C₉⋊C₉

C₃²

semidirect

27. C₃×C₉×C₃₇

3₋¹⁺²

semidirect

32. C₃×#66524

3₋¹⁺²

semidirect

46. C₉×C₃₇

C₃×3₋¹⁺²

semidirect

49. #66524

C₃×3₋¹⁺²

semidirect

55. C₃×C₉⋊C₉

C₃₇⋊C₃

semidirect

57. C₃×C₃₇

#6900

semidirect

73. C₉⋊C₉

C₃×C₃₇⋊C₃

semidirect

75. C₃₇

C₃×#6900

semidirect

89. C₃×C₉

#11758606

semidirect

102. C₉

C₃×#11758606

semidirect

104. C₃

#22582604

direct (N × H)

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

subgroup	index
C₃²×#11758606	3
#22582604	3
C₃×#15688505	3
C₃×#15688504	3
C₃×C₉×#66524	3
C₃×C₃₇×C₉⋊C₉	3
C₃×#6900	37

subgroup

index

C₃×C₉×#66524

C₃×C₃₇×C₉⋊C₉

C₃×#6900

Subgroups — C₃×#22582604 #41007625

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#22582604 #41007625

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#22582604 #41007625