Group #41004687 — subgroups

Subgroups — 3₋¹⁺²²×C₃₇ #41004687

Order 26973.

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	C₃²×3₋¹⁺²×C₃₇	C₃	1	2	yes	maximal
2	C₃₇×#6900	C₃	1	9	yes	maximal, complemented
3	C₃₇×#6896	C₃	1	9	yes	maximal, complemented
4	C₃₇×#6894	C₃	1	12	yes	maximal, complemented
5	C₃₇×#6891	C₃	1	2	yes	maximal
6	C₉×3₋¹⁺²×C₃₇	C₃	1	6	yes	maximal, complemented
7	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
8	C₃×3₋¹⁺²×C₃₇	—	3	18	no	—
9	C₃×3₋¹⁺²×C₃₇	C₃²	1	4	yes	—
10	C₃×3₋¹⁺²×C₃₇	C₃²	1	6	yes	—
11	C₃×3₋¹⁺²×C₃₇	C₃²	1	6	yes	—
12	C₃×3₋¹⁺²×C₃₇	C₃²	1	6	yes	—
13	C₃×3₋¹⁺²×C₃₇	C₃²	1	2	yes	—
14	C₃²×C₉×C₃₇	C₃²	1	6	yes	—
15	C₃₇×C₉⋊C₉	C₃²	1	36	yes	complemented
16	C₃₇×C₉⋊C₉	C₃²	1	36	yes	complemented
17	C₃₇×C₃²⋊C₉	C₃²	1	12	yes	—
18	C₃₇×C₃²⋊C₉	C₃²	1	6	yes	—
19	C₉²×C₃₇	C₃²	1	9	yes	complemented
20	C₃³×C₃₇	—	3	6	no	—
21	C₃³×C₃₇	—	9	2	no	—
22	C₃³×C₃₇	C₃³	1	1	yes	characteristic
23	C₃³×C₃₇	C₃³	1	1	yes	characteristic
24	C₃³×C₃₇	C₃³	1	2	yes	—
25	3₋¹⁺²×C₃₇	—	3	12	no	—
26	3₋¹⁺²×C₃₇	—	3	18	no	—
27	3₋¹⁺²×C₃₇	—	3	18	no	—
28	3₋¹⁺²×C₃₇	—	9	6	no	—
29	3₋¹⁺²×C₃₇	3₋¹⁺²	1	18	yes	direct product factor
30	C₃×C₉×C₃₇	—	3	18	no	—
31	C₃×C₉×C₃₇	C₃³	1	18	yes	—
32	C₃×C₉×C₃₇	C₃³	1	12	yes	—
33	C₃×C₉×C₃₇	C₃³	1	6	yes	—
34	3₋¹⁺²²	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₃²×C₃₇	—	3	2	no	—
36	C₃²×C₃₇	—	3	4	no	—
37	C₃²×C₃₇	—	3	2	no	—
38	C₃²×C₃₇	—	3	2	no	—
39	C₃²×C₃₇	—	3	2	no	—
40	C₃²×C₃₇	—	3	2	no	—
41	C₃²×C₃₇	—	9	9	no	—
42	C₃²×C₃₇	C₃×3₋¹⁺²	1	6	yes	—
43	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
44	C₉×C₃₇	—	3	12	no	—
45	C₉×C₃₇	—	3	18	no	—
46	C₉×C₃₇	C₃×3₋¹⁺²	1	18	yes	complemented
47	C₃²×3₋¹⁺²	C₃×C₃₇	1	2	yes	—
48	#6900	C₃×C₃₇	1	9	yes	complemented
49	#6896	C₃×C₃₇	1	9	yes	complemented
50	#6894	C₃×C₃₇	1	12	yes	complemented
51	#6891	C₃×C₃₇	1	2	yes	—
52	C₉×3₋¹⁺²	C₃×C₃₇	1	6	yes	complemented
53	C₃×C₃₇	—	3	6	no	—
54	C₃×C₃₇	—	9	1	no	—
55	C₃×C₃₇	—	9	1	no	—
56	C₃×C₃₇	C₃²×3₋¹⁺²	1	2	yes	—
57	C₃×C₃₇	3₋¹⁺⁴	1	2	yes	—
58	C₃⁴	C₃²×C₃₇	1	1	yes	characteristic
59	C₃×3₋¹⁺²	—	3	18	no	—
60	C₃×3₋¹⁺²	C₃²×C₃₇	1	4	yes	—
61	C₃×3₋¹⁺²	C₃²×C₃₇	1	6	yes	—
62	C₃×3₋¹⁺²	C₃²×C₃₇	1	6	yes	—
63	C₃×3₋¹⁺²	C₃²×C₃₇	1	6	yes	—
64	C₃×3₋¹⁺²	C₃²×C₃₇	1	2	yes	—
65	C₃²×C₉	C₃²×C₃₇	1	6	yes	—
66	C₉⋊C₉	C₃²×C₃₇	1	36	yes	complemented
67	C₉⋊C₉	C₃²×C₃₇	1	36	yes	complemented
68	C₃²⋊C₉	C₃²×C₃₇	1	12	yes	—
69	C₃²⋊C₉	C₃²×C₃₇	1	6	yes	—
70	C₉²	C₃²×C₃₇	1	9	yes	complemented
71	C₃₇	3₋¹⁺²²	1	1	yes	characteristic, direct product factor, Sylow
72	C₃³	—	3	6	no	—
73	C₃³	—	9	2	no	—
74	C₃³	C₃³×C₃₇	1	1	yes	characteristic
75	C₃³	C₃³×C₃₇	1	1	yes	characteristic
76	C₃³	C₃³×C₃₇	1	2	yes	—
77	3₋¹⁺²	—	3	12	no	—
78	3₋¹⁺²	—	3	18	no	—
79	3₋¹⁺²	—	3	18	no	—
80	3₋¹⁺²	—	9	6	no	—
81	3₋¹⁺²	3₋¹⁺²×C₃₇	1	18	yes	direct product factor
82	C₃×C₉	—	3	18	no	—
83	C₃×C₉	C₃³×C₃₇	1	18	yes	—
84	C₃×C₉	C₃³×C₃₇	1	12	yes	—
85	C₃×C₉	C₃³×C₃₇	1	6	yes	—
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	C₃×3₋¹⁺²×C₃₇	1	6	yes	—
94	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
95	C₉	—	3	12	no	—
96	C₉	—	3	18	no	—
97	C₉	C₃×3₋¹⁺²×C₃₇	1	18	yes	complemented
98	C₃	—	3	6	no	—
99	C₃	—	9	1	no	—
100	C₃	—	9	1	no	—
101	C₃	C₃²×3₋¹⁺²×C₃₇	1	2	yes	—
102	C₃	C₃₇×3₋¹⁺⁴	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×3₋¹⁺²×C₃₇

C₃

yes

maximal

C₃₇×#6900

C₃

yes

maximal, complemented

C₃₇×#6896

C₃

yes

maximal, complemented

C₃₇×#6894

C₃

yes

maximal, complemented

C₃₇×#6891

C₃

yes

maximal

C₉×3₋¹⁺²×C₃₇

C₃

yes

maximal, complemented

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₃₇

—

C₃×3₋¹⁺²×C₃₇

C₃²

yes

—

C₃×3₋¹⁺²×C₃₇

C₃²

yes

—

C₃×3₋¹⁺²×C₃₇

C₃²

yes

—

C₃×3₋¹⁺²×C₃₇

C₃²

yes

—

C₃×3₋¹⁺²×C₃₇

C₃²

yes

—

C₃²×C₉×C₃₇

C₃²

yes

—

C₃₇×C₉⋊C₉

C₃²

yes

complemented

C₃₇×C₉⋊C₉

C₃²

yes

complemented

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₉²×C₃₇

C₃²

yes

complemented

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃³×C₃₇

C₃³

yes

characteristic

C₃³×C₃₇

C₃³

yes

—

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

3₋¹⁺²

yes

direct product factor

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

C₃³

yes

—

C₃×C₉×C₃₇

C₃³

yes

—

C₃×C₉×C₃₇

C₃³

yes

—

3₋¹⁺²²

C₃₇

yes

maximal, characteristic, direct product factor, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×3₋¹⁺²

yes

—

C₃²×C₃₇

C₃⁴

yes

characteristic

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

C₃×3₋¹⁺²

yes

complemented

C₃²×3₋¹⁺²

C₃×C₃₇

yes

—

#6900

C₃×C₃₇

yes

complemented

#6896

C₃×C₃₇

yes

complemented

#6894

C₃×C₃₇

yes

complemented

#6891

C₃×C₃₇

yes

—

C₉×3₋¹⁺²

C₃×C₃₇

yes

complemented

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃²×3₋¹⁺²

yes

—

C₃×C₃₇

3₋¹⁺⁴

yes

—

C₃⁴

C₃²×C₃₇

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃²×C₃₇

yes

—

C₃×3₋¹⁺²

C₃²×C₃₇

yes

—

C₃×3₋¹⁺²

C₃²×C₃₇

yes

—

C₃×3₋¹⁺²

C₃²×C₃₇

yes

—

C₃×3₋¹⁺²

C₃²×C₃₇

yes

—

C₃²×C₉

C₃²×C₃₇

yes

—

C₉⋊C₉

C₃²×C₃₇

yes

complemented

C₉⋊C₉

C₃²×C₃₇

yes

complemented

C₃²⋊C₉

C₃²×C₃₇

yes

—

C₃²⋊C₉

C₃²×C₃₇

yes

—

C₉²

C₃²×C₃₇

yes

complemented

C₃₇

3₋¹⁺²²

yes

characteristic, direct product factor, Sylow

C₃³

—

C₃³

—

C₃³

C₃³×C₃₇

yes

characteristic

C₃³

C₃³×C₃₇

yes

characteristic

C₃³

C₃³×C₃₇

yes

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

3₋¹⁺²×C₃₇

yes

direct product factor

C₃×C₉

—

C₃×C₉

C₃³×C₃₇

yes

—

C₃×C₉

C₃³×C₃₇

yes

—

C₃×C₉

C₃³×C₃₇

yes

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃×3₋¹⁺²×C₃₇

yes

—

C₃²

C₃⁴×C₃₇

yes

characteristic

C₉

—

C₉

—

C₉

C₃×3₋¹⁺²×C₃₇

yes

complemented

C₃

—

C₃

—

100

C₃

—

101

C₃

C₃²×3₋¹⁺²×C₃₇

yes

—

102

C₃

C₃₇×3₋¹⁺⁴

yes

—

	subgroup	G/H
center, socle	C₃²×C₃₇	C₃⁴
commutator, Frattini, cosocle	C₃²	C₃⁴×C₃₇
prime radical	C₃⁴×C₃₇	C₃²
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×C₃₇ ◁ G

subgroup

G/H

center, socle

C₃²×C₃₇

C₃⁴

commutator, Frattini, cosocle

C₃²

C₃⁴×C₃₇

prime radical

C₃⁴×C₃₇

C₃²

LCS

G ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₃²×C₃₇ ◁ G

N	H ≅ G/N	type
2. C₃₇×#6900	C₃	semidirect
3. C₃₇×#6896	C₃	semidirect
4. C₃₇×#6894	C₃	semidirect
6. C₉×3₋¹⁺²×C₃₇	C₃	semidirect
15. C₃₇×C₉⋊C₉	C₃²	semidirect
16. C₃₇×C₉⋊C₉	C₃²	semidirect
19. C₉²×C₃₇	C₃²	semidirect
29. 3₋¹⁺²×C₃₇	3₋¹⁺²	direct (N × H)
34. 3₋¹⁺²²	C₃₇	direct (N × H)
46. C₉×C₃₇	C₃×3₋¹⁺²	semidirect
48. #6900	C₃×C₃₇	semidirect
49. #6896	C₃×C₃₇	semidirect
50. #6894	C₃×C₃₇	semidirect
52. C₉×3₋¹⁺²	C₃×C₃₇	semidirect
66. C₉⋊C₉	C₃²×C₃₇	semidirect
67. C₉⋊C₉	C₃²×C₃₇	semidirect
70. C₉²	C₃²×C₃₇	semidirect
71. C₃₇	3₋¹⁺²²	direct (N × H)
81. 3₋¹⁺²	3₋¹⁺²×C₃₇	direct (N × H)
97. C₉	C₃×3₋¹⁺²×C₃₇	semidirect

H ≅ G/N

type

2. C₃₇×#6900

C₃

semidirect

3. C₃₇×#6896

C₃

semidirect

4. C₃₇×#6894

C₃

semidirect

6. C₉×3₋¹⁺²×C₃₇

C₃

semidirect

15. C₃₇×C₉⋊C₉

C₃²

semidirect

16. C₃₇×C₉⋊C₉

C₃²

semidirect

19. C₉²×C₃₇

C₃²

semidirect

29. 3₋¹⁺²×C₃₇

3₋¹⁺²

direct (N × H)

34. 3₋¹⁺²²

C₃₇

direct (N × H)

46. C₉×C₃₇

C₃×3₋¹⁺²

semidirect

48. #6900

C₃×C₃₇

semidirect

49. #6896

C₃×C₃₇

semidirect

50. #6894

C₃×C₃₇

semidirect

52. C₉×3₋¹⁺²

C₃×C₃₇

semidirect

66. C₉⋊C₉

C₃²×C₃₇

semidirect

67. C₉⋊C₉

C₃²×C₃₇

semidirect

70. C₉²

C₃²×C₃₇

semidirect

71. C₃₇

3₋¹⁺²²

direct (N × H)

81. 3₋¹⁺²

3₋¹⁺²×C₃₇

direct (N × H)

97. C₉

C₃×3₋¹⁺²×C₃₇

semidirect

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

subgroup	index
C₃²×3₋¹⁺²×C₃₇	3
C₃₇×#6900	3
C₃₇×#6896	3
C₃₇×#6894	3
C₃₇×#6891	3
C₉×3₋¹⁺²×C₃₇	3
3₋¹⁺²²	37

subgroup

index

C₃²×3₋¹⁺²×C₃₇

C₃₇×#6900

C₃₇×#6896

C₃₇×#6894

C₃₇×#6891

C₉×3₋¹⁺²×C₃₇

3₋¹⁺²²

Subgroups — 3₋¹⁺²²×C₃₇ #41004687

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3−1+22×C37 #41004687

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²²×C₃₇ #41004687