Group #41004649 — subgroups

Subgroups — C₃×C₉×3₋¹⁺²×C₃₇ #41004649

Order 26973.

1148 conjugacy classes of subgroups; only the 872 normal subgroups are shown.

normal subgroups only

This group has 1148 conjugacy classes of subgroups — too many to draw. Only its 872 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	1	yes	maximal, characteristic
2	C₃³×C₉×C₃₇	C₃	1	1	yes	maximal, characteristic
3	C₉×3₋¹⁺²×C₃₇	C₃	1	27	yes	maximal, direct product factor
4	C₃×C₃₇×C₉⋊C₉	C₃	1	6	yes	maximal, complemented
5	C₃×C₃₇×C₃²⋊C₉	C₃	1	2	yes	maximal
6	C₃×C₉²×C₃₇	C₃	1	3	yes	maximal, complemented
7	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
8	C₃×3₋¹⁺²×C₃₇	C₉	1	27	yes	direct product factor
9	C₃×3₋¹⁺²×C₃₇	C₃²	1	9	yes	—
10	C₃²×C₉×C₃₇	—	3	9	no	—
11	C₃²×C₉×C₃₇	C₃²	1	6	yes	—
12	C₃²×C₉×C₃₇	C₃²	1	2	yes	—
13	C₃²×C₉×C₃₇	C₃²	1	9	yes	—
14	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
15	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
16	C₃₇×C₉⋊C₉	C₃²	1	54	yes	complemented
17	C₃₇×C₃²⋊C₉	C₃²	1	18	yes	—
18	C₉²×C₃₇	C₃²	1	27	yes	complemented
19	C₃³×C₃₇	—	3	3	no	—
20	C₃³×C₃₇	—	3	6	no	—
21	C₃³×C₃₇	C₃×C₉	1	9	yes	—
22	C₃³×C₃₇	C₃³	1	3	yes	—
23	C₃³×C₃₇	C₃³	1	1	yes	characteristic
24	3₋¹⁺²×C₃₇	C₃×C₉	1	81	yes	direct product factor
25	C₃×C₉×C₃₇	—	3	6	no	—
26	C₃×C₉×C₃₇	—	3	18	no	—
27	C₃×C₉×C₃₇	—	3	27	no	—
28	C₃×C₉×C₃₇	C₃×C₉	1	27	yes	complemented
29	C₃×C₉×C₃₇	3₋¹⁺²	1	9	yes	direct product factor
30	C₃×C₉×C₃₇	C₃³	1	18	yes	—
31	C₃×C₉×C₃₇	C₃³	1	6	yes	—
32	C₃×C₉×C₃₇	C₃³	1	9	yes	—
33	C₃×C₉×C₃₇	C₃³	1	3	yes	—
34	C₃×C₉×3₋¹⁺²	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₃²×C₃₇	—	3	3	no	—
36	C₃²×C₃₇	—	3	6	no	—
37	C₃²×C₃₇	—	3	27	no	—
38	C₃²×C₃₇	C₃²×C₉	1	9	yes	—
39	C₃²×C₃₇	C₃²×C₉	1	3	yes	—
40	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
41	C₃²×C₃₇	#737	1	6	yes	—
42	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
43	C₉×C₃₇	—	3	6	no	—
44	C₉×C₃₇	—	3	18	no	—
45	C₉×C₃₇	C₃²×C₉	1	27	yes	complemented
46	C₉×C₃₇	C₃×3₋¹⁺²	1	9	yes	direct product factor
47	C₃²×3₋¹⁺²	C₃×C₃₇	1	1	yes	characteristic
48	C₃³×C₉	C₃×C₃₇	1	1	yes	characteristic
49	C₉×3₋¹⁺²	C₃×C₃₇	1	27	yes	direct product factor
50	C₃×C₉⋊C₉	C₃×C₃₇	1	6	yes	complemented
51	C₃×C₃²⋊C₉	C₃×C₃₇	1	2	yes	—
52	C₃×C₉²	C₃×C₃₇	1	3	yes	complemented
53	C₃×C₃₇	—	3	9	no	—
54	C₃×C₃₇	C₉×3₋¹⁺²	1	9	yes	direct product factor
55	C₃×C₃₇	C₃³×C₉	1	1	yes	characteristic
56	C₃×C₃₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
57	C₃×C₃₇	C₃×#737	1	2	yes	—
58	C₃⁴	C₃²×C₃₇	1	1	yes	characteristic
59	C₃×3₋¹⁺²	C₉×C₃₇	1	27	yes	direct product factor
60	C₃×3₋¹⁺²	C₃²×C₃₇	1	9	yes	—
61	C₃²×C₉	—	3	9	no	—
62	C₃²×C₉	C₃²×C₃₇	1	2	yes	—
63	C₃²×C₉	C₃²×C₃₇	1	9	yes	—
64	C₃²×C₉	C₃²×C₃₇	1	6	yes	—
65	C₃²×C₉	C₃²×C₃₇	1	3	yes	—
66	C₃²×C₉	C₃²×C₃₇	1	1	yes	characteristic
67	C₉⋊C₉	C₃²×C₃₇	1	54	yes	complemented
68	C₃²⋊C₉	C₃²×C₃₇	1	18	yes	—
69	C₉²	C₃²×C₃₇	1	27	yes	complemented
70	C₃₇	C₃×C₉×3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
71	C₃³	—	3	3	no	—
72	C₃³	—	3	6	no	—
73	C₃³	C₃×C₉×C₃₇	1	9	yes	—
74	C₃³	C₃³×C₃₇	1	3	yes	—
75	C₃³	C₃³×C₃₇	1	1	yes	characteristic
76	3₋¹⁺²	C₃×C₉×C₃₇	1	81	yes	direct product factor
77	C₃×C₉	—	3	6	no	—
78	C₃×C₉	—	3	18	no	—
79	C₃×C₉	—	3	27	no	—
80	C₃×C₉	C₃×C₉×C₃₇	1	27	yes	complemented
81	C₃×C₉	3₋¹⁺²×C₃₇	1	9	yes	direct product factor
82	C₃×C₉	C₃³×C₃₇	1	18	yes	—
83	C₃×C₉	C₃³×C₃₇	1	6	yes	—
84	C₃×C₉	C₃³×C₃₇	1	9	yes	—
85	C₃×C₉	C₃³×C₃₇	1	3	yes	—
86	C₃²	—	3	27	no	—
87	C₃²	—	3	3	no	—
88	C₃²	—	3	6	no	—
89	C₃²	C₃²×C₉×C₃₇	1	9	yes	—
90	C₃²	C₃²×C₉×C₃₇	1	3	yes	—
91	C₃²	C₃×3₋¹⁺²×C₃₇	1	3	yes	—
92	C₃²	C₃₇×#737	1	6	yes	—
93	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
94	C₉	—	3	6	no	—
95	C₉	—	3	18	no	—
96	C₉	C₃²×C₉×C₃₇	1	27	yes	complemented
97	C₉	C₃×3₋¹⁺²×C₃₇	1	9	yes	direct product factor
98	C₃	—	3	9	no	—
99	C₃	C₉×3₋¹⁺²×C₃₇	1	9	yes	direct product factor
100	C₃	C₃³×C₉×C₃₇	1	1	yes	characteristic
101	C₃	C₃²×3₋¹⁺²×C₃₇	1	1	yes	characteristic
102	C₃	C₃×C₃₇×#737	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×3₋¹⁺²×C₃₇

C₃

yes

maximal, characteristic

C₃³×C₉×C₃₇

C₃

yes

maximal, characteristic

C₉×3₋¹⁺²×C₃₇

C₃

yes

maximal, direct product factor

C₃×C₃₇×C₉⋊C₉

C₃

yes

maximal, complemented

C₃×C₃₇×C₃²⋊C₉

C₃

yes

maximal

C₃×C₉²×C₃₇

C₃

yes

maximal, complemented

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₃₇

C₉

yes

direct product factor

C₃×3₋¹⁺²×C₃₇

C₃²

yes

—

C₃²×C₉×C₃₇

—

C₃²×C₉×C₃₇

C₃²

yes

—

C₃²×C₉×C₃₇

C₃²

yes

—

C₃²×C₉×C₃₇

C₃²

yes

—

C₃²×C₉×C₃₇

C₃²

yes

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×C₉⋊C₉

C₃²

yes

complemented

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₉²×C₃₇

C₃²

yes

complemented

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

C₃×C₉

yes

—

C₃³×C₃₇

C₃³

yes

—

C₃³×C₃₇

C₃³

yes

characteristic

3₋¹⁺²×C₃₇

C₃×C₉

yes

direct product factor

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

C₃×C₉

yes

complemented

C₃×C₉×C₃₇

3₋¹⁺²

yes

direct product factor

C₃×C₉×C₃₇

C₃³

yes

—

C₃×C₉×C₃₇

C₃³

yes

—

C₃×C₉×C₃₇

C₃³

yes

—

C₃×C₉×C₃₇

C₃³

yes

—

C₃×C₉×3₋¹⁺²

C₃₇

yes

maximal, characteristic, direct product factor, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃²×C₉

yes

—

C₃²×C₃₇

C₃²×C₉

yes

—

C₃²×C₃₇

C₃×3₋¹⁺²

yes

—

C₃²×C₃₇

#737

yes

—

C₃²×C₃₇

C₃⁴

yes

characteristic

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

C₃²×C₉

yes

complemented

C₉×C₃₇

C₃×3₋¹⁺²

yes

direct product factor

C₃²×3₋¹⁺²

C₃×C₃₇

yes

characteristic

C₃³×C₉

C₃×C₃₇

yes

characteristic

C₉×3₋¹⁺²

C₃×C₃₇

yes

direct product factor

C₃×C₉⋊C₉

C₃×C₃₇

yes

complemented

C₃×C₃²⋊C₉

C₃×C₃₇

yes

—

C₃×C₉²

C₃×C₃₇

yes

complemented

C₃×C₃₇

—

C₃×C₃₇

C₉×3₋¹⁺²

yes

direct product factor

C₃×C₃₇

C₃³×C₉

yes

characteristic

C₃×C₃₇

C₃²×3₋¹⁺²

yes

characteristic

C₃×C₃₇

C₃×#737

yes

—

C₃⁴

C₃²×C₃₇

yes

characteristic

C₃×3₋¹⁺²

C₉×C₃₇

yes

direct product factor

C₃×3₋¹⁺²

C₃²×C₃₇

yes

—

C₃²×C₉

—

C₃²×C₉

C₃²×C₃₇

yes

—

C₃²×C₉

C₃²×C₃₇

yes

—

C₃²×C₉

C₃²×C₃₇

yes

—

C₃²×C₉

C₃²×C₃₇

yes

—

C₃²×C₉

C₃²×C₃₇

yes

characteristic

C₉⋊C₉

C₃²×C₃₇

yes

complemented

C₃²⋊C₉

C₃²×C₃₇

yes

—

C₉²

C₃²×C₃₇

yes

complemented

C₃₇

C₃×C₉×3₋¹⁺²

yes

characteristic, direct product factor, Sylow

C₃³

—

C₃³

—

C₃³

C₃×C₉×C₃₇

yes

—

C₃³

C₃³×C₃₇

yes

—

C₃³

C₃³×C₃₇

yes

characteristic

3₋¹⁺²

C₃×C₉×C₃₇

yes

direct product factor

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃×C₉×C₃₇

yes

complemented

C₃×C₉

3₋¹⁺²×C₃₇

yes

direct product factor

C₃×C₉

C₃³×C₃₇

yes

—

C₃×C₉

C₃³×C₃₇

yes

—

C₃×C₉

C₃³×C₃₇

yes

—

C₃×C₉

C₃³×C₃₇

yes

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃²×C₉×C₃₇

yes

—

C₃²

C₃²×C₉×C₃₇

yes

—

C₃²

C₃×3₋¹⁺²×C₃₇

yes

—

C₃²

C₃₇×#737

yes

—

C₃²

C₃⁴×C₃₇

yes

characteristic

C₉

—

C₉

—

C₉

C₃²×C₉×C₃₇

yes

complemented

C₉

C₃×3₋¹⁺²×C₃₇

yes

direct product factor

C₃

—

C₃

C₉×3₋¹⁺²×C₃₇

yes

direct product factor

100

C₃

C₃³×C₉×C₃₇

yes

characteristic

101

C₃

C₃²×3₋¹⁺²×C₃₇

yes

characteristic

102

C₃

C₃×C₃₇×#737

yes

—

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

subgroup

G/H

center

C₃²×C₉×C₃₇

C₃²

commutator

C₃

C₃³×C₉×C₃₇

Frattini, cosocle

C₃²

C₃⁴×C₃₇

socle

C₃³×C₃₇

C₃³

prime radical

C₃⁴×C₃₇

C₃²

LCS

G ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃²×C₉×C₃₇ ◁ G

N	H ≅ G/N	type
3. C₉×3₋¹⁺²×C₃₇	C₃	direct (N × H)
4. C₃×C₃₇×C₉⋊C₉	C₃	semidirect
6. C₃×C₉²×C₃₇	C₃	semidirect
8. C₃×3₋¹⁺²×C₃₇	C₉	direct (N × H)
16. C₃₇×C₉⋊C₉	C₃²	semidirect
18. C₉²×C₃₇	C₃²	semidirect
24. 3₋¹⁺²×C₃₇	C₃×C₉	direct (N × H)
28. C₃×C₉×C₃₇	C₃×C₉	semidirect
29. C₃×C₉×C₃₇	3₋¹⁺²	direct (N × H)
34. C₃×C₉×3₋¹⁺²	C₃₇	direct (N × H)
45. C₉×C₃₇	C₃²×C₉	semidirect
46. C₉×C₃₇	C₃×3₋¹⁺²	direct (N × H)
49. C₉×3₋¹⁺²	C₃×C₃₇	direct (N × H)
50. C₃×C₉⋊C₉	C₃×C₃₇	semidirect
52. C₃×C₉²	C₃×C₃₇	semidirect
54. C₃×C₃₇	C₉×3₋¹⁺²	direct (N × H)
59. C₃×3₋¹⁺²	C₉×C₃₇	direct (N × H)
67. C₉⋊C₉	C₃²×C₃₇	semidirect
69. C₉²	C₃²×C₃₇	semidirect
70. C₃₇	C₃×C₉×3₋¹⁺²	direct (N × H)
76. 3₋¹⁺²	C₃×C₉×C₃₇	direct (N × H)
80. C₃×C₉	C₃×C₉×C₃₇	semidirect
81. C₃×C₉	3₋¹⁺²×C₃₇	direct (N × H)
96. C₉	C₃²×C₉×C₃₇	semidirect
97. C₉	C₃×3₋¹⁺²×C₃₇	direct (N × H)
99. C₃	C₉×3₋¹⁺²×C₃₇	direct (N × H)

H ≅ G/N

type

3. C₉×3₋¹⁺²×C₃₇

C₃

direct (N × H)

4. C₃×C₃₇×C₉⋊C₉

C₃

semidirect

6. C₃×C₉²×C₃₇

C₃

semidirect

8. C₃×3₋¹⁺²×C₃₇

C₉

direct (N × H)

16. C₃₇×C₉⋊C₉

C₃²

semidirect

18. C₉²×C₃₇

C₃²

semidirect

24. 3₋¹⁺²×C₃₇

C₃×C₉

direct (N × H)

28. C₃×C₉×C₃₇

C₃×C₉

semidirect

29. C₃×C₉×C₃₇

3₋¹⁺²

direct (N × H)

34. C₃×C₉×3₋¹⁺²

C₃₇

direct (N × H)

45. C₉×C₃₇

C₃²×C₉

semidirect

46. C₉×C₃₇

C₃×3₋¹⁺²

direct (N × H)

49. C₉×3₋¹⁺²

C₃×C₃₇

direct (N × H)

50. C₃×C₉⋊C₉

C₃×C₃₇

semidirect

52. C₃×C₉²

C₃×C₃₇

semidirect

54. C₃×C₃₇

C₉×3₋¹⁺²

direct (N × H)

59. C₃×3₋¹⁺²

C₉×C₃₇

direct (N × H)

67. C₉⋊C₉

C₃²×C₃₇

semidirect

69. C₉²

C₃²×C₃₇

semidirect

70. C₃₇

C₃×C₉×3₋¹⁺²

direct (N × H)

76. 3₋¹⁺²

C₃×C₉×C₃₇

direct (N × H)

80. C₃×C₉

C₃×C₉×C₃₇

semidirect

81. C₃×C₉

3₋¹⁺²×C₃₇

direct (N × H)

96. C₉

C₃²×C₉×C₃₇

semidirect

97. C₉

C₃×3₋¹⁺²×C₃₇

direct (N × H)

99. C₃

C₉×3₋¹⁺²×C₃₇

direct (N × H)

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
C₃³×C₉×C₃₇	3
C₉×3₋¹⁺²×C₃₇	3
C₃×C₃₇×C₉⋊C₉	3
C₃×C₃₇×C₃²⋊C₉	3
C₃×C₉²×C₃₇	3
C₃×C₉×3₋¹⁺²	37

subgroup

index

C₃²×3₋¹⁺²×C₃₇

C₃³×C₉×C₃₇

C₉×3₋¹⁺²×C₃₇

C₃×C₃₇×C₉⋊C₉

C₃×C₃₇×C₃²⋊C₉

C₃×C₉²×C₃₇

C₃×C₉×3₋¹⁺²

Subgroups — C₃×C₉×3₋¹⁺²×C₃₇ #41004649

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×C9×3−1+2×C37 #41004649

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₉×3₋¹⁺²×C₃₇ #41004649