Group #41008244 — subgroups

Subgroups — C₃²×3₋¹⁺²×C₃₇⋊C₃ #41008244

Order 26973.

7208 conjugacy classes of subgroups; only the 3082 normal subgroups are shown.

normal subgroups only

This group has 7208 conjugacy classes of subgroups — too many to draw. Only its 3082 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	C₃²×#11758605	C₃	1	3	yes	maximal, complemented
4	C₃²×#11758606	C₃	1	3	yes	maximal, complemented
5	C₃×3₋¹⁺²×C₃₇⋊C₃	C₃	1	108	yes	maximal, direct product factor
6	C₃²×C₉×C₃₇⋊C₃	C₃	1	3	yes	maximal, complemented
7	C₃²×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃³×C₃₇⋊C₃	—	3	27	no	—
9	C₃³×C₃₇⋊C₃	C₃²	1	36	yes	—
10	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
11	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
12	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
13	C₃×#11758608	C₃²	1	72	yes	complemented
14	C₃×#11758605	C₃²	1	108	yes	complemented
15	C₃×#11758606	C₃²	1	108	yes	complemented
16	3₋¹⁺²×C₃₇⋊C₃	C₃²	1	729	yes	direct product factor
17	C₃²×#66524	C₃²	1	6	yes	complemented
18	C₃×C₉×C₃₇⋊C₃	C₃²	1	108	yes	complemented
19	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
20	C₃×3₋¹⁺²×C₃₇	C₃²	1	36	yes	complemented
21	C₃²×C₉×C₃₇	C₃²	1	3	yes	complemented
22	C₃³×C₃₇	—	3	9	no	—
23	C₃³×C₃₇	C₃³	1	12	yes	—
24	C₃³×C₃₇	C₃³	1	1	yes	characteristic
25	C₃²×C₃₇⋊C₃	—	3	9	no	—
26	C₃²×C₃₇⋊C₃	—	3	9	no	—
27	C₃²×C₃₇⋊C₃	—	3	324	no	—
28	C₃²×C₃₇⋊C₃	3₋¹⁺²	1	27	yes	direct product factor
29	C₃²×C₃₇⋊C₃	C₃³	1	81	yes	—
30	C₃²×C₃₇⋊C₃	C₃³	1	12	yes	—
31	C₃²×C₃₇⋊C₃	C₃³	1	12	yes	—
32	C₃²×C₃₇⋊C₃	C₃³	1	12	yes	—
33	3₋¹⁺²×C₃₇	C₃³	1	81	yes	complemented
34	C₃×C₉×C₃₇	C₃³	1	36	yes	complemented
35	#11758608	C₃³	1	162	yes	complemented
36	C₃×#66524	C₃³	1	72	yes	complemented
37	#11758606	C₃³	1	243	yes	complemented
38	#11758605	C₃³	1	243	yes	complemented
39	C₉×C₃₇⋊C₃	C₃³	1	243	yes	complemented
40	C₃³×3₋¹⁺²	—	37	1	no	maximal, Sylow
41	C₃²×C₃₇	—	3	36	no	—
42	C₃²×C₃₇	C₃×3₋¹⁺²	1	9	yes	complemented
43	C₃²×C₃₇	C₃⁴	1	9	yes	—
44	C₃²×C₃₇	C₃⁴	1	4	yes	—
45	C₃×C₃₇⋊C₃	—	3	36	no	—
46	C₃×C₃₇⋊C₃	—	3	36	no	—
47	C₃×C₃₇⋊C₃	—	3	243	no	—
48	C₃×C₃₇⋊C₃	C₃×3₋¹⁺²	1	108	yes	direct product factor
49	C₃×C₃₇⋊C₃	C₃⁴	1	9	yes	—
50	C₃×C₃₇⋊C₃	C₃⁴	1	9	yes	—
51	C₃×C₃₇⋊C₃	C₃⁴	1	9	yes	—
52	C₉×C₃₇	C₃⁴	1	27	yes	complemented
53	#66524	C₃⁴	1	54	yes	complemented
54	C₃⁵	—	37	1	no	—
55	C₃²×3₋¹⁺²	—	37	2	no	—
56	C₃²×3₋¹⁺²	—	37	108	no	—
57	C₃²×3₋¹⁺²	—	37	3	no	—
58	C₃²×3₋¹⁺²	—	37	3	no	—
59	C₃²×3₋¹⁺²	C₃₇⋊C₃	1	1	yes	characteristic, direct product factor
60	C₃³×C₉	—	37	3	no	—
61	C₃×C₃₇	—	3	9	no	—
62	C₃×C₃₇	C₃²×3₋¹⁺²	1	12	yes	complemented
63	C₃×C₃₇	C₃⁵	1	1	yes	characteristic
64	C₃₇⋊C₃	—	3	9	no	—
65	C₃₇⋊C₃	—	3	9	no	—
66	C₃₇⋊C₃	C₃²×3₋¹⁺²	1	27	yes	direct product factor
67	C₃⁴	—	37	1	no	—
68	C₃⁴	—	37	36	no	—
69	C₃⁴	—	37	1	no	—
70	C₃⁴	—	37	1	no	—
71	C₃⁴	—	111	27	no	—
72	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
73	C₃×3₋¹⁺²	—	37	72	no	—
74	C₃×3₋¹⁺²	—	37	729	no	—
75	C₃×3₋¹⁺²	—	37	108	no	—
76	C₃×3₋¹⁺²	—	37	108	no	—
77	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	36	yes	direct product factor
78	C₃²×C₉	—	37	6	no	—
79	C₃²×C₉	—	37	108	no	—
80	C₃²×C₉	C₃×C₃₇⋊C₃	1	3	yes	complemented
81	C₃₇	C₃³×3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
82	C₃³	—	3	9	no	—
83	C₃³	—	37	27	no	—
84	C₃³	—	37	12	no	—
85	C₃³	—	37	81	no	—
86	C₃³	—	37	12	no	—
87	C₃³	—	37	12	no	—
88	C₃³	—	111	324	no	—
89	C₃³	—	111	9	no	—
90	C₃³	—	111	9	no	—
91	C₃³	C₃²×C₃₇⋊C₃	1	12	yes	—
92	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
93	3₋¹⁺²	—	37	162	no	—
94	3₋¹⁺²	—	37	243	no	—
95	3₋¹⁺²	—	37	243	no	—
96	3₋¹⁺²	C₃²×C₃₇⋊C₃	1	81	yes	direct product factor
97	C₃×C₉	—	37	72	no	—
98	C₃×C₉	—	37	243	no	—
99	C₃×C₉	C₃²×C₃₇⋊C₃	1	36	yes	complemented
100	C₃²	—	3	36	no	—
101	C₃²	—	37	108	no	—
102	C₃²	—	37	9	no	—
103	C₃²	—	37	9	no	—
104	C₃²	—	37	9	no	—
105	C₃²	—	111	243	no	—
106	C₃²	—	111	36	no	—
107	C₃²	—	111	36	no	—
108	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	9	yes	direct product factor
109	C₃²	C₃³×C₃₇⋊C₃	1	9	yes	—
110	C₃²	C₃³×C₃₇⋊C₃	1	4	yes	—
111	C₉	—	37	54	no	—
112	C₉	C₃³×C₃₇⋊C₃	1	27	yes	complemented
113	C₃	—	3	9	no	—
114	C₃	—	37	27	no	—
115	C₃	—	111	9	no	—
116	C₃	—	111	9	no	—
117	C₃	C₃×3₋¹⁺²×C₃₇⋊C₃	1	12	yes	direct product factor
118	C₃	C₃⁴×C₃₇⋊C₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃⁴×C₃₇⋊C₃

C₃

yes

maximal, characteristic

C₃²×#11758608

C₃

yes

maximal, complemented

C₃²×#11758605

C₃

yes

maximal, complemented

C₃²×#11758606

C₃

yes

maximal, complemented

C₃×3₋¹⁺²×C₃₇⋊C₃

C₃

108

yes

maximal, direct product factor

C₃²×C₉×C₃₇⋊C₃

C₃

yes

maximal, complemented

C₃²×3₋¹⁺²×C₃₇

C₃

yes

maximal, characteristic, complemented

C₃³×C₃₇⋊C₃

—

C₃³×C₃₇⋊C₃

C₃²

yes

—

C₃³×C₃₇⋊C₃

C₃²

yes

characteristic

C₃³×C₃₇⋊C₃

C₃²

yes

characteristic

C₃³×C₃₇⋊C₃

C₃²

yes

characteristic

C₃×#11758608

C₃²

yes

complemented

C₃×#11758605

C₃²

108

yes

complemented

C₃×#11758606

C₃²

108

yes

complemented

3₋¹⁺²×C₃₇⋊C₃

C₃²

729

yes

direct product factor

C₃²×#66524

C₃²

yes

complemented

C₃×C₉×C₃₇⋊C₃

C₃²

108

yes

complemented

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₃₇

C₃²

yes

complemented

C₃²×C₉×C₃₇

C₃²

yes

complemented

C₃³×C₃₇

—

C₃³×C₃₇

C₃³

yes

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

324

—

C₃²×C₃₇⋊C₃

3₋¹⁺²

yes

direct product factor

C₃²×C₃₇⋊C₃

C₃³

yes

—

C₃²×C₃₇⋊C₃

C₃³

yes

—

C₃²×C₃₇⋊C₃

C₃³

yes

—

C₃²×C₃₇⋊C₃

C₃³

yes

—

3₋¹⁺²×C₃₇

C₃³

yes

complemented

C₃×C₉×C₃₇

C₃³

yes

complemented

#11758608

C₃³

162

yes

complemented

C₃×#66524

C₃³

yes

complemented

#11758606

C₃³

243

yes

complemented

#11758605

C₃³

243

yes

complemented

C₉×C₃₇⋊C₃

C₃³

243

yes

complemented

C₃³×3₋¹⁺²

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

C₃×3₋¹⁺²

yes

complemented

C₃²×C₃₇

C₃⁴

yes

—

C₃²×C₃₇

C₃⁴

yes

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

243

—

C₃×C₃₇⋊C₃

C₃×3₋¹⁺²

108

yes

direct product factor

C₃×C₃₇⋊C₃

C₃⁴

yes

—

C₃×C₃₇⋊C₃

C₃⁴

yes

—

C₃×C₃₇⋊C₃

C₃⁴

yes

—

C₉×C₃₇

C₃⁴

yes

complemented

#66524

C₃⁴

yes

complemented

C₃⁵

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

—

108

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

C₃₇⋊C₃

yes

characteristic, direct product factor

C₃³×C₉

—

C₃×C₃₇

—

C₃×C₃₇

C₃²×3₋¹⁺²

yes

complemented

C₃×C₃₇

C₃⁵

yes

characteristic

C₃₇⋊C₃

—

C₃₇⋊C₃

—

C₃₇⋊C₃

C₃²×3₋¹⁺²

yes

direct product factor

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

111

—

C₃⁴

C₃×C₃₇⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

729

—

C₃×3₋¹⁺²

—

108

—

C₃×3₋¹⁺²

—

108

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

direct product factor

C₃²×C₉

—

C₃²×C₉

—

108

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

complemented

C₃₇

C₃³×3₋¹⁺²

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

111

324

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

C₃²×C₃₇⋊C₃

yes

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

162

—

3₋¹⁺²

—

243

—

3₋¹⁺²

—

243

—

3₋¹⁺²

C₃²×C₃₇⋊C₃

yes

direct product factor

C₃×C₉

—

C₃×C₉

—

243

—

C₃×C₉

C₃²×C₃₇⋊C₃

yes

complemented

100

C₃²

—

101

C₃²

—

108

—

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

111

243

—

106

C₃²

—

111

—

107

C₃²

—

111

—

108

C₃²

3₋¹⁺²×C₃₇⋊C₃

yes

direct product factor

109

C₃²

C₃³×C₃₇⋊C₃

yes

—

110

C₃²

C₃³×C₃₇⋊C₃

yes

—

111

C₉

—

112

C₉

C₃³×C₃₇⋊C₃

yes

complemented

113

C₃

—

114

C₃

—

115

C₃

—

111

—

116

C₃

—

111

—

117

C₃

C₃×3₋¹⁺²×C₃₇⋊C₃

yes

direct product factor

118

C₃

C₃⁴×C₃₇⋊C₃

yes

characteristic

	subgroup	G/H
center	C₃³	C₃²×C₃₇⋊C₃
commutator, cosocle	C₃×C₃₇	C₃⁵
Frattini	C₃	C₃⁴×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	C₃³×3₋¹⁺²
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃³ ◁ C₃²×3₋¹⁺²

subgroup

G/H

center

C₃³

C₃²×C₃₇⋊C₃

commutator, cosocle

C₃×C₃₇

C₃⁵

Frattini

C₃

C₃⁴×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

C₃³×3₋¹⁺²

LCS

G ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃³ ◁ C₃²×3₋¹⁺²

N	H ≅ G/N	type
2. C₃²×#11758608	C₃	semidirect
3. C₃²×#11758605	C₃	semidirect
4. C₃²×#11758606	C₃	semidirect
5. C₃×3₋¹⁺²×C₃₇⋊C₃	C₃	direct (N × H)
6. C₃²×C₉×C₃₇⋊C₃	C₃	semidirect
7. C₃²×3₋¹⁺²×C₃₇	C₃	semidirect
13. C₃×#11758608	C₃²	semidirect
14. C₃×#11758605	C₃²	semidirect
15. C₃×#11758606	C₃²	semidirect
16. 3₋¹⁺²×C₃₇⋊C₃	C₃²	direct (N × H)
17. C₃²×#66524	C₃²	semidirect
18. C₃×C₉×C₃₇⋊C₃	C₃²	semidirect
20. C₃×3₋¹⁺²×C₃₇	C₃²	semidirect
21. C₃²×C₉×C₃₇	C₃²	semidirect
28. C₃²×C₃₇⋊C₃	3₋¹⁺²	direct (N × H)
33. 3₋¹⁺²×C₃₇	C₃³	semidirect
34. C₃×C₉×C₃₇	C₃³	semidirect
35. #11758608	C₃³	semidirect
36. C₃×#66524	C₃³	semidirect
37. #11758606	C₃³	semidirect
38. #11758605	C₃³	semidirect
39. C₉×C₃₇⋊C₃	C₃³	semidirect
42. C₃²×C₃₇	C₃×3₋¹⁺²	semidirect
48. C₃×C₃₇⋊C₃	C₃×3₋¹⁺²	direct (N × H)
52. C₉×C₃₇	C₃⁴	semidirect
53. #66524	C₃⁴	semidirect
59. C₃²×3₋¹⁺²	C₃₇⋊C₃	direct (N × H)
62. C₃×C₃₇	C₃²×3₋¹⁺²	semidirect
66. C₃₇⋊C₃	C₃²×3₋¹⁺²	direct (N × H)
77. C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	direct (N × H)
80. C₃²×C₉	C₃×C₃₇⋊C₃	semidirect
81. C₃₇	C₃³×3₋¹⁺²	semidirect
96. 3₋¹⁺²	C₃²×C₃₇⋊C₃	direct (N × H)
99. C₃×C₉	C₃²×C₃₇⋊C₃	semidirect
108. C₃²	3₋¹⁺²×C₃₇⋊C₃	direct (N × H)
112. C₉	C₃³×C₃₇⋊C₃	semidirect
117. C₃	C₃×3₋¹⁺²×C₃₇⋊C₃	direct (N × H)

H ≅ G/N

type

2. C₃²×#11758608

C₃

semidirect

3. C₃²×#11758605

C₃

semidirect

4. C₃²×#11758606

C₃

semidirect

5. C₃×3₋¹⁺²×C₃₇⋊C₃

C₃

direct (N × H)

6. C₃²×C₉×C₃₇⋊C₃

C₃

semidirect

7. C₃²×3₋¹⁺²×C₃₇

C₃

semidirect

13. C₃×#11758608

C₃²

semidirect

14. C₃×#11758605

C₃²

semidirect

15. C₃×#11758606

C₃²

semidirect

16. 3₋¹⁺²×C₃₇⋊C₃

C₃²

direct (N × H)

17. C₃²×#66524

C₃²

semidirect

18. C₃×C₉×C₃₇⋊C₃

C₃²

semidirect

20. C₃×3₋¹⁺²×C₃₇

C₃²

semidirect

21. C₃²×C₉×C₃₇

C₃²

semidirect

28. C₃²×C₃₇⋊C₃

3₋¹⁺²

direct (N × H)

33. 3₋¹⁺²×C₃₇

C₃³

semidirect

34. C₃×C₉×C₃₇

C₃³

semidirect

35. #11758608

C₃³

semidirect

36. C₃×#66524

C₃³

semidirect

37. #11758606

C₃³

semidirect

38. #11758605

C₃³

semidirect

39. C₉×C₃₇⋊C₃

C₃³

semidirect

42. C₃²×C₃₇

C₃×3₋¹⁺²

semidirect

48. C₃×C₃₇⋊C₃

C₃×3₋¹⁺²

direct (N × H)

52. C₉×C₃₇

C₃⁴

semidirect

53. #66524

C₃⁴

semidirect

59. C₃²×3₋¹⁺²

C₃₇⋊C₃

direct (N × H)

62. C₃×C₃₇

C₃²×3₋¹⁺²

semidirect

66. C₃₇⋊C₃

C₃²×3₋¹⁺²

direct (N × H)

77. C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

direct (N × H)

80. C₃²×C₉

C₃×C₃₇⋊C₃

semidirect

81. C₃₇

C₃³×3₋¹⁺²

semidirect

96. 3₋¹⁺²

C₃²×C₃₇⋊C₃

direct (N × H)

99. C₃×C₉

C₃²×C₃₇⋊C₃

semidirect

108. C₃²

3₋¹⁺²×C₃₇⋊C₃

direct (N × H)

112. C₉

C₃³×C₃₇⋊C₃

semidirect

117. C₃

C₃×3₋¹⁺²×C₃₇⋊C₃

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
C₃²×#11758605	3
C₃²×#11758606	3
C₃×3₋¹⁺²×C₃₇⋊C₃	3
C₃²×C₉×C₃₇⋊C₃	3
C₃²×3₋¹⁺²×C₃₇	3
C₃³×3₋¹⁺²	37

subgroup

index

C₃⁴×C₃₇⋊C₃

C₃²×#11758608

C₃²×#11758605

C₃²×#11758606

C₃×3₋¹⁺²×C₃₇⋊C₃

C₃²×C₉×C₃₇⋊C₃

C₃²×3₋¹⁺²×C₃₇

C₃³×3₋¹⁺²

Subgroups — C₃²×3₋¹⁺²×C₃₇⋊C₃ #41008244

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C32×3−1+2×C37⋊C3 #41008244

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×3₋¹⁺²×C₃₇⋊C₃ #41008244