Group #41007643 — subgroups

Subgroups — #41007643

Order 26973.

1124 conjugacy classes of subgroups; only the 292 normal subgroups are shown.

normal subgroups only

This group has 1124 conjugacy classes of subgroups — too many to draw. Only its 292 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×#11758608	C₃	1	2	yes	maximal, complemented
2	C₃²×C₃²⋊(C₃₇⋊C₃)	C₃	1	1	yes	maximal, characteristic
3	#22582565	C₃	1	18	yes	maximal, complemented
4	#22582564	C₃	1	9	yes	maximal, complemented
5	#22582551	C₃	1	9	yes	maximal, complemented
6	C₃²×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃³×C₃₇⋊C₃	C₃²	1	3	yes	—
8	C₃×#11758608	—	3	18	no	—
9	C₃×#11758608	C₃²	1	6	yes	—
10	C₃×#11758608	C₃²	1	6	yes	complemented
11	C₃×#11758608	C₃²	1	6	yes	complemented
12	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	9	no	—
13	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	3	yes	—
14	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	6	yes	—
15	C₃²×#66524	C₃²	1	6	yes	complemented
16	#15688502	C₃²	1	81	yes	complemented
17	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
18	C₃×3₋¹⁺²×C₃₇	—	3	9	no	—
19	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
20	C₃×3₋¹⁺²×C₃₇	C₃²	1	6	yes	complemented
21	C₃²×C₉×C₃₇	C₃²	1	3	yes	complemented
22	C₃³×C₃₇	—	9	3	no	—
23	C₃³×C₃₇	H(3,𝔽₃)	1	9	yes	—
24	C₃³×C₃₇	C₃³	1	1	yes	characteristic
25	C₃³×C₃₇	C₃³	1	1	yes	characteristic
26	C₃³×C₃₇	C₃³	1	2	yes	—
27	C₃²×C₃₇⋊C₃	—	3	9	no	—
28	C₃²×C₃₇⋊C₃	—	9	9	no	—
29	C₃²×C₃₇⋊C₃	C₃³	1	9	yes	—
30	3₋¹⁺²×C₃₇	—	3	9	no	—
31	3₋¹⁺²×C₃₇	—	3	18	no	—
32	C₃×C₉×C₃₇	—	3	9	no	—
33	C₃×C₉×C₃₇	C₃³	1	9	yes	—
34	C₃²⋊(C₃₇⋊C₃)	—	3	18	no	—
35	C₃²⋊(C₃₇⋊C₃)	—	3	9	no	—
36	#11758608	—	3	18	no	—
37	#11758608	—	3	18	no	—
38	#11758608	—	3	18	no	—
39	C₃×#66524	—	3	18	no	—
40	C₃×#66524	C₃³	1	18	yes	—
41	#10624881	—	37	1	no	maximal, Sylow
42	C₃²×C₃₇	—	3	3	no	—
43	C₃²×C₃₇	—	3	3	no	—
44	C₃²×C₃₇	—	3	6	no	—
45	C₃²×C₃₇	—	9	9	no	—
46	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	6	yes	—
47	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	3	yes	—
48	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	3	yes	—
49	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
50	C₃×C₃₇⋊C₃	—	3	9	no	—
51	C₃×C₃₇⋊C₃	—	3	27	no	—
52	C₃×C₃₇⋊C₃	—	9	27	no	—
53	C₉×C₃₇	—	3	9	no	—
54	#66524	—	3	18	no	—
55	C₃²×3₋¹⁺²	—	37	2	no	—
56	C₃²×3₋¹⁺²	C₃₇⋊C₃	1	1	yes	characteristic, complemented
57	C₃²×H(3,𝔽₃)	—	37	1	no	—
58	#6893	—	37	18	no	—
59	#6893	—	37	9	no	—
60	#6891	—	37	9	no	—
61	C₃×C₃₇	—	3	3	no	—
62	C₃×C₃₇	—	3	6	no	—
63	C₃×C₃₇	—	3	3	no	—
64	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
65	C₃×C₃₇	3₋¹⁺⁴	1	3	yes	—
66	C₃₇⋊C₃	—	9	9	no	—
67	C₃⁴	—	37	3	no	—
68	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
69	C₃×3₋¹⁺²	—	3	9	no	—
70	C₃×3₋¹⁺²	—	37	6	no	—
71	C₃×3₋¹⁺²	—	37	6	no	—
72	C₃×3₋¹⁺²	—	37	6	no	—
73	C₃×3₋¹⁺²	—	111	18	no	—
74	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	3	yes	—
75	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	6	yes	complemented
76	C₃×H(3,𝔽₃)	—	37	6	no	—
77	C₃×H(3,𝔽₃)	—	37	3	no	—
78	C₃×H(3,𝔽₃)	—	111	9	no	—
79	C₃²×C₉	—	37	6	no	—
80	C₃²×C₉	C₃×C₃₇⋊C₃	1	3	yes	complemented
81	C₃²⋊C₉	—	37	81	no	—
82	C₃₇	#10624881	1	1	yes	characteristic, complemented, Sylow
83	C₃³	—	9	3	no	—
84	C₃³	—	37	9	no	—
85	C₃³	—	333	9	no	—
86	C₃³	—	111	9	no	—
87	C₃³	C₃²⋊(C₃₇⋊C₃)	1	9	yes	—
88	C₃³	C₃²×C₃₇⋊C₃	1	2	yes	—
89	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
90	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
91	3₋¹⁺²	—	3	9	no	—
92	3₋¹⁺²	—	3	18	no	—
93	3₋¹⁺²	—	111	18	no	—
94	3₋¹⁺²	—	111	18	no	—
95	3₋¹⁺²	—	111	18	no	—
96	H(3,𝔽₃)	—	111	9	no	—
97	H(3,𝔽₃)	—	111	18	no	—
98	C₃×C₉	—	3	9	no	—
99	C₃×C₉	—	37	18	no	—
100	C₃×C₉	—	111	18	no	—
101	C₃×C₉	C₃²×C₃₇⋊C₃	1	9	yes	—
102	C₃²	—	3	3	no	—
103	C₃²	—	3	6	no	—
104	C₃²	—	3	3	no	—
105	C₃²	—	9	9	no	—
106	C₃²	—	333	27	no	—
107	C₃²	—	111	9	no	—
108	C₃²	—	111	27	no	—
109	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
110	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	6	yes	—
111	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
112	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
113	C₉	—	3	9	no	—
114	C₉	—	111	18	no	—
115	C₃	—	3	3	no	—
116	C₃	—	3	3	no	—
117	C₃	—	3	6	no	—
118	C₃	—	333	9	no	—
119	C₃	C₃²×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
120	C₃	#22582679	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#11758608

C₃

yes

maximal, complemented

C₃²×C₃²⋊(C₃₇⋊C₃)

C₃

yes

maximal, characteristic

#22582565

C₃

yes

maximal, complemented

#22582564

C₃

yes

maximal, complemented

#22582551

C₃

yes

maximal, complemented

C₃²×3₋¹⁺²×C₃₇

C₃

yes

maximal, characteristic, complemented

C₃³×C₃₇⋊C₃

C₃²

yes

—

C₃×#11758608

—

C₃×#11758608

C₃²

yes

—

C₃×#11758608

C₃²

yes

complemented

C₃×#11758608

C₃²

yes

complemented

C₃×C₃²⋊(C₃₇⋊C₃)

—

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

yes

—

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

yes

—

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

—

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

characteristic

C₃³×C₃₇

C₃³

yes

characteristic

C₃³×C₃₇

C₃³

yes

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

C₃³

yes

—

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

C₃³

yes

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

—

—

—

—

C₃³

yes

—

#10624881

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

—

C₃²×C₃₇

C₃⁴

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

#66524

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

C₃₇⋊C₃

yes

characteristic, complemented

C₃²×H(3,𝔽₃)

—

#6893

—

#6893

—

#6891

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃²×H(3,𝔽₃)

yes

characteristic

C₃×C₃₇

3₋¹⁺⁴

yes

—

C₃₇⋊C₃

—

C₃⁴

—

C₃⁴

C₃×C₃₇⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

complemented

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

111

—

C₃²×C₉

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

complemented

C₃²⋊C₉

—

C₃₇

#10624881

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

333

—

C₃³

—

111

—

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

—

C₃³

C₃²×C₃₇⋊C₃

yes

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

H(3,𝔽₃)

—

111

—

H(3,𝔽₃)

—

111

—

C₃×C₉

—

C₃×C₉

—

100

C₃×C₉

—

111

—

101

C₃×C₉

C₃²×C₃₇⋊C₃

yes

—

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

—

333

—

107

C₃²

—

111

—

108

C₃²

—

111

—

109

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

—

110

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

—

111

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

—

112

C₃²

C₃³×C₃₇⋊C₃

yes

characteristic

113

C₉

—

114

C₉

—

111

—

115

C₃

—

116

C₃

—

117

C₃

—

118

C₃

—

333

—

119

C₃

C₃²×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

120

C₃

#22582679

yes

—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₇⋊C₃
commutator, socle, cosocle	C₃²×C₃₇	C₃⁴
Fitting	C₃²×3₋¹⁺²×C₃₇	C₃
prime radical	C₃²×C₃²⋊(C₃₇⋊C₃)	C₃
hypercenter	C₃²×3₋¹⁺²	C₃₇⋊C₃
hypocenter	C₃₇	#10624881
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃²×3₋¹⁺²

subgroup

G/H

center, Frattini

C₃²

C₃³×C₃₇⋊C₃

commutator, socle, cosocle

C₃²×C₃₇

C₃⁴

Fitting

C₃²×3₋¹⁺²×C₃₇

C₃

prime radical

C₃²×C₃²⋊(C₃₇⋊C₃)

C₃

hypercenter

C₃²×3₋¹⁺²

C₃₇⋊C₃

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃₇

C₁ ◁ C₃² ◁ C₃²×3₋¹⁺²

N	H ≅ G/N	type
1. C₃²×#11758608	C₃	semidirect
3. #22582565	C₃	semidirect
4. #22582564	C₃	semidirect
5. #22582551	C₃	semidirect
6. C₃²×3₋¹⁺²×C₃₇	C₃	semidirect
10. C₃×#11758608	C₃²	semidirect
11. C₃×#11758608	C₃²	semidirect
15. C₃²×#66524	C₃²	semidirect
16. #15688502	C₃²	semidirect
20. C₃×3₋¹⁺²×C₃₇	C₃²	semidirect
21. C₃²×C₉×C₃₇	C₃²	semidirect
56. C₃²×3₋¹⁺²	C₃₇⋊C₃	semidirect
75. C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	semidirect
80. C₃²×C₉	C₃×C₃₇⋊C₃	semidirect
82. C₃₇	#10624881	semidirect

H ≅ G/N

type

1. C₃²×#11758608

C₃

semidirect

3. #22582565

C₃

semidirect

4. #22582564

C₃

semidirect

5. #22582551

C₃

semidirect

6. C₃²×3₋¹⁺²×C₃₇

C₃

semidirect

10. C₃×#11758608

C₃²

semidirect

11. C₃×#11758608

C₃²

semidirect

15. C₃²×#66524

C₃²

semidirect

16. #15688502

C₃²

semidirect

20. C₃×3₋¹⁺²×C₃₇

C₃²

semidirect

21. C₃²×C₉×C₃₇

C₃²

semidirect

56. C₃²×3₋¹⁺²

C₃₇⋊C₃

semidirect

75. C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

semidirect

80. C₃²×C₉

C₃×C₃₇⋊C₃

semidirect

82. C₃₇

#10624881

semidirect

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

subgroup	index
C₃²×#11758608	3
C₃²×C₃²⋊(C₃₇⋊C₃)	3
#22582565	3
#22582564	3
#22582551	3
C₃²×3₋¹⁺²×C₃₇	3
#10624881	37

subgroup

index

C₃²×C₃²⋊(C₃₇⋊C₃)

C₃²×3₋¹⁺²×C₃₇

Subgroups — #41007643

Conjugacy classes of subgroups

Complements

Maximal subgroups