Group #41007644 — subgroups

Subgroups — #41007644

Order 26973.

1124 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 1124 conjugacy classes of subgroups — too many to draw. Only its 274 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	C₃×3₋¹⁺²×C₃₇⋊C₃	C₃	1	3	yes	maximal, complemented
2	C₃²×C₃²⋊(C₃₇⋊C₃)	C₃	1	1	yes	maximal, characteristic
3	#22582563	C₃	1	27	yes	maximal, complemented
4	#22582554	C₃	1	2	yes	maximal, complemented
5	#22582551	C₃	1	6	yes	maximal, complemented
6	C₃₇×#6891	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃³×C₃₇⋊C₃	C₃²	1	2	yes	—
8	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
9	C₃×#11758608	C₃²	1	6	yes	—
10	C₃×#11758605	C₃²	1	9	yes	complemented
11	C₃×#11758606	C₃²	1	9	yes	complemented
12	3₋¹⁺²×C₃₇⋊C₃	—	3	27	no	—
13	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	9	no	—
14	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	9	yes	—
15	C₃×C₉×C₃₇⋊C₃	C₃²	1	9	yes	complemented
16	C₃²⋊#66524	C₃²	1	18	yes	complemented
17	#15688502	C₃²	1	54	yes	complemented
18	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
19	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
20	C₃₇×C₃²⋊C₉	C₃²	1	9	yes	complemented
21	C₃³×C₃₇	—	3	3	no	—
22	C₃³×C₃₇	—	9	3	no	—
23	C₃³×C₃₇	C₃³	1	3	yes	—
24	C₃³×C₃₇	C₃³	1	1	yes	characteristic
25	C₃²×C₃₇⋊C₃	—	3	6	no	—
26	C₃²×C₃₇⋊C₃	—	9	3	no	—
27	C₃²×C₃₇⋊C₃	—	9	6	no	—
28	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	9	yes	—
29	C₃²×C₃₇⋊C₃	C₃³	1	6	yes	—
30	C₃²×C₃₇⋊C₃	C₃³	1	3	yes	—
31	3₋¹⁺²×C₃₇	—	3	9	no	—
32	C₃×C₉×C₃₇	C₃³	1	9	yes	—
33	C₃²⋊(C₃₇⋊C₃)	—	3	27	no	—
34	#11758608	—	3	18	no	—
35	C₃×#66524	C₃³	1	18	yes	—
36	#11758606	—	3	27	no	—
37	#11758605	—	3	27	no	—
38	C₉×C₃₇⋊C₃	—	3	27	no	—
39	#10624881	—	37	1	no	maximal, Sylow
40	C₃²×C₃₇	—	3	3	no	—
41	C₃²×C₃₇	—	3	3	no	—
42	C₃²×C₃₇	—	3	9	no	—
43	C₃²×C₃₇	—	9	9	no	—
44	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	3	yes	—
45	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
46	C₃×C₃₇⋊C₃	—	3	9	no	—
47	C₃×C₃₇⋊C₃	—	3	6	no	—
48	C₃×C₃₇⋊C₃	—	3	18	no	—
49	C₃×C₃₇⋊C₃	—	9	9	no	—
50	C₃×C₃₇⋊C₃	—	9	18	no	—
51	C₃×C₃₇⋊C₃	C₃×H(3,𝔽₃)	1	9	yes	—
52	C₉×C₃₇	—	3	9	no	—
53	#66524	—	3	18	no	—
54	C₃²×3₋¹⁺²	—	37	3	no	—
55	C₃²×H(3,𝔽₃)	—	37	1	no	—
56	#6893	—	37	27	no	—
57	#6891	—	37	2	no	—
58	#6891	—	37	6	no	—
59	#6891	C₃₇⋊C₃	1	1	yes	characteristic, complemented
60	C₃×C₃₇	—	3	3	no	—
61	C₃×C₃₇	—	9	3	no	—
62	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
63	C₃×C₃₇	3₋¹⁺⁴	1	3	yes	—
64	C₃₇⋊C₃	—	3	9	no	—
65	C₃₇⋊C₃	—	9	6	no	—
66	C₃⁴	—	37	1	no	—
67	C₃⁴	—	37	2	no	—
68	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
69	C₃×3₋¹⁺²	—	37	6	no	—
70	C₃×3₋¹⁺²	—	37	9	no	—
71	C₃×3₋¹⁺²	—	37	9	no	—
72	C₃×3₋¹⁺²	—	111	27	no	—
73	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	3	yes	—
74	C₃×H(3,𝔽₃)	—	37	9	no	—
75	C₃×H(3,𝔽₃)	—	111	9	no	—
76	C₃²×C₉	—	37	9	no	—
77	C₃²⋊C₉	—	37	18	no	—
78	C₃²⋊C₉	—	37	54	no	—
79	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	9	yes	complemented
80	C₃₇	#10624881	1	1	yes	characteristic, complemented, Sylow
81	C₃³	—	3	3	no	—
82	C₃³	—	9	3	no	—
83	C₃³	—	37	9	no	—
84	C₃³	—	37	3	no	—
85	C₃³	—	37	6	no	—
86	C₃³	—	333	6	no	—
87	C₃³	—	333	3	no	—
88	C₃³	—	111	6	no	—
89	C₃³	C₃²×C₃₇⋊C₃	1	3	yes	—
90	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
91	3₋¹⁺²	—	3	9	no	—
92	3₋¹⁺²	—	111	27	no	—
93	3₋¹⁺²	—	111	27	no	—
94	3₋¹⁺²	—	111	18	no	—
95	H(3,𝔽₃)	—	111	27	no	—
96	C₃×C₉	—	37	18	no	—
97	C₃×C₉	—	111	27	no	—
98	C₃×C₉	C₃²×C₃₇⋊C₃	1	9	yes	—
99	C₃²	—	3	3	no	—
100	C₃²	—	3	3	no	—
101	C₃²	—	3	9	no	—
102	C₃²	—	9	9	no	—
103	C₃²	—	37	9	no	—
104	C₃²	—	333	18	no	—
105	C₃²	—	333	9	no	—
106	C₃²	—	111	6	no	—
107	C₃²	—	111	18	no	—
108	C₃²	—	111	9	no	—
109	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	3	yes	—
110	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
111	C₉	—	3	9	no	—
112	C₉	—	111	18	no	—
113	C₃	—	3	3	no	—
114	C₃	—	9	3	no	—
115	C₃	—	333	6	no	—
116	C₃	—	111	9	no	—
117	C₃	C₃×H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
118	C₃	#22582679	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×3₋¹⁺²×C₃₇⋊C₃

C₃

yes

maximal, complemented

C₃²×C₃²⋊(C₃₇⋊C₃)

C₃

yes

maximal, characteristic

#22582563

C₃

yes

maximal, complemented

#22582554

C₃

yes

maximal, complemented

#22582551

C₃

yes

maximal, complemented

C₃₇×#6891

C₃

yes

maximal, characteristic, complemented

C₃³×C₃₇⋊C₃

C₃²

yes

—

C₃³×C₃₇⋊C₃

C₃²

yes

characteristic

C₃×#11758608

C₃²

yes

—

C₃×#11758605

C₃²

yes

complemented

C₃×#11758606

C₃²

yes

complemented

3₋¹⁺²×C₃₇⋊C₃

—

C₃×C₃²⋊(C₃₇⋊C₃)

—

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

yes

—

C₃×C₉×C₃₇⋊C₃

C₃²

yes

complemented

C₃²⋊#66524

C₃²

yes

complemented

#15688502

C₃²

yes

complemented

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₃₇

C₃²

yes

—

C₃₇×C₃²⋊C₉

C₃²

yes

complemented

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

C₃³

yes

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

H(3,𝔽₃)

yes

—

C₃²×C₃₇⋊C₃

C₃³

yes

—

C₃²×C₃₇⋊C₃

C₃³

yes

—

3₋¹⁺²×C₃₇

—

C₃×C₉×C₃₇

C₃³

yes

—

C₃²⋊(C₃₇⋊C₃)

—

#11758608

—

C₃×#66524

C₃³

yes

—

#11758606

—

#11758605

—

C₉×C₃₇⋊C₃

—

#10624881

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

—

C₃²×C₃₇

C₃⁴

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

C₃×H(3,𝔽₃)

yes

—

C₉×C₃₇

—

#66524

—

C₃²×3₋¹⁺²

—

C₃²×H(3,𝔽₃)

—

—

—

—

C₃₇⋊C₃

yes

characteristic, complemented

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃²×H(3,𝔽₃)

yes

characteristic

C₃×C₃₇

3₋¹⁺⁴

yes

—

C₃₇⋊C₃

—

C₃₇⋊C₃

—

C₃⁴

—

C₃⁴

—

C₃⁴

C₃×C₃₇⋊C₃

yes

characteristic

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×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

111

—

C₃²×C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

complemented

C₃₇

#10624881

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

333

—

C₃³

—

333

—

C₃³

—

111

—

C₃³

C₃²×C₃₇⋊C₃

yes

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

H(3,𝔽₃)

—

111

—

C₃×C₉

—

C₃×C₉

—

111

—

C₃×C₉

C₃²×C₃₇⋊C₃

yes

—

C₃²

—

100

C₃²

—

101

C₃²

—

102

C₃²

—

103

C₃²

—

104

C₃²

—

333

—

105

C₃²

—

333

—

106

C₃²

—

111

—

107

C₃²

—

111

—

108

C₃²

—

111

—

109

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

—

110

C₃²

C₃³×C₃₇⋊C₃

yes

characteristic

111

C₉

—

112

C₉

—

111

—

113

C₃

—

114

C₃

—

115

C₃

—

333

—

116

C₃

—

111

—

117

C₃

C₃×H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

118

C₃

#22582679

yes

—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₇⋊C₃
commutator, socle, cosocle	C₃²×C₃₇	C₃⁴
Fitting	C₃₇×#6891	C₃
prime radical	C₃²×C₃²⋊(C₃₇⋊C₃)	C₃
hypercenter	#6891	C₃₇⋊C₃
hypocenter	C₃₇	#10624881
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ #6891

subgroup

G/H

center, Frattini

C₃²

C₃³×C₃₇⋊C₃

commutator, socle, cosocle

C₃²×C₃₇

C₃⁴

Fitting

C₃₇×#6891

C₃

prime radical

C₃²×C₃²⋊(C₃₇⋊C₃)

C₃

C₃₇⋊C₃

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃₇

C₁ ◁ C₃² ◁ #6891

N	H ≅ G/N	type
1. C₃×3₋¹⁺²×C₃₇⋊C₃	C₃	semidirect
3. #22582563	C₃	semidirect
4. #22582554	C₃	semidirect
5. #22582551	C₃	semidirect
6. C₃₇×#6891	C₃	semidirect
10. C₃×#11758605	C₃²	semidirect
11. C₃×#11758606	C₃²	semidirect
15. C₃×C₉×C₃₇⋊C₃	C₃²	semidirect
16. C₃²⋊#66524	C₃²	semidirect
17. #15688502	C₃²	semidirect
20. C₃₇×C₃²⋊C₉	C₃²	semidirect
59. #6891	C₃₇⋊C₃	semidirect
79. C₃²⋊C₉	C₃×C₃₇⋊C₃	semidirect
80. C₃₇	#10624881	semidirect

H ≅ G/N

type

1. C₃×3₋¹⁺²×C₃₇⋊C₃

C₃

semidirect

3. #22582563

C₃

semidirect

4. #22582554

C₃

semidirect

5. #22582551

C₃

semidirect

6. C₃₇×#6891

C₃

semidirect

10. C₃×#11758605

C₃²

semidirect

11. C₃×#11758606

C₃²

semidirect

15. C₃×C₉×C₃₇⋊C₃

C₃²

semidirect

16. C₃²⋊#66524

C₃²

semidirect

17. #15688502

C₃²

semidirect

20. C₃₇×C₃²⋊C₉

C₃²

semidirect

59. #6891

C₃₇⋊C₃

semidirect

79. C₃²⋊C₉

C₃×C₃₇⋊C₃

semidirect

80. C₃₇

#10624881

semidirect

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

subgroup	index
C₃×3₋¹⁺²×C₃₇⋊C₃	3
C₃²×C₃²⋊(C₃₇⋊C₃)	3
#22582563	3
#22582554	3
#22582551	3
C₃₇×#6891	3
#10624881	37

subgroup

index

C₃×3₋¹⁺²×C₃₇⋊C₃

C₃²×C₃²⋊(C₃₇⋊C₃)

C₃₇×#6891

Subgroups — #41007644

Conjugacy classes of subgroups

Complements

Maximal subgroups