Group #41006489 — subgroups

Subgroups — #41006489

Order 26973.

2072 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 2072 conjugacy classes of subgroups — too many to draw. Only its 85 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, complemented
2	C₃×#15688514	C₃	1	9	yes	maximal, complemented
3	#22582554	C₃	1	1	yes	maximal, characteristic, complemented
4	#22582550	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃₇×#6891	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃³×C₃₇⋊C₃	—	3	3	no	—
7	C₃³×C₃₇⋊C₃	—	3	9	no	—
8	C₃³×C₃₇⋊C₃	—	3	27	no	—
9	C₃³×C₃₇⋊C₃	C₃²	1	3	yes	—
10	C₃×#11758608	C₃²	1	3	yes	complemented
11	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
12	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	3	yes	complemented
13	#15688514	—	3	27	no	—
14	C₃²⋊#66524	—	3	3	no	—
15	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
16	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	complemented
17	C₃₇×C₃²⋊C₉	—	3	3	no	—
18	C₃³×C₃₇	—	3	9	no	—
19	C₃³×C₃₇	—	3	3	no	—
20	C₃³×C₃₇	H(3,𝔽₃)	1	3	yes	—
21	C₃³×C₃₇	C₃³	1	1	yes	characteristic
22	C₃²×C₃₇⋊C₃	—	3	243	no	—
23	C₃²×C₃₇⋊C₃	—	3	27	no	—
24	C₃²×C₃₇⋊C₃	—	3	54	no	—
25	C₃²×C₃₇⋊C₃	—	3	27	no	—
26	C₃²×C₃₇⋊C₃	—	3	6	no	—
27	C₃²×C₃₇⋊C₃	—	3	3	no	—
28	C₃²×C₃₇⋊C₃	—	3	27	no	—
29	C₃²×C₃₇⋊C₃	—	3	3	no	—
30	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	9	yes	complemented
31	3₋¹⁺²×C₃₇	—	3	9	no	—
32	C₃×C₉×C₃₇	—	3	3	no	—
33	C₃²⋊(C₃₇⋊C₃)	—	3	9	no	—
34	C₃²⋊(C₃₇⋊C₃)	—	9	9	no	—
35	#11758608	—	3	9	no	—
36	C₃×#66524	—	3	3	no	—
37	#10624730	—	37	1	no	maximal, Sylow
38	C₃²×C₃₇	—	3	27	no	—
39	C₃²×C₃₇	—	3	3	no	—
40	C₃²×C₃₇	—	3	6	no	—
41	C₃²×C₃₇	—	3	3	no	—
42	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	9	yes	complemented
43	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	3	yes	—
44	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
45	C₃×C₃₇⋊C₃	—	3	243	no	—
46	C₃×C₃₇⋊C₃	—	3	54	no	—
47	C₃×C₃₇⋊C₃	—	3	9	no	—
48	C₃×C₃₇⋊C₃	—	3	27	no	—
49	C₃×C₃₇⋊C₃	—	3	27	no	—
50	C₃×C₃₇⋊C₃	—	9	3	no	—
51	C₃×C₃₇⋊C₃	—	9	9	no	—
52	C₉×C₃₇	—	9	3	no	—
53	#66524	—	9	3	no	—
54	C₃⁵	—	37	1	no	—
55	C₃×AΣL(1,𝔽₂₇)	—	37	9	no	—
56	#6891	—	37	1	no	—
57	#6891	C₃₇⋊C₃	1	1	yes	characteristic, complemented
58	C₃²⋊H(3,𝔽₃)	—	37	1	no	—
59	C₃×C₃₇	—	3	9	no	—
60	C₃×C₃₇	—	3	3	no	—
61	C₃×C₃₇	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
62	C₃×C₃₇	C₃×AΣL(1,𝔽₂₇)	1	3	yes	—
63	C₃₇⋊C₃	—	3	27	no	—
64	C₃₇⋊C₃	—	27	3	no	—
65	C₃⁴	—	37	3	no	—
66	C₃⁴	—	111	9	no	—
67	C₃⁴	—	111	27	no	—
68	C₃⁴	—	111	3	no	—
69	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
70	C₃×3₋¹⁺²	—	37	3	no	—
71	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	3	yes	complemented
72	C₃×H(3,𝔽₃)	—	37	3	no	—
73	C₃×H(3,𝔽₃)	—	111	3	no	—
74	AΣL(1,𝔽₂₇)	—	111	27	no	—
75	C₃²⋊C₉	—	3	3	no	—
76	C₃²⋊C₉	—	111	3	no	—
77	C₃₇	#10624730	1	1	yes	characteristic, complemented, Sylow
78	C₃³	—	3	9	no	—
79	C₃³	—	3	3	no	—
80	C₃³	—	37	9	no	—
81	C₃³	—	111	27	no	—
82	C₃³	—	111	243	no	—
83	C₃³	—	111	27	no	—
84	C₃³	—	111	54	no	—
85	C₃³	—	111	3	no	—
86	C₃³	—	111	27	no	—
87	C₃³	—	111	6	no	—
88	C₃³	—	111	3	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	9	no	—
93	H(3,𝔽₃)	—	333	9	no	—
94	H(3,𝔽₃)	—	111	9	no	—
95	C₃×C₉	—	3	3	no	—
96	C₃×C₉	—	111	3	no	—
97	C₃²	—	3	27	no	—
98	C₃²	—	3	3	no	—
99	C₃²	—	3	6	no	—
100	C₃²	—	3	3	no	—
101	C₃²	—	333	3	no	—
102	C₃²	—	333	9	no	—
103	C₃²	—	111	27	no	—
104	C₃²	—	111	243	no	—
105	C₃²	—	111	54	no	—
106	C₃²	—	111	9	no	—
107	C₃²	—	111	27	no	—
108	C₃²	#15688514	1	9	yes	complemented
109	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	3	yes	—
110	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
111	C₉	—	9	3	no	—
112	C₉	—	333	3	no	—
113	C₃	—	3	9	no	—
114	C₃	—	3	3	no	—
115	C₃	—	111	27	no	—
116	C₃	—	999	3	no	—
117	C₃	(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)	1	1	yes	characteristic
118	C₃	C₃×#15688514	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃⁴×C₃₇⋊C₃

C₃

yes

maximal, characteristic, complemented

C₃×#15688514

C₃

yes

maximal, complemented

#22582554

C₃

yes

maximal, characteristic, complemented

#22582550

C₃

yes

maximal, characteristic, complemented

C₃₇×#6891

C₃

yes

maximal, characteristic, complemented

C₃³×C₃₇⋊C₃

—

C₃³×C₃₇⋊C₃

—

C₃³×C₃₇⋊C₃

—

C₃³×C₃₇⋊C₃

C₃²

yes

—

C₃×#11758608

C₃²

yes

complemented

C₃×C₃²⋊(C₃₇⋊C₃)

—

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

yes

complemented

#15688514

—

C₃²⋊#66524

—

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₃₇

C₃²

yes

complemented

C₃₇×C₃²⋊C₉

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

H(3,𝔽₃)

yes

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

243

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

H(3,𝔽₃)

yes

complemented

3₋¹⁺²×C₃₇

—

C₃×C₉×C₃₇

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

—

—

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

AΣL(1,𝔽₂₇)

yes

complemented

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₃₇⋊C₃

—

243

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

#66524

—

C₃⁵

—

C₃×AΣL(1,𝔽₂₇)

—

#6891

—

#6891

C₃₇⋊C₃

yes

characteristic, complemented

C₃²⋊H(3,𝔽₃)

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃²⋊H(3,𝔽₃)

yes

characteristic

C₃×C₃₇

C₃×AΣL(1,𝔽₂₇)

yes

—

C₃₇⋊C₃

—

C₃₇⋊C₃

—

C₃⁴

—

C₃⁴

—

111

—

C₃⁴

—

111

—

C₃⁴

—

111

—

C₃⁴

C₃×C₃₇⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

complemented

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

111

—

AΣL(1,𝔽₂₇)

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

C₃₇

#10624730

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

—

111

—

C₃³

—

111

243

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

111

—

H(3,𝔽₃)

—

333

—

H(3,𝔽₃)

—

111

—

C₃×C₉

—

C₃×C₉

—

111

—

C₃²

—

C₃²

—

C₃²

—

100

C₃²

—

101

C₃²

—

333

—

102

C₃²

—

333

—

103

C₃²

—

111

—

104

C₃²

—

111

243

—

105

C₃²

—

111

—

106

C₃²

—

111

—

107

C₃²

—

111

—

108

C₃²

#15688514

yes

complemented

109

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

—

110

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

111

C₉

—

112

C₉

—

333

—

113

C₃

—

114

C₃

—

115

C₃

—

111

—

116

C₃

—

999

—

117

C₃

(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)

yes

characteristic

118

C₃

C₃×#15688514

yes

—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6891	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
hypercenter	#6891	C₃₇⋊C₃
hypocenter	C₃₇	#10624730
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ #6891

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃₇×#6891

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

C₃₇⋊C₃

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ #6891

N	H ≅ G/N	type
1. C₃⁴×C₃₇⋊C₃	C₃	semidirect
2. C₃×#15688514	C₃	semidirect
3. #22582554	C₃	semidirect
4. #22582550	C₃	semidirect
5. C₃₇×#6891	C₃	semidirect
10. C₃×#11758608	C₃²	semidirect
12. C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	semidirect
16. C₃×3₋¹⁺²×C₃₇	C₃²	semidirect
30. C₃²×C₃₇⋊C₃	H(3,𝔽₃)	semidirect
42. C₃²×C₃₇	AΣL(1,𝔽₂₇)	semidirect
57. #6891	C₃₇⋊C₃	semidirect
71. C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	semidirect
77. C₃₇	#10624730	semidirect
108. C₃²	#15688514	semidirect

H ≅ G/N

type

1. C₃⁴×C₃₇⋊C₃

C₃

semidirect

2. C₃×#15688514

C₃

semidirect

3. #22582554

C₃

semidirect

4. #22582550

C₃

semidirect

5. C₃₇×#6891

C₃

semidirect

10. C₃×#11758608

C₃²

semidirect

12. C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

semidirect

16. C₃×3₋¹⁺²×C₃₇

C₃²

semidirect

30. C₃²×C₃₇⋊C₃

H(3,𝔽₃)

semidirect

42. C₃²×C₃₇

AΣL(1,𝔽₂₇)

semidirect

57. #6891

C₃₇⋊C₃

semidirect

71. C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

semidirect

77. C₃₇

#10624730

semidirect

108. C₃²

#15688514

semidirect

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

subgroup	index
C₃⁴×C₃₇⋊C₃	3
C₃×#15688514	3
#22582554	3
#22582550	3
C₃₇×#6891	3
#10624730	37

subgroup

index

C₃⁴×C₃₇⋊C₃

C₃₇×#6891

Subgroups — #41006489

Conjugacy classes of subgroups

Complements

Maximal subgroups