Group #41008166 — subgroups

Subgroups — C₃×#22582649 #41008166

Order 26973.

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

normal subgroups only

This group has 980 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	C₃²×#11758605	C₃	1	1	yes	maximal, characteristic
2	C₃×3₋¹⁺²×C₃₇⋊C₃	C₃	1	3	yes	maximal
3	#22582649	C₃	1	27	yes	maximal, direct product factor
4	C₃×#15688521	C₃	1	2	yes	maximal, complemented
5	C₃×#15688520	C₃	1	6	yes	maximal, complemented
6	C₃×C₃₇×#733	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
8	C₃×#11758608	C₃²	1	6	yes	—
9	C₃×#11758605	—	3	3	no	—
10	C₃×#11758605	—	3	3	no	—
11	C₃×#11758605	—	3	6	no	—
12	C₃×#11758605	C₃²	1	9	yes	—
13	C₃×#11758606	—	3	3	no	—
14	3₋¹⁺²×C₃₇⋊C₃	—	3	18	no	—
15	3₋¹⁺²×C₃₇⋊C₃	C₃²	1	27	yes	—
16	C₃²×#66524	C₃²	1	2	yes	—
17	C₃×C₉×C₃₇⋊C₃	—	3	3	no	—
18	#15688521	C₃²	1	18	yes	complemented
19	#15688520	C₃²	1	54	yes	complemented
20	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
21	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
22	C₃₇×#733	C₃²	1	9	yes	complemented
23	C₃³×C₃₇	C₃³	1	1	yes	characteristic
24	C₃²×C₃₇⋊C₃	—	9	3	no	—
25	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	3	yes	—
26	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	6	yes	—
27	C₃²×C₃₇⋊C₃	C₃³	1	3	yes	—
28	3₋¹⁺²×C₃₇	—	3	6	no	—
29	3₋¹⁺²×C₃₇	C₃³	1	9	yes	—
30	C₃×C₉×C₃₇	—	3	2	no	—
31	C₃×C₉×C₃₇	—	3	1	no	—
32	C₃×C₉×C₃₇	—	3	3	no	—
33	C₃×C₉×C₃₇	C₃³	1	3	yes	—
34	#11758608	—	3	12	no	—
35	#11758608	C₃³	1	18	yes	—
36	C₃×#66524	—	3	6	no	—
37	C₃×#66524	—	3	4	no	—
38	C₃×#66524	—	3	2	no	—
39	C₃×#66524	C₃³	1	6	yes	—
40	#11758606	—	3	27	no	—
41	#11758605	—	3	27	no	—
42	#11758605	—	3	27	no	—
43	C₉×C₃₇⋊C₃	—	3	27	no	—
44	C₃×#6912	—	37	1	no	maximal, Sylow
45	C₃²×C₃₇	—	3	3	no	—
46	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	2	yes	—
47	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
48	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
49	C₃×C₃₇⋊C₃	—	3	9	no	—
50	C₃×C₃₇⋊C₃	—	9	6	no	—
51	C₃×C₃₇⋊C₃	—	9	3	no	—
52	C₃×C₃₇⋊C₃	C₃×H(3,𝔽₃)	1	9	yes	—
53	C₉×C₃₇	—	3	9	no	—
54	C₉×C₃₇	—	3	3	no	—
55	#66524	—	3	18	no	—
56	#66524	—	3	6	no	—
57	C₃²×3₋¹⁺²	—	37	1	no	—
58	C₃²×3₋¹⁺²	—	37	3	no	—
59	#6912	—	37	27	no	—
60	C₃×#733	—	37	6	no	—
61	C₃×#733	—	37	2	no	—
62	C₃×#733	C₃₇⋊C₃	1	1	yes	characteristic, complemented
63	C₃×C₃₇	—	3	2	no	—
64	C₃×C₃₇	—	3	1	no	—
65	C₃×C₃₇	#6912	1	3	yes	complemented
66	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
67	C₃₇⋊C₃	—	3	9	no	—
68	C₃⁴	—	37	1	no	—
69	C₃×3₋¹⁺²	—	37	9	no	—
70	C₃×3₋¹⁺²	—	37	6	no	—
71	C₃×3₋¹⁺²	—	37	27	no	—
72	C₃×3₋¹⁺²	—	111	3	no	—
73	C₃×3₋¹⁺²	—	111	6	no	—
74	C₃×3₋¹⁺²	—	111	3	no	—
75	C₃×3₋¹⁺²	—	111	3	no	—
76	C₃×3₋¹⁺²	—	111	18	no	—
77	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	3	yes	—
78	C₃²×C₉	—	37	2	no	—
79	C₃²×C₉	—	111	3	no	—
80	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
81	#733	—	37	54	no	—
82	#733	—	37	18	no	—
83	#733	C₃×C₃₇⋊C₃	1	9	yes	complemented
84	C₃₇	C₃×#6912	1	1	yes	characteristic, complemented, Sylow
85	C₃³	—	37	6	no	—
86	C₃³	—	37	3	no	—
87	C₃³	—	37	3	no	—
88	C₃³	—	333	3	no	—
89	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
90	3₋¹⁺²	—	3	6	no	—
91	3₋¹⁺²	—	37	18	no	—
92	3₋¹⁺²	—	111	27	no	—
93	3₋¹⁺²	—	111	27	no	—
94	3₋¹⁺²	—	111	27	no	—
95	3₋¹⁺²	—	111	12	no	—
96	3₋¹⁺²	C₃²×C₃₇⋊C₃	1	9	yes	—
97	C₃×C₉	—	3	2	no	—
98	C₃×C₉	—	3	3	no	—
99	C₃×C₉	—	3	1	no	—
100	C₃×C₉	—	37	6	no	—
101	C₃×C₉	—	111	27	no	—
102	C₃×C₉	—	111	4	no	—
103	C₃×C₉	—	111	2	no	—
104	C₃×C₉	—	111	6	no	—
105	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
106	C₃²	—	3	3	no	—
107	C₃²	—	37	9	no	—
108	C₃²	—	333	6	no	—
109	C₃²	—	333	3	no	—
110	C₃²	—	111	9	no	—
111	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	2	yes	—
112	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
113	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
114	C₉	—	3	3	no	—
115	C₉	—	3	9	no	—
116	C₉	—	111	6	no	—
117	C₉	—	111	18	no	—
118	C₃	—	3	2	no	—
119	C₃	—	3	1	no	—
120	C₃	—	111	9	no	—
121	C₃	#22582649	1	3	yes	direct product factor
122	C₃	C₃×H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#11758605

C₃

yes

maximal, characteristic

C₃×3₋¹⁺²×C₃₇⋊C₃

C₃

yes

maximal

#22582649

C₃

yes

maximal, direct product factor

C₃×#15688521

C₃

yes

maximal, complemented

C₃×#15688520

C₃

yes

maximal, complemented

C₃×C₃₇×#733

C₃

yes

maximal, characteristic, complemented

C₃³×C₃₇⋊C₃

C₃²

yes

characteristic

C₃×#11758608

C₃²

yes

—

—

—

—

C₃²

yes

—

C₃×#11758606

—

3₋¹⁺²×C₃₇⋊C₃

—

3₋¹⁺²×C₃₇⋊C₃

C₃²

yes

—

C₃²×#66524

C₃²

yes

—

C₃×C₉×C₃₇⋊C₃

—

#15688521

C₃²

yes

complemented

#15688520

C₃²

yes

complemented

C₃×3₋¹⁺²×C₃₇

C₃²

yes

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×#733

C₃²

yes

complemented

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

H(3,𝔽₃)

yes

—

C₃²×C₃₇⋊C₃

H(3,𝔽₃)

yes

—

C₃²×C₃₇⋊C₃

C₃³

yes

—

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

C₃³

yes

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

C₃³

yes

—

#11758608

—

#11758608

C₃³

yes

—

—

—

—

C₃³

yes

—

#11758606

—

#11758605

—

#11758605

—

C₉×C₃₇⋊C₃

—

C₃×#6912

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃⁴

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

C₃×H(3,𝔽₃)

yes

—

C₉×C₃₇

—

C₉×C₃₇

—

#66524

—

#66524

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

—

—

—

—

C₃₇⋊C₃

yes

characteristic, complemented

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

#6912

yes

complemented

C₃×C₃₇

C₃²×H(3,𝔽₃)

yes

characteristic

C₃₇⋊C₃

—

C₃⁴

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

—

C₃²×C₉

—

C₃²×C₉

—

111

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

—

—

C₃×C₃₇⋊C₃

yes

complemented

C₃₇

C₃×#6912

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

—

333

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

3₋¹⁺²

C₃²×C₃₇⋊C₃

yes

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

100

C₃×C₉

—

101

C₃×C₉

—

111

—

102

C₃×C₉

—

111

—

103

C₃×C₉

—

111

—

104

C₃×C₉

—

111

—

105

C₃×C₉

C₃²×C₃₇⋊C₃

yes

—

106

C₃²

—

107

C₃²

—

108

C₃²

—

333

—

109

C₃²

—

333

—

110

C₃²

—

111

—

111

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

—

112

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

113

C₃²

C₃³×C₃₇⋊C₃

yes

characteristic

114

C₉

—

115

C₉

—

116

C₉

—

111

—

117

C₉

—

111

—

118

C₃

—

119

C₃

—

120

C₃

—

111

—

121

C₃

#22582649

yes

direct product factor

122

C₃

C₃×H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₃₇⋊C₃
commutator, cosocle	C₃²×C₃₇	C₃⁴
Frattini	C₃²	C₃³×C₃₇⋊C₃
Fitting	C₃×C₃₇×#733	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃³×C₃₇⋊C₃	C₃²
hypercenter	C₃×#733	C₃₇⋊C₃
hypocenter	C₃₇	C₃×#6912
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×#733

subgroup

G/H

center

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

commutator, cosocle

C₃²×C₃₇

C₃⁴

Frattini

C₃²

C₃³×C₃₇⋊C₃

Fitting

C₃×C₃₇×#733

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

prime radical

C₃³×C₃₇⋊C₃

C₃²

C₃₇⋊C₃

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ C₃×#733

N	H ≅ G/N	type
3. #22582649	C₃	direct (N × H)
4. C₃×#15688521	C₃	semidirect
5. C₃×#15688520	C₃	semidirect
6. C₃×C₃₇×#733	C₃	semidirect
18. #15688521	C₃²	semidirect
19. #15688520	C₃²	semidirect
22. C₃₇×#733	C₃²	semidirect
62. C₃×#733	C₃₇⋊C₃	semidirect
65. C₃×C₃₇	#6912	semidirect
83. #733	C₃×C₃₇⋊C₃	semidirect
84. C₃₇	C₃×#6912	semidirect
121. C₃	#22582649	direct (N × H)

H ≅ G/N

type

3. #22582649

C₃

direct (N × H)

4. C₃×#15688521

C₃

semidirect

5. C₃×#15688520

C₃

semidirect

6. C₃×C₃₇×#733

C₃

semidirect

18. #15688521

C₃²

semidirect

19. #15688520

C₃²

semidirect

22. C₃₇×#733

C₃²

semidirect

62. C₃×#733

C₃₇⋊C₃

semidirect

65. C₃×C₃₇

#6912

semidirect

83. #733

C₃×C₃₇⋊C₃

semidirect

84. C₃₇

C₃×#6912

semidirect

121. C₃

#22582649

direct (N × H)

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

subgroup	index
C₃²×#11758605	3
C₃×3₋¹⁺²×C₃₇⋊C₃	3
#22582649	3
C₃×#15688521	3
C₃×#15688520	3
C₃×C₃₇×#733	3
C₃×#6912	37

subgroup

index

C₃×3₋¹⁺²×C₃₇⋊C₃

C₃×C₃₇×#733

Subgroups — C₃×#22582649 #41008166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#22582649 #41008166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#22582649 #41008166