Group #41008282 — subgroups

Subgroups — #41008282

Order 26973.

350 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 48 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×C₃²⋊#66526	C₃	1	3	yes	maximal
2	C₃×3₋¹⁺²×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
3	C₃²×#66526	—	3	3	no	—
4	C₃²⋊#66526	—	3	9	no	—
5	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
6	C₃×#11758608	C₉	1	2	yes	complemented
7	C₃×#11758605	—	3	1	no	—
8	C₃×#11758606	—	3	1	no	—
9	3₋¹⁺²×C₃₇⋊C₃	—	3	9	no	—
10	C₃×C₉×C₃₇⋊C₃	—	3	1	no	—
11	C₃×3₋¹⁺²×C₃₇	C₉	1	1	yes	characteristic, complemented
12	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
13	C₃²×C₃₇⋊C₃	—	3	3	no	—
14	C₃²×C₃₇⋊C₃	—	3	3	no	—
15	C₃²×C₃₇⋊C₃	—	3	3	no	—
16	C₃²×C₃₇⋊C₃	—	3	3	no	—
17	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
18	C₃²×C₃₇⋊C₃	3₋¹⁺²	1	1	yes	characteristic
19	C₃²×C₃₇⋊C₃	3₋¹⁺²	1	1	yes	characteristic
20	C₃×#66526	—	3	9	no	—
21	C₃×#66526	—	9	3	no	—
22	C₃×#66526	—	9	3	no	—
23	C₃×#66526	—	9	3	no	—
24	3₋¹⁺²×C₃₇	—	3	3	no	—
25	C₃×C₉×C₃₇	—	3	1	no	—
26	#11758608	—	3	6	no	—
27	C₃×#66524	—	3	2	no	—
28	#11758606	—	3	9	no	—
29	#11758605	—	3	9	no	—
30	C₉×C₃₇⋊C₃	—	3	9	no	—
31	#10624452	—	37	1	no	maximal, Sylow
32	C₃²×C₃₇	—	3	1	no	—
33	C₃²×C₃₇	—	3	3	no	—
34	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
35	C₃×C₃₇⋊C₃	—	3	1	no	—
36	C₃×C₃₇⋊C₃	—	3	1	no	—
37	C₃×C₃₇⋊C₃	—	3	3	no	—
38	C₃×C₃₇⋊C₃	—	3	3	no	—
39	C₃×C₃₇⋊C₃	—	9	3	no	—
40	C₃×C₃₇⋊C₃	—	9	3	no	—
41	C₃×C₃₇⋊C₃	—	9	3	no	—
42	C₃×C₃₇⋊C₃	AΣL(1,𝔽₂₇)	1	3	yes	—
43	C₃×C₃₇⋊C₃	AΣL(1,𝔽₂₇)	1	3	yes	—
44	C₃×C₃₇⋊C₃	#731	1	3	yes	—
45	C₃×C₃₇⋊C₃	#733	1	3	yes	—
46	#66526	—	9	9	no	—
47	C₉×C₃₇	—	3	3	no	—
48	#66524	—	3	6	no	—
49	C₃²×3₋¹⁺²	—	37	1	no	—
50	C₃×C₃²⋊C₉	—	37	3	no	—
51	C₃×C₃₇	—	9	1	no	—
52	C₃×C₃₇	#6866	1	1	yes	characteristic
53	C₃×C₃₇	3₋¹⁺²⋊C₉	1	3	yes	—
54	C₃₇⋊C₃	—	9	1	no	—
55	C₃₇⋊C₃	—	9	1	no	—
56	C₃₇⋊C₃	#6857	1	3	yes	—
57	C₃₇⋊C₃	#6858	1	3	yes	—
58	C₃₇⋊C₃	#6860	1	3	yes	—
59	C₃⁴	—	37	1	no	—
60	C₃×3₋¹⁺²	—	37	2	no	—
61	C₃×3₋¹⁺²	—	111	1	no	—
62	C₃×3₋¹⁺²	—	111	1	no	—
63	C₃×3₋¹⁺²	—	111	9	no	—
64	C₃×3₋¹⁺²	#66526	1	1	yes	characteristic, complemented
65	C₃²×C₉	—	111	1	no	—
66	C₃²×C₉	—	111	3	no	—
67	C₃²⋊C₉	—	111	9	no	—
68	C₃₇	#10624452	1	1	yes	characteristic, complemented, Sylow
69	C₃³	—	37	1	no	—
70	C₃³	—	37	1	no	—
71	C₃³	—	37	1	no	—
72	C₃³	—	111	3	no	—
73	C₃³	—	111	3	no	—
74	C₃³	—	111	3	no	—
75	C₃³	—	111	3	no	—
76	C₃³	C₃×#66526	1	1	yes	characteristic
77	3₋¹⁺²	—	3	3	no	—
78	3₋¹⁺²	—	111	9	no	—
79	3₋¹⁺²	—	111	9	no	—
80	3₋¹⁺²	—	111	6	no	—
81	C₃×C₉	—	3	1	no	—
82	C₃×C₉	—	333	3	no	—
83	C₃×C₉	—	333	3	no	—
84	C₃×C₉	—	333	3	no	—
85	C₃×C₉	—	111	9	no	—
86	C₃×C₉	—	111	9	no	—
87	C₃×C₉	—	111	2	no	—
88	C₃²	—	3	1	no	—
89	C₃²	—	3	3	no	—
90	C₃²	—	37	3	no	—
91	C₃²	—	37	3	no	—
92	C₃²	—	37	3	no	—
93	C₃²	—	37	3	no	—
94	C₃²	—	333	3	no	—
95	C₃²	—	333	3	no	—
96	C₃²	—	333	3	no	—
97	C₃²	—	111	1	no	—
98	C₃²	—	111	1	no	—
99	C₃²	—	111	3	no	—
100	C₃²	—	111	3	no	—
101	C₃²	C₃²⋊#66526	1	1	yes	characteristic
102	C₉	—	3	3	no	—
103	C₉	—	333	9	no	—
104	C₉	—	111	6	no	—
105	C₃	—	9	1	no	—
106	C₃	—	37	3	no	—
107	C₃	—	37	3	no	—
108	C₃	—	37	3	no	—
109	C₃	—	333	1	no	—
110	C₃	—	333	1	no	—
111	C₃	#22582691	1	1	yes	characteristic
112	C₃	3₋¹⁺²⋊#66526	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₃²⋊#66526

C₃

yes

maximal

C₃×3₋¹⁺²×C₃₇⋊C₃

C₃

yes

maximal, characteristic

C₃²×#66526

—

C₃²⋊#66526

—

C₃³×C₃₇⋊C₃

C₃²

yes

characteristic

C₃×#11758608

C₉

yes

complemented

C₃×#11758605

—

C₃×#11758606

—

3₋¹⁺²×C₃₇⋊C₃

—

C₃×C₉×C₃₇⋊C₃

—

C₃×3₋¹⁺²×C₃₇

C₉

yes

characteristic, complemented

C₃³×C₃₇

C₃×C₉

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇⋊C₃

3₋¹⁺²

yes

characteristic

C₃²×C₃₇⋊C₃

3₋¹⁺²

yes

—

—

—

—

3₋¹⁺²×C₃₇

—

C₃×C₉×C₃₇

—

—

—

—

—

C₉×C₃₇⋊C₃

—

#10624452

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃²⋊C₉

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

AΣL(1,𝔽₂₇)

yes

—

C₃×C₃₇⋊C₃

AΣL(1,𝔽₂₇)

yes

—

C₃×C₃₇⋊C₃

#731

yes

—

C₃×C₃₇⋊C₃

#733

yes

—

#66526

—

C₉×C₃₇

—

#66524

—

C₃²×3₋¹⁺²

—

C₃×C₃²⋊C₉

—

C₃×C₃₇

—

C₃×C₃₇

#6866

yes

characteristic

C₃×C₃₇

3₋¹⁺²⋊C₉

yes

—

C₃₇⋊C₃

—

C₃₇⋊C₃

—

C₃₇⋊C₃

#6857

yes

—

C₃₇⋊C₃

#6858

yes

—

C₃₇⋊C₃

#6860

yes

—

C₃⁴

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

#66526

yes

characteristic, complemented

C₃²×C₉

—

111

—

C₃²×C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃₇

#10624452

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

C₃×#66526

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

C₃×C₉

—

C₃×C₉

—

333

—

C₃×C₉

—

333

—

C₃×C₉

—

333

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

333

—

C₃²

—

333

—

C₃²

—

333

—

C₃²

—

111

—

C₃²

—

111

—

C₃²

—

111

—

100

C₃²

—

111

—

101

C₃²

C₃²⋊#66526

yes

characteristic

102

C₉

—

103

C₉

—

333

—

104

C₉

—

111

—

105

C₃

—

106

C₃

—

107

C₃

—

108

C₃

—

109

C₃

—

333

—

110

C₃

—

333

—

111

C₃

#22582691

yes

characteristic

112

C₃

3₋¹⁺²⋊#66526

yes

—

	subgroup	G/H
center	C₃²	C₃²⋊#66526
commutator	C₃³×C₃₇	C₃×C₉
Frattini	C₃³	C₃×#66526
Fitting	C₃×3₋¹⁺²×C₃₇	C₉
socle	C₃²×C₃₇	C₃²⋊C₉
prime radical, cosocle	C₃³×C₃₇⋊C₃	C₃²
hypercenter	C₃×3₋¹⁺²	#66526
hypocenter	C₃₇	#10624452
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×3₋¹⁺²

subgroup

G/H

center

C₃²

C₃²⋊#66526

commutator

C₃³×C₃₇

C₃×C₉

Frattini

C₃³

C₃×#66526

Fitting

C₃×3₋¹⁺²×C₃₇

C₉

socle

C₃²×C₃₇

C₃²⋊C₉

prime radical, cosocle

C₃³×C₃₇⋊C₃

C₃²

C₃×3₋¹⁺²

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ C₃×3₋¹⁺²

N	H ≅ G/N	type
6. C₃×#11758608	C₉	semidirect
11. C₃×3₋¹⁺²×C₃₇	C₉	semidirect
64. C₃×3₋¹⁺²	#66526	semidirect
68. C₃₇	#10624452	semidirect

H ≅ G/N

type

6. C₃×#11758608

C₉

semidirect

11. C₃×3₋¹⁺²×C₃₇

C₉

semidirect

64. C₃×3₋¹⁺²

#66526

semidirect

68. C₃₇

#10624452

semidirect

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

subgroup	index
C₃×C₃²⋊#66526	3
C₃×3₋¹⁺²×C₃₇⋊C₃	3
#10624452	37

subgroup

index

C₃×C₃²⋊#66526

C₃×3₋¹⁺²×C₃₇⋊C₃

#10624452

Subgroups — #41008282

Conjugacy classes of subgroups

Complements

Maximal subgroups