Group #41005032 — subgroups

Subgroups — #41005032

Order 26973.

130 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 130 conjugacy classes of subgroups — too many to draw. Only its 37 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×#15688521	C₃	1	1	yes	maximal, characteristic
2	C₃×#15688505	C₃	1	1	yes	maximal, characteristic
3	#22582472	C₃	1	1	yes	maximal, characteristic
4	C₃₇×#6868	C₃	1	1	yes	maximal, characteristic
5	C₃×#11758608	—	3	1	no	—
6	C₃²×#66524	—	3	1	no	—
7	#15688521	—	3	1	no	—
8	#15688521	—	3	1	no	—
9	#15688521	C₉	1	3	yes	complemented
10	#15688505	—	3	3	no	—
11	C₃²⋊#66524	—	3	1	no	—
12	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
13	C₃₇×C₃²⋊C₉	—	3	1	no	—
14	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
15	C₃×C₉×C₃₇	—	3	1	no	—
16	C₃×C₉×C₃₇	—	3	1	no	—
17	C₃×C₉×C₃₇	—	3	1	no	—
18	C₃×C₉×C₃₇	—	9	1	no	—
19	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
20	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
21	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
22	#11758608	—	3	3	no	—
23	#11758608	—	9	1	no	—
24	#11758608	—	9	1	no	—
25	C₃×#66524	—	3	3	no	—
26	C₃×#66524	—	9	1	no	—
27	C₃×#66524	—	9	1	no	—
28	C₃×#66524	—	9	1	no	—
29	C₃×#66524	—	9	1	no	—
30	C₃×#66524	—	9	1	no	—
31	#10624528	—	37	1	no	maximal, Sylow
32	C₃²×C₃₇	—	3	1	no	—
33	C₃²×C₃₇	—	3	1	no	—
34	C₃²×C₃₇	—	3	1	no	—
35	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
36	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
37	C₃²×C₃₇	#731	1	1	yes	characteristic
38	C₃²×C₃₇	#732	1	1	yes	characteristic
39	C₉×C₃₇	—	3	1	no	—
40	C₉×C₃₇	—	3	1	no	—
41	C₉×C₃₇	—	3	1	no	—
42	C₉×C₃₇	—	27	1	no	—
43	#66524	—	9	3	no	—
44	#66524	—	9	3	no	—
45	#66524	—	27	1	no	—
46	C₃×#733	—	37	1	no	—
47	C₃×C₉⋊C₉	—	37	1	no	—
48	#6868	—	37	1	no	—
49	#6868	C₃₇⋊C₃	1	1	yes	characteristic
50	C₃×C₃₇	—	3	1	no	—
51	C₃×C₃₇	—	3	1	no	—
52	C₃×C₃₇	—	3	1	no	—
53	C₃×C₃₇	H(3,𝔽₃)⋊C₉	1	1	yes	characteristic
54	C₃×C₃₇	#6881	1	1	yes	characteristic
55	C₃×C₃₇	#6882	1	1	yes	characteristic
56	C₃×C₃₇	#6882	1	1	yes	characteristic
57	C₃×3₋¹⁺²	—	111	1	no	—
58	C₃²×C₉	—	111	1	no	—
59	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
60	#733	—	37	3	no	—
61	#733	—	111	1	no	—
62	#733	—	111	1	no	—
63	C₉⋊C₉	—	111	3	no	—
64	C₃²⋊C₉	—	3	1	no	—
65	C₃²⋊C₉	—	111	1	no	—
66	C₃₇	#10624528	1	1	yes	characteristic, complemented, Sylow
67	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
68	3₋¹⁺²	—	333	1	no	—
69	3₋¹⁺²	—	333	1	no	—
70	3₋¹⁺²	—	111	3	no	—
71	C₃×C₉	—	3	1	no	—
72	C₃×C₉	—	3	1	no	—
73	C₃×C₉	—	3	1	no	—
74	C₃×C₉	—	9	1	no	—
75	C₃×C₉	—	333	1	no	—
76	C₃×C₉	—	333	1	no	—
77	C₃×C₉	—	333	1	no	—
78	C₃×C₉	—	333	1	no	—
79	C₃×C₉	—	333	1	no	—
80	C₃×C₉	—	111	3	no	—
81	C₃×C₉	C₉×C₃₇⋊C₃	1	1	yes	characteristic
82	C₃×C₉	#11758605	1	1	yes	characteristic
83	C₃×C₉	#11758606	1	1	yes	characteristic
84	C₃²	—	3	1	no	—
85	C₃²	—	3	1	no	—
86	C₃²	—	3	1	no	—
87	C₃²	#15688502	1	1	yes	characteristic
88	C₃²	#15688514	1	1	yes	characteristic
89	C₃²	#15688517	1	1	yes	characteristic
90	C₃²	#15688518	1	1	yes	characteristic
91	C₉	—	3	1	no	—
92	C₉	—	3	1	no	—
93	C₉	—	3	1	no	—
94	C₉	—	27	1	no	—
95	C₉	—	333	3	no	—
96	C₉	—	333	3	no	—
97	C₉	—	999	1	no	—
98	C₃	—	3	1	no	—
99	C₃	—	3	1	no	—
100	C₃	—	3	1	no	—
101	C₃	#22582475	1	1	yes	characteristic
102	C₃	#22582512	1	1	yes	characteristic
103	C₃	#22582517	1	1	yes	characteristic
104	C₃	#22582520	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688521

C₃

yes

maximal, characteristic

C₃×#15688505

C₃

yes

maximal, characteristic

#22582472

C₃

yes

maximal, characteristic

C₃₇×#6868

C₃

yes

maximal, characteristic

C₃×#11758608

—

C₃²×#66524

—

#15688521

—

#15688521

—

#15688521

C₉

yes

complemented

#15688505

—

C₃²⋊#66524

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×C₃²⋊C₉

—

C₃³×C₃₇

H(3,𝔽₃)

yes

characteristic

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

C₃×C₉

yes

characteristic

C₃×C₉×C₃₇

3₋¹⁺²

yes

characteristic

C₃×C₉×C₃₇

3₋¹⁺²

yes

—

—

—

—

—

—

—

—

—

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃²⋊C₉

yes

characteristic

C₃²×C₃₇

AΣL(1,𝔽₂₇)

yes

C₃²×C₃₇

yes

C₃²×C₃₇

yes

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

—

—

—

—

C₃×C₉⋊C₉

—

#6868

—

#6868

C₃₇⋊C₃

yes

characteristic

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

H(3,𝔽₃)⋊C₉

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃×3₋¹⁺²

—

111

—

C₃²×C₉

—

111

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

#733

—

#733

—

111

—

#733

—

111

—

C₉⋊C₉

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

C₃₇

#10624528

yes

characteristic, complemented, Sylow

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

3₋¹⁺²

—

333

—

3₋¹⁺²

—

333

—

3₋¹⁺²

—

111

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

333

—

C₃×C₉

—

333

—

C₃×C₉

—

333

—

C₃×C₉

—

333

—

C₃×C₉

—

333

—

C₃×C₉

—

111

—

C₃×C₉

C₉×C₃₇⋊C₃

yes

C₃×C₉

yes

C₃×C₉

yes

C₃²

—

C₃²

—

C₃²

—

C₃²

yes

C₃²

yes

C₃²

yes

C₃²

yes

C₉

—

C₉

—

C₉

—

C₉

—

C₉

—

333

—

C₉

—

333

—

C₉

—

999

—

C₃

—

C₃

—

100

C₃

—

101

C₃

#22582475

yes

characteristic

102

C₃

#22582512

yes

characteristic

103

C₃

#22582517

yes

characteristic

104

C₃

#22582520

yes

characteristic

	subgroup	G/H
center	C₃²	#15688518
commutator	C₃×C₉×C₃₇	C₃×C₉
Frattini	C₃²×C₉	C₃×C₃₇⋊C₃
Fitting	C₃₇×#6868	C₃
socle	C₃²×C₃₇	#732
prime radical	C₃³×C₃₇	H(3,𝔽₃)
cosocle	C₃²×C₉×C₃₇	C₃²
hypercenter	#6868	C₃₇⋊C₃
hypocenter	C₃₇	#10624528
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃²×C₉ ◁ #6868

subgroup

G/H

center

C₃²

#15688518

commutator

C₃×C₉×C₃₇

C₃×C₉

Frattini

C₃²×C₉

C₃×C₃₇⋊C₃

Fitting

C₃₇×#6868

C₃

C₃²×C₃₇

C₃³×C₃₇

C₃²×C₉×C₃₇

C₃²

C₃₇⋊C₃

C₃₇

G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ C₃²×C₉ ◁ #6868

N	H ≅ G/N	type
9. #15688521	C₉	semidirect
66. C₃₇	#10624528	semidirect

H ≅ G/N

type

9. #15688521

C₉

semidirect

66. C₃₇

#10624528

semidirect

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

subgroup	index
C₃×#15688521	3
C₃×#15688505	3
#22582472	3
C₃₇×#6868	3
#10624528	37

subgroup

index

C₃₇×#6868

Subgroups — #41005032

Conjugacy classes of subgroups

Complements

Maximal subgroups