Group #41006272 — subgroups

Subgroups — #41006272

Order 26973.

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

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 130 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	#22582538	C₃	1	2	yes	maximal, complemented
2	C₃×C₉×#66524	C₃	1	1	yes	maximal, characteristic, complemented
3	3₋¹⁺²⋊#66524	C₃	1	6	yes	maximal, complemented
4	H(3,𝔽₃)⋊#66524	C₃	1	3	yes	maximal
5	C₃×C₃₇×#737	C₃	1	1	yes	maximal, characteristic
6	C₃²×#66524	C₃²	1	3	yes	—
7	#15688506	—	3	2	no	—
8	#15688505	—	3	2	no	—
9	C₃²⋊#66524	C₃²	1	6	yes	—
10	C₉×#66524	—	3	2	no	—
11	C₉×#66524	—	3	2	no	—
12	C₉×#66524	—	3	1	no	—
13	C₃₇×#737	—	3	6	no	—
14	C₃₇×#737	C₉	1	9	yes	complemented
15	C₃×3₋¹⁺²×C₃₇	—	3	2	no	—
16	C₃×3₋¹⁺²×C₃₇	C₃²	1	2	yes	—
17	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
18	C₃²×C₉×C₃₇	—	3	1	no	—
19	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
20	C₃³×C₃₇	—	3	1	no	—
21	C₃³×C₃₇	C₃³	1	1	yes	characteristic
22	3₋¹⁺²×C₃₇	—	3	4	no	—
23	3₋¹⁺²×C₃₇	—	3	18	no	—
24	3₋¹⁺²×C₃₇	C₃×C₉	1	6	yes	—
25	H(3,𝔽₃)×C₃₇	—	3	2	no	—
26	H(3,𝔽₃)×C₃₇	C₃×C₉	1	3	yes	—
27	C₃×C₉×C₃₇	—	3	2	no	—
28	C₃×C₉×C₃₇	—	3	9	no	—
29	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	—
30	C₃×C₉×C₃₇	H(3,𝔽₃)	1	2	yes	—
31	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
32	C₃×C₉×C₃₇	3₋¹⁺²	1	6	yes	—
33	C₃×#66524	—	3	3	no	—
34	C₃×#66524	—	3	9	no	—
35	C₃×#66524	—	3	6	no	—
36	#10624697	—	37	1	no	maximal, Sylow
37	C₃²×C₃₇	—	3	3	no	—
38	C₃²×C₃₇	—	3	3	no	—
39	C₃²×C₃₇	—	9	3	no	—
40	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
41	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
42	C₃²×C₃₇	C₃×3₋¹⁺²	1	2	yes	—
43	C₉×C₃₇	—	3	6	no	—
44	C₉×C₃₇	C₃²⋊C₉	1	3	yes	—
45	C₉×C₃₇	C₃²⋊C₉	1	6	yes	—
46	#66524	—	3	9	no	—
47	#66524	—	9	6	no	—
48	C₃×#737	C₃₇⋊C₃	1	1	yes	characteristic
49	#6887	—	37	2	no	—
50	C₃×C₉²	—	37	1	no	—
51	3₋¹⁺²⋊C₉	—	37	6	no	—
52	H(3,𝔽₃)⋊C₉	—	37	3	no	—
53	C₃×C₃₇	—	3	1	no	—
54	C₃×C₃₇	—	3	2	no	—
55	C₃×C₃₇	—	9	3	no	—
56	C₃×C₃₇	C₃×C₃²⋊C₉	1	1	yes	characteristic
57	C₃×C₃₇	#6908	1	3	yes	—
58	#737	—	3	6	no	—
59	#737	#66524	1	9	yes	complemented
60	C₃×3₋¹⁺²	—	3	2	no	—
61	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	2	yes	—
62	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
63	C₃²×C₉	—	3	1	no	—
64	C₃²×C₉	—	37	3	no	—
65	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
66	C₉⋊C₉	—	111	2	no	—
67	C₉⋊C₉	—	111	2	no	—
68	C₃²⋊C₉	—	37	6	no	—
69	C₉²	—	111	2	no	—
70	C₉²	—	111	1	no	—
71	C₉²	—	111	2	no	—
72	C₃₇	#10624697	1	1	yes	characteristic, complemented, Sylow
73	C₃³	—	3	1	no	—
74	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
75	3₋¹⁺²	—	3	4	no	—
76	3₋¹⁺²	—	3	18	no	—
77	3₋¹⁺²	C₃×#66524	1	6	yes	—
78	H(3,𝔽₃)	—	3	2	no	—
79	H(3,𝔽₃)	C₃×#66524	1	3	yes	—
80	C₃×C₉	—	3	2	no	—
81	C₃×C₉	—	3	9	no	—
82	C₃×C₉	—	111	6	no	—
83	C₃×C₉	—	111	3	no	—
84	C₃×C₉	—	111	9	no	—
85	C₃×C₉	C₃×#66524	1	3	yes	—
86	C₃×C₉	#11758608	1	6	yes	—
87	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
88	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
89	C₃²	—	3	3	no	—
90	C₃²	—	3	3	no	—
91	C₃²	—	9	3	no	—
92	C₃²	C₃²×#66524	1	1	yes	characteristic
93	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
94	C₃²	C₃×#11758608	1	2	yes	—
95	C₉	—	3	6	no	—
96	C₉	—	333	6	no	—
97	C₉	—	111	9	no	—
98	C₉	C₃²⋊#66524	1	3	yes	—
99	C₉	C₃²⋊#66524	1	6	yes	—
100	C₃	—	3	1	no	—
101	C₃	—	3	2	no	—
102	C₃	—	9	3	no	—
103	C₃	C₃×C₃²⋊#66524	1	1	yes	characteristic
104	C₃	#22582630	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22582538

C₃

yes

maximal, complemented

C₃×C₉×#66524

C₃

yes

maximal, characteristic, complemented

3₋¹⁺²⋊#66524

C₃

yes

maximal, complemented

H(3,𝔽₃)⋊#66524

C₃

yes

maximal

C₃×C₃₇×#737

C₃

yes

maximal, characteristic

C₃²×#66524

C₃²

yes

—

#15688506

—

#15688505

—

C₃²⋊#66524

C₃²

yes

—

C₉×#66524

—

C₉×#66524

—

C₉×#66524

—

C₃₇×#737

—

C₃₇×#737

C₉

yes

complemented

C₃×3₋¹⁺²×C₃₇

—

C₃×3₋¹⁺²×C₃₇

C₃²

yes

—

C₃×H(3,𝔽₃)×C₃₇

C₃²

yes

characteristic

C₃²×C₉×C₃₇

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃³×C₃₇

—

C₃³×C₃₇

C₃³

yes

characteristic

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

C₃×C₉

yes

—

H(3,𝔽₃)×C₃₇

—

H(3,𝔽₃)×C₃₇

C₃×C₉

yes

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

C₃×C₉

yes

—

C₃×C₉×C₃₇

H(3,𝔽₃)

yes

—

C₃×C₉×C₃₇

H(3,𝔽₃)

yes

characteristic

C₃×C₉×C₃₇

3₋¹⁺²

yes

—

—

—

—

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃²×C₉

yes

characteristic

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

—

C₉×C₃₇

—

C₉×C₃₇

C₃²⋊C₉

yes

—

C₉×C₃₇

C₃²⋊C₉

yes

—

#66524

—

#66524

—

C₃×#737

C₃₇⋊C₃

yes

characteristic

#6887

—

C₃×C₉²

—

3₋¹⁺²⋊C₉

—

H(3,𝔽₃)⋊C₉

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃×C₃²⋊C₉

yes

characteristic

C₃×C₃₇

#6908

yes

—

—

yes

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

—

C₃×H(3,𝔽₃)

C₃×C₃₇⋊C₃

yes

characteristic

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₃²⋊C₉

—

C₉²

—

111

—

C₉²

—

111

—

C₉²

—

111

—

C₃₇

#10624697

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

C₃×#66524

yes

—

H(3,𝔽₃)

—

H(3,𝔽₃)

C₃×#66524

yes

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

C₃×#66524

yes

—

C₃×C₉

#11758608

yes

—

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

—

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃²×#66524

yes

characteristic

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

C₃²

C₃×#11758608

yes

—

C₉

—

C₉

—

333

—

C₉

—

111

—

C₉

C₃²⋊#66524

yes

—

C₉

C₃²⋊#66524

yes

—

100

C₃

—

101

C₃

—

102

C₃

—

103

C₃

C₃×C₃²⋊#66524

yes

characteristic

104

C₃

#22582630

yes

—

	subgroup	G/H
center	C₃×C₉	C₃²⋊(C₃₇⋊C₃)
commutator	C₃²×C₃₇	C₃²×C₉
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃×C₃₇×#737	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃×H(3,𝔽₃)×C₃₇	C₃²
cosocle	C₃³×C₃₇	C₃³
hypercenter	C₃×#737	C₃₇⋊C₃
hypocenter	C₃₇	#10624697
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ C₃×#737

subgroup

G/H

center

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

commutator

C₃²×C₃₇

C₃²×C₉

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃×C₃₇×#737

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

prime radical

C₃×H(3,𝔽₃)×C₃₇

C₃²

cosocle

C₃³×C₃₇

C₃³

C₃₇⋊C₃

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ C₃×#737

N	H ≅ G/N	type
1. #22582538	C₃	semidirect
2. C₃×C₉×#66524	C₃	semidirect
3. 3₋¹⁺²⋊#66524	C₃	semidirect
14. C₃₇×#737	C₉	semidirect
59. #737	#66524	semidirect
72. C₃₇	#10624697	semidirect

H ≅ G/N

type

1. #22582538

C₃

semidirect

2. C₃×C₉×#66524

C₃

semidirect

3. 3₋¹⁺²⋊#66524

C₃

semidirect

14. C₃₇×#737

C₉

semidirect

59. #737

#66524

semidirect

72. C₃₇

#10624697

semidirect

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

subgroup	index
#22582538	3
C₃×C₉×#66524	3
3₋¹⁺²⋊#66524	3
H(3,𝔽₃)⋊#66524	3
C₃×C₃₇×#737	3
#10624697	37

subgroup

index

C₃×C₉×#66524

C₃×C₃₇×#737

Subgroups — #41006272

Conjugacy classes of subgroups

Complements

Maximal subgroups