Group #41008656 — subgroups

Subgroups — #41008656

Order 26973.

828 conjugacy classes of subgroups; only the 426 normal subgroups are shown.

normal subgroups only

This group has 828 conjugacy classes of subgroups — too many to draw. Only its 426 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	#22582732	C₃	1	27	yes	maximal, complemented
2	(C₃₇⋊C₃)⋊(C₂₇⋊C₃)	C₃	1	9	yes	maximal, complemented
3	C₃×#15688541	C₃	1	3	yes	maximal, complemented
4	C₃×#15688532	C₃	1	1	yes	maximal, characteristic
5	#15688543	C₃²	1	27	yes	complemented
6	#15688542	C₃²	1	27	yes	complemented
7	#15688541	C₃²	1	9	yes	—
8	#15688541	C₃²	1	18	yes	complemented
9	(C₃₇⋊C₃)⋊C₂₇	C₃²	1	27	yes	complemented
10	C₃×#11758603	C₃²	1	9	yes	complemented
11	#15688532	C₉	1	18	yes	—
12	#15688532	C₃²	1	9	yes	—
13	C₃×#11758608	C₉	1	1	yes	characteristic
14	C₃×#11758605	C₉	1	3	yes	—
15	C₃×#11758606	C₉	1	3	yes	—
16	C₃×C₃²⋊(C₃₇⋊C₃)	C₉	1	1	yes	characteristic
17	C₃²×#66524	C₃²	1	1	yes	characteristic
18	C₃×C₉×C₃₇⋊C₃	C₃²	1	3	yes	—
19	C₃×3₋¹⁺²×C₃₇	C₉	1	1	yes	characteristic
20	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
21	C₃²×C₃₇⋊C₃	C₃×C₉	1	3	yes	—
22	3₋¹⁺²×C₃₇	C₃×C₉	1	9	yes	—
23	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	—
24	C₃²⋊(C₃₇⋊C₃)	C₃×C₉	1	9	yes	—
25	#11758608	C₃×C₉	1	9	yes	—
26	C₃×#66524	C₃×C₉	1	3	yes	—
27	C₃×#66524	C₃×C₉	1	1	yes	characteristic
28	C₃×#66524	C₃×C₉	1	1	yes	characteristic
29	C₃×#66524	C₃×C₉	1	6	yes	—
30	C₃×#66524	C₃³	1	1	yes	characteristic
31	C₃×#66524	C₃³	1	3	yes	—
32	#11758606	C₃×C₉	1	27	yes	—
33	#11758605	C₃×C₉	1	27	yes	—
34	C₉×C₃₇⋊C₃	C₃×C₉	1	18	yes	—
35	C₉×C₃₇⋊C₃	C₃³	1	9	yes	—
36	#11758603	C₃³	1	27	yes	—
37	#10624918	—	37	1	no	maximal, Sylow
38	C₃²×C₃₇	—	9	1	no	—
39	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
40	C₃²×C₃₇	C₃²×C₉	1	3	yes	—
41	C₃×C₃₇⋊C₃	—	9	3	no	—
42	C₃×C₃₇⋊C₃	C₃²×C₉	1	9	yes	—
43	C₉×C₃₇	C₃²×C₉	1	9	yes	—
44	#66524	C₃²×C₉	1	2	yes	—
45	#66524	C₃²×C₉	1	9	yes	—
46	#66524	C₃²×C₉	1	3	yes	—
47	#66524	C₃²×C₉	1	3	yes	—
48	#66524	C₃⁴	1	1	yes	characteristic
49	C₃×#737	—	37	1	no	—
50	#6903	—	37	27	no	—
51	C₃×C₂₇⋊C₃	—	37	3	no	—
52	C₃×C₂₇⋊C₃	—	37	9	no	—
53	C₃×C₃₇	—	3	1	no	—
54	C₃×C₃₇	—	3	3	no	—
55	C₃×C₃₇	C₃³×C₉	1	1	yes	characteristic
56	C₃₇⋊C₃	—	3	9	no	—
57	#737	—	37	18	no	—
58	#737	—	37	9	no	—
59	C₃×3₋¹⁺²	—	37	1	no	—
60	C₃×3₋¹⁺²	—	37	3	no	—
61	C₃×3₋¹⁺²	—	37	3	no	—
62	C₃×3₋¹⁺²	#66526	1	1	yes	characteristic
63	C₃×H(3,𝔽₃)	—	37	1	no	—
64	C₃²×C₉	—	37	1	no	—
65	C₃²×C₉	—	37	3	no	—
66	C₂₇⋊C₃	—	37	9	no	—
67	C₂₇⋊C₃	—	37	18	no	—
68	C₂₇⋊C₃	—	37	27	no	—
69	C₂₇⋊C₃	—	37	27	no	—
70	C₃×C₂₇	—	37	9	no	—
71	C₃×C₂₇	—	37	27	no	—
72	C₃₇	#10624918	1	1	yes	characteristic, complemented, Sylow
73	C₃³	—	37	3	no	—
74	C₃³	C₃×#66526	1	1	yes	characteristic
75	3₋¹⁺²	—	37	9	no	—
76	3₋¹⁺²	—	37	27	no	—
77	3₋¹⁺²	—	37	27	no	—
78	3₋¹⁺²	C₃×#66526	1	9	yes	—
79	H(3,𝔽₃)	—	37	9	no	—
80	C₃×C₉	—	37	3	no	—
81	C₃×C₉	—	37	1	no	—
82	C₃×C₉	—	37	1	no	—
83	C₃×C₉	—	37	6	no	—
84	C₃×C₉	—	37	3	no	—
85	C₃×C₉	—	37	1	no	—
86	C₃×C₉	—	37	18	no	—
87	C₃×C₉	—	37	9	no	—
88	C₃×C₉	C₃×#66526	1	3	yes	—
89	C₂₇	—	37	27	no	—
90	C₃²	—	9	1	no	—
91	C₃²	—	37	9	no	—
92	C₃²	—	333	3	no	—
93	C₃²	C₃²×#66526	1	1	yes	characteristic
94	C₃²	C₃²×#66526	1	3	yes	—
95	C₉	—	37	1	no	—
96	C₉	—	37	2	no	—
97	C₉	—	37	3	no	—
98	C₉	—	37	3	no	—
99	C₉	—	37	9	no	—
100	C₉	C₃²×#66526	1	9	yes	—
101	C₃	—	3	1	no	—
102	C₃	—	3	3	no	—
103	C₃	—	111	9	no	—
104	C₃	C₃³×#66526	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22582732

C₃

yes

maximal, complemented

(C₃₇⋊C₃)⋊(C₂₇⋊C₃)

C₃

yes

maximal, complemented

C₃×#15688541

C₃

yes

maximal, complemented

C₃×#15688532

C₃

yes

maximal, characteristic

#15688543

C₃²

yes

complemented

#15688542

C₃²

yes

complemented

#15688541

C₃²

yes

—

#15688541

C₃²

yes

complemented

(C₃₇⋊C₃)⋊C₂₇

C₃²

yes

complemented

C₃×#11758603

C₃²

yes

complemented

#15688532

C₉

yes

—

#15688532

C₃²

yes

—

C₃×#11758608

C₉

yes

characteristic

C₃×#11758605

C₉

yes

—

C₃×#11758606

C₉

yes

—

C₃×C₃²⋊(C₃₇⋊C₃)

C₉

yes

characteristic

C₃²×#66524

C₃²

yes

characteristic

C₃×C₉×C₃₇⋊C₃

C₃²

yes

—

C₃×3₋¹⁺²×C₃₇

C₉

yes

characteristic

C₃³×C₃₇

C₃×C₉

yes

characteristic

C₃²×C₃₇⋊C₃

C₃×C₉

yes

—

3₋¹⁺²×C₃₇

C₃×C₉

yes

—

C₃×C₉×C₃₇

C₃×C₉

yes

—

C₃²⋊(C₃₇⋊C₃)

C₃×C₉

yes

—

#11758608

C₃×C₉

yes

—

C₃×#66524

C₃×C₉

yes

—

C₃×#66524

C₃×C₉

yes

characteristic

C₃×#66524

C₃×C₉

yes

characteristic

C₃×#66524

C₃×C₉

yes

—

C₃×#66524

C₃³

yes

characteristic

C₃×#66524

C₃³

yes

—

#11758606

C₃×C₉

yes

—

#11758605

C₃×C₉

yes

—

C₉×C₃₇⋊C₃

C₃×C₉

yes

—

C₉×C₃₇⋊C₃

C₃³

yes

—

#11758603

C₃³

yes

—

#10624918

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

C₃²×C₉

yes

characteristic

C₃²×C₃₇

C₃²×C₉

yes

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

C₃²×C₉

yes

—

C₉×C₃₇

C₃²×C₉

yes

—

#66524

C₃²×C₉

yes

—

#66524

C₃²×C₉

yes

—

#66524

C₃²×C₉

yes

—

#66524

C₃²×C₉

yes

—

#66524

C₃⁴

yes

characteristic

C₃×#737

—

#6903

—

C₃×C₂₇⋊C₃

—

C₃×C₂₇⋊C₃

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃³×C₉

yes

characteristic

C₃₇⋊C₃

—

#737

—

#737

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

#66526

yes

characteristic

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃₇

#10624918

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

C₃×#66526

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

C₃×#66526

yes

—

H(3,𝔽₃)

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃×#66526

yes

—

C₂₇

—

C₃²

—

C₃²

—

C₃²

—

333

—

C₃²

C₃²×#66526

yes

characteristic

C₃²

C₃²×#66526

yes

—

C₉

—

C₉

—

C₉

—

C₉

—

C₉

—

100

C₉

C₃²×#66526

yes

—

101

C₃

—

102

C₃

—

103

C₃

—

111

—

104

C₃

C₃³×#66526

yes

characteristic

	subgroup	G/H
center, Frattini	C₃	C₃³×#66526
commutator, socle	C₃×C₃₇	C₃³×C₉
Fitting	C₃×3₋¹⁺²×C₃₇	C₉
prime radical	C₃×C₃²⋊(C₃₇⋊C₃)	C₉
cosocle	#66524	C₃⁴
hypercenter	C₃×3₋¹⁺²	#66526
hypocenter	C₃₇	#10624918
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃×3₋¹⁺²

subgroup

G/H

center, Frattini

C₃

C₃³×#66526

commutator, socle

C₃×C₃₇

C₃³×C₉

Fitting

C₃×3₋¹⁺²×C₃₇

C₉

prime radical

C₃×C₃²⋊(C₃₇⋊C₃)

C₉

C₃⁴

C₃×3₋¹⁺²

C₃₇

G ▷ C₃×C₃₇ ▷ C₃₇

C₁ ◁ C₃ ◁ C₃×3₋¹⁺²

N	H ≅ G/N	type
1. #22582732	C₃	semidirect
2. (C₃₇⋊C₃)⋊(C₂₇⋊C₃)	C₃	semidirect
3. C₃×#15688541	C₃	semidirect
5. #15688543	C₃²	semidirect
6. #15688542	C₃²	semidirect
8. #15688541	C₃²	semidirect
9. (C₃₇⋊C₃)⋊C₂₇	C₃²	semidirect
10. C₃×#11758603	C₃²	semidirect
72. C₃₇	#10624918	semidirect

H ≅ G/N

type

1. #22582732

C₃

semidirect

2. (C₃₇⋊C₃)⋊(C₂₇⋊C₃)

C₃

semidirect

3. C₃×#15688541

C₃

semidirect

5. #15688543

C₃²

semidirect

6. #15688542

C₃²

semidirect

8. #15688541

C₃²

semidirect

9. (C₃₇⋊C₃)⋊C₂₇

C₃²

semidirect

10. C₃×#11758603

C₃²

semidirect

72. C₃₇

#10624918

semidirect

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

subgroup	index
#22582732	3
(C₃₇⋊C₃)⋊(C₂₇⋊C₃)	3
C₃×#15688541	3
C₃×#15688532	3
#10624918	37

subgroup

index

(C₃₇⋊C₃)⋊(C₂₇⋊C₃)

Subgroups — #41008656

Conjugacy classes of subgroups

Complements

Maximal subgroups