Group #41005146 — subgroups

Subgroups — #41005146

Order 26973.

608 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 175 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	C₃³×#66524	C₃	1	1	yes	maximal, characteristic
2	C₃×C₃²⋊#66524	C₃	1	2	yes	maximal
3	#22582441	C₃	1	9	yes	maximal, complemented
4	C₃²×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic
5	C₃²×#66524	—	3	3	no	—
6	C₃²×#66524	—	3	9	no	—
7	C₃²×#66524	C₃²	1	6	yes	—
8	C₃²×#66524	C₃²	1	3	yes	—
9	C₃²⋊#66524	—	3	6	no	—
10	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
11	C₃×3₋¹⁺²×C₃₇	—	3	3	no	—
12	C₃×3₋¹⁺²×C₃₇	—	3	6	no	—
13	C₃×3₋¹⁺²×C₃₇	C₉	1	9	yes	complemented
14	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
15	C₃³×C₃₇	—	3	3	no	—
16	C₃³×C₃₇	—	3	6	no	—
17	C₃³×C₃₇	C₃×C₉	1	3	yes	—
18	C₃³×C₃₇	H(3,𝔽₃)	1	3	yes	—
19	C₃³×C₃₇	3₋¹⁺²	1	6	yes	—
20	C₃³×C₃₇	C₃³	1	1	yes	characteristic
21	3₋¹⁺²×C₃₇	—	3	27	no	—
22	C₃×C₉×C₃₇	—	3	3	no	—
23	C₃×C₉×C₃₇	—	3	6	no	—
24	C₃×C₉×C₃₇	C₃×C₉	1	9	yes	complemented
25	C₃×#66524	—	3	27	no	—
26	C₃×#66524	—	3	3	no	—
27	C₃×#66524	—	3	6	no	—
28	C₃×#66524	—	3	6	no	—
29	C₃×#66524	—	3	18	no	—
30	C₃×#66524	3₋¹⁺²	1	9	yes	complemented
31	#10624558	—	37	1	no	maximal, Sylow
32	C₃²×C₃₇	—	3	9	no	—
33	C₃²×C₃₇	—	3	3	no	—
34	C₃²×C₃₇	—	3	6	no	—
35	C₃²×C₃₇	—	3	18	no	—
36	C₃²×C₃₇	C₃²⋊C₉	1	9	yes	—
37	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
38	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
39	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
40	C₃²×C₃₇	C₃×3₋¹⁺²	1	2	yes	—
41	C₃²×C₃₇	#737	1	6	yes	—
42	C₉×C₃₇	—	3	9	no	—
43	#66524	—	3	9	no	—
44	#66524	—	9	6	no	—
45	C₃²×3₋¹⁺²	C₃₇⋊C₃	1	1	yes	characteristic
46	C₃³×C₉	—	37	1	no	—
47	C₃×C₃²⋊C₉	—	37	2	no	—
48	#6855	—	37	9	no	—
49	C₃×C₃₇	—	3	9	no	—
50	C₃×C₃₇	C₃×C₃²⋊C₉	1	1	yes	characteristic
51	C₃×C₃₇	C₉×3₋¹⁺²	1	3	yes	—
52	C₃×C₃₇	#6891	1	1	yes	characteristic
53	C₃×C₃₇	#6893	1	2	yes	—
54	C₃×C₃₇	#6894	1	6	yes	—
55	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
56	C₃×3₋¹⁺²	—	3	6	no	—
57	C₃×3₋¹⁺²	—	3	3	no	—
58	C₃×3₋¹⁺²	#66524	1	9	yes	complemented
59	C₃²×C₉	—	37	3	no	—
60	C₃²×C₉	—	37	6	no	—
61	C₃²×C₉	—	111	9	no	—
62	C₃²×C₉	—	111	3	no	—
63	C₃²×C₉	C₃×C₃₇⋊C₃	1	3	yes	—
64	C₃²⋊C₉	—	111	6	no	—
65	C₃₇	#10624558	1	1	yes	characteristic, complemented, Sylow
66	C₃³	—	3	3	no	—
67	C₃³	—	3	6	no	—
68	C₃³	C₃×#66524	1	3	yes	—
69	C₃³	#11758608	1	6	yes	—
70	C₃³	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
71	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
72	3₋¹⁺²	—	3	27	no	—
73	C₃×C₉	—	3	3	no	—
74	C₃×C₉	—	3	6	no	—
75	C₃×C₉	—	37	9	no	—
76	C₃×C₉	—	111	6	no	—
77	C₃×C₉	—	111	18	no	—
78	C₃×C₉	—	111	27	no	—
79	C₃×C₉	—	111	3	no	—
80	C₃×C₉	—	111	6	no	—
81	C₃×C₉	C₃×#66524	1	9	yes	complemented
82	C₃²	—	3	9	no	—
83	C₃²	—	3	3	no	—
84	C₃²	—	3	6	no	—
85	C₃²	—	3	18	no	—
86	C₃²	C₃²⋊#66524	1	9	yes	—
87	C₃²	C₃²×#66524	1	1	yes	characteristic
88	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
89	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	3	yes	—
90	C₃²	C₃×#11758608	1	2	yes	—
91	C₃²	#15688532	1	6	yes	—
92	C₉	—	3	9	no	—
93	C₉	—	333	6	no	—
94	C₉	—	111	9	no	—
95	C₃	—	3	9	no	—
96	C₃	C₃×C₃²⋊#66524	1	1	yes	characteristic
97	C₃	3₋¹⁺²×#66524	1	3	yes	—
98	C₃	#22582551	1	1	yes	characteristic
99	C₃	#22582565	1	2	yes	—
100	C₃	#22582568	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃³×#66524

C₃

yes

maximal, characteristic

C₃×C₃²⋊#66524

C₃

yes

maximal

#22582441

C₃

yes

maximal, complemented

C₃²×3₋¹⁺²×C₃₇

C₃

yes

maximal, characteristic

C₃²×#66524

—

C₃²×#66524

—

C₃²×#66524

C₃²

yes

—

C₃²×#66524

C₃²

yes

—

C₃²⋊#66524

—

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₃₇

—

C₃×3₋¹⁺²×C₃₇

—

C₃×3₋¹⁺²×C₃₇

C₉

yes

complemented

C₃²×C₉×C₃₇

C₃²

yes

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

C₃×C₉

yes

—

C₃³×C₃₇

H(3,𝔽₃)

yes

—

C₃³×C₃₇

3₋¹⁺²

yes

—

C₃³×C₃₇

C₃³

yes

characteristic

3₋¹⁺²×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

C₃×C₉

yes

—

—

—

—

—

3₋¹⁺²

yes

complemented

#10624558

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃²⋊C₉

yes

—

C₃²×C₃₇

C₃²×C₉

yes

characteristic

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

—

C₃²×C₃₇

C₃×3₋¹⁺²

yes

—

C₃²×C₃₇

#737

yes

—

C₉×C₃₇

—

#66524

—

#66524

—

C₃²×3₋¹⁺²

C₃₇⋊C₃

yes

characteristic

C₃³×C₉

—

C₃×C₃²⋊C₉

—

#6855

—

C₃×C₃₇

—

C₃×C₃₇

C₃×C₃²⋊C₉

yes

characteristic

C₃×C₃₇

C₉×3₋¹⁺²

yes

—

C₃×C₃₇

#6891

yes

characteristic

C₃×C₃₇

#6893

yes

—

C₃×C₃₇

#6894

yes

—

C₃⁴

C₃×C₃₇⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

#66524

yes

complemented

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

111

—

C₃²×C₉

—

111

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

—

C₃²⋊C₉

—

111

—

C₃₇

#10624558

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

C₃×#66524

yes

—

C₃³

#11758608

yes

—

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

C₃×#66524

yes

complemented

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃²⋊#66524

yes

—

C₃²

C₃²×#66524

yes

characteristic

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

C₃²

3₋¹⁺²×C₃₇⋊C₃

yes

—

C₃²

C₃×#11758608

yes

—

C₃²

#15688532

yes

—

C₉

—

C₉

—

333

—

C₉

—

111

—

C₃

—

C₃

C₃×C₃²⋊#66524

yes

characteristic

C₃

3₋¹⁺²×#66524

yes

—

C₃

#22582551

yes

characteristic

C₃

#22582565

yes

—

100

C₃

#22582568

yes

—

	subgroup	G/H
center, Frattini	C₃³	C₃²×C₃₇⋊C₃
commutator	C₃²×C₃₇	C₃²×C₉
Fitting	C₃²×3₋¹⁺²×C₃₇	C₃
socle, cosocle	C₃³×C₃₇	C₃³
prime radical	C₃⁴×C₃₇	C₃²
hypercenter	C₃²×3₋¹⁺²	C₃₇⋊C₃
hypocenter	C₃₇	#10624558
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃³ ◁ C₃²×3₋¹⁺²

subgroup

G/H

center, Frattini

C₃³

C₃²×C₃₇⋊C₃

commutator

C₃²×C₃₇

C₃²×C₉

Fitting

C₃²×3₋¹⁺²×C₃₇

C₃

socle, cosocle

C₃³×C₃₇

C₃³

prime radical

C₃⁴×C₃₇

C₃²

hypercenter

C₃²×3₋¹⁺²

C₃₇⋊C₃

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃₇

C₁ ◁ C₃³ ◁ C₃²×3₋¹⁺²

N	H ≅ G/N	type
3. #22582441	C₃	semidirect
13. C₃×3₋¹⁺²×C₃₇	C₉	semidirect
24. C₃×C₉×C₃₇	C₃×C₉	semidirect
30. C₃×#66524	3₋¹⁺²	semidirect
58. C₃×3₋¹⁺²	#66524	semidirect
65. C₃₇	#10624558	semidirect
81. C₃×C₉	C₃×#66524	semidirect

H ≅ G/N

type

3. #22582441

C₃

semidirect

13. C₃×3₋¹⁺²×C₃₇

C₉

semidirect

24. C₃×C₉×C₃₇

C₃×C₉

semidirect

30. C₃×#66524

3₋¹⁺²

semidirect

58. C₃×3₋¹⁺²

#66524

semidirect

65. C₃₇

#10624558

semidirect

81. C₃×C₉

C₃×#66524

semidirect

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

subgroup	index
C₃³×#66524	3
C₃×C₃²⋊#66524	3
#22582441	3
C₃²×3₋¹⁺²×C₃₇	3
#10624558	37

subgroup

index

C₃³×#66524

C₃×C₃²⋊#66524

#22582441

C₃²×3₋¹⁺²×C₃₇

#10624558

Subgroups — #41005146

Conjugacy classes of subgroups

Complements

Maximal subgroups