Group #41005145 — subgroups

Subgroups — #41005145

Order 26973.

608 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 166 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	C₃²×#11758608	C₃	1	1	yes	maximal, characteristic
2	C₃×C₃²⋊#66524	C₃	1	2	yes	maximal
3	#22582441	C₃	1	9	yes	maximal, complemented
4	C₃³×C₉×C₃₇	C₃	1	1	yes	maximal, characteristic
5	C₃×#11758608	—	3	3	no	—
6	C₃×#11758608	—	3	3	no	—
7	C₃×#11758608	—	3	3	no	—
8	C₃×#11758608	C₉	1	9	yes	complemented
9	C₃²×#66524	C₃²	1	6	yes	—
10	C₃²×#66524	C₃²	1	3	yes	—
11	C₃²⋊#66524	—	3	6	no	—
12	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
13	C₃²×C₉×C₃₇	—	3	3	no	—
14	C₃²×C₉×C₃₇	—	3	9	no	—
15	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
16	C₃³×C₃₇	—	3	3	no	—
17	C₃³×C₃₇	—	3	6	no	—
18	C₃³×C₃₇	C₃×C₉	1	3	yes	—
19	C₃³×C₃₇	H(3,𝔽₃)	1	3	yes	—
20	C₃³×C₃₇	3₋¹⁺²	1	3	yes	—
21	C₃³×C₃₇	3₋¹⁺²	1	3	yes	—
22	C₃³×C₃₇	C₃³	1	1	yes	characteristic
23	C₃×C₉×C₃₇	—	3	27	no	—
24	C₃×C₉×C₃₇	—	3	3	no	—
25	C₃×C₉×C₃₇	—	3	6	no	—
26	C₃×C₉×C₃₇	3₋¹⁺²	1	9	yes	complemented
27	#11758608	—	3	27	no	—
28	C₃×#66524	—	3	3	no	—
29	C₃×#66524	—	3	3	no	—
30	C₃×#66524	—	3	3	no	—
31	C₃×#66524	—	3	6	no	—
32	C₃×#66524	—	3	18	no	—
33	C₃×#66524	C₃×C₉	1	9	yes	complemented
34	#10624558	—	37	1	no	maximal, Sylow
35	C₃²×C₃₇	—	3	9	no	—
36	C₃²×C₃₇	—	3	3	no	—
37	C₃²×C₃₇	—	3	6	no	—
38	C₃²×C₃₇	—	3	9	no	—
39	C₃²×C₃₇	—	3	9	no	—
40	C₃²×C₃₇	C₃²⋊C₉	1	9	yes	—
41	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
42	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
43	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
44	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
45	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
46	C₃²×C₃₇	#737	1	6	yes	—
47	C₉×C₃₇	—	3	9	no	—
48	#66524	—	3	9	no	—
49	#66524	—	9	6	no	—
50	C₃²×3₋¹⁺²	—	37	1	no	—
51	C₃³×C₉	C₃₇⋊C₃	1	1	yes	characteristic
52	C₃×C₃²⋊C₉	—	37	2	no	—
53	#6855	—	37	9	no	—
54	C₃×C₃₇	—	3	9	no	—
55	C₃×C₃₇	C₃×C₃²⋊C₉	1	1	yes	characteristic
56	C₃×C₃₇	C₉×3₋¹⁺²	1	3	yes	—
57	C₃×C₃₇	#6891	1	1	yes	characteristic
58	C₃×C₃₇	#6893	1	2	yes	—
59	C₃×C₃₇	#6894	1	3	yes	—
60	C₃×C₃₇	#6894	1	3	yes	—
61	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
62	C₃×3₋¹⁺²	—	37	9	no	—
63	C₃×3₋¹⁺²	—	111	3	no	—
64	C₃×3₋¹⁺²	—	111	3	no	—
65	C₃×3₋¹⁺²	—	111	3	no	—
66	C₃²×C₉	—	3	3	no	—
67	C₃²×C₉	—	3	9	no	—
68	C₃²×C₉	—	37	3	no	—
69	C₃²×C₉	—	37	6	no	—
70	C₃²×C₉	C₃×C₃₇⋊C₃	1	3	yes	—
71	C₃²⋊C₉	—	111	6	no	—
72	C₃₇	#10624558	1	1	yes	characteristic, complemented, Sylow
73	C₃³	—	3	3	no	—
74	C₃³	—	3	6	no	—
75	C₃³	C₉×C₃₇⋊C₃	1	3	yes	—
76	C₃³	#11758605	1	3	yes	—
77	C₃³	#11758606	1	3	yes	—
78	C₃³	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
79	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
80	3₋¹⁺²	—	111	27	no	—
81	C₃×C₉	—	3	27	no	—
82	C₃×C₉	—	3	3	no	—
83	C₃×C₉	—	3	6	no	—
84	C₃×C₉	—	37	9	no	—
85	C₃×C₉	—	111	6	no	—
86	C₃×C₉	—	111	18	no	—
87	C₃×C₉	—	111	3	no	—
88	C₃×C₉	—	111	3	no	—
89	C₃×C₉	—	111	3	no	—
90	C₃×C₉	#11758608	1	9	yes	complemented
91	C₃²	—	3	9	no	—
92	C₃²	—	3	3	no	—
93	C₃²	—	3	6	no	—
94	C₃²	—	3	9	no	—
95	C₃²	—	3	9	no	—
96	C₃²	#15688502	1	9	yes	—
97	C₃²	C₃×C₉×C₃₇⋊C₃	1	1	yes	characteristic
98	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
99	C₃²	C₃×#11758606	1	1	yes	characteristic
100	C₃²	C₃×#11758605	1	1	yes	characteristic
101	C₃²	C₃×#11758608	1	3	yes	—
102	C₃²	#15688531	1	6	yes	—
103	C₉	—	3	9	no	—
104	C₉	—	333	6	no	—
105	C₉	—	111	9	no	—
106	C₃	—	3	9	no	—
107	C₃	C₃×#15688502	1	1	yes	characteristic
108	C₃	C₉×#11758608	1	3	yes	—
109	C₃	#22582554	1	1	yes	characteristic
110	C₃	#22582564	1	2	yes	—
111	C₃	#22582571	1	3	yes	—
112	C₃	#22582572	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#11758608

C₃

yes

maximal, characteristic

C₃×C₃²⋊#66524

C₃

yes

maximal

#22582441

C₃

yes

maximal, complemented

C₃³×C₉×C₃₇

C₃

yes

maximal, characteristic

—

—

—

C₉

yes

complemented

C₃²×#66524

C₃²

yes

—

C₃²×#66524

C₃²

yes

—

C₃²⋊#66524

—

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃²×C₉×C₃₇

—

C₃²×C₉×C₃₇

—

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

3₋¹⁺²

yes

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

3₋¹⁺²

yes

—

—

—

—

—

—

C₃×C₉

yes

complemented

#10624558

—

maximal, Sylow

C₃²×C₃₇

—

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

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

—

C₃²×C₃₇

#737

yes

—

C₉×C₃₇

—

#66524

—

#66524

—

C₃²×3₋¹⁺²

—

C₃³×C₉

C₃₇⋊C₃

yes

characteristic

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

—

C₃×C₃₇

#6894

yes

—

C₃⁴

C₃×C₃₇⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

—

C₃²⋊C₉

—

111

—

C₃₇

#10624558

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

C₉×C₃₇⋊C₃

yes

—

C₃³

#11758605

yes

—

C₃³

#11758606

yes

—

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

111

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

#11758608

yes

complemented

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

#15688502

yes

—

C₃²

C₃×C₉×C₃₇⋊C₃

yes

characteristic

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

C₃²

C₃×#11758606

yes

characteristic

100

C₃²

C₃×#11758605

yes

characteristic

101

C₃²

C₃×#11758608

yes

—

102

C₃²

#15688531

yes

—

103

C₉

—

104

C₉

—

333

—

105

C₉

—

111

—

106

C₃

—

107

C₃

C₃×#15688502

yes

characteristic

108

C₃

C₉×#11758608

yes

—

109

C₃

#22582554

yes

characteristic

110

C₃

#22582564

yes

—

111

C₃

#22582571

yes

—

112

C₃

#22582572

yes

—

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

subgroup

G/H

center, Frattini

C₃³

C₃²×C₃₇⋊C₃

commutator

C₃²×C₃₇

C₃²×C₉

Fitting

C₃³×C₉×C₃₇

C₃

socle, cosocle

C₃³×C₃₇

C₃³

prime radical

C₃⁴×C₃₇

C₃²

hypercenter

C₃³×C₉

C₃₇⋊C₃

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃₇

C₁ ◁ C₃³ ◁ C₃³×C₉

N	H ≅ G/N	type
3. #22582441	C₃	semidirect
8. C₃×#11758608	C₉	semidirect
26. C₃×C₉×C₃₇	3₋¹⁺²	semidirect
33. C₃×#66524	C₃×C₉	semidirect
72. C₃₇	#10624558	semidirect
90. C₃×C₉	#11758608	semidirect

H ≅ G/N

type

3. #22582441

C₃

semidirect

8. C₃×#11758608

C₉

semidirect

26. C₃×C₉×C₃₇

3₋¹⁺²

semidirect

33. C₃×#66524

C₃×C₉

semidirect

72. C₃₇

#10624558

semidirect

90. C₃×C₉

#11758608

semidirect

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

subgroup	index
C₃²×#11758608	3
C₃×C₃²⋊#66524	3
#22582441	3
C₃³×C₉×C₃₇	3
#10624558	37

subgroup

index

C₃²×#11758608

C₃×C₃²⋊#66524

#22582441

C₃³×C₉×C₃₇

#10624558

Subgroups — #41005145

Conjugacy classes of subgroups

Complements

Maximal subgroups