Group #41006520 — subgroups

Subgroups — #41006520

Order 26973.

446 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 67 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×#11758605	C₃	1	1	yes	maximal, characteristic
2	C₃×#15688521	C₃	1	2	yes	maximal, complemented
3	C₃×#15688520	C₃	1	6	yes	maximal, complemented
4	#22582551	C₃	1	3	yes	maximal
5	C₃×C₃₇×#733	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
7	C₃×#11758608	C₃²	1	6	yes	—
8	C₃×#11758605	—	3	3	no	—
9	C₃×#11758605	—	3	3	no	—
10	C₃×#11758605	—	3	6	no	—
11	C₃²×#66524	C₃²	1	2	yes	—
12	#15688521	—	3	6	no	—
13	#15688520	—	3	18	no	—
14	#15688502	—	3	9	no	—
15	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
16	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
17	C₃₇×#733	—	3	3	no	—
18	C₃³×C₃₇	C₃³	1	1	yes	characteristic
19	C₃²×C₃₇⋊C₃	—	3	2	no	—
20	C₃²×C₃₇⋊C₃	—	3	1	no	—
21	C₃²×C₃₇⋊C₃	—	9	3	no	—
22	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	3	yes	—
23	3₋¹⁺²×C₃₇	—	3	3	no	—
24	3₋¹⁺²×C₃₇	—	3	6	no	—
25	C₃×C₉×C₃₇	—	3	2	no	—
26	C₃×C₉×C₃₇	—	3	1	no	—
27	C₃×C₉×C₃₇	—	3	3	no	—
28	C₃×C₉×C₃₇	H(3,𝔽₃)	1	3	yes	—
29	#11758608	—	3	12	no	—
30	#11758608	—	3	6	no	—
31	C₃×#66524	—	3	2	no	—
32	C₃×#66524	—	3	2	no	—
33	C₃×#66524	—	3	2	no	—
34	C₃×#66524	—	3	6	no	—
35	C₃×#66524	H(3,𝔽₃)	1	6	yes	—
36	#11758605	—	3	27	no	—
37	#10624738	—	37	1	no	maximal, Sylow
38	C₃²×C₃₇	—	3	3	no	—
39	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	2	yes	—
40	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
41	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
42	C₃×C₃₇⋊C₃	—	3	9	no	—
43	C₃×C₃₇⋊C₃	—	3	3	no	—
44	C₃×C₃₇⋊C₃	—	9	6	no	—
45	C₃×C₃₇⋊C₃	—	9	3	no	—
46	C₉×C₃₇	—	3	3	no	—
47	C₉×C₃₇	—	9	3	no	—
48	#66524	—	3	6	no	—
49	#66524	—	9	6	no	—
50	C₃²×3₋¹⁺²	—	37	1	no	—
51	C₃×#733	—	37	6	no	—
52	C₃×#733	—	37	2	no	—
53	C₃×#733	C₃₇⋊C₃	1	1	yes	characteristic, complemented
54	#6891	—	37	3	no	—
55	C₃×C₃₇	—	3	2	no	—
56	C₃×C₃₇	—	3	1	no	—
57	C₃×C₃₇	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
58	C₃×C₃₇	#6912	1	3	yes	—
59	C₃₇⋊C₃	—	9	3	no	—
60	C₃⁴	—	37	1	no	—
61	C₃×3₋¹⁺²	—	37	6	no	—
62	C₃×3₋¹⁺²	—	111	6	no	—
63	C₃×3₋¹⁺²	—	111	3	no	—
64	C₃×3₋¹⁺²	—	111	3	no	—
65	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	3	yes	—
66	C₃²×C₉	—	37	2	no	—
67	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
68	#733	—	3	3	no	—
69	#733	—	111	18	no	—
70	#733	—	111	6	no	—
71	C₃²⋊C₉	—	111	9	no	—
72	C₃₇	#10624738	1	1	yes	characteristic, complemented, Sylow
73	C₃³	—	37	3	no	—
74	C₃³	—	333	3	no	—
75	C₃³	—	111	2	no	—
76	C₃³	—	111	1	no	—
77	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
78	3₋¹⁺²	—	3	6	no	—
79	3₋¹⁺²	—	3	3	no	—
80	3₋¹⁺²	—	111	27	no	—
81	3₋¹⁺²	—	111	12	no	—
82	3₋¹⁺²	—	111	6	no	—
83	C₃×C₉	—	3	2	no	—
84	C₃×C₉	—	3	1	no	—
85	C₃×C₉	—	3	3	no	—
86	C₃×C₉	—	37	6	no	—
87	C₃×C₉	—	111	6	no	—
88	C₃×C₉	—	111	2	no	—
89	C₃×C₉	—	111	2	no	—
90	C₃×C₉	—	111	2	no	—
91	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
92	C₃²	—	3	3	no	—
93	C₃²	—	333	6	no	—
94	C₃²	—	333	3	no	—
95	C₃²	—	111	3	no	—
96	C₃²	—	111	9	no	—
97	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	2	yes	—
98	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
99	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
100	C₉	—	3	3	no	—
101	C₉	—	9	3	no	—
102	C₉	—	333	6	no	—
103	C₉	—	111	6	no	—
104	C₃	—	3	2	no	—
105	C₃	—	3	1	no	—
106	C₃	—	333	3	no	—
107	C₃	(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)	1	1	yes	characteristic
108	C₃	#22582649	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#11758605

C₃

yes

maximal, characteristic

C₃×#15688521

C₃

yes

maximal, complemented

C₃×#15688520

C₃

yes

maximal, complemented

#22582551

C₃

yes

maximal

C₃×C₃₇×#733

C₃

yes

maximal, characteristic, complemented

C₃³×C₃₇⋊C₃

C₃²

yes

characteristic

C₃×#11758608

C₃²

yes

—

C₃×#11758605

—

C₃×#11758605

—

C₃×#11758605

—

C₃²×#66524

C₃²

yes

—

#15688521

—

#15688520

—

#15688502

—

C₃×3₋¹⁺²×C₃₇

C₃²

yes

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×#733

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

H(3,𝔽₃)

yes

—

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

H(3,𝔽₃)

yes

—

—

—

—

—

—

—

yes

—

#11758605

—

#10624738

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

C₉×C₃₇

—

#66524

—

#66524

—

C₃²×3₋¹⁺²

—

C₃×#733

—

C₃×#733

—

C₃×#733

C₃₇⋊C₃

yes

characteristic, complemented

#6891

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃²⋊H(3,𝔽₃)

yes

characteristic

C₃×C₃₇

#6912

yes

—

C₃₇⋊C₃

—

C₃⁴

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

—

C₃²×C₉

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

#733

—

#733

—

111

—

#733

—

111

—

C₃²⋊C₉

—

111

—

C₃₇

#10624738

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

333

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

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

C₃²⋊(C₃₇⋊C₃)

yes

—

C₃²

—

C₃²

—

333

—

C₃²

—

333

—

C₃²

—

111

—

C₃²

—

111

—

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

—

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

100

C₉

—

101

C₉

—

102

C₉

—

333

—

103

C₉

—

111

—

104

C₃

—

105

C₃

—

106

C₃

—

333

—

107

C₃

(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)

yes

characteristic

108

C₃

#22582649

yes

—

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₃₇⋊C₃
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃×C₃₇×#733	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃³×C₃₇⋊C₃	C₃²
hypercenter	C₃×#733	C₃₇⋊C₃
hypocenter	C₃₇	#10624738
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×#733

subgroup

G/H

center

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃×C₃₇×#733

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

prime radical

C₃³×C₃₇⋊C₃

C₃²

C₃₇⋊C₃

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ C₃×#733

N	H ≅ G/N	type
2. C₃×#15688521	C₃	semidirect
3. C₃×#15688520	C₃	semidirect
5. C₃×C₃₇×#733	C₃	semidirect
53. C₃×#733	C₃₇⋊C₃	semidirect
72. C₃₇	#10624738	semidirect

H ≅ G/N

type

2. C₃×#15688521

C₃

semidirect

3. C₃×#15688520

C₃

semidirect

5. C₃×C₃₇×#733

C₃

semidirect

53. C₃×#733

C₃₇⋊C₃

semidirect

72. C₃₇

#10624738

semidirect

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

subgroup	index
C₃²×#11758605	3
C₃×#15688521	3
C₃×#15688520	3
#22582551	3
C₃×C₃₇×#733	3
#10624738	37

subgroup

index

C₃×C₃₇×#733

Subgroups — #41006520

Conjugacy classes of subgroups

Complements

Maximal subgroups