Group #41006554 — subgroups

Subgroups — #41006554

Order 26973.

458 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 82 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	C₃×#15688520	C₃	1	9	yes	maximal, complemented
2	#22582554	C₃	1	2	yes	maximal
3	C₃×C₃²⋊#66524	C₃	1	1	yes	maximal, characteristic
4	C₃²×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic
5	C₃×#11758608	C₃²	1	6	yes	—
6	C₃²×#66524	C₃²	1	3	yes	—
7	#15688520	—	3	27	no	—
8	C₃²⋊#66524	—	3	2	no	—
9	C₃²⋊#66524	—	3	1	no	—
10	C₃²⋊#66524	—	3	4	no	—
11	C₃²⋊#66524	—	3	2	no	—
12	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
13	C₃×3₋¹⁺²×C₃₇	—	3	2	no	—
14	C₃×3₋¹⁺²×C₃₇	—	3	9	no	—
15	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
16	C₃²×C₉×C₃₇	—	3	1	no	—
17	C₃³×C₃₇	—	3	3	no	—
18	C₃³×C₃₇	—	3	3	no	—
19	C₃³×C₃₇	—	9	2	no	—
20	C₃³×C₃₇	H(3,𝔽₃)	1	2	yes	—
21	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
22	C₃³×C₃₇	C₃³	1	1	yes	characteristic
23	3₋¹⁺²×C₃₇	—	3	18	no	—
24	3₋¹⁺²×C₃₇	—	3	6	no	—
25	3₋¹⁺²×C₃₇	H(3,𝔽₃)	1	9	yes	—
26	C₃×C₉×C₃₇	—	3	9	no	—
27	C₃×C₉×C₃₇	—	3	3	no	—
28	#11758608	—	3	18	no	—
29	C₃×#66524	—	3	3	no	—
30	C₃×#66524	—	3	6	no	—
31	C₃×#66524	—	3	9	no	—
32	#10624744	—	37	1	no	maximal, Sylow
33	C₃²×C₃₇	—	3	2	no	—
34	C₃²×C₃₇	—	3	2	no	—
35	C₃²×C₃₇	—	3	1	no	—
36	C₃²×C₃₇	—	3	1	no	—
37	C₃²×C₃₇	—	3	4	no	—
38	C₃²×C₃₇	—	3	2	no	—
39	C₃²×C₃₇	—	3	2	no	—
40	C₃²×C₃₇	—	9	9	no	—
41	C₃²×C₃₇	#733	1	3	yes	complemented
42	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	3	yes	—
43	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
44	C₉×C₃₇	—	3	9	no	—
45	#66524	—	9	3	no	—
46	#66524	—	9	6	no	—
47	C₃²×3₋¹⁺²	C₃₇⋊C₃	1	1	yes	characteristic
48	C₃×#733	—	37	9	no	—
49	#6891	—	37	2	no	—
50	C₃×C₃²⋊C₉	—	37	1	no	—
51	C₃×C₃₇	—	3	3	no	—
52	C₃×C₃₇	—	3	3	no	—
53	C₃×C₃₇	—	9	2	no	—
54	C₃×C₃₇	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
55	C₃×C₃₇	C₃×#733	1	1	yes	characteristic
56	C₃×C₃₇	#6912	1	2	yes	—
57	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
58	C₃×3₋¹⁺²	—	3	2	no	—
59	C₃×3₋¹⁺²	—	3	9	no	—
60	C₃×3₋¹⁺²	—	37	6	no	—
61	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	3	yes	—
62	C₃²×C₉	—	3	1	no	—
63	C₃²×C₉	—	37	3	no	—
64	#733	—	111	27	no	—
65	C₃²⋊C₉	—	111	2	no	—
66	C₃²⋊C₉	—	111	1	no	—
67	C₃²⋊C₉	—	111	4	no	—
68	C₃²⋊C₉	—	111	2	no	—
69	C₃₇	#10624744	1	1	yes	characteristic, complemented, Sylow
70	C₃³	—	3	3	no	—
71	C₃³	—	3	3	no	—
72	C₃³	—	9	2	no	—
73	C₃³	C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
74	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
75	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
76	3₋¹⁺²	—	3	18	no	—
77	3₋¹⁺²	—	3	6	no	—
78	3₋¹⁺²	—	111	18	no	—
79	3₋¹⁺²	C₃²⋊(C₃₇⋊C₃)	1	9	yes	—
80	C₃×C₉	—	3	9	no	—
81	C₃×C₉	—	3	3	no	—
82	C₃×C₉	—	111	9	no	—
83	C₃×C₉	—	111	3	no	—
84	C₃×C₉	—	111	6	no	—
85	C₃²	—	3	1	no	—
86	C₃²	—	3	1	no	—
87	C₃²	—	3	2	no	—
88	C₃²	—	3	2	no	—
89	C₃²	—	3	2	no	—
90	C₃²	—	3	2	no	—
91	C₃²	—	3	4	no	—
92	C₃²	—	9	9	no	—
93	C₃²	#15688520	1	3	yes	complemented
94	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
95	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
96	C₉	—	3	9	no	—
97	C₉	—	333	3	no	—
98	C₉	—	333	6	no	—
99	C₃	—	3	3	no	—
100	C₃	—	3	3	no	—
101	C₃	—	9	2	no	—
102	C₃	#22582550	1	1	yes	characteristic
103	C₃	C₃×#15688520	1	1	yes	characteristic
104	C₃	#22582651	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688520

C₃

yes

maximal, complemented

#22582554

C₃

yes

maximal

C₃×C₃²⋊#66524

C₃

yes

maximal, characteristic

C₃²×3₋¹⁺²×C₃₇

C₃

yes

maximal, characteristic

C₃×#11758608

C₃²

yes

—

C₃²×#66524

C₃²

yes

—

—

—

—

—

—

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₃₇

—

C₃×3₋¹⁺²×C₃₇

—

C₃×3₋¹⁺²×C₃₇

C₃²

yes

—

C₃²×C₉×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

H(3,𝔽₃)

yes

—

C₃³×C₃₇

H(3,𝔽₃)

yes

characteristic

C₃³×C₃₇

C₃³

yes

characteristic

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

H(3,𝔽₃)

yes

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

—

—

—

—

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

#733

yes

complemented

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₃₇

—

#66524

—

#66524

—

C₃²×3₋¹⁺²

C₃₇⋊C₃

yes

characteristic

C₃×#733

—

#6891

—

C₃×C₃²⋊C₉

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃²⋊H(3,𝔽₃)

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

—

C₃⁴

C₃×C₃₇⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

—

C₃²×C₉

—

C₃²×C₉

—

#733

—

111

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃₇

#10624744

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

—

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

111

—

3₋¹⁺²

C₃²⋊(C₃₇⋊C₃)

yes

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

#15688520

yes

complemented

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

—

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

C₉

—

C₉

—

333

—

C₉

—

333

—

C₃

—

100

C₃

—

101

C₃

—

102

C₃

#22582550

yes

characteristic

103

C₃

C₃×#15688520

yes

characteristic

104

C₃

#22582651

yes

—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃²×3₋¹⁺²×C₃₇	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃⁴×C₃₇	C₃²
hypercenter	C₃²×3₋¹⁺²	C₃₇⋊C₃
hypocenter	C₃₇	#10624744
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃²×3₋¹⁺²

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃²×3₋¹⁺²×C₃₇

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

prime radical

C₃⁴×C₃₇

C₃²

hypercenter

C₃²×3₋¹⁺²

C₃₇⋊C₃

hypocenter

C₃₇

#10624744

LCS

G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃²×3₋¹⁺²

N	H ≅ G/N	type
1. C₃×#15688520	C₃	semidirect
41. C₃²×C₃₇	#733	semidirect
69. C₃₇	#10624744	semidirect
93. C₃²	#15688520	semidirect

H ≅ G/N

type

1. C₃×#15688520

C₃

semidirect

41. C₃²×C₃₇

#733

semidirect

69. C₃₇

#10624744

semidirect

93. C₃²

#15688520

semidirect

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

subgroup	index
C₃×#15688520	3
#22582554	3
C₃×C₃²⋊#66524	3
C₃²×3₋¹⁺²×C₃₇	3
#10624744	37

subgroup

index

C₃×#15688520

#22582554

C₃×C₃²⋊#66524

C₃²×3₋¹⁺²×C₃₇

#10624744

Subgroups — #41006554

Conjugacy classes of subgroups

Complements

Maximal subgroups