Group #41006756 — subgroups

Subgroups — #41006756

Order 26973.

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

normal subgroups only

This group has 356 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×#11758608	C₃	1	1	yes	maximal, characteristic
2	#22582554	C₃	1	2	yes	maximal
3	#22582469	C₃	1	9	yes	maximal, complemented
4	C₃₇×#6891	C₃	1	1	yes	maximal, characteristic
5	C₃×#11758608	—	3	2	no	—
6	C₃×#11758608	—	3	1	no	—
7	C₃×#11758608	—	3	2	no	—
8	C₃×#11758608	—	3	9	no	—
9	C₃²×#66524	C₃²	1	3	yes	—
10	C₃²⋊#66524	—	3	2	no	—
11	C₃²⋊#66524	—	3	2	no	—
12	C₃²⋊#66524	C₃²	1	6	yes	—
13	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
14	C₃×3₋¹⁺²×C₃₇	—	3	1	no	—
15	C₃₇×C₃²⋊C₉	—	3	2	no	—
16	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
17	C₃³×C₃₇	—	3	3	no	—
18	C₃³×C₃₇	—	9	2	no	—
19	C₃³×C₃₇	—	9	1	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	—	9	3	no	—
24	C₃×C₉×C₃₇	—	3	3	no	—
25	#11758608	—	3	18	no	—
26	#11758608	—	3	9	no	—
27	#11758608	—	9	6	no	—
28	C₃×#66524	—	3	3	no	—
29	C₃×#66524	—	3	6	no	—
30	C₃×#66524	—	3	6	no	—
31	C₃×#66524	3₋¹⁺²	1	9	yes	complemented
32	#10624775	—	37	1	no	maximal, Sylow
33	C₃²×C₃₇	—	3	1	no	—
34	C₃²×C₃₇	—	3	2	no	—
35	C₃²×C₃₇	—	3	2	no	—
36	C₃²×C₃₇	—	3	1	no	—
37	C₃²×C₃₇	—	3	2	no	—
38	C₃²×C₃₇	—	3	1	no	—
39	C₃²×C₃₇	—	3	2	no	—
40	C₃²×C₃₇	—	3	2	no	—
41	C₃²×C₃₇	—	3	2	no	—
42	C₃²×C₃₇	—	9	9	no	—
43	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
44	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
45	C₉×C₃₇	—	9	3	no	—
46	#66524	—	3	9	no	—
47	#66524	—	9	6	no	—
48	C₃²×3₋¹⁺²	—	37	1	no	—
49	#6891	—	37	2	no	—
50	#6891	C₃₇⋊C₃	1	1	yes	characteristic
51	#6867	—	37	9	no	—
52	C₃×C₃₇	—	3	3	no	—
53	C₃×C₃₇	—	9	2	no	—
54	C₃×C₃₇	—	9	1	no	—
55	C₃×C₃₇	#6891	1	1	yes	characteristic
56	C₃×C₃₇	#6910	1	1	yes	characteristic
57	C₃×C₃₇	#6910	1	2	yes	—
58	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
59	C₃×3₋¹⁺²	—	3	1	no	—
60	C₃×3₋¹⁺²	—	111	9	no	—
61	C₃×3₋¹⁺²	—	111	2	no	—
62	C₃×3₋¹⁺²	—	111	2	no	—
63	C₃×3₋¹⁺²	—	111	1	no	—
64	C₃²×C₉	—	37	3	no	—
65	C₃²⋊C₉	—	3	2	no	—
66	C₃²⋊C₉	—	37	6	no	—
67	C₃²⋊C₉	—	111	2	no	—
68	C₃²⋊C₉	—	111	2	no	—
69	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
70	C₃₇	#10624775	1	1	yes	characteristic, complemented, Sylow
71	C₃³	—	3	3	no	—
72	C₃³	—	9	1	no	—
73	C₃³	—	9	2	no	—
74	C₃³	C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
75	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
76	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
77	3₋¹⁺²	—	9	3	no	—
78	3₋¹⁺²	—	333	6	no	—
79	3₋¹⁺²	—	111	18	no	—
80	3₋¹⁺²	—	111	9	no	—
81	C₃×C₉	—	3	3	no	—
82	C₃×C₉	—	37	9	no	—
83	C₃×C₉	—	111	6	no	—
84	C₃×C₉	—	111	3	no	—
85	C₃×C₉	—	111	6	no	—
86	C₃²	—	3	1	no	—
87	C₃²	—	3	2	no	—
88	C₃²	—	3	2	no	—
89	C₃²	—	3	1	no	—
90	C₃²	—	3	1	no	—
91	C₃²	—	3	2	no	—
92	C₃²	—	3	2	no	—
93	C₃²	—	3	2	no	—
94	C₃²	—	3	2	no	—
95	C₃²	—	9	9	no	—
96	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
97	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	3	yes	—
98	C₉	—	9	3	no	—
99	C₉	—	333	6	no	—
100	C₉	—	111	9	no	—
101	C₃	—	3	3	no	—
102	C₃	—	9	1	no	—
103	C₃	—	9	2	no	—
104	C₃	#22582551	1	1	yes	characteristic
105	C₃	#22582642	1	1	yes	characteristic
106	C₃	#22582644	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#11758608

C₃

yes

maximal, characteristic

#22582554

C₃

yes

maximal

#22582469

C₃

yes

maximal, complemented

C₃₇×#6891

C₃

yes

maximal, characteristic

—

—

—

—

C₃²×#66524

C₃²

yes

—

C₃²⋊#66524

—

C₃²⋊#66524

—

C₃²⋊#66524

C₃²

yes

—

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₃₇

—

C₃₇×C₃²⋊C₉

—

C₃₇×C₃²⋊C₉

C₃²

yes

—

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

—

C₃×C₉×C₃₇

—

—

—

—

—

—

—

3₋¹⁺²

yes

complemented

#10624775

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

—

C₉×C₃₇

—

#66524

—

#66524

—

C₃²×3₋¹⁺²

—

#6891

—

#6891

C₃₇⋊C₃

yes

characteristic

#6867

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃×C₃₇

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

—

C₃×3₋¹⁺²

—

111

—

C₃²×C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

C₃₇

#10624775

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

—

333

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

C₃²

3₋¹⁺²×C₃₇⋊C₃

yes

—

C₉

—

C₉

—

333

—

100

C₉

—

111

—

101

C₃

—

102

C₃

—

103

C₃

—

104

C₃

#22582551

yes

characteristic

105

C₃

#22582642

yes

characteristic

106

C₃

#22582644

yes

—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6891	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃⁴×C₃₇	C₃²
hypercenter	#6891	C₃₇⋊C₃
hypocenter	C₃₇	#10624775
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ #6891

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃₇×#6891

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

prime radical

C₃⁴×C₃₇

C₃²

C₃₇⋊C₃

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ #6891

N	H ≅ G/N	type
3. #22582469	C₃	semidirect
31. C₃×#66524	3₋¹⁺²	semidirect
70. C₃₇	#10624775	semidirect

H ≅ G/N

type

3. #22582469

C₃

semidirect

31. C₃×#66524

3₋¹⁺²

semidirect

70. C₃₇

#10624775

semidirect

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

subgroup	index
C₃²×#11758608	3
#22582554	3
#22582469	3
C₃₇×#6891	3
#10624775	37

subgroup

index

C₃₇×#6891

Subgroups — #41006756

Conjugacy classes of subgroups

Complements

Maximal subgroups