Group #41006741 — subgroups

Subgroups — #41006741

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×#11758605	C₃	1	1	yes	maximal, characteristic
2	#22582553	C₃	1	3	yes	maximal
3	#22582472	C₃	1	2	yes	maximal, complemented
4	#22582471	C₃	1	6	yes	maximal, complemented
5	C₃₇×#6868	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
7	C₃×#11758605	—	3	3	no	—
8	C₃×#11758605	—	3	3	no	—
9	C₃×#11758605	—	3	6	no	—
10	C₃×#11758605	—	3	3	no	—
11	C₃²×#66524	C₃²	1	2	yes	—
12	C₃²⋊#66524	C₃²	1	6	yes	—
13	#15688502	—	3	6	no	—
14	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
15	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
16	C₃³×C₃₇	C₃³	1	1	yes	characteristic
17	C₃²×C₃₇⋊C₃	—	3	2	no	—
18	C₃²×C₃₇⋊C₃	—	3	1	no	—
19	C₃²×C₃₇⋊C₃	—	9	2	no	—
20	C₃²×C₃₇⋊C₃	—	9	1	no	—
21	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	3	yes	—
22	C₃×C₉×C₃₇	—	3	1	no	—
23	C₃×C₉×C₃₇	—	3	2	no	—
24	C₃×C₉×C₃₇	—	3	3	no	—
25	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	complemented
26	C₃×#66524	—	3	2	no	—
27	C₃×#66524	—	3	2	no	—
28	C₃×#66524	—	3	2	no	—
29	C₃×#66524	—	3	6	no	—
30	C₃×#66524	3₋¹⁺²	1	6	yes	complemented
31	#11758605	—	3	27	no	—
32	#11758605	—	9	9	no	—
33	#10624772	—	37	1	no	maximal, Sylow
34	C₃²×C₃₇	—	3	2	no	—
35	C₃²×C₃₇	—	3	1	no	—
36	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
37	C₃²×C₃₇	C₃×3₋¹⁺²	1	2	yes	—
38	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
39	C₃×C₃₇⋊C₃	—	3	6	no	—
40	C₃×C₃₇⋊C₃	—	3	3	no	—
41	C₃×C₃₇⋊C₃	—	3	3	no	—
42	C₃×C₃₇⋊C₃	—	9	6	no	—
43	C₃×C₃₇⋊C₃	—	9	3	no	—
44	C₉×C₃₇	—	3	3	no	—
45	C₉×C₃₇	—	9	3	no	—
46	#66524	—	3	6	no	—
47	#66524	—	9	6	no	—
48	C₃²×3₋¹⁺²	—	37	1	no	—
49	#6891	—	37	3	no	—
50	#6868	—	37	2	no	—
51	#6868	—	37	6	no	—
52	#6868	C₃₇⋊C₃	1	1	yes	characteristic, complemented
53	C₃×C₃₇	—	3	2	no	—
54	C₃×C₃₇	—	3	1	no	—
55	C₃×C₃₇	#6891	1	1	yes	characteristic
56	C₃×C₃₇	#6911	1	1	yes	characteristic
57	C₃×C₃₇	#6912	1	2	yes	—
58	C₃₇⋊C₃	—	9	3	no	—
59	C₃⁴	—	37	1	no	—
60	C₃×3₋¹⁺²	—	111	6	no	—
61	C₃×3₋¹⁺²	—	111	3	no	—
62	C₃×3₋¹⁺²	—	111	3	no	—
63	C₃×3₋¹⁺²	—	111	3	no	—
64	C₃²×C₉	—	37	2	no	—
65	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
66	C₃²⋊C₉	—	37	6	no	—
67	C₃²⋊C₉	—	111	6	no	—
68	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
69	C₃₇	#10624772	1	1	yes	characteristic, complemented, Sylow
70	C₃³	—	37	3	no	—
71	C₃³	—	333	1	no	—
72	C₃³	—	333	2	no	—
73	C₃³	—	111	2	no	—
74	C₃³	—	111	1	no	—
75	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
76	3₋¹⁺²	—	333	9	no	—
77	3₋¹⁺²	—	111	27	no	—
78	C₃×C₉	—	3	1	no	—
79	C₃×C₉	—	3	2	no	—
80	C₃×C₉	—	3	3	no	—
81	C₃×C₉	—	37	6	no	—
82	C₃×C₉	—	111	6	no	—
83	C₃×C₉	—	111	2	no	—
84	C₃×C₉	—	111	2	no	—
85	C₃×C₉	—	111	2	no	—
86	C₃×C₉	#11758605	1	3	yes	complemented
87	C₃²	—	3	1	no	—
88	C₃²	—	3	2	no	—
89	C₃²	—	333	6	no	—
90	C₃²	—	333	3	no	—
91	C₃²	—	111	3	no	—
92	C₃²	—	111	3	no	—
93	C₃²	—	111	6	no	—
94	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
95	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	2	yes	—
96	C₃²	C₃×#11758605	1	1	yes	characteristic
97	C₉	—	3	3	no	—
98	C₉	—	9	3	no	—
99	C₉	—	333	6	no	—
100	C₉	—	111	6	no	—
101	C₃	—	3	2	no	—
102	C₃	—	3	1	no	—
103	C₃	—	333	3	no	—
104	C₃	#22582553	1	1	yes	characteristic
105	C₃	#22582645	1	1	yes	characteristic
106	C₃	#22582649	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#11758605

C₃

yes

maximal, characteristic

#22582553

C₃

yes

maximal

#22582472

C₃

yes

maximal, complemented

#22582471

C₃

yes

maximal, complemented

C₃₇×#6868

C₃

yes

maximal, characteristic, complemented

C₃³×C₃₇⋊C₃

C₃²

yes

—

—

—

—

C₃²×#66524

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

—

#15688502

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

H(3,𝔽₃)

yes

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

3₋¹⁺²

yes

—

—

—

—

3₋¹⁺²

yes

—

—

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

—

C₃²×C₃₇

C₃×3₋¹⁺²

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

C₉×C₃₇

—

#66524

—

#66524

—

C₃²×3₋¹⁺²

—

—

—

—

C₃₇⋊C₃

yes

characteristic, complemented

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

—

C₃₇⋊C₃

—

C₃⁴

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃²×C₉

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

C₃₇

#10624772

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

333

—

C₃³

—

333

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

333

—

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

#11758605

yes

complemented

C₃²

—

C₃²

—

C₃²

—

333

—

C₃²

—

333

—

C₃²

—

111

—

C₃²

—

111

—

C₃²

—

111

—

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

C₃²

3₋¹⁺²×C₃₇⋊C₃

yes

—

C₃²

C₃×#11758605

yes

characteristic

C₉

—

C₉

—

C₉

—

333

—

100

C₉

—

111

—

101

C₃

—

102

C₃

—

103

C₃

—

333

—

104

C₃

#22582553

yes

characteristic

105

C₃

#22582645

yes

characteristic

106

C₃

#22582649

yes

—

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₃₇⋊C₃
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6868	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃³×C₃₇⋊C₃	C₃²
hypercenter	#6868	C₃₇⋊C₃
hypocenter	C₃₇	#10624772
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ #6868

subgroup

G/H

center

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃₇×#6868

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 ◁ #6868

N	H ≅ G/N	type
3. #22582472	C₃	semidirect
4. #22582471	C₃	semidirect
5. C₃₇×#6868	C₃	semidirect
25. C₃×C₉×C₃₇	3₋¹⁺²	semidirect
30. C₃×#66524	3₋¹⁺²	semidirect
52. #6868	C₃₇⋊C₃	semidirect
69. C₃₇	#10624772	semidirect
86. C₃×C₉	#11758605	semidirect

H ≅ G/N

type

3. #22582472

C₃

semidirect

4. #22582471

C₃

semidirect

5. C₃₇×#6868

C₃

semidirect

25. C₃×C₉×C₃₇

3₋¹⁺²

semidirect

30. C₃×#66524

3₋¹⁺²

semidirect

52. #6868

C₃₇⋊C₃

semidirect

69. C₃₇

#10624772

semidirect

86. C₃×C₉

#11758605

semidirect

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

subgroup	index
C₃²×#11758605	3
#22582553	3
#22582472	3
#22582471	3
C₃₇×#6868	3
#10624772	37

subgroup

index

C₃₇×#6868

Subgroups — #41006741

Conjugacy classes of subgroups

Complements

Maximal subgroups