Group #41008590 — subgroups

Subgroups — #41008590

Order 26973.

250 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 69 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	3₋¹⁺²×#66526	C₃	1	3	yes	maximal
2	C₃×#15688538	C₃	1	1	yes	maximal, characteristic
3	3₋¹⁺²⋊#66526	C₃	1	6	yes	maximal
4	#22582693	C₃	1	2	yes	maximal
5	#733×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
6	C₃²×#66526	C₃²	1	2	yes	—
7	C₃²×#66526	C₃²	1	1	yes	characteristic
8	#15688538	—	3	3	no	—
9	#15688538	—	3	3	no	—
10	#15688537	—	3	3	no	—
11	C₃²⋊#66526	C₃²	1	6	yes	—
12	C₉×#66526	—	3	3	no	—
13	3₋¹⁺²×C₃₇⋊C₃	C₃²	1	3	yes	—
14	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
15	#15688521	C₉	1	2	yes	complemented
16	#15688520	C₉	1	6	yes	complemented
17	C₃₇×#733	C₉	1	1	yes	characteristic, complemented
18	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
19	C₃×#66526	—	3	6	no	—
20	C₃×#66526	—	3	2	no	—
21	C₃×#66526	—	9	1	no	—
22	C₃×#66526	—	9	2	no	—
23	C₃×#66526	H(3,𝔽₃)	1	3	yes	—
24	3₋¹⁺²×C₃₇	C₃×C₉	1	3	yes	—
25	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
26	#11758608	C₃×C₉	1	6	yes	—
27	C₃×#66524	C₃×C₉	1	2	yes	—
28	#11758606	—	3	3	no	—
29	#11758605	—	3	3	no	—
30	C₉×C₃₇⋊C₃	—	3	1	no	—
31	C₉×C₃₇⋊C₃	—	3	3	no	—
32	C₉×C₃₇⋊C₃	—	3	1	no	—
33	C₉×C₃₇⋊C₃	—	3	1	no	—
34	#10624723	—	37	1	no	maximal, Sylow
35	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
36	C₃×C₃₇⋊C₃	—	3	1	no	—
37	C₃×C₃₇⋊C₃	—	3	2	no	—
38	C₃×C₃₇⋊C₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
39	C₃×C₃₇⋊C₃	#737	1	1	yes	characteristic
40	C₃×C₃₇⋊C₃	#737	1	1	yes	characteristic
41	#66526	—	3	3	no	—
42	#66526	—	9	6	no	—
43	#66526	—	9	2	no	—
44	C₉×C₃₇	—	3	1	no	—
45	C₉×C₃₇	—	3	3	no	—
46	#66524	—	3	2	no	—
47	#66524	—	3	6	no	—
48	C₃×#733	—	37	1	no	—
49	C₉×3₋¹⁺²	—	37	3	no	—
50	C₃×C₉⋊C₉	—	37	1	no	—
51	3₋¹⁺²⋊C₉	—	37	6	no	—
52	#6868	—	37	2	no	—
53	C₃×C₃₇	—	3	1	no	—
54	C₃×C₃₇	C₉×H(3,𝔽₃)	1	1	yes	characteristic
55	C₃₇⋊C₃	—	3	1	no	—
56	C₃₇⋊C₃	—	3	1	no	—
57	C₃₇⋊C₃	#6912	1	1	yes	characteristic
58	C₃₇⋊C₃	#6913	1	2	yes	—
59	C₃×3₋¹⁺²	—	37	3	no	—
60	C₃²×C₉	—	37	1	no	—
61	C₃²×C₉	—	37	1	no	—
62	C₃²×C₉	—	37	2	no	—
63	#733	—	37	6	no	—
64	#733	—	37	2	no	—
65	#733	#66526	1	1	yes	characteristic, complemented
66	C₉⋊C₉	—	111	3	no	—
67	C₉⋊C₉	—	111	3	no	—
68	C₉⋊C₉	—	111	3	no	—
69	C₃²⋊C₉	—	37	6	no	—
70	C₉²	—	111	3	no	—
71	C₃₇	#10624723	1	1	yes	characteristic, complemented, Sylow
72	C₃³	—	37	1	no	—
73	3₋¹⁺²	—	37	6	no	—
74	3₋¹⁺²	—	111	3	no	—
75	3₋¹⁺²	—	111	3	no	—
76	3₋¹⁺²	C₃×#66526	1	3	yes	—
77	C₃×C₉	—	37	3	no	—
78	C₃×C₉	—	37	2	no	—
79	C₃×C₉	—	333	2	no	—
80	C₃×C₉	—	333	1	no	—
81	C₃×C₉	—	111	3	no	—
82	C₃×C₉	—	111	1	no	—
83	C₃×C₉	—	111	1	no	—
84	C₃×C₉	—	111	1	no	—
85	C₃×C₉	—	111	6	no	—
86	C₃×C₉	—	111	2	no	—
87	C₃×C₉	C₃×#66526	1	1	yes	characteristic
88	C₃²	—	37	1	no	—
89	C₃²	—	37	1	no	—
90	C₃²	—	37	1	no	—
91	C₃²	—	111	2	no	—
92	C₃²	—	111	1	no	—
93	C₃²	C₃²×#66526	1	1	yes	characteristic
94	C₉	—	3	1	no	—
95	C₉	—	3	3	no	—
96	C₉	—	333	2	no	—
97	C₉	—	333	6	no	—
98	C₉	—	111	3	no	—
99	C₉	—	111	2	no	—
100	C₉	—	111	6	no	—
101	C₃	—	3	1	no	—
102	C₃	—	37	2	no	—
103	C₃	—	37	1	no	—
104	C₃	—	111	1	no	—
105	C₃	—	111	1	no	—
106	C₃	H(3,𝔽₃)×#66526	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

3₋¹⁺²×#66526

C₃

yes

maximal

C₃×#15688538

C₃

yes

maximal, characteristic

3₋¹⁺²⋊#66526

C₃

yes

maximal

#22582693

C₃

yes

maximal

#733×C₃₇⋊C₃

C₃

yes

maximal, characteristic

C₃²×#66526

C₃²

yes

—

C₃²×#66526

C₃²

yes

—

—

—

C₃²

yes

—

C₉×#66526

—

3₋¹⁺²×C₃₇⋊C₃

C₃²

yes

—

C₃×C₉×C₃₇⋊C₃

C₃²

yes

characteristic

#15688521

C₉

yes

complemented

#15688520

C₉

yes

complemented

C₃₇×#733

C₉

yes

characteristic, complemented

C₃²×C₃₇⋊C₃

C₃³

yes

—

—

—

—

yes

—

3₋¹⁺²×C₃₇

C₃×C₉

yes

—

C₃×C₉×C₃₇

C₃×C₉

yes

characteristic

#11758608

C₃×C₉

yes

—

C₃×#66524

C₃×C₉

yes

—

#11758606

—

#11758605

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

#10624723

—

maximal, Sylow

C₃²×C₃₇

C₃²×C₉

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

C₃×H(3,𝔽₃)

yes

C₃×C₃₇⋊C₃

yes

C₃×C₃₇⋊C₃

yes

—

—

—

C₉×C₃₇

—

C₉×C₃₇

—

#66524

—

#66524

—

C₃×#733

—

C₉×3₋¹⁺²

—

C₃×C₉⋊C₉

—

3₋¹⁺²⋊C₉

—

#6868

—

C₃×C₃₇

—

C₃×C₃₇

C₉×H(3,𝔽₃)

yes

characteristic

C₃₇⋊C₃

—

C₃₇⋊C₃

—

C₃₇⋊C₃

#6912

yes

characteristic

C₃₇⋊C₃

#6913

yes

—

C₃×3₋¹⁺²

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

—

—

yes

characteristic, complemented

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₃²⋊C₉

—

C₉²

—

111

—

C₃₇

#10624723

yes

characteristic, complemented, Sylow

C₃³

—

3₋¹⁺²

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

3₋¹⁺²

C₃×#66526

yes

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

333

—

C₃×C₉

—

333

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

C₃×#66526

yes

characteristic

C₃²

—

C₃²

—

C₃²

—

C₃²

—

111

—

C₃²

—

111

—

C₃²

C₃²×#66526

yes

characteristic

C₉

—

C₉

—

C₉

—

333

—

C₉

—

333

—

C₉

—

111

—

C₉

—

111

—

100

C₉

—

111

—

101

C₃

—

102

C₃

—

103

C₃

—

104

C₃

—

111

—

105

C₃

—

111

—

106

C₃

H(3,𝔽₃)×#66526

yes

characteristic

	subgroup	G/H
center	C₃	H(3,𝔽₃)×#66526
commutator	C₃²×C₃₇	C₃²×C₉
Frattini	C₃²	C₃²×#66526
Fitting	C₃₇×#733	C₉
socle	C₃×C₃₇	C₉×H(3,𝔽₃)
prime radical, cosocle	C₃²×C₃₇⋊C₃	C₃³
hypercenter	#733	#66526
hypocenter	C₃₇	#10624723
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃² ◁ #733

subgroup

G/H

center

C₃

H(3,𝔽₃)×#66526

commutator

C₃²×C₃₇

C₃²×C₉

Frattini

C₃²

C₃²×#66526

Fitting

C₃₇×#733

C₉

socle

C₃×C₃₇

C₉×H(3,𝔽₃)

prime radical, cosocle

C₃²×C₃₇⋊C₃

C₃³

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃ ◁ C₃² ◁ #733

N	H ≅ G/N	type
15. #15688521	C₉	semidirect
16. #15688520	C₉	semidirect
17. C₃₇×#733	C₉	semidirect
65. #733	#66526	semidirect
71. C₃₇	#10624723	semidirect

H ≅ G/N

type

15. #15688521

C₉

semidirect

16. #15688520

C₉

semidirect

17. C₃₇×#733

C₉

semidirect

65. #733

#66526

semidirect

71. C₃₇

#10624723

semidirect

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

subgroup	index
3₋¹⁺²×#66526	3
C₃×#15688538	3
3₋¹⁺²⋊#66526	3
#22582693	3
#733×C₃₇⋊C₃	3
#10624723	37

subgroup

index

3₋¹⁺²×#66526

#733×C₃₇⋊C₃

Subgroups — #41008590

Conjugacy classes of subgroups

Complements

Maximal subgroups