Group #41008486 — subgroups

Subgroups — #41008486

Order 26973.

332 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 90 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×#15688541	C₃	1	2	yes	maximal, complemented
2	C₃²×#11758603	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582690	C₃	1	9	yes	maximal, complemented
4	C₃×#15688532	C₃	1	1	yes	maximal, characteristic
5	#15688541	—	3	6	no	—
6	C₃×#11758603	—	3	3	no	—
7	C₃×#11758603	C₃²	1	3	yes	—
8	C₃×#11758603	C₃²	1	6	yes	—
9	#15688532	—	3	3	no	—
10	#15688532	—	3	6	no	—
11	C₃×#11758608	C₉	1	1	yes	characteristic
12	C₃×#11758605	C₉	1	3	yes	—
13	C₃×#11758606	C₉	1	3	yes	—
14	C₃×C₃²⋊(C₃₇⋊C₃)	C₉	1	1	yes	characteristic
15	C₃²×#66524	C₃²	1	1	yes	characteristic
16	C₃×C₉×C₃₇⋊C₃	C₃²	1	3	yes	—
17	C₃×3₋¹⁺²×C₃₇	C₉	1	1	yes	characteristic
18	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
19	C₃²×C₃₇⋊C₃	C₃×C₉	1	3	yes	—
20	3₋¹⁺²×C₃₇	—	3	3	no	—
21	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	—
22	C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
23	#11758608	—	3	3	no	—
24	C₃×#66524	C₃×C₉	1	3	yes	—
25	C₃×#66524	C₃×C₉	1	1	yes	characteristic
26	C₃×#66524	C₃×C₉	1	1	yes	characteristic
27	C₃×#66524	H(3,𝔽₃)	1	3	yes	—
28	C₃×#66524	3₋¹⁺²	1	6	yes	—
29	C₃×#66524	C₃³	1	1	yes	characteristic
30	#11758606	—	3	9	no	—
31	#11758605	—	3	9	no	—
32	C₉×C₃₇⋊C₃	—	3	3	no	—
33	C₉×C₃₇⋊C₃	—	3	6	no	—
34	#11758603	—	3	3	no	—
35	#11758603	—	3	6	no	—
36	#10624684	—	37	1	no	maximal, Sylow
37	C₃²×C₃₇	—	3	3	no	—
38	C₃²×C₃₇	C₃²⋊C₉	1	3	yes	—
39	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
40	C₃×C₃₇⋊C₃	—	3	3	no	—
41	C₃×C₃₇⋊C₃	—	3	9	no	—
42	C₉×C₃₇	—	3	3	no	—
43	#66524	—	3	3	no	—
44	#66524	C₃²⋊C₉	1	3	yes	—
45	#66524	C₃²⋊C₉	1	3	yes	—
46	#66524	C₃×H(3,𝔽₃)	1	1	yes	characteristic
47	#66524	C₃×3₋¹⁺²	1	2	yes	—
48	C₃×#737	—	37	1	no	—
49	C₃×C₂₇⋊C₃	—	37	2	no	—
50	C₃²×C₂₇	—	37	1	no	—
51	C₃²⋊C₂₇	—	37	9	no	—
52	C₃×C₃₇	—	3	3	no	—
53	C₃×C₃₇	C₃×C₃²⋊C₉	1	1	yes	characteristic
54	C₃×C₃₇	#6903	1	3	yes	—
55	C₃₇⋊C₃	—	9	3	no	—
56	#737	—	111	3	no	—
57	#737	—	111	6	no	—
58	C₃×3₋¹⁺²	—	37	1	no	—
59	C₃×3₋¹⁺²	—	37	3	no	—
60	C₃×3₋¹⁺²	—	37	3	no	—
61	C₃×3₋¹⁺²	#66526	1	1	yes	characteristic
62	C₃×H(3,𝔽₃)	—	37	1	no	—
63	C₃²×C₉	—	37	1	no	—
64	C₃²×C₉	—	37	3	no	—
65	C₂₇⋊C₃	—	111	6	no	—
66	C₃×C₂₇	—	37	3	no	—
67	C₃×C₂₇	—	37	6	no	—
68	C₃×C₂₇	—	111	3	no	—
69	C₃₇	#10624684	1	1	yes	characteristic, complemented, Sylow
70	C₃³	—	37	3	no	—
71	C₃³	C₃×#66526	1	1	yes	characteristic
72	3₋¹⁺²	—	3	3	no	—
73	3₋¹⁺²	—	111	9	no	—
74	3₋¹⁺²	—	111	9	no	—
75	3₋¹⁺²	—	111	3	no	—
76	H(3,𝔽₃)	—	111	3	no	—
77	C₃×C₉	—	37	3	no	—
78	C₃×C₉	—	37	6	no	—
79	C₃×C₉	—	37	1	no	—
80	C₃×C₉	—	37	1	no	—
81	C₃×C₉	—	37	1	no	—
82	C₃×C₉	—	37	3	no	—
83	C₃×C₉	—	111	3	no	—
84	C₃×C₉	—	111	6	no	—
85	C₃×C₉	C₃×#66526	1	3	yes	—
86	C₂₇	—	111	3	no	—
87	C₂₇	—	111	6	no	—
88	C₃²	—	3	3	no	—
89	C₃²	—	111	3	no	—
90	C₃²	—	111	9	no	—
91	C₃²	C₃²⋊#66526	1	3	yes	—
92	C₃²	C₃²×#66526	1	1	yes	characteristic
93	C₉	—	3	3	no	—
94	C₉	—	37	1	no	—
95	C₉	—	37	2	no	—
96	C₉	—	37	3	no	—
97	C₉	—	37	3	no	—
98	C₉	—	111	3	no	—
99	C₃	—	3	3	no	—
100	C₃	—	333	3	no	—
101	C₃	C₃×C₃²⋊#66526	1	1	yes	characteristic
102	C₃	#22582732	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688541

C₃

yes

maximal, complemented

C₃²×#11758603

C₃

yes

maximal, characteristic, complemented

#22582690

C₃

yes

maximal, complemented

C₃×#15688532

C₃

yes

maximal, characteristic

#15688541

—

C₃×#11758603

—

C₃×#11758603

C₃²

yes

—

C₃×#11758603

C₃²

yes

—

#15688532

—

#15688532

—

C₃×#11758608

C₉

yes

characteristic

C₃×#11758605

C₉

yes

—

C₃×#11758606

C₉

yes

—

C₃×C₃²⋊(C₃₇⋊C₃)

C₉

yes

characteristic

C₃²×#66524

C₃²

yes

characteristic

C₃×C₉×C₃₇⋊C₃

C₃²

yes

—

C₃×3₋¹⁺²×C₃₇

C₉

yes

characteristic

C₃³×C₃₇

C₃×C₉

yes

characteristic

C₃²×C₃₇⋊C₃

C₃×C₉

yes

—

3₋¹⁺²×C₃₇

—

C₃×C₉×C₃₇

C₃×C₉

yes

—

C₃²⋊(C₃₇⋊C₃)

—

#11758608

—

C₃×#66524

C₃×C₉

yes

—

C₃×#66524

C₃×C₉

yes

characteristic

C₃×#66524

C₃×C₉

yes

characteristic

C₃×#66524

H(3,𝔽₃)

yes

—

C₃×#66524

3₋¹⁺²

yes

—

C₃×#66524

C₃³

yes

characteristic

#11758606

—

#11758605

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

—

—

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃²⋊C₉

yes

—

C₃²×C₃₇

C₃²×C₉

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

#66524

—

#66524

C₃²⋊C₉

yes

—

#66524

C₃²⋊C₉

yes

—

#66524

C₃×H(3,𝔽₃)

yes

characteristic

#66524

C₃×3₋¹⁺²

yes

—

C₃×#737

—

C₃×C₂₇⋊C₃

—

C₃²×C₂₇

—

C₃²⋊C₂₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃×C₃²⋊C₉

yes

characteristic

C₃×C₃₇

#6903

yes

—

C₃₇⋊C₃

—

#737

—

111

—

#737

—

111

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

#66526

yes

characteristic

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

—

C₂₇⋊C₃

—

111

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

111

—

C₃₇

#10624684

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

C₃×#66526

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

H(3,𝔽₃)

—

111

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

C₃×#66526

yes

—

C₂₇

—

111

—

C₂₇

—

111

—

C₃²

—

C₃²

—

111

—

C₃²

—

111

—

C₃²

C₃²⋊#66526

yes

—

C₃²

C₃²×#66526

yes

characteristic

C₉

—

C₉

—

C₉

—

C₉

—

C₉

—

C₉

—

111

—

C₃

—

100

C₃

—

333

—

101

C₃

C₃×C₃²⋊#66526

yes

characteristic

102

C₃

#22582732

yes

—

	subgroup	G/H
center, Frattini	C₃²	C₃²×#66526
commutator, socle	C₃²×C₃₇	C₃²×C₉
Fitting	C₃×3₋¹⁺²×C₃₇	C₉
prime radical	C₃×C₃²⋊(C₃₇⋊C₃)	C₉
cosocle	C₃×#66524	C₃³
hypercenter	C₃×3₋¹⁺²	#66526
hypocenter	C₃₇	#10624684
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃×3₋¹⁺²

subgroup

G/H

center, Frattini

C₃²

C₃²×#66526

commutator, socle

C₃²×C₃₇

C₃²×C₉

Fitting

C₃×3₋¹⁺²×C₃₇

C₉

prime radical

C₃×C₃²⋊(C₃₇⋊C₃)

C₉

C₃³

C₃×3₋¹⁺²

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃₇

C₁ ◁ C₃² ◁ C₃×3₋¹⁺²

N	H ≅ G/N	type
1. C₃×#15688541	C₃	semidirect
2. C₃²×#11758603	C₃	semidirect
3. #22582690	C₃	semidirect
69. C₃₇	#10624684	semidirect

H ≅ G/N

type

1. C₃×#15688541

C₃

semidirect

2. C₃²×#11758603

C₃

semidirect

3. #22582690

C₃

semidirect

69. C₃₇

#10624684

semidirect

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

subgroup	index
C₃×#15688541	3
C₃²×#11758603	3
#22582690	3
C₃×#15688532	3
#10624684	37

subgroup

index

Subgroups — #41008486

Conjugacy classes of subgroups

Complements

Maximal subgroups