Group #41008269 — subgroups

Subgroups — #41008269

Order 26973.

316 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 39 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⋊#11758603	C₃	1	3	yes	maximal, complemented
2	C₃²×C₉×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
3	C₃×#11758603	—	3	3	no	—
4	C₃³×C₃₇⋊C₃	C₉	1	1	yes	characteristic
5	C₃²×#66524	C₉	1	1	yes	characteristic
6	C₃²×#66524	C₃²	1	1	yes	characteristic
7	C₃×C₉×C₃₇⋊C₃	—	3	1	no	—
8	C₃×C₉×C₃₇⋊C₃	—	3	2	no	—
9	C₃×C₉×C₃₇⋊C₃	—	3	9	no	—
10	C₃²×C₉×C₃₇	C₉	1	1	yes	characteristic
11	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
12	C₃²×C₃₇⋊C₃	—	3	1	no	—
13	C₃²×C₃₇⋊C₃	—	3	1	no	—
14	C₃²×C₃₇⋊C₃	—	3	1	no	—
15	C₃²×C₃₇⋊C₃	—	3	3	no	—
16	C₃²×C₃₇⋊C₃	—	3	3	no	—
17	C₃²×C₃₇⋊C₃	—	3	3	no	—
18	C₃²×C₃₇⋊C₃	C₂₇	1	3	yes	complemented
19	C₃×C₉×C₃₇	—	3	1	no	—
20	C₃×C₉×C₃₇	—	3	3	no	—
21	C₃×#66524	—	3	1	no	—
22	C₃×#66524	—	3	3	no	—
23	C₃×#66524	—	3	3	no	—
24	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
25	C₃×#66524	3₋¹⁺²	1	2	yes	—
26	C₉×C₃₇⋊C₃	—	3	18	no	—
27	C₉×C₃₇⋊C₃	—	3	9	no	—
28	#11758603	—	9	3	no	—
29	#10624444	—	37	1	no	maximal, Sylow
30	C₃²×C₃₇	—	3	1	no	—
31	C₃²×C₃₇	—	3	1	no	—
32	C₃²×C₃₇	—	3	1	no	—
33	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
34	C₃²×C₃₇	C₃×C₂₇	1	1	yes	characteristic
35	C₃²×C₃₇	C₂₇⋊C₃	1	1	yes	characteristic
36	C₃²×C₃₇	C₂₇⋊C₃	1	1	yes	characteristic
37	C₃×C₃₇⋊C₃	—	3	9	no	—
38	C₃×C₃₇⋊C₃	—	3	9	no	—
39	C₃×C₃₇⋊C₃	—	3	3	no	—
40	C₃×C₃₇⋊C₃	—	3	3	no	—
41	C₃×C₃₇⋊C₃	—	3	3	no	—
42	C₃×C₃₇⋊C₃	—	3	3	no	—
43	C₃×C₃₇⋊C₃	—	3	9	no	—
44	C₉×C₃₇	—	3	3	no	—
45	#66524	—	3	3	no	—
46	#66524	—	3	2	no	—
47	#66524	AΣL(1,𝔽₂₇)	1	3	yes	—
48	C₃³×C₉	—	37	1	no	—
49	C₃²⋊C₂₇	—	37	3	no	—
50	C₃×C₃₇	—	3	1	no	—
51	C₃×C₃₇	—	3	1	no	—
52	C₃×C₃₇	—	3	1	no	—
53	C₃×C₃₇	C₃²⋊C₂₇	1	1	yes	characteristic
54	C₃×C₃₇	#6866	1	1	yes	characteristic
55	C₃×C₃₇	#6869	1	1	yes	characteristic
56	C₃×C₃₇	#6869	1	1	yes	characteristic
57	C₃₇⋊C₃	—	3	9	no	—
58	C₃⁴	—	37	1	no	—
59	C₃²×C₉	—	37	1	no	—
60	C₃²×C₉	—	37	1	no	—
61	C₃²×C₉	—	111	1	no	—
62	C₃²×C₉	—	111	2	no	—
63	C₃²×C₉	—	111	9	no	—
64	C₃²×C₉	#66526	1	1	yes	characteristic
65	C₃×C₂₇	—	111	3	no	—
66	C₃₇	#10624444	1	1	yes	characteristic, complemented, Sylow
67	C₃³	—	37	3	no	—
68	C₃³	—	111	3	no	—
69	C₃³	—	111	3	no	—
70	C₃³	—	111	3	no	—
71	C₃³	—	111	1	no	—
72	C₃³	—	111	1	no	—
73	C₃³	—	111	1	no	—
74	C₃³	C₃×#66526	1	1	yes	characteristic
75	C₃×C₉	—	3	1	no	—
76	C₃×C₉	—	3	3	no	—
77	C₃×C₉	—	37	1	no	—
78	C₃×C₉	—	37	2	no	—
79	C₃×C₉	—	111	9	no	—
80	C₃×C₉	—	111	18	no	—
81	C₃×C₉	—	111	3	no	—
82	C₃×C₉	—	111	1	no	—
83	C₃×C₉	—	111	3	no	—
84	C₂₇	—	333	3	no	—
85	C₃²	—	3	1	no	—
86	C₃²	—	3	1	no	—
87	C₃²	—	3	1	no	—
88	C₃²	—	111	9	no	—
89	C₃²	—	111	9	no	—
90	C₃²	—	111	9	no	—
91	C₃²	—	111	3	no	—
92	C₃²	—	111	3	no	—
93	C₃²	—	111	3	no	—
94	C₃²	—	111	3	no	—
95	C₃²	C₃²⋊#66526	1	1	yes	characteristic
96	C₃²	(C₃₇⋊C₃)⋊C₂₇	1	1	yes	characteristic
97	C₃²	#15688542	1	1	yes	characteristic
98	C₃²	#15688543	1	1	yes	characteristic
99	C₉	—	3	3	no	—
100	C₉	—	37	3	no	—
101	C₉	—	111	2	no	—
102	C₉	—	111	3	no	—
103	C₃	—	3	1	no	—
104	C₃	—	3	1	no	—
105	C₃	—	3	1	no	—
106	C₃	—	111	9	no	—
107	C₃	#22582690	1	1	yes	characteristic
108	C₃	#22582691	1	1	yes	characteristic
109	C₃	#22582695	1	1	yes	characteristic
110	C₃	#22582696	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²⋊#11758603

C₃

yes

maximal, complemented

C₃²×C₉×C₃₇⋊C₃

C₃

yes

maximal, characteristic

C₃×#11758603

—

C₃³×C₃₇⋊C₃

C₉

yes

characteristic

C₃²×#66524

C₉

yes

characteristic

C₃²×#66524

C₃²

yes

characteristic

C₃×C₉×C₃₇⋊C₃

—

C₃×C₉×C₃₇⋊C₃

—

C₃×C₉×C₃₇⋊C₃

—

C₃²×C₉×C₃₇

C₉

yes

characteristic

C₃³×C₃₇

C₃×C₉

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

C₂₇

yes

complemented

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

—

—

—

yes

3₋¹⁺²

yes

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

#11758603

—

#10624444

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃²⋊C₉

yes

characteristic

C₃²×C₃₇

C₃×C₂₇

yes

characteristic

C₃²×C₃₇

C₂₇⋊C₃

yes

characteristic

C₃²×C₃₇

C₂₇⋊C₃

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

#66524

—

#66524

—

#66524

AΣL(1,𝔽₂₇)

yes

—

C₃³×C₉

—

C₃²⋊C₂₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃²⋊C₂₇

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃₇⋊C₃

—

C₃⁴

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

111

—

C₃²×C₉

—

111

—

C₃²×C₉

—

111

—

C₃²×C₉

#66526

yes

characteristic

C₃×C₂₇

—

111

—

C₃₇

#10624444

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

C₃×#66526

yes

characteristic

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₂₇

—

333

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

111

—

C₃²

—

111

—

C₃²

—

111

—

C₃²

—

111

—

C₃²

—

111

—

C₃²

—

111

—

C₃²

—

111

—

C₃²

C₃²⋊#66526

yes

characteristic

C₃²

(C₃₇⋊C₃)⋊C₂₇

yes

C₃²

yes

C₃²

yes

C₉

—

100

C₉

—

101

C₉

—

111

—

102

C₉

—

111

—

103

C₃

—

104

C₃

—

105

C₃

—

106

C₃

—

111

—

107

C₃

#22582690

yes

characteristic

108

C₃

#22582691

yes

characteristic

109

C₃

#22582695

yes

characteristic

110

C₃

#22582696

yes

characteristic

	subgroup	G/H
center	C₃²	C₃²⋊#66526
commutator	C₃²×C₃₇	C₃×C₂₇
Frattini	C₃³	C₃×#66526
Fitting	C₃²×C₉×C₃₇	C₉
socle	C₃²×C₃₇	C₃²⋊C₉
prime radical	C₃³×C₃₇⋊C₃	C₉
cosocle	C₃²×#66524	C₃²
hypercenter	C₃²×C₉	#66526
hypocenter	C₃₇	#10624444
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃²×C₉

subgroup

G/H

center

C₃²

C₃²⋊#66526

commutator

C₃²×C₃₇

C₃×C₂₇

Frattini

C₃³

C₃×#66526

Fitting

C₃²×C₉×C₃₇

C₉

socle

C₃²×C₃₇

C₃²⋊C₉

prime radical

C₃³×C₃₇⋊C₃

C₉

cosocle

C₃²×#66524

C₃²

C₃²×C₉

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ C₃²×C₉

N	H ≅ G/N	type
1. C₃²⋊#11758603	C₃	semidirect
18. C₃²×C₃₇⋊C₃	C₂₇	semidirect
66. C₃₇	#10624444	semidirect

H ≅ G/N

type

1. C₃²⋊#11758603

C₃

semidirect

18. C₃²×C₃₇⋊C₃

C₂₇

semidirect

66. C₃₇

#10624444

semidirect

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

subgroup	index
C₃²⋊#11758603	3
C₃²×C₉×C₃₇⋊C₃	3
#10624444	37

subgroup

index

C₃²⋊#11758603

C₃²×C₉×C₃₇⋊C₃

#10624444

Subgroups — #41008269

Conjugacy classes of subgroups

Complements

Maximal subgroups