Group #41006850 — subgroups

Subgroups — #41006850

Order 26973.

238 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 64 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×#15688521	C₃	1	1	yes	maximal, characteristic
2	#22582573	C₃	1	3	yes	maximal
3	C₃×C₉×#66524	C₃	1	1	yes	maximal, characteristic
4	#22582481	C₃	1	3	yes	maximal
5	#22582478	C₃	1	3	yes	maximal
6	#22582472	C₃	1	1	yes	maximal, characteristic
7	C₃₇×#6868	C₃	1	1	yes	maximal, characteristic
8	C₃×#11758608	C₃²	1	3	yes	—
9	C₃²×#66524	C₃²	1	1	yes	characteristic
10	C₃²×#66524	C₃²	1	1	yes	characteristic
11	C₃²×#66524	C₃²	1	1	yes	characteristic
12	#15688521	—	3	3	no	—
13	#15688506	—	3	3	no	—
14	#15688504	—	3	3	no	—
15	#15688504	—	3	3	no	—
16	C₃²⋊#66524	C₃²	1	3	yes	—
17	C₉×#66524	—	3	1	no	—
18	C₉×#66524	—	3	1	no	—
19	C₉×#66524	—	3	1	no	—
20	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
21	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
22	C₃³×C₃₇	C₃³	1	1	yes	characteristic
23	C₃×C₉×C₃₇	—	3	1	no	—
24	C₃×C₉×C₃₇	—	3	2	no	—
25	C₃×C₉×C₃₇	—	3	3	no	—
26	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	—
27	#11758608	—	3	3	no	—
28	#11758608	—	3	6	no	—
29	C₃×#66524	—	3	1	no	—
30	C₃×#66524	—	3	1	no	—
31	C₃×#66524	—	3	3	no	—
32	C₃×#66524	—	3	3	no	—
33	C₃×#66524	—	3	3	no	—
34	C₃×#66524	—	3	3	no	—
35	C₃×#66524	—	3	3	no	—
36	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
37	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
38	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
39	#10624787	—	37	1	no	maximal, Sylow
40	C₃²×C₃₇	—	3	1	no	—
41	C₃²×C₃₇	—	3	1	no	—
42	C₃²×C₃₇	—	3	1	no	—
43	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
44	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
45	C₃²×C₃₇	#737	1	2	yes	—
46	C₉×C₃₇	—	3	3	no	—
47	C₉×C₃₇	—	9	3	no	—
48	#66524	—	3	1	no	—
49	#66524	—	3	1	no	—
50	#66524	—	3	3	no	—
51	#66524	—	3	3	no	—
52	#66524	—	9	3	no	—
53	#66524	—	9	3	no	—
54	#66524	#733	1	3	yes	complemented
55	C₃×#733	—	37	1	no	—
56	#6894	—	37	3	no	—
57	C₃×C₉²	—	37	1	no	—
58	3₋¹⁺²⋊C₉	—	37	3	no	—
59	3₋¹⁺²⋊C₉	—	37	3	no	—
60	#6868	—	37	1	no	—
61	#6868	C₃₇⋊C₃	1	1	yes	characteristic
62	C₃×C₃₇	—	3	1	no	—
63	C₃×C₃₇	—	3	2	no	—
64	C₃×C₃₇	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
65	C₃×C₃₇	C₃×#733	1	1	yes	characteristic
66	C₃×C₃₇	#6908	1	1	yes	characteristic
67	C₃×C₃₇	#6908	1	1	yes	characteristic
68	C₃×3₋¹⁺²	—	37	3	no	—
69	C₃²×C₉	—	37	1	no	—
70	C₃²×C₉	—	37	1	no	—
71	C₃²×C₉	—	37	1	no	—
72	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
73	#733	—	111	3	no	—
74	C₉⋊C₉	—	111	3	no	—
75	C₉⋊C₉	—	111	3	no	—
76	C₉⋊C₉	—	111	3	no	—
77	C₃²⋊C₉	—	37	3	no	—
78	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
79	C₉²	—	111	1	no	—
80	C₉²	—	111	1	no	—
81	C₉²	—	111	1	no	—
82	C₃₇	#10624787	1	1	yes	characteristic, complemented, Sylow
83	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
84	3₋¹⁺²	—	111	3	no	—
85	3₋¹⁺²	—	111	6	no	—
86	C₃×C₉	—	3	1	no	—
87	C₃×C₉	—	3	2	no	—
88	C₃×C₉	—	3	3	no	—
89	C₃×C₉	—	37	1	no	—
90	C₃×C₉	—	37	1	no	—
91	C₃×C₉	—	37	1	no	—
92	C₃×C₉	—	111	3	no	—
93	C₃×C₉	—	111	3	no	—
94	C₃×C₉	—	111	3	no	—
95	C₃×C₉	—	111	1	no	—
96	C₃×C₉	—	111	1	no	—
97	C₃×C₉	—	111	3	no	—
98	C₃×C₉	—	111	3	no	—
99	C₃×C₉	#11758606	1	3	yes	—
100	C₃²	—	3	1	no	—
101	C₃²	—	3	1	no	—
102	C₃²	—	3	1	no	—
103	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
104	C₃²	C₃×#11758606	1	1	yes	characteristic
105	C₃²	#15688532	1	2	yes	—
106	C₉	—	3	3	no	—
107	C₉	—	9	3	no	—
108	C₉	—	37	3	no	—
109	C₉	—	333	3	no	—
110	C₉	—	333	3	no	—
111	C₉	—	111	1	no	—
112	C₉	—	111	1	no	—
113	C₉	—	111	3	no	—
114	C₉	—	111	3	no	—
115	C₃	—	3	1	no	—
116	C₃	—	3	2	no	—
117	C₃	#22582560	1	1	yes	characteristic
118	C₃	#733×C₃₇⋊C₃	1	1	yes	characteristic
119	C₃	#22582631	1	1	yes	characteristic
120	C₃	#22582634	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688521

C₃

yes

maximal, characteristic

#22582573

C₃

yes

maximal

C₃×C₉×#66524

C₃

yes

maximal, characteristic

#22582481

C₃

yes

maximal

#22582478

C₃

yes

maximal

#22582472

C₃

yes

maximal, characteristic

C₃₇×#6868

C₃

yes

maximal, characteristic

C₃×#11758608

C₃²

yes

—

C₃²×#66524

C₃²

yes

characteristic

C₃²×#66524

C₃²

yes

characteristic

C₃²×#66524

C₃²

yes

—

—

—

—

C₃²

yes

—

C₉×#66524

—

C₉×#66524

—

C₉×#66524

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

3₋¹⁺²

yes

—

—

—

—

—

—

—

—

—

—

yes

yes

yes

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₃₇

#737

yes

—

C₉×C₃₇

—

C₉×C₃₇

—

—

—

—

—

—

—

yes

—

—

C₃×C₉²

—

3₋¹⁺²⋊C₉

—

3₋¹⁺²⋊C₉

—

#6868

—

#6868

C₃₇⋊C₃

yes

characteristic

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₉⋊H(3,𝔽₃)

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃×3₋¹⁺²

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

#733

—

111

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

C₉²

—

111

—

C₉²

—

111

—

C₉²

—

111

—

C₃₇

#10624787

yes

characteristic, complemented, Sylow

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

#11758606

yes

—

100

C₃²

—

101

C₃²

—

102

C₃²

—

103

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

104

C₃²

C₃×#11758606

yes

characteristic

105

C₃²

#15688532

yes

—

106

C₉

—

107

C₉

—

108

C₉

—

109

C₉

—

333

—

110

C₉

—

333

—

111

C₉

—

111

—

112

C₉

—

111

—

113

C₉

—

111

—

114

C₉

—

111

—

115

C₃

—

116

C₃

—

117

C₃

#22582560

yes

characteristic

118

C₃

#733×C₃₇⋊C₃

yes

characteristic

119

C₃

#22582631

yes

characteristic

120

C₃

#22582634

yes

characteristic

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₃₇⋊C₃
commutator, prime radical, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6868	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
hypercenter	#6868	C₃₇⋊C₃
hypocenter	C₃₇	#10624787
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ #6868

subgroup

G/H

center

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

commutator, prime radical, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃₇×#6868

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

C₃₇⋊C₃

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ #6868

N	H ≅ G/N	type
54. #66524	#733	semidirect
82. C₃₇	#10624787	semidirect

H ≅ G/N

type

54. #66524

#733

semidirect

82. C₃₇

#10624787

semidirect

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

subgroup	index
C₃×#15688521	3
#22582573	3
C₃×C₉×#66524	3
#22582481	3
#22582478	3
#22582472	3
C₃₇×#6868	3
#10624787	37

subgroup

index

C₃×C₉×#66524

C₃₇×#6868

Subgroups — #41006850

Conjugacy classes of subgroups

Complements

Maximal subgroups