Group #41006553 — subgroups

Subgroups — #41006553

Order 26973.

458 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 73 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×#11758608	C₃	1	1	yes	maximal, characteristic
2	C₃×#15688520	C₃	1	9	yes	maximal, complemented
3	#22582554	C₃	1	1	yes	maximal, characteristic
4	C₃×C₃²⋊#66524	C₃	1	1	yes	maximal, characteristic
5	C₃₇×#6891	C₃	1	1	yes	maximal, characteristic
6	C₃×#11758608	—	3	2	no	—
7	C₃×#11758608	—	3	9	no	—
8	C₃×#11758608	C₃²	1	3	yes	—
9	C₃×#11758608	C₃²	1	3	yes	—
10	C₃²×#66524	—	3	1	no	—
11	C₃²×#66524	C₃²	1	3	yes	—
12	#15688520	—	3	27	no	—
13	C₃²⋊#66524	—	3	2	no	—
14	C₃²⋊#66524	—	3	1	no	—
15	C₃²⋊#66524	—	3	2	no	—
16	C₃²⋊#66524	—	3	1	no	—
17	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
18	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
19	C₃₇×C₃²⋊C₉	—	3	2	no	—
20	C₃₇×C₃²⋊C₉	—	3	1	no	—
21	C₃³×C₃₇	—	3	3	no	—
22	C₃³×C₃₇	—	3	3	no	—
23	C₃³×C₃₇	—	9	2	no	—
24	C₃³×C₃₇	H(3,𝔽₃)	1	2	yes	—
25	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
26	C₃³×C₃₇	C₃³	1	1	yes	characteristic
27	3₋¹⁺²×C₃₇	—	3	9	no	—
28	C₃×C₉×C₃₇	—	3	3	no	—
29	#11758608	—	3	18	no	—
30	#11758608	—	3	6	no	—
31	#11758608	—	3	9	no	—
32	#11758608	H(3,𝔽₃)	1	9	yes	—
33	C₃×#66524	—	3	9	no	—
34	C₃×#66524	—	3	3	no	—
35	C₃×#66524	—	3	3	no	—
36	C₃×#66524	—	3	3	no	—
37	C₃×#66524	—	3	9	no	—
38	#10624744	—	37	1	no	maximal, Sylow
39	C₃²×C₃₇	—	3	2	no	—
40	C₃²×C₃₇	—	3	2	no	—
41	C₃²×C₃₇	—	3	1	no	—
42	C₃²×C₃₇	—	3	1	no	—
43	C₃²×C₃₇	—	3	4	no	—
44	C₃²×C₃₇	—	3	2	no	—
45	C₃²×C₃₇	—	3	2	no	—
46	C₃²×C₃₇	—	9	9	no	—
47	C₃²×C₃₇	#733	1	3	yes	complemented
48	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	3	yes	—
49	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
50	C₉×C₃₇	—	9	3	no	—
51	#66524	—	3	9	no	—
52	#66524	—	9	3	no	—
53	#66524	—	9	3	no	—
54	C₃²×3₋¹⁺²	—	37	1	no	—
55	C₃×#733	—	37	9	no	—
56	#6891	—	37	1	no	—
57	#6891	C₃₇⋊C₃	1	1	yes	characteristic
58	C₃×C₃²⋊C₉	—	37	1	no	—
59	C₃×C₃₇	—	3	3	no	—
60	C₃×C₃₇	—	3	3	no	—
61	C₃×C₃₇	—	9	2	no	—
62	C₃×C₃₇	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
63	C₃×C₃₇	C₃×#733	1	1	yes	characteristic
64	C₃×C₃₇	#6912	1	2	yes	—
65	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
66	C₃×3₋¹⁺²	—	37	3	no	—
67	C₃×3₋¹⁺²	—	37	3	no	—
68	C₃×3₋¹⁺²	—	111	9	no	—
69	C₃×3₋¹⁺²	—	111	2	no	—
70	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	3	yes	—
71	C₃²×C₉	—	37	3	no	—
72	C₃²×C₉	—	111	1	no	—
73	#733	—	111	27	no	—
74	C₃²⋊C₉	—	3	2	no	—
75	C₃²⋊C₉	—	3	1	no	—
76	C₃²⋊C₉	—	111	2	no	—
77	C₃²⋊C₉	—	111	1	no	—
78	C₃²⋊C₉	—	111	2	no	—
79	C₃²⋊C₉	—	111	1	no	—
80	C₃₇	#10624744	1	1	yes	characteristic, complemented, Sylow
81	C₃³	—	3	3	no	—
82	C₃³	—	3	3	no	—
83	C₃³	—	9	2	no	—
84	C₃³	C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
85	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
86	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
87	3₋¹⁺²	—	3	9	no	—
88	3₋¹⁺²	—	37	9	no	—
89	3₋¹⁺²	—	111	9	no	—
90	3₋¹⁺²	—	111	18	no	—
91	3₋¹⁺²	—	111	6	no	—
92	C₃×C₉	—	3	3	no	—
93	C₃×C₉	—	111	9	no	—
94	C₃×C₉	—	111	3	no	—
95	C₃×C₉	—	111	3	no	—
96	C₃×C₉	—	111	9	no	—
97	C₃×C₉	—	111	3	no	—
98	C₃²	—	3	1	no	—
99	C₃²	—	3	1	no	—
100	C₃²	—	3	2	no	—
101	C₃²	—	3	2	no	—
102	C₃²	—	3	2	no	—
103	C₃²	—	3	2	no	—
104	C₃²	—	3	4	no	—
105	C₃²	—	9	9	no	—
106	C₃²	#15688520	1	3	yes	complemented
107	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	3	yes	—
108	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
109	C₉	—	9	3	no	—
110	C₉	—	333	3	no	—
111	C₉	—	333	3	no	—
112	C₉	—	111	9	no	—
113	C₃	—	3	3	no	—
114	C₃	—	3	3	no	—
115	C₃	—	9	2	no	—
116	C₃	(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)	1	1	yes	characteristic
117	C₃	C₃×#15688520	1	1	yes	characteristic
118	C₃	#22582651	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#11758608

C₃

yes

maximal, characteristic

C₃×#15688520

C₃

yes

maximal, complemented

#22582554

C₃

yes

maximal, characteristic

C₃×C₃²⋊#66524

C₃

yes

maximal, characteristic

C₃₇×#6891

C₃

yes

maximal, characteristic

C₃×#11758608

—

C₃×#11758608

—

C₃×#11758608

C₃²

yes

—

C₃×#11758608

C₃²

yes

—

C₃²×#66524

—

C₃²×#66524

C₃²

yes

—

—

—

—

—

—

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₃₇

C₃²

yes

—

C₃₇×C₃²⋊C₉

—

C₃₇×C₃²⋊C₉

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

H(3,𝔽₃)

yes

—

C₃³×C₃₇

H(3,𝔽₃)

yes

characteristic

C₃³×C₃₇

C₃³

yes

characteristic

3₋¹⁺²×C₃₇

—

C₃×C₉×C₃₇

—

—

—

—

yes

—

—

—

—

—

—

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

#733

yes

complemented

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₃₇

—

#66524

—

#66524

—

#66524

—

C₃²×3₋¹⁺²

—

C₃×#733

—

#6891

—

#6891

C₃₇⋊C₃

yes

characteristic

C₃×C₃²⋊C₉

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃²⋊H(3,𝔽₃)

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

—

C₃⁴

C₃×C₃₇⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

—

C₃²×C₉

—

C₃²×C₉

—

111

—

#733

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃₇

#10624744

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

—

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

C₃×C₉

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃²

—

C₃²

—

100

C₃²

—

101

C₃²

—

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

#15688520

yes

complemented

107

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

—

108

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

109

C₉

—

110

C₉

—

333

—

111

C₉

—

333

—

112

C₉

—

111

—

113

C₃

—

114

C₃

—

115

C₃

—

116

C₃

(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)

yes

characteristic

117

C₃

C₃×#15688520

yes

characteristic

118

C₃

#22582651

yes

—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6891	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃⁴×C₃₇	C₃²
hypercenter	#6891	C₃₇⋊C₃
hypocenter	C₃₇	#10624744
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ #6891

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃₇×#6891

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

prime radical

C₃⁴×C₃₇

C₃²

C₃₇⋊C₃

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ #6891

N	H ≅ G/N	type
2. C₃×#15688520	C₃	semidirect
47. C₃²×C₃₇	#733	semidirect
80. C₃₇	#10624744	semidirect
106. C₃²	#15688520	semidirect

H ≅ G/N

type

2. C₃×#15688520

C₃

semidirect

47. C₃²×C₃₇

#733

semidirect

80. C₃₇

#10624744

semidirect

106. C₃²

#15688520

semidirect

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

subgroup	index
C₃²×#11758608	3
C₃×#15688520	3
#22582554	3
C₃×C₃²⋊#66524	3
C₃₇×#6891	3
#10624744	37

subgroup

index

C₃²×#11758608

C₃×#15688520

#22582554

C₃×C₃²⋊#66524

C₃₇×#6891

#10624744

Subgroups — #41006553

Conjugacy classes of subgroups

Complements

Maximal subgroups