Group #41006242 — subgroups

Subgroups — #41006242

Order 26973.

404 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 112 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×C₃²⋊#66524	C₃	1	2	yes	maximal
3	#22582471	C₃	1	9	yes	maximal, complemented
4	C₃×C₃₇×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
5	C₃×#11758608	—	3	2	no	—
6	C₃×#11758608	—	3	1	no	—
7	C₃×#11758608	—	3	3	no	—
8	C₃×#11758608	—	3	3	no	—
9	C₃×#11758608	C₉	1	9	yes	complemented
10	C₃²×#66524	—	3	2	no	—
11	C₃²×#66524	C₃²	1	3	yes	—
12	C₃²⋊#66524	—	3	2	no	—
13	C₃²⋊#66524	—	3	2	no	—
14	C₃²⋊#66524	C₃²	1	6	yes	—
15	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
16	C₃²×C₉×C₃₇	—	3	1	no	—
17	C₃₇×C₃²⋊C₉	—	3	2	no	—
18	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
19	C₃³×C₃₇	—	9	2	no	—
20	C₃³×C₃₇	—	9	1	no	—
21	C₃³×C₃₇	C₃×C₉	1	3	yes	—
22	C₃³×C₃₇	H(3,𝔽₃)	1	2	yes	—
23	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
24	C₃³×C₃₇	3₋¹⁺²	1	3	yes	—
25	C₃³×C₃₇	3₋¹⁺²	1	3	yes	—
26	C₃³×C₃₇	C₃³	1	1	yes	characteristic
27	C₃×C₉×C₃₇	—	3	3	no	—
28	C₃×C₉×C₃₇	—	9	3	no	—
29	#11758608	—	3	18	no	—
30	#11758608	—	3	9	no	—
31	C₃×#66524	—	3	3	no	—
32	C₃×#66524	—	3	3	no	—
33	C₃×#66524	—	3	3	no	—
34	C₃×#66524	—	3	6	no	—
35	C₃×#66524	—	9	6	no	—
36	C₃×#66524	C₃×C₉	1	9	yes	complemented
37	#10624691	—	37	1	no	maximal, Sylow
38	C₃²×C₃₇	—	3	4	no	—
39	C₃²×C₃₇	—	3	2	no	—
40	C₃²×C₃₇	—	3	2	no	—
41	C₃²×C₃₇	—	3	2	no	—
42	C₃²×C₃₇	—	3	1	no	—
43	C₃²×C₃₇	—	3	1	no	—
44	C₃²×C₃₇	—	9	3	no	—
45	C₃²×C₃₇	—	9	3	no	—
46	C₃²×C₃₇	—	9	3	no	—
47	C₃²×C₃₇	C₃²⋊C₉	1	6	yes	—
48	C₃²×C₃₇	C₃²⋊C₉	1	3	yes	—
49	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
50	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
51	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
52	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
53	C₉×C₃₇	—	9	3	no	—
54	#66524	—	3	9	no	—
55	#66524	—	9	6	no	—
56	C₃²×3₋¹⁺²	—	37	1	no	—
57	C₃×C₃²⋊C₉	—	37	2	no	—
58	C₃×C₃²⋊C₉	C₃₇⋊C₃	1	1	yes	characteristic
59	#6868	—	37	9	no	—
60	C₃×C₃₇	—	3	6	no	—
61	C₃×C₃₇	—	3	3	no	—
62	C₃×C₃₇	—	3	1	no	—
63	C₃×C₃₇	—	3	1	no	—
64	C₃×C₃₇	—	3	1	no	—
65	C₃×C₃₇	C₃×C₃²⋊C₉	1	1	yes	characteristic
66	C₃×C₃₇	#6911	1	1	yes	characteristic
67	C₃×C₃₇	#6912	1	2	yes	—
68	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
69	C₃×3₋¹⁺²	—	37	9	no	—
70	C₃×3₋¹⁺²	—	111	3	no	—
71	C₃×3₋¹⁺²	—	111	3	no	—
72	C₃×3₋¹⁺²	—	111	2	no	—
73	C₃×3₋¹⁺²	—	111	1	no	—
74	C₃²×C₉	—	3	1	no	—
75	C₃²×C₉	—	37	3	no	—
76	C₃²×C₉	—	111	2	no	—
77	C₃²⋊C₉	—	3	2	no	—
78	C₃²⋊C₉	—	37	6	no	—
79	C₃²⋊C₉	—	111	2	no	—
80	C₃²⋊C₉	—	111	2	no	—
81	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
82	C₃₇	#10624691	1	1	yes	characteristic, complemented, Sylow
83	C₃³	—	9	1	no	—
84	C₃³	—	9	2	no	—
85	C₃³	C₉×C₃₇⋊C₃	1	3	yes	—
86	C₃³	#11758605	1	3	yes	—
87	C₃³	#11758606	1	3	yes	—
88	C₃³	C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
89	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
90	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
91	3₋¹⁺²	—	111	18	no	—
92	3₋¹⁺²	—	111	9	no	—
93	C₃×C₉	—	3	3	no	—
94	C₃×C₉	—	9	3	no	—
95	C₃×C₉	—	37	9	no	—
96	C₃×C₉	—	333	6	no	—
97	C₃×C₉	—	111	6	no	—
98	C₃×C₉	—	111	3	no	—
99	C₃×C₉	—	111	3	no	—
100	C₃×C₉	—	111	3	no	—
101	C₃²	—	3	1	no	—
102	C₃²	—	3	2	no	—
103	C₃²	—	3	1	no	—
104	C₃²	—	3	4	no	—
105	C₃²	—	3	2	no	—
106	C₃²	—	3	2	no	—
107	C₃²	—	9	3	no	—
108	C₃²	—	9	3	no	—
109	C₃²	—	9	3	no	—
110	C₃²	#15688502	1	6	yes	—
111	C₃²	#15688502	1	3	yes	—
112	C₃²	C₃×C₉×C₃₇⋊C₃	1	1	yes	characteristic
113	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
114	C₃²	C₃×#11758606	1	1	yes	characteristic
115	C₃²	C₃×#11758605	1	1	yes	characteristic
116	C₉	—	9	3	no	—
117	C₉	—	333	6	no	—
118	C₉	—	111	9	no	—
119	C₃	—	3	3	no	—
120	C₃	—	3	6	no	—
121	C₃	—	3	1	no	—
122	C₃	—	3	1	no	—
123	C₃	—	3	1	no	—
124	C₃	C₃×#15688502	1	1	yes	characteristic
125	C₃	#22582647	1	1	yes	characteristic
126	C₃	#22582651	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#11758608

C₃

yes

maximal, characteristic

C₃×C₃²⋊#66524

C₃

yes

maximal

#22582471

C₃

yes

maximal, complemented

C₃×C₃₇×C₃²⋊C₉

C₃

yes

maximal, characteristic

—

—

—

—

C₉

yes

complemented

C₃²×#66524

—

C₃²×#66524

C₃²

yes

—

C₃²⋊#66524

—

C₃²⋊#66524

—

C₃²⋊#66524

C₃²

yes

—

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃²×C₉×C₃₇

—

C₃₇×C₃²⋊C₉

—

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

C₃×C₉

yes

—

C₃³×C₃₇

H(3,𝔽₃)

yes

—

C₃³×C₃₇

H(3,𝔽₃)

yes

characteristic

C₃³×C₃₇

3₋¹⁺²

yes

—

C₃³×C₃₇

3₋¹⁺²

yes

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

—

—

—

—

—

—

—

C₃×C₉

yes

complemented

#10624691

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃²⋊C₉

yes

—

C₃²×C₃₇

C₃²⋊C₉

yes

—

C₃²×C₃₇

C₃²×C₉

yes

characteristic

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

characteristic

C₉×C₃₇

—

#66524

—

#66524

—

C₃²×3₋¹⁺²

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

C₃₇⋊C₃

yes

characteristic

#6868

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃×C₃²⋊C₉

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

—

C₃⁴

C₃×C₃₇⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

C₃₇

#10624691

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

C₉×C₃₇⋊C₃

yes

—

C₃³

#11758605

yes

—

C₃³

#11758606

yes

—

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

—

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

333

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

100

C₃×C₉

—

111

—

101

C₃²

—

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

—

108

C₃²

—

109

C₃²

—

110

C₃²

#15688502

yes

—

111

C₃²

#15688502

yes

—

112

C₃²

C₃×C₉×C₃₇⋊C₃

yes

characteristic

113

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

114

C₃²

C₃×#11758606

yes

characteristic

115

C₃²

C₃×#11758605

yes

characteristic

116

C₉

—

117

C₉

—

333

—

118

C₉

—

111

—

119

C₃

—

120

C₃

—

121

C₃

—

122

C₃

—

123

C₃

—

124

C₃

C₃×#15688502

yes

characteristic

125

C₃

#22582647

yes

characteristic

126

C₃

#22582651

yes

—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator	C₃²×C₃₇	C₃²×C₉
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃×C₃₇×C₃²⋊C₉	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃⁴×C₃₇	C₃²
cosocle	C₃³×C₃₇	C₃³
hypercenter	C₃×C₃²⋊C₉	C₃₇⋊C₃
hypocenter	C₃₇	#10624691
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃×C₃²⋊C₉

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

commutator

C₃²×C₃₇

C₃²×C₉

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃×C₃₇×C₃²⋊C₉

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

prime radical

C₃⁴×C₃₇

C₃²

cosocle

C₃³×C₃₇

C₃³

hypercenter

C₃×C₃²⋊C₉

C₃₇⋊C₃

hypocenter

C₃₇

#10624691

LCS

G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃×C₃²⋊C₉

N	H ≅ G/N	type
3. #22582471	C₃	semidirect
9. C₃×#11758608	C₉	semidirect
36. C₃×#66524	C₃×C₉	semidirect
82. C₃₇	#10624691	semidirect

H ≅ G/N

type

3. #22582471

C₃

semidirect

9. C₃×#11758608

C₉

semidirect

36. C₃×#66524

C₃×C₉

semidirect

82. C₃₇

#10624691

semidirect

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

subgroup	index
C₃²×#11758608	3
C₃×C₃²⋊#66524	3
#22582471	3
C₃×C₃₇×C₃²⋊C₉	3
#10624691	37

subgroup

index

C₃²×#11758608

C₃×C₃²⋊#66524

#22582471

C₃×C₃₇×C₃²⋊C₉

#10624691

Subgroups — #41006242

Conjugacy classes of subgroups

Complements

Maximal subgroups