Group #41008168 — subgroups

Subgroups — C₃×#22582651 #41008168

Order 26973.

980 conjugacy classes of subgroups; only the 292 normal subgroups are shown.

normal subgroups only

This group has 980 conjugacy classes of subgroups — too many to draw. Only its 292 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	2	yes	maximal
2	C₃²×#11758608	C₃	1	1	yes	maximal, characteristic
3	#22582651	C₃	1	27	yes	maximal, direct product factor
4	C₃×#15688520	C₃	1	9	yes	maximal, complemented
5	C₃²×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic
6	C₃×#11758608	—	3	2	no	—
7	C₃×#11758608	—	3	2	no	—
8	C₃×#11758608	—	3	12	no	—
9	C₃×#11758608	—	3	1	no	—
10	C₃×#11758608	—	3	2	no	—
11	C₃×#11758608	—	3	6	no	—
12	C₃×#11758608	C₃²	1	6	yes	—
13	C₃×#11758608	C₃²	1	18	yes	—
14	C₃×#11758608	C₃²	1	9	yes	—
15	C₃²×#66524	—	3	2	no	—
16	C₃²×#66524	C₃²	1	3	yes	—
17	#15688520	C₃²	1	81	yes	complemented
18	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
19	C₃×3₋¹⁺²×C₃₇	—	3	2	no	—
20	C₃×3₋¹⁺²×C₃₇	—	3	6	no	—
21	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
22	C₃×3₋¹⁺²×C₃₇	C₃²	1	9	yes	—
23	C₃²×C₉×C₃₇	—	3	1	no	—
24	C₃³×C₃₇	—	9	3	no	—
25	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
26	C₃³×C₃₇	H(3,𝔽₃)	1	2	yes	—
27	C₃³×C₃₇	H(3,𝔽₃)	1	6	yes	—
28	C₃³×C₃₇	C₃³	1	1	yes	characteristic
29	C₃³×C₃₇	C₃³	1	3	yes	—
30	3₋¹⁺²×C₃₇	—	3	6	no	—
31	3₋¹⁺²×C₃₇	—	3	18	no	—
32	3₋¹⁺²×C₃₇	C₃³	1	9	yes	—
33	C₃×C₉×C₃₇	—	3	9	no	—
34	C₃×C₉×C₃₇	—	3	3	no	—
35	#11758608	—	3	12	no	—
36	#11758608	—	3	18	no	—
37	#11758608	—	3	18	no	—
38	#11758608	—	3	18	no	—
39	#11758608	—	3	9	no	—
40	#11758608	C₃³	1	18	yes	—
41	C₃×#66524	—	3	18	no	—
42	C₃×#66524	—	3	6	no	—
43	C₃×#66524	—	3	6	no	—
44	C₃×#66524	—	3	3	no	—
45	C₃×#66524	C₃³	1	9	yes	—
46	C₃×#6912	—	37	1	no	maximal, Sylow
47	C₃²×C₃₇	—	3	3	no	—
48	C₃²×C₃₇	—	3	6	no	—
49	C₃²×C₃₇	—	3	3	no	—
50	C₃²×C₃₇	—	9	6	no	—
51	C₃²×C₃₇	—	9	3	no	—
52	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	6	yes	—
53	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	2	yes	—
54	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	3	yes	—
55	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
56	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
57	C₉×C₃₇	—	3	9	no	—
58	#66524	—	3	18	no	—
59	#66524	—	3	9	no	—
60	C₃²×3₋¹⁺²	—	37	1	no	—
61	C₃²×3₋¹⁺²	—	37	2	no	—
62	C₃²×3₋¹⁺²	C₃₇⋊C₃	1	1	yes	characteristic
63	#6912	—	37	27	no	—
64	C₃×#733	—	37	9	no	—
65	C₃×C₃₇	—	3	6	no	—
66	C₃×C₃₇	—	3	3	no	—
67	C₃×C₃₇	—	3	2	no	—
68	C₃×C₃₇	—	3	1	no	—
69	C₃×C₃₇	#6912	1	3	yes	complemented
70	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
71	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
72	C₃×3₋¹⁺²	—	3	6	no	—
73	C₃×3₋¹⁺²	—	3	2	no	—
74	C₃×3₋¹⁺²	—	37	9	no	—
75	C₃×3₋¹⁺²	—	37	18	no	—
76	C₃×3₋¹⁺²	—	37	6	no	—
77	C₃×3₋¹⁺²	—	111	6	no	—
78	C₃×3₋¹⁺²	—	111	12	no	—
79	C₃×3₋¹⁺²	—	111	1	no	—
80	C₃×3₋¹⁺²	—	111	2	no	—
81	C₃×3₋¹⁺²	—	111	2	no	—
82	C₃×3₋¹⁺²	—	111	2	no	—
83	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	9	yes	—
84	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	3	yes	—
85	C₃²×C₉	—	3	1	no	—
86	C₃²×C₉	—	37	3	no	—
87	C₃²×C₉	—	111	2	no	—
88	#733	—	37	81	no	—
89	C₃₇	C₃×#6912	1	1	yes	characteristic, complemented, Sylow
90	C₃³	—	9	3	no	—
91	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
92	C₃³	C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
93	C₃³	C₃²⋊(C₃₇⋊C₃)	1	6	yes	—
94	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
95	C₃³	C₃²×C₃₇⋊C₃	1	3	yes	—
96	3₋¹⁺²	—	3	18	no	—
97	3₋¹⁺²	—	3	6	no	—
98	3₋¹⁺²	—	37	18	no	—
99	3₋¹⁺²	—	111	9	no	—
100	3₋¹⁺²	—	111	18	no	—
101	3₋¹⁺²	—	111	18	no	—
102	3₋¹⁺²	—	111	18	no	—
103	3₋¹⁺²	—	111	12	no	—
104	3₋¹⁺²	C₃²×C₃₇⋊C₃	1	9	yes	—
105	C₃×C₉	—	3	9	no	—
106	C₃×C₉	—	3	3	no	—
107	C₃×C₉	—	37	9	no	—
108	C₃×C₉	—	111	6	no	—
109	C₃×C₉	—	111	18	no	—
110	C₃×C₉	—	111	3	no	—
111	C₃×C₉	—	111	6	no	—
112	C₃²	—	3	3	no	—
113	C₃²	—	3	6	no	—
114	C₃²	—	3	3	no	—
115	C₃²	—	9	6	no	—
116	C₃²	—	9	3	no	—
117	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
118	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
119	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	6	yes	—
120	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
121	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
122	C₉	—	3	9	no	—
123	C₉	—	111	9	no	—
124	C₉	—	111	18	no	—
125	C₃	—	3	3	no	—
126	C₃	—	3	6	no	—
127	C₃	—	3	2	no	—
128	C₃	—	3	1	no	—
129	C₃	#22582651	1	3	yes	direct product factor
130	C₃	C₃²×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#11758608

C₃

yes

maximal

C₃²×#11758608

C₃

yes

maximal, characteristic

#22582651

C₃

yes

maximal, direct product factor

C₃×#15688520

C₃

yes

maximal, complemented

C₃²×3₋¹⁺²×C₃₇

C₃

yes

maximal, characteristic

—

—

—

—

—

—

C₃²

yes

—

C₃×#11758608

C₃²

yes

—

C₃×#11758608

C₃²

yes

—

C₃²×#66524

—

C₃²×#66524

C₃²

yes

—

#15688520

C₃²

yes

complemented

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₃₇

—

C₃×3₋¹⁺²×C₃₇

—

C₃×3₋¹⁺²×C₃₇

C₃²

yes

—

C₃×3₋¹⁺²×C₃₇

C₃²

yes

—

C₃²×C₉×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

H(3,𝔽₃)

yes

characteristic

C₃³×C₃₇

H(3,𝔽₃)

yes

—

C₃³×C₃₇

H(3,𝔽₃)

yes

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃³×C₃₇

C₃³

yes

—

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

C₃³

yes

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

—

—

—

—

—

C₃³

yes

—

—

—

—

—

C₃³

yes

—

C₃×#6912

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃⁴

yes

characteristic

C₉×C₃₇

—

#66524

—

#66524

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

C₃₇⋊C₃

yes

characteristic

#6912

—

C₃×#733

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

#6912

yes

complemented

C₃×C₃₇

C₃²×H(3,𝔽₃)

yes

characteristic

C₃⁴

C₃×C₃₇⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

111

—

#733

—

C₃₇

C₃×#6912

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

—

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

C₃³

C₃²×C₃₇⋊C₃

yes

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

111

—

100

3₋¹⁺²

—

111

—

101

3₋¹⁺²

—

111

—

102

3₋¹⁺²

—

111

—

103

3₋¹⁺²

—

111

—

104

3₋¹⁺²

C₃²×C₃₇⋊C₃

yes

—

105

C₃×C₉

—

106

C₃×C₉

—

107

C₃×C₉

—

108

C₃×C₉

—

111

—

109

C₃×C₉

—

111

—

110

C₃×C₉

—

111

—

111

C₃×C₉

—

111

—

112

C₃²

—

113

C₃²

—

114

C₃²

—

115

C₃²

—

116

C₃²

—

117

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

—

118

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

—

119

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

—

120

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

121

C₃²

C₃³×C₃₇⋊C₃

yes

characteristic

122

C₉

—

123

C₉

—

111

—

124

C₉

—

111

—

125

C₃

—

126

C₃

—

127

C₃

—

128

C₃

—

129

C₃

#22582651

yes

direct product factor

130

C₃

C₃²×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃²×C₃₇	C₃⁴
Frattini	C₃²	C₃³×C₃₇⋊C₃
Fitting	C₃²×3₋¹⁺²×C₃₇	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃⁴×C₃₇	C₃²
hypercenter	C₃²×3₋¹⁺²	C₃₇⋊C₃
hypocenter	C₃₇	C₃×#6912
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃²×3₋¹⁺²

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

commutator, cosocle

C₃²×C₃₇

C₃⁴

Frattini

C₃²

C₃³×C₃₇⋊C₃

Fitting

C₃²×3₋¹⁺²×C₃₇

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

prime radical

C₃⁴×C₃₇

C₃²

hypercenter

C₃²×3₋¹⁺²

C₃₇⋊C₃

hypocenter

C₃₇

C₃×#6912

LCS

G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃²×3₋¹⁺²

N	H ≅ G/N	type
3. #22582651	C₃	direct (N × H)
4. C₃×#15688520	C₃	semidirect
17. #15688520	C₃²	semidirect
69. C₃×C₃₇	#6912	semidirect
89. C₃₇	C₃×#6912	semidirect
129. C₃	#22582651	direct (N × H)

H ≅ G/N

type

3. #22582651

C₃

direct (N × H)

4. C₃×#15688520

C₃

semidirect

17. #15688520

C₃²

semidirect

69. C₃×C₃₇

#6912

semidirect

89. C₃₇

C₃×#6912

semidirect

129. C₃

#22582651

direct (N × H)

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

subgroup	index
C₃²×#11758608	3
#22582651	3
C₃×#15688520	3
C₃²×3₋¹⁺²×C₃₇	3
C₃×#6912	37

subgroup

index

C₃²×3₋¹⁺²×C₃₇

Subgroups — C₃×#22582651 #41008168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#22582651 #41008168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#22582651 #41008168