Group #41008144 — subgroups

Subgroups — C₃×#22582631 #41008144

Order 26973.

1064 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 1064 conjugacy classes of subgroups — too many to draw. Only its 274 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×#15688530	C₃	1	3	yes	maximal, complemented
2	C₃²×C₉×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582631	C₃	1	27	yes	maximal, direct product factor
4	C₃×#15688521	C₃	1	2	yes	maximal, complemented
5	C₃×#15688515	C₃	1	3	yes	maximal, complemented
6	C₃×#15688512	C₃	1	3	yes	maximal, complemented
7	C₃×C₃₇×#732	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
9	#15688530	—	3	18	no	—
10	#15688530	C₃²	1	27	yes	complemented
11	C₃×#11758608	—	3	6	no	—
12	C₃×#11758608	C₃²	1	6	yes	—
13	C₃²×#66524	—	3	3	no	—
14	C₃²×#66524	C₃²	1	1	yes	characteristic
15	C₃²×#66524	C₃²	1	1	yes	characteristic
16	C₃×C₉×C₃₇⋊C₃	—	3	1	no	—
17	C₃×C₉×C₃₇⋊C₃	—	3	2	no	—
18	C₃×C₉×C₃₇⋊C₃	—	3	6	no	—
19	C₃×C₉×C₃₇⋊C₃	C₃²	1	9	yes	complemented
20	#15688521	C₃²	1	18	yes	complemented
21	#15688515	C₃²	1	27	yes	complemented
22	#15688512	C₃²	1	27	yes	complemented
23	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	3	yes	—
24	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
25	C₃₇×#732	C₃²	1	9	yes	complemented
26	C₃³×C₃₇	—	3	3	no	—
27	C₃³×C₃₇	C₃³	1	1	yes	characteristic
28	C₃²×C₃₇⋊C₃	—	3	2	no	—
29	C₃²×C₃₇⋊C₃	—	3	1	no	—
30	C₃²×C₃₇⋊C₃	—	3	9	no	—
31	C₃²×C₃₇⋊C₃	C₃³	1	3	yes	—
32	H(3,𝔽₃)×C₃₇	—	3	6	no	—
33	H(3,𝔽₃)×C₃₇	C₃³	1	9	yes	—
34	C₃×C₉×C₃₇	—	3	2	no	—
35	C₃×C₉×C₃₇	—	3	1	no	—
36	C₃×C₉×C₃₇	C₃³	1	3	yes	—
37	#11758608	—	3	12	no	—
38	#11758608	—	3	54	no	—
39	#11758608	C₃³	1	18	yes	—
40	C₃×#66524	—	3	2	no	—
41	C₃×#66524	—	3	1	no	—
42	C₃×#66524	—	3	6	no	—
43	C₃×#66524	—	3	27	no	—
44	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
45	C₃×#66524	H(3,𝔽₃)	1	2	yes	—
46	C₃×#66524	H(3,𝔽₃)	1	6	yes	—
47	C₃×#66524	C₃³	1	3	yes	—
48	C₃×#66524	C₃³	1	3	yes	—
49	C₉×C₃₇⋊C₃	—	3	9	no	—
50	C₉×C₃₇⋊C₃	—	3	18	no	—
51	C₃×#6908	—	37	1	no	maximal, Sylow
52	C₃²×C₃₇	—	3	3	no	—
53	C₃²×C₃₇	—	3	9	no	—
54	C₃²×C₃₇	—	9	9	no	—
55	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	2	yes	—
56	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
57	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
58	C₃×C₃₇⋊C₃	—	3	18	no	—
59	C₃×C₃₇⋊C₃	—	3	3	no	—
60	C₃×C₃₇⋊C₃	—	3	9	no	—
61	C₃×C₃₇⋊C₃	—	3	9	no	—
62	C₉×C₃₇	—	3	3	no	—
63	#66524	—	3	3	no	—
64	#66524	—	3	18	no	—
65	#66524	C₃×H(3,𝔽₃)	1	3	yes	—
66	#66524	C₃×H(3,𝔽₃)	1	6	yes	—
67	C₃×#737	—	37	3	no	—
68	C₃³×C₉	—	37	1	no	—
69	#6908	—	37	27	no	—
70	C₃×#733	—	37	2	no	—
71	C₃×#732	C₃₇⋊C₃	1	1	yes	characteristic, complemented
72	C₃×#731	—	37	3	no	—
73	C₃×AΣL(1,𝔽₂₇)	—	37	3	no	—
74	C₃×C₃₇	—	3	2	no	—
75	C₃×C₃₇	—	3	1	no	—
76	C₃×C₃₇	—	9	9	no	—
77	C₃×C₃₇	#6908	1	3	yes	complemented
78	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
79	C₃₇⋊C₃	—	3	9	no	—
80	C₃⁴	—	37	1	no	—
81	#737	—	37	27	no	—
82	#737	—	111	18	no	—
83	C₃×3₋¹⁺²	—	37	6	no	—
84	C₃×3₋¹⁺²	—	111	6	no	—
85	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	3	yes	—
86	C₃²×C₉	—	37	1	no	—
87	C₃²×C₉	—	37	1	no	—
88	C₃²×C₉	—	37	9	no	—
89	C₃²×C₉	—	111	6	no	—
90	C₃²×C₉	—	111	1	no	—
91	C₃²×C₉	—	111	2	no	—
92	C₃²×C₉	—	111	3	no	—
93	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
94	#733	—	37	18	no	—
95	#732	C₃×C₃₇⋊C₃	1	9	yes	complemented
96	#731	—	37	27	no	—
97	AΣL(1,𝔽₂₇)	—	37	27	no	—
98	C₃₇	C₃×#6908	1	1	yes	characteristic, complemented, Sylow
99	C₃³	—	3	3	no	—
100	C₃³	—	37	3	no	—
101	C₃³	—	111	9	no	—
102	C₃³	—	111	2	no	—
103	C₃³	—	111	1	no	—
104	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
105	3₋¹⁺²	—	37	18	no	—
106	3₋¹⁺²	—	111	54	no	—
107	3₋¹⁺²	—	111	12	no	—
108	H(3,𝔽₃)	—	3	6	no	—
109	H(3,𝔽₃)	C₃²×C₃₇⋊C₃	1	9	yes	—
110	C₃×C₉	—	3	2	no	—
111	C₃×C₉	—	3	1	no	—
112	C₃×C₉	—	37	6	no	—
113	C₃×C₉	—	37	1	no	—
114	C₃×C₉	—	37	2	no	—
115	C₃×C₉	—	37	3	no	—
116	C₃×C₉	—	37	3	no	—
117	C₃×C₉	—	111	9	no	—
118	C₃×C₉	—	111	18	no	—
119	C₃×C₉	—	111	27	no	—
120	C₃×C₉	—	111	2	no	—
121	C₃×C₉	—	111	1	no	—
122	C₃×C₉	—	111	6	no	—
123	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
124	C₃²	—	3	3	no	—
125	C₃²	—	3	9	no	—
126	C₃²	—	9	9	no	—
127	C₃²	—	111	9	no	—
128	C₃²	—	111	18	no	—
129	C₃²	—	111	9	no	—
130	C₃²	—	111	3	no	—
131	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	2	yes	—
132	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
133	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
134	C₉	—	3	3	no	—
135	C₉	—	37	3	no	—
136	C₉	—	37	6	no	—
137	C₉	—	111	3	no	—
138	C₉	—	111	18	no	—
139	C₃	—	3	2	no	—
140	C₃	—	3	1	no	—
141	C₃	—	9	9	no	—
142	C₃	—	111	9	no	—
143	C₃	#22582631	1	3	yes	direct product factor
144	C₃	C₃×H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688530

C₃

yes

maximal, complemented

C₃²×C₉×C₃₇⋊C₃

C₃

yes

maximal, characteristic, complemented

#22582631

C₃

yes

maximal, direct product factor

C₃×#15688521

C₃

yes

maximal, complemented

C₃×#15688515

C₃

yes

maximal, complemented

C₃×#15688512

C₃

yes

maximal, complemented

C₃×C₃₇×#732

C₃

yes

maximal, characteristic, complemented

C₃³×C₃₇⋊C₃

C₃²

yes

characteristic

#15688530

—

#15688530

C₃²

yes

complemented

C₃×#11758608

—

C₃×#11758608

C₃²

yes

—

C₃²×#66524

—

C₃²×#66524

C₃²

yes

characteristic

C₃²×#66524

C₃²

yes

characteristic

C₃×C₉×C₃₇⋊C₃

—

C₃×C₉×C₃₇⋊C₃

—

C₃×C₉×C₃₇⋊C₃

—

C₃×C₉×C₃₇⋊C₃

C₃²

yes

complemented

#15688521

C₃²

yes

complemented

#15688515

C₃²

yes

complemented

#15688512

C₃²

yes

complemented

C₃×H(3,𝔽₃)×C₃₇

C₃²

yes

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×#732

C₃²

yes

complemented

C₃³×C₃₇

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

C₃³

yes

—

H(3,𝔽₃)×C₃₇

—

H(3,𝔽₃)×C₃₇

C₃³

yes

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

C₃³

yes

—

#11758608

—

#11758608

—

#11758608

C₃³

yes

—

—

—

—

—

yes

yes

—

C₃×#66524

H(3,𝔽₃)

yes

—

C₃×#66524

C₃³

yes

—

C₃×#66524

C₃³

yes

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

C₃×#6908

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃⁴

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

#66524

—

#66524

—

#66524

C₃×H(3,𝔽₃)

yes

—

#66524

C₃×H(3,𝔽₃)

yes

—

C₃×#737

—

C₃³×C₉

—

#6908

—

C₃×#733

—

C₃×#732

C₃₇⋊C₃

yes

characteristic, complemented

C₃×#731

—

C₃×AΣL(1,𝔽₂₇)

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

#6908

yes

complemented

C₃×C₃₇

C₃²×H(3,𝔽₃)

yes

characteristic

C₃₇⋊C₃

—

C₃⁴

—

#737

—

#737

—

111

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃×H(3,𝔽₃)

C₃×C₃₇⋊C₃

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

111

—

C₃²×C₉

—

111

—

C₃²×C₉

—

111

—

C₃²×C₉

—

111

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

#733

—

#732

C₃×C₃₇⋊C₃

yes

complemented

#731

—

AΣL(1,𝔽₂₇)

—

C₃₇

C₃×#6908

yes

characteristic, complemented, Sylow

C₃³

—

100

C₃³

—

101

C₃³

—

111

—

102

C₃³

—

111

—

103

C₃³

—

111

—

104

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

105

3₋¹⁺²

—

106

3₋¹⁺²

—

111

—

107

3₋¹⁺²

—

111

—

108

H(3,𝔽₃)

—

109

H(3,𝔽₃)

C₃²×C₃₇⋊C₃

yes

—

110

C₃×C₉

—

111

C₃×C₉

—

112

C₃×C₉

—

113

C₃×C₉

—

114

C₃×C₉

—

115

C₃×C₉

—

116

C₃×C₉

—

117

C₃×C₉

—

111

—

118

C₃×C₉

—

111

—

119

C₃×C₉

—

111

—

120

C₃×C₉

—

111

—

121

C₃×C₉

—

111

—

122

C₃×C₉

—

111

—

123

C₃×C₉

C₃²×C₃₇⋊C₃

yes

—

124

C₃²

—

125

C₃²

—

126

C₃²

—

127

C₃²

—

111

—

128

C₃²

—

111

—

129

C₃²

—

111

—

130

C₃²

—

111

—

131

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

—

132

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

133

C₃²

C₃³×C₃₇⋊C₃

yes

characteristic

134

C₉

—

135

C₉

—

136

C₉

—

137

C₉

—

111

—

138

C₉

—

111

—

139

C₃

—

140

C₃

—

141

C₃

—

142

C₃

—

111

—

143

C₃

#22582631

yes

direct product factor

144

C₃

C₃×H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₃₇⋊C₃
commutator, cosocle	C₃²×C₃₇	C₃⁴
Frattini	C₃²	C₃³×C₃₇⋊C₃
Fitting	C₃×C₃₇×#732	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
hypercenter	C₃×#732	C₃₇⋊C₃
hypocenter	C₃₇	C₃×#6908
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×#732

subgroup

G/H

center

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

commutator, cosocle

C₃²×C₃₇

C₃⁴

Frattini

C₃²

C₃³×C₃₇⋊C₃

Fitting

C₃×C₃₇×#732

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

C₃₇⋊C₃

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ C₃×#732

N	H ≅ G/N	type
1. C₃×#15688530	C₃	semidirect
2. C₃²×C₉×C₃₇⋊C₃	C₃	semidirect
3. #22582631	C₃	direct (N × H)
4. C₃×#15688521	C₃	semidirect
5. C₃×#15688515	C₃	semidirect
6. C₃×#15688512	C₃	semidirect
7. C₃×C₃₇×#732	C₃	semidirect
10. #15688530	C₃²	semidirect
19. C₃×C₉×C₃₇⋊C₃	C₃²	semidirect
20. #15688521	C₃²	semidirect
21. #15688515	C₃²	semidirect
22. #15688512	C₃²	semidirect
25. C₃₇×#732	C₃²	semidirect
71. C₃×#732	C₃₇⋊C₃	semidirect
77. C₃×C₃₇	#6908	semidirect
95. #732	C₃×C₃₇⋊C₃	semidirect
98. C₃₇	C₃×#6908	semidirect
143. C₃	#22582631	direct (N × H)

H ≅ G/N

type

1. C₃×#15688530

C₃

semidirect

2. C₃²×C₉×C₃₇⋊C₃

C₃

semidirect

3. #22582631

C₃

direct (N × H)

4. C₃×#15688521

C₃

semidirect

5. C₃×#15688515

C₃

semidirect

6. C₃×#15688512

C₃

semidirect

7. C₃×C₃₇×#732

C₃

semidirect

10. #15688530

C₃²

semidirect

19. C₃×C₉×C₃₇⋊C₃

C₃²

semidirect

20. #15688521

C₃²

semidirect

21. #15688515

C₃²

semidirect

22. #15688512

C₃²

semidirect

25. C₃₇×#732

C₃²

semidirect

71. C₃×#732

C₃₇⋊C₃

semidirect

77. C₃×C₃₇

#6908

semidirect

95. #732

C₃×C₃₇⋊C₃

semidirect

98. C₃₇

C₃×#6908

semidirect

143. C₃

#22582631

direct (N × H)

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

subgroup	index
C₃×#15688530	3
C₃²×C₉×C₃₇⋊C₃	3
#22582631	3
C₃×#15688521	3
C₃×#15688515	3
C₃×#15688512	3
C₃×C₃₇×#732	3
C₃×#6908	37

subgroup

index

C₃²×C₉×C₃₇⋊C₃

C₃×C₃₇×#732

Subgroups — C₃×#22582631 #41008144

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#22582631 #41008144

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#22582631 #41008144