Group #41006270 — subgroups

Subgroups — #41006270

Order 26973.

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

normal subgroups only

This group has 368 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	C₃×#15688531	C₃	1	1	yes	maximal, characteristic
2	#22582538	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₉×#66524	C₃	1	1	yes	maximal, characteristic, complemented
4	#22582481	C₃	1	3	yes	maximal, complemented
5	#22582478	C₃	1	3	yes	maximal, complemented
6	#22582475	C₃	1	3	yes	maximal
7	C₃₇×#6887	C₃	1	1	yes	maximal, characteristic, complemented
8	#15688531	—	3	3	no	—
9	#15688531	—	3	3	no	—
10	#15688531	C₉	1	9	yes	complemented
11	C₃×#11758608	C₃²	1	1	yes	characteristic
12	C₃×#11758608	C₃²	1	1	yes	characteristic
13	C₃×#11758605	—	3	1	no	—
14	C₃×#11758606	—	3	1	no	—
15	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	1	yes	characteristic
16	C₃²×#66524	C₃²	1	3	yes	—
17	C₃×C₉×C₃₇⋊C₃	—	3	1	no	—
18	#15688506	—	3	1	no	—
19	#15688505	—	3	1	no	—
20	C₃²⋊#66524	C₃²	1	3	yes	—
21	C₉×#66524	—	3	1	no	—
22	C₉×#66524	—	3	1	no	—
23	C₉×#66524	—	3	1	no	—
24	C₉×#66524	—	3	1	no	—
25	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
26	C₃₇×C₉⋊C₉	—	3	1	no	—
27	C₃₇×C₉⋊C₉	—	3	1	no	—
28	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
29	C₉²×C₃₇	—	3	1	no	—
30	C₃³×C₃₇	C₃³	1	1	yes	characteristic
31	C₃²×C₃₇⋊C₃	—	3	1	no	—
32	C₃×C₉×C₃₇	—	3	3	no	—
33	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	—
34	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
35	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
36	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
37	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	—
38	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	—
39	C₃²⋊(C₃₇⋊C₃)	—	3	1	no	—
40	C₃²⋊(C₃₇⋊C₃)	—	3	1	no	—
41	C₃²⋊(C₃₇⋊C₃)	C₃×C₉	1	3	yes	—
42	#11758608	—	3	1	no	—
43	#11758608	—	3	1	no	—
44	#11758608	—	3	1	no	—
45	#11758608	—	3	1	no	—
46	#11758608	C₃×C₉	1	3	yes	—
47	#11758608	C₃×C₉	1	3	yes	—
48	C₃×#66524	—	3	3	no	—
49	C₃×#66524	—	3	1	no	—
50	C₃×#66524	—	3	1	no	—
51	C₃×#66524	—	3	9	no	—
52	C₃×#66524	—	3	3	no	—
53	#11758606	—	3	9	no	—
54	#11758605	—	3	9	no	—
55	C₉×C₃₇⋊C₃	—	3	9	no	—
56	#10624697	—	37	1	no	maximal, Sylow
57	C₃²×C₃₇	—	3	3	no	—
58	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
59	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
60	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
61	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
62	C₃×C₃₇⋊C₃	—	3	3	no	—
63	C₃×C₃₇⋊C₃	—	9	3	no	—
64	C₉×C₃₇	—	9	3	no	—
65	C₉×C₃₇	C₃²⋊C₉	1	3	yes	—
66	C₉×C₃₇	C₃²⋊C₉	1	3	yes	—
67	C₉×C₃₇	C₃²⋊C₉	1	3	yes	—
68	#66524	—	3	3	no	—
69	#66524	—	3	3	no	—
70	#66524	—	3	9	no	—
71	#66524	—	9	3	no	—
72	C₃×#737	—	37	1	no	—
73	#6887	—	37	1	no	—
74	#6887	C₃₇⋊C₃	1	1	yes	characteristic, complemented
75	C₃×C₉²	—	37	1	no	—
76	3₋¹⁺²⋊C₉	—	37	3	no	—
77	3₋¹⁺²⋊C₉	—	37	3	no	—
78	H(3,𝔽₃)⋊C₉	—	37	3	no	—
79	C₃×C₃₇	—	3	1	no	—
80	C₃×C₃₇	—	3	1	no	—
81	C₃×C₃₇	—	3	1	no	—
82	C₃×C₃₇	C₃×C₃²⋊C₉	1	1	yes	characteristic
83	C₃×C₃₇	#6908	1	3	yes	—
84	C₃₇⋊C₃	—	9	3	no	—
85	#737	—	37	9	no	—
86	#737	—	111	3	no	—
87	#737	—	111	3	no	—
88	C₃×3₋¹⁺²	—	37	1	no	—
89	C₃×3₋¹⁺²	—	37	1	no	—
90	C₃×3₋¹⁺²	—	111	1	no	—
91	C₃×3₋¹⁺²	—	111	1	no	—
92	C₃×H(3,𝔽₃)	—	37	1	no	—
93	C₃²×C₉	—	37	3	no	—
94	C₃²×C₉	—	111	1	no	—
95	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
96	C₉⋊C₉	—	3	1	no	—
97	C₉⋊C₉	—	3	1	no	—
98	C₉⋊C₉	—	111	1	no	—
99	C₉⋊C₉	—	111	1	no	—
100	C₃²⋊C₉	—	37	3	no	—
101	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
102	C₉²	—	3	1	no	—
103	C₉²	—	111	1	no	—
104	C₉²	—	111	1	no	—
105	C₉²	—	111	1	no	—
106	C₉²	—	111	1	no	—
107	C₃₇	#10624697	1	1	yes	characteristic, complemented, Sylow
108	C₃³	—	111	1	no	—
109	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
110	3₋¹⁺²	—	37	3	no	—
111	3₋¹⁺²	—	37	3	no	—
112	3₋¹⁺²	—	111	1	no	—
113	3₋¹⁺²	—	111	1	no	—
114	3₋¹⁺²	—	111	1	no	—
115	3₋¹⁺²	—	111	1	no	—
116	3₋¹⁺²	—	111	9	no	—
117	3₋¹⁺²	—	111	9	no	—
118	H(3,𝔽₃)	—	37	3	no	—
119	H(3,𝔽₃)	—	111	1	no	—
120	H(3,𝔽₃)	—	111	1	no	—
121	C₃×C₉	—	3	3	no	—
122	C₃×C₉	—	111	3	no	—
123	C₃×C₉	—	111	3	no	—
124	C₃×C₉	—	111	1	no	—
125	C₃×C₉	—	111	1	no	—
126	C₃×C₉	—	111	9	no	—
127	C₃×C₉	—	111	9	no	—
128	C₃×C₉	C₉×C₃₇⋊C₃	1	3	yes	—
129	C₃×C₉	#11758605	1	3	yes	—
130	C₃×C₉	#11758606	1	3	yes	—
131	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
132	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
133	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
134	C₃²	—	3	3	no	—
135	C₃²	—	333	3	no	—
136	C₃²	—	111	3	no	—
137	C₃²	C₃×C₉×C₃₇⋊C₃	1	1	yes	characteristic
138	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
139	C₃²	C₃×#11758606	1	1	yes	characteristic
140	C₃²	C₃×#11758605	1	1	yes	characteristic
141	C₉	—	9	3	no	—
142	C₉	—	333	3	no	—
143	C₉	—	111	3	no	—
144	C₉	—	111	3	no	—
145	C₉	—	111	9	no	—
146	C₉	#15688502	1	3	yes	—
147	C₉	#15688502	1	3	yes	—
148	C₉	#15688502	1	3	yes	—
149	C₃	—	3	1	no	—
150	C₃	—	3	1	no	—
151	C₃	—	3	1	no	—
152	C₃	—	333	3	no	—
153	C₃	C₃×#15688502	1	1	yes	characteristic
154	C₃	#22582630	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688531

C₃

yes

maximal, characteristic

#22582538

C₃

yes

maximal, characteristic, complemented

C₃×C₉×#66524

C₃

yes

maximal, characteristic, complemented

#22582481

C₃

yes

maximal, complemented

#22582478

C₃

yes

maximal, complemented

#22582475

C₃

yes

maximal

C₃₇×#6887

C₃

yes

maximal, characteristic, complemented

#15688531

—

#15688531

—

#15688531

C₉

yes

complemented

C₃×#11758608

C₃²

yes

characteristic

C₃×#11758608

C₃²

yes

characteristic

C₃×#11758605

—

C₃×#11758606

—

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

yes

characteristic

C₃²×#66524

C₃²

yes

—

C₃×C₉×C₃₇⋊C₃

—

#15688506

—

#15688505

—

C₃²⋊#66524

C₃²

yes

—

—

—

—

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×C₉⋊C₉

—

C₃₇×C₉⋊C₉

—

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₉²×C₃₇

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

C₃×C₉

yes

—

C₃×C₉×C₃₇

yes

C₃×C₉×C₃₇

yes

C₃×C₉×C₃₇

yes

C₃×C₉×C₃₇

3₋¹⁺²

yes

—

C₃×C₉×C₃₇

3₋¹⁺²

yes

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

C₃×C₉

yes

—

—

—

—

—

C₃×C₉

yes

—

#11758608

C₃×C₉

yes

—

—

—

—

—

—

—

—

C₉×C₃₇⋊C₃

—

#10624697

—

maximal, Sylow

C₃²×C₃₇

—

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

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

C₉×C₃₇

C₃²⋊C₉

yes

—

C₉×C₃₇

C₃²⋊C₉

yes

—

C₉×C₃₇

C₃²⋊C₉

yes

—

—

—

—

—

—

—

C₃₇⋊C₃

yes

characteristic, complemented

C₃×C₉²

—

3₋¹⁺²⋊C₉

—

3₋¹⁺²⋊C₉

—

H(3,𝔽₃)⋊C₉

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃×C₃²⋊C₉

yes

characteristic

C₃×C₃₇

#6908

yes

—

C₃₇⋊C₃

—

#737

—

#737

—

111

—

#737

—

111

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

—

111

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

100

C₃²⋊C₉

—

101

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

102

C₉²

—

103

C₉²

—

111

—

104

C₉²

—

111

—

105

C₉²

—

111

—

106

C₉²

—

111

—

107

C₃₇

#10624697

yes

characteristic, complemented, Sylow

108

C₃³

—

111

—

109

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

110

3₋¹⁺²

—

111

3₋¹⁺²

—

112

3₋¹⁺²

—

111

—

113

3₋¹⁺²

—

111

—

114

3₋¹⁺²

—

111

—

115

3₋¹⁺²

—

111

—

116

3₋¹⁺²

—

111

—

117

3₋¹⁺²

—

111

—

118

H(3,𝔽₃)

—

119

H(3,𝔽₃)

—

111

—

120

H(3,𝔽₃)

—

111

—

121

C₃×C₉

—

122

C₃×C₉

—

111

—

123

C₃×C₉

—

111

—

124

C₃×C₉

—

111

—

125

C₃×C₉

—

111

—

126

C₃×C₉

—

111

—

127

C₃×C₉

—

111

—

128

C₃×C₉

C₉×C₃₇⋊C₃

yes

—

129

C₃×C₉

#11758605

yes

—

130

C₃×C₉

#11758606

yes

—

131

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

132

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

133

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

134

C₃²

—

135

C₃²

—

333

—

136

C₃²

—

111

—

137

C₃²

C₃×C₉×C₃₇⋊C₃

yes

characteristic

138

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

139

C₃²

C₃×#11758606

yes

characteristic

140

C₃²

C₃×#11758605

yes

characteristic

141

C₉

—

142

C₉

—

333

—

143

C₉

—

111

—

144

C₉

—

111

—

145

C₉

—

111

—

146

C₉

#15688502

yes

—

147

C₉

#15688502

yes

—

148

C₉

#15688502

yes

—

149

C₃

—

150

C₃

—

151

C₃

—

152

C₃

—

333

—

153

C₃

C₃×#15688502

yes

characteristic

154

C₃

#22582630

yes

—

	subgroup	G/H
center	C₃×C₉	C₃²⋊(C₃₇⋊C₃)
commutator	C₃²×C₃₇	C₃²×C₉
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6887	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²
cosocle	C₃³×C₃₇	C₃³
hypercenter	#6887	C₃₇⋊C₃
hypocenter	C₃₇	#10624697
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ #6887

subgroup

G/H

center

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

commutator

C₃²×C₃₇

C₃²×C₉

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃₇×#6887

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

prime radical

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

cosocle

C₃³×C₃₇

C₃³

C₃₇⋊C₃

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ #6887

N	H ≅ G/N	type
2. #22582538	C₃	semidirect
3. C₃×C₉×#66524	C₃	semidirect
4. #22582481	C₃	semidirect
5. #22582478	C₃	semidirect
7. C₃₇×#6887	C₃	semidirect
10. #15688531	C₉	semidirect
74. #6887	C₃₇⋊C₃	semidirect
107. C₃₇	#10624697	semidirect

H ≅ G/N

type

2. #22582538

C₃

semidirect

3. C₃×C₉×#66524

C₃

semidirect

4. #22582481

C₃

semidirect

5. #22582478

C₃

semidirect

7. C₃₇×#6887

C₃

semidirect

10. #15688531

C₉

semidirect

74. #6887

C₃₇⋊C₃

semidirect

107. C₃₇

#10624697

semidirect

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

subgroup	index
C₃×#15688531	3
#22582538	3
C₃×C₉×#66524	3
#22582481	3
#22582478	3
#22582475	3
C₃₇×#6887	3
#10624697	37

subgroup

index

C₃×C₉×#66524

C₃₇×#6887

Subgroups — #41006270

Conjugacy classes of subgroups

Complements

Maximal subgroups