Group #41006799 — subgroups

Subgroups — #41006799

Order 26973.

464 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 82 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	#22582554	C₃	1	1	yes	maximal, characteristic
3	#22582554	C₃	1	1	yes	maximal, characteristic
4	3₋¹⁺²⋊#66524	C₃	1	9	yes	maximal, complemented
5	C₃²×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic
6	C₃×#11758608	—	3	1	no	—
7	C₃×#11758608	—	3	1	no	—
8	C₃×#11758608	—	3	1	no	—
9	C₃×#11758608	—	3	9	no	—
10	C₃×#11758608	—	3	1	no	—
11	C₃×#11758608	—	3	1	no	—
12	C₃²×#66524	C₃²	1	3	yes	—
13	C₃²⋊#66524	—	3	1	no	—
14	C₃²⋊#66524	—	3	1	no	—
15	C₃²⋊#66524	—	3	1	no	—
16	C₃²⋊#66524	—	3	1	no	—
17	C₃²⋊#66524	C₃²	1	3	yes	—
18	C₃²⋊#66524	C₃²	1	3	yes	—
19	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
20	C₃×3₋¹⁺²×C₃₇	—	3	1	no	—
21	C₃×3₋¹⁺²×C₃₇	—	3	1	no	—
22	C₃×3₋¹⁺²×C₃₇	—	3	9	no	—
23	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
24	C₃²×C₉×C₃₇	—	3	1	no	—
25	C₃³×C₃₇	—	3	3	no	—
26	C₃³×C₃₇	—	3	3	no	—
27	C₃³×C₃₇	—	9	1	no	—
28	C₃³×C₃₇	—	9	1	no	—
29	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
30	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
31	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
32	C₃³×C₃₇	C₃³	1	1	yes	characteristic
33	3₋¹⁺²×C₃₇	—	3	9	no	—
34	3₋¹⁺²×C₃₇	—	3	9	no	—
35	3₋¹⁺²×C₃₇	—	3	6	no	—
36	3₋¹⁺²×C₃₇	3₋¹⁺²	1	9	yes	complemented
37	C₃×C₉×C₃₇	—	3	9	no	—
38	C₃×C₉×C₃₇	—	3	3	no	—
39	#11758608	—	3	9	no	—
40	#11758608	—	3	9	no	—
41	#11758608	—	3	9	no	—
42	#11758608	—	9	3	no	—
43	#11758608	—	9	3	no	—
44	C₃×#66524	—	3	3	no	—
45	C₃×#66524	—	3	9	no	—
46	C₃×#66524	—	3	3	no	—
47	C₃×#66524	—	3	3	no	—
48	#10624781	—	37	1	no	maximal, Sylow
49	C₃²×C₃₇	—	3	1	no	—
50	C₃²×C₃₇	—	3	1	no	—
51	C₃²×C₃₇	—	3	1	no	—
52	C₃²×C₃₇	—	3	1	no	—
53	C₃²×C₃₇	—	3	1	no	—
54	C₃²×C₃₇	—	3	1	no	—
55	C₃²×C₃₇	—	3	1	no	—
56	C₃²×C₃₇	—	3	1	no	—
57	C₃²×C₃₇	—	3	1	no	—
58	C₃²×C₃₇	—	3	1	no	—
59	C₃²×C₃₇	—	3	1	no	—
60	C₃²×C₃₇	—	3	1	no	—
61	C₃²×C₃₇	—	3	1	no	—
62	C₃²×C₃₇	—	3	1	no	—
63	C₃²×C₃₇	—	9	9	no	—
64	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	3	yes	—
65	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
66	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
67	C₉×C₃₇	—	3	9	no	—
68	#66524	—	3	9	no	—
69	#66524	—	9	3	no	—
70	#66524	—	9	3	no	—
71	C₃²×3₋¹⁺²	—	37	1	no	—
72	C₃²×3₋¹⁺²	C₃₇⋊C₃	1	1	yes	characteristic
73	#6891	—	37	1	no	—
74	#6891	—	37	1	no	—
75	3₋¹⁺²⋊C₉	—	37	9	no	—
76	C₃×C₃₇	—	3	3	no	—
77	C₃×C₃₇	—	3	3	no	—
78	C₃×C₃₇	—	9	1	no	—
79	C₃×C₃₇	—	9	1	no	—
80	C₃×C₃₇	#6891	1	1	yes	characteristic
81	C₃×C₃₇	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
82	C₃×C₃₇	#6910	1	1	yes	characteristic
83	C₃×C₃₇	#6912	1	1	yes	characteristic
84	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
85	C₃×3₋¹⁺²	—	3	1	no	—
86	C₃×3₋¹⁺²	—	3	1	no	—
87	C₃×3₋¹⁺²	—	3	9	no	—
88	C₃×3₋¹⁺²	—	111	1	no	—
89	C₃×3₋¹⁺²	—	111	1	no	—
90	C₃×3₋¹⁺²	—	111	1	no	—
91	C₃×3₋¹⁺²	—	111	1	no	—
92	C₃×3₋¹⁺²	—	111	1	no	—
93	C₃×3₋¹⁺²	—	111	9	no	—
94	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	3	yes	—
95	C₃²×C₉	—	3	1	no	—
96	C₃²×C₉	—	37	3	no	—
97	C₃²⋊C₉	—	37	3	no	—
98	C₃²⋊C₉	—	37	3	no	—
99	C₃²⋊C₉	—	111	1	no	—
100	C₃²⋊C₉	—	111	1	no	—
101	C₃²⋊C₉	—	111	1	no	—
102	C₃²⋊C₉	—	111	1	no	—
103	C₃₇	#10624781	1	1	yes	characteristic, complemented, Sylow
104	C₃³	—	3	3	no	—
105	C₃³	—	3	3	no	—
106	C₃³	—	9	1	no	—
107	C₃³	—	9	1	no	—
108	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
109	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
110	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
111	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
112	3₋¹⁺²	—	3	9	no	—
113	3₋¹⁺²	—	3	9	no	—
114	3₋¹⁺²	—	3	6	no	—
115	3₋¹⁺²	—	333	3	no	—
116	3₋¹⁺²	—	333	3	no	—
117	3₋¹⁺²	—	111	9	no	—
118	3₋¹⁺²	—	111	9	no	—
119	3₋¹⁺²	—	111	9	no	—
120	3₋¹⁺²	#11758608	1	9	yes	complemented
121	C₃×C₉	—	3	9	no	—
122	C₃×C₉	—	3	3	no	—
123	C₃×C₉	—	111	3	no	—
124	C₃×C₉	—	111	3	no	—
125	C₃×C₉	—	111	3	no	—
126	C₃×C₉	—	111	9	no	—
127	C₃²	—	3	1	no	—
128	C₃²	—	3	1	no	—
129	C₃²	—	3	1	no	—
130	C₃²	—	3	1	no	—
131	C₃²	—	3	1	no	—
132	C₃²	—	3	1	no	—
133	C₃²	—	3	1	no	—
134	C₃²	—	3	1	no	—
135	C₃²	—	3	1	no	—
136	C₃²	—	3	1	no	—
137	C₃²	—	3	1	no	—
138	C₃²	—	3	1	no	—
139	C₃²	—	3	1	no	—
140	C₃²	—	3	1	no	—
141	C₃²	—	9	9	no	—
142	C₃²	#15688514	1	3	yes	—
143	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
144	C₃²	C₃×#11758608	1	3	yes	—
145	C₉	—	3	9	no	—
146	C₉	—	333	3	no	—
147	C₉	—	333	3	no	—
148	C₉	—	111	9	no	—
149	C₃	—	3	3	no	—
150	C₃	—	3	3	no	—
151	C₃	—	9	1	no	—
152	C₃	—	9	1	no	—
153	C₃	#22582554	1	1	yes	characteristic
154	C₃	C₃×#15688514	1	1	yes	characteristic
155	C₃	#22582644	1	1	yes	characteristic
156	C₃	#22582651	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#11758608

C₃

yes

maximal, characteristic

#22582554

C₃

yes

maximal, characteristic

#22582554

C₃

yes

maximal, characteristic

3₋¹⁺²⋊#66524

C₃

yes

maximal, complemented

C₃²×3₋¹⁺²×C₃₇

C₃

yes

maximal, characteristic

—

—

—

—

—

—

C₃²×#66524

C₃²

yes

—

—

—

—

—

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

—

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₃₇

—

C₃×3₋¹⁺²×C₃₇

—

C₃×3₋¹⁺²×C₃₇

—

C₃×3₋¹⁺²×C₃₇

C₃²

yes

—

C₃²×C₉×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

yes

C₃³×C₃₇

yes

C₃³×C₃₇

yes

C₃³×C₃₇

C₃³

yes

characteristic

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

3₋¹⁺²

yes

complemented

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

—

—

—

—

—

—

—

—

—

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

AΣL(1,𝔽₂₇)

yes

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

—

C₉×C₃₇

—

#66524

—

#66524

—

#66524

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

C₃₇⋊C₃

yes

characteristic

#6891

—

#6891

—

3₋¹⁺²⋊C₉

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

#6891

yes

characteristic

C₃×C₃₇

C₃×AΣL(1,𝔽₂₇)

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

—

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

—

C₃²×C₉

—

C₃²×C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

100

C₃²⋊C₉

—

111

—

101

C₃²⋊C₉

—

111

—

102

C₃²⋊C₉

—

111

—

103

C₃₇

#10624781

yes

characteristic, complemented, Sylow

104

C₃³

—

105

C₃³

—

106

C₃³

—

107

C₃³

—

108

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

109

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

110

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

111

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

112

3₋¹⁺²

—

113

3₋¹⁺²

—

114

3₋¹⁺²

—

115

3₋¹⁺²

—

333

—

116

3₋¹⁺²

—

333

—

117

3₋¹⁺²

—

111

—

118

3₋¹⁺²

—

111

—

119

3₋¹⁺²

—

111

—

120

3₋¹⁺²

#11758608

yes

complemented

121

C₃×C₉

—

122

C₃×C₉

—

123

C₃×C₉

—

111

—

124

C₃×C₉

—

111

—

125

C₃×C₉

—

111

—

126

C₃×C₉

—

111

—

127

C₃²

—

128

C₃²

—

129

C₃²

—

130

C₃²

—

131

C₃²

—

132

C₃²

—

133

C₃²

—

134

C₃²

—

135

C₃²

—

136

C₃²

—

137

C₃²

—

138

C₃²

—

139

C₃²

—

140

C₃²

—

141

C₃²

—

142

C₃²

#15688514

yes

—

143

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

144

C₃²

C₃×#11758608

yes

—

145

C₉

—

146

C₉

—

333

—

147

C₉

—

333

—

148

C₉

—

111

—

149

C₃

—

150

C₃

—

151

C₃

—

152

C₃

—

153

C₃

#22582554

yes

characteristic

154

C₃

C₃×#15688514

yes

characteristic

155

C₃

#22582644

yes

characteristic

156

C₃

#22582651

yes

characteristic

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×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	#10624781
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃²×3₋¹⁺²

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×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

#10624781

LCS

G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃²×3₋¹⁺²

N	H ≅ G/N	type
4. 3₋¹⁺²⋊#66524	C₃	semidirect
36. 3₋¹⁺²×C₃₇	3₋¹⁺²	semidirect
103. C₃₇	#10624781	semidirect
120. 3₋¹⁺²	#11758608	semidirect

H ≅ G/N

type

4. 3₋¹⁺²⋊#66524

C₃

semidirect

36. 3₋¹⁺²×C₃₇

3₋¹⁺²

semidirect

103. C₃₇

#10624781

semidirect

120. 3₋¹⁺²

#11758608

semidirect

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

subgroup	index
C₃²×#11758608	3
#22582554	3
3₋¹⁺²⋊#66524	3
C₃²×3₋¹⁺²×C₃₇	3
#10624781	37

subgroup

index

C₃²×3₋¹⁺²×C₃₇

Subgroups — #41006799

Conjugacy classes of subgroups

Complements

Maximal subgroups