Group #41006412 — subgroups

Subgroups — #41006412

Order 26973.

268 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 79 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×#15688515	C₃	1	1	yes	maximal, characteristic
2	#22582548	C₃	1	2	yes	maximal, complemented
3	H(3,𝔽₃)×#66524	C₃	1	1	yes	maximal, characteristic
4	C₃×#15688504	C₃	1	1	yes	maximal, characteristic, complemented
5	#22582480	C₃	1	2	yes	maximal, complemented
6	#22582478	C₃	1	2	yes	maximal, complemented
7	H(3,𝔽₃)⋊#66524	C₃	1	1	yes	maximal, characteristic
8	#22582469	C₃	1	2	yes	maximal, complemented
9	C₃₇×H(3,𝔽₃)⋊C₉	C₃	1	1	yes	maximal, characteristic
10	C₃×#11758608	C₃²	1	2	yes	—
11	C₃²×#66524	—	3	1	no	—
12	C₃²×#66524	C₃²	1	1	yes	characteristic
13	C₃²×#66524	C₃²	1	1	yes	characteristic
14	C₃²×#66524	C₃²	1	1	yes	characteristic
15	#15688515	C₉	1	9	yes	complemented
16	#15688504	—	3	2	no	—
17	#15688504	—	3	1	no	—
18	#15688504	—	3	2	no	—
19	#15688504	—	3	2	no	—
20	C₃²⋊#66524	—	3	2	no	—
21	C₃²⋊#66524	C₃²	1	2	yes	—
22	C₃²⋊#66524	C₃²	1	2	yes	—
23	C₉×#66524	—	3	2	no	—
24	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
25	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
26	C₃₇×C₃²⋊C₉	C₃²	1	2	yes	—
27	C₃³×C₃₇	—	3	1	no	—
28	C₃³×C₃₇	C₃³	1	1	yes	characteristic
29	H(3,𝔽₃)×C₃₇	—	3	2	no	—
30	H(3,𝔽₃)×C₃₇	C₃×C₉	1	3	yes	—
31	C₃×C₉×C₃₇	—	3	2	no	—
32	C₃×C₉×C₃₇	—	3	1	no	—
33	C₃×C₉×C₃₇	—	9	1	no	—
34	#11758608	—	3	4	no	—
35	#11758608	C₃×C₉	1	6	yes	complemented
36	C₃×#66524	—	3	2	no	—
37	C₃×#66524	—	3	1	no	—
38	C₃×#66524	—	3	2	no	—
39	C₃×#66524	—	3	2	no	—
40	C₃×#66524	—	3	2	no	—
41	C₃×#66524	—	3	1	no	—
42	C₃×#66524	—	9	1	no	—
43	C₃×#66524	—	9	3	no	—
44	C₃×#66524	—	9	1	no	—
45	C₃×#66524	C₃×C₉	1	3	yes	complemented
46	C₃×#66524	H(3,𝔽₃)	1	2	yes	—
47	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
48	#731⋊C₉	—	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	3	no	—
53	C₃²×C₃₇	—	9	1	no	—
54	C₃²×C₃₇	—	9	1	no	—
55	C₃²×C₃₇	—	9	1	no	—
56	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
57	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
58	C₃²×C₃₇	#737	1	2	yes	—
59	C₉×C₃₇	—	9	2	no	—
60	C₉×C₃₇	—	9	1	no	—
61	#66524	—	3	3	no	—
62	#66524	—	3	6	no	—
63	#66524	—	3	1	no	—
64	#66524	—	3	2	no	—
65	#66524	—	9	2	no	—
66	#66524	—	9	1	no	—
67	#66524	—	9	2	no	—
68	C₃×#731	—	37	1	no	—
69	C₉×3₋¹⁺²	—	37	2	no	—
70	C₉×H(3,𝔽₃)	—	37	1	no	—
71	C₃×C₉⋊C₉	—	37	1	no	—
72	3₋¹⁺²⋊C₉	—	37	2	no	—
73	3₋¹⁺²⋊C₉	—	37	2	no	—
74	H(3,𝔽₃)⋊C₉	—	37	1	no	—
75	H(3,𝔽₃)⋊C₉	C₃₇⋊C₃	1	1	yes	characteristic
76	#6867	—	37	2	no	—
77	C₃×C₃₇	—	3	1	no	—
78	C₃×C₃₇	—	3	2	no	—
79	C₃×C₃₇	—	9	3	no	—
80	C₃×C₃₇	C₉×H(3,𝔽₃)	1	1	yes	characteristic
81	C₃×C₃₇	#6910	1	1	yes	characteristic
82	C₃×C₃₇	#6913	1	1	yes	characteristic
83	C₃×C₃₇	#6913	1	1	yes	characteristic
84	C₃×3₋¹⁺²	—	37	2	no	—
85	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
86	C₃²×C₉	—	37	1	no	—
87	C₃²×C₉	—	37	1	no	—
88	C₃²×C₉	—	37	1	no	—
89	C₃²×C₉	—	111	1	no	—
90	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
91	#731	—	37	9	no	—
92	C₉⋊C₉	—	111	2	no	—
93	C₉⋊C₉	—	111	1	no	—
94	C₉⋊C₉	—	111	2	no	—
95	C₉⋊C₉	—	111	2	no	—
96	C₃²⋊C₉	—	37	2	no	—
97	C₃²⋊C₉	—	37	2	no	—
98	C₃²⋊C₉	—	111	2	no	—
99	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	2	yes	—
100	C₉²	—	111	2	no	—
101	C₃₇	#731⋊C₉	1	1	yes	characteristic, complemented, Sylow
102	C₃³	—	3	1	no	—
103	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
104	3₋¹⁺²	—	37	6	no	—
105	3₋¹⁺²	—	111	4	no	—
106	H(3,𝔽₃)	—	3	2	no	—
107	H(3,𝔽₃)	C₉×C₃₇⋊C₃	1	3	yes	—
108	C₃×C₉	—	3	2	no	—
109	C₃×C₉	—	3	1	no	—
110	C₃×C₉	—	9	1	no	—
111	C₃×C₉	—	37	3	no	—
112	C₃×C₉	—	37	2	no	—
113	C₃×C₉	—	37	1	no	—
114	C₃×C₉	—	333	3	no	—
115	C₃×C₉	—	333	1	no	—
116	C₃×C₉	—	333	1	no	—
117	C₃×C₉	—	111	2	no	—
118	C₃×C₉	—	111	2	no	—
119	C₃×C₉	—	111	1	no	—
120	C₃×C₉	—	111	2	no	—
121	C₃×C₉	—	111	1	no	—
122	C₃×C₉	—	111	2	no	—
123	C₃²	—	3	1	no	—
124	C₃²	—	3	1	no	—
125	C₃²	—	3	1	no	—
126	C₃²	—	3	3	no	—
127	C₃²	—	9	1	no	—
128	C₃²	—	9	1	no	—
129	C₃²	—	9	1	no	—
130	C₃²	C₃×C₉×C₃₇⋊C₃	1	1	yes	characteristic
131	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
132	C₃²	#15688532	1	2	yes	—
133	C₉	—	9	1	no	—
134	C₉	—	9	2	no	—
135	C₉	—	333	2	no	—
136	C₉	—	333	2	no	—
137	C₉	—	333	1	no	—
138	C₉	—	111	3	no	—
139	C₉	—	111	6	no	—
140	C₉	—	111	1	no	—
141	C₉	—	111	2	no	—
142	C₃	—	3	1	no	—
143	C₃	—	3	2	no	—
144	C₃	—	9	3	no	—
145	C₃	#22582539	1	1	yes	characteristic
146	C₃	#22582640	1	1	yes	characteristic
147	C₃	#22582653	1	1	yes	characteristic
148	C₃	#22582657	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688515

C₃

yes

maximal, characteristic

#22582548

C₃

yes

maximal, complemented

H(3,𝔽₃)×#66524

C₃

yes

maximal, characteristic

C₃×#15688504

C₃

yes

maximal, characteristic, complemented

#22582480

C₃

yes

maximal, complemented

#22582478

C₃

yes

maximal, complemented

H(3,𝔽₃)⋊#66524

C₃

yes

maximal, characteristic

#22582469

C₃

yes

maximal, complemented

C₃₇×H(3,𝔽₃)⋊C₉

C₃

yes

maximal, characteristic

C₃×#11758608

C₃²

yes

—

C₃²×#66524

—

C₃²×#66524

C₃²

yes

characteristic

C₃²×#66524

C₃²

yes

characteristic

C₃²×#66524

C₃²

yes

characteristic

#15688515

C₉

yes

—

—

—

—

—

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

—

C₉×#66524

—

C₃×H(3,𝔽₃)×C₃₇

C₃²

yes

characteristic

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₃³×C₃₇

—

C₃³×C₃₇

C₃³

yes

characteristic

H(3,𝔽₃)×C₃₇

—

H(3,𝔽₃)×C₃₇

C₃×C₉

yes

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

#11758608

—

#11758608

C₃×C₉

yes

—

—

—

—

—

—

—

—

—

C₃×C₉

yes

complemented

C₃×#66524

H(3,𝔽₃)

yes

—

yes

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃²×C₉

yes

characteristic

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

#737

yes

—

C₉×C₃₇

—

C₉×C₃₇

—

—

—

—

—

—

—

—

—

C₉×3₋¹⁺²

—

C₉×H(3,𝔽₃)

—

C₃×C₉⋊C₉

—

3₋¹⁺²⋊C₉

—

3₋¹⁺²⋊C₉

—

H(3,𝔽₃)⋊C₉

—

H(3,𝔽₃)⋊C₉

C₃₇⋊C₃

yes

characteristic

#6867

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₉×H(3,𝔽₃)

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

C₃×C₃₇⋊C₃

yes

characteristic

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

111

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

#731

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

100

C₉²

—

111

—

101

C₃₇

#731⋊C₉

yes

characteristic, complemented, Sylow

102

C₃³

—

103

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

104

3₋¹⁺²

—

105

3₋¹⁺²

—

111

—

106

H(3,𝔽₃)

—

107

H(3,𝔽₃)

C₉×C₃₇⋊C₃

yes

—

108

C₃×C₉

—

109

C₃×C₉

—

110

C₃×C₉

—

111

C₃×C₉

—

112

C₃×C₉

—

113

C₃×C₉

—

114

C₃×C₉

—

333

—

115

C₃×C₉

—

333

—

116

C₃×C₉

—

333

—

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

C₃²

—

124

C₃²

—

125

C₃²

—

126

C₃²

—

127

C₃²

—

128

C₃²

—

129

C₃²

—

130

C₃²

C₃×C₉×C₃₇⋊C₃

yes

characteristic

131

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

132

C₃²

#15688532

yes

—

133

C₉

—

134

C₉

—

135

C₉

—

333

—

136

C₉

—

333

—

137

C₉

—

333

—

138

C₉

—

111

—

139

C₉

—

111

—

140

C₉

—

111

—

141

C₉

—

111

—

142

C₃

—

143

C₃

—

144

C₃

—

145

C₃

#22582539

yes

characteristic

146

C₃

#22582640

yes

characteristic

147

C₃

#22582653

yes

characteristic

148

C₃

#22582657

yes

characteristic

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₃₇⋊C₃
commutator	C₃²×C₃₇	C₃²×C₉
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×H(3,𝔽₃)⋊C₉	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃×H(3,𝔽₃)×C₃₇	C₃²
cosocle	C₃³×C₃₇	C₃³
hypercenter	H(3,𝔽₃)⋊C₉	C₃₇⋊C₃
hypocenter	C₃₇	#731⋊C₉
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ H(3,𝔽₃)⋊C₉

subgroup

G/H

center

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

commutator

C₃²×C₃₇

C₃²×C₉

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃₇×H(3,𝔽₃)⋊C₉

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

prime radical

C₃×H(3,𝔽₃)×C₃₇

C₃²

cosocle

C₃³×C₃₇

C₃³

hypercenter

H(3,𝔽₃)⋊C₉

C₃₇⋊C₃

hypocenter

C₃₇

#731⋊C₉

LCS

G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ H(3,𝔽₃)⋊C₉

N	H ≅ G/N	type
2. #22582548	C₃	semidirect
4. C₃×#15688504	C₃	semidirect
5. #22582480	C₃	semidirect
6. #22582478	C₃	semidirect
8. #22582469	C₃	semidirect
15. #15688515	C₉	semidirect
35. #11758608	C₃×C₉	semidirect
45. C₃×#66524	C₃×C₉	semidirect
101. C₃₇	#731⋊C₉	semidirect

H ≅ G/N

type

2. #22582548

C₃

semidirect

4. C₃×#15688504

C₃

semidirect

5. #22582480

C₃

semidirect

6. #22582478

C₃

semidirect

8. #22582469

C₃

semidirect

15. #15688515

C₉

semidirect

35. #11758608

C₃×C₉

semidirect

45. C₃×#66524

C₃×C₉

semidirect

101. C₃₇

#731⋊C₉

semidirect

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

subgroup	index
C₃×#15688515	3
#22582548	3
H(3,𝔽₃)×#66524	3
C₃×#15688504	3
#22582480	3
#22582478	3
H(3,𝔽₃)⋊#66524	3
#22582469	3
C₃₇×H(3,𝔽₃)⋊C₉	3
#731⋊C₉	37

subgroup

index

C₃₇×H(3,𝔽₃)⋊C₉

Subgroups — #41006412

Conjugacy classes of subgroups

Complements

Maximal subgroups