Group #41006381 — subgroups

Subgroups — #41006381

Order 26973.

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

normal subgroups only

This group has 292 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	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×C₉×#66524	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	4	no	—
17	C₃²⋊#66524	—	3	2	no	—
18	C₃²⋊#66524	C₃²	1	2	yes	—
19	C₃²⋊#66524	C₃²	1	2	yes	—
20	C₉×#66524	—	3	2	no	—
21	C₉×#66524	—	3	2	no	—
22	C₉×#66524	—	3	1	no	—
23	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
24	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
25	C₃₇×C₃²⋊C₉	C₃²	1	2	yes	—
26	C₃³×C₃₇	—	3	1	no	—
27	C₃³×C₃₇	C₃³	1	1	yes	characteristic
28	H(3,𝔽₃)×C₃₇	—	3	2	no	—
29	H(3,𝔽₃)×C₃₇	C₃×C₉	1	3	yes	—
30	C₃×C₉×C₃₇	—	3	1	no	—
31	C₃×C₉×C₃₇	—	3	3	no	—
32	C₃×C₉×C₃₇	—	3	2	no	—
33	#11758608	—	3	4	no	—
34	#11758608	C₃×C₉	1	6	yes	complemented
35	C₃×#66524	—	3	1	no	—
36	C₃×#66524	—	3	1	no	—
37	C₃×#66524	—	3	2	no	—
38	C₃×#66524	—	3	2	no	—
39	C₃×#66524	—	3	3	no	—
40	C₃×#66524	—	3	3	no	—
41	C₃×#66524	—	3	2	no	—
42	C₃×#66524	—	3	2	no	—
43	C₃×#66524	—	9	3	no	—
44	C₃×#66524	C₃×C₉	1	3	yes	complemented
45	C₃×#66524	H(3,𝔽₃)	1	2	yes	—
46	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
47	C₉×#731	—	37	1	no	maximal, Sylow
48	C₃²×C₃₇	—	3	1	no	—
49	C₃²×C₃₇	—	3	1	no	—
50	C₃²×C₃₇	—	3	1	no	—
51	C₃²×C₃₇	—	3	3	no	—
52	C₃²×C₃₇	—	9	1	no	—
53	C₃²×C₃₇	—	9	1	no	—
54	C₃²×C₃₇	—	9	1	no	—
55	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
56	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
57	C₃²×C₃₇	#737	1	2	yes	—
58	C₉×C₃₇	—	3	3	no	—
59	C₉×C₃₇	—	9	2	no	—
60	#66524	—	3	3	no	—
61	#66524	—	3	6	no	—
62	#66524	—	3	2	no	—
63	#66524	—	3	3	no	—
64	#66524	—	9	2	no	—
65	#66524	—	9	2	no	—
66	#66524	#731	1	3	yes	complemented
67	C₃×#731	—	37	1	no	—
68	C₉×3₋¹⁺²	—	37	2	no	—
69	C₉×H(3,𝔽₃)	—	37	1	no	—
70	C₃×C₉²	—	37	1	no	—
71	3₋¹⁺²⋊C₉	—	37	2	no	—
72	3₋¹⁺²⋊C₉	—	37	2	no	—
73	H(3,𝔽₃)⋊C₉	—	37	1	no	—
74	H(3,𝔽₃)⋊C₉	C₃₇⋊C₃	1	1	yes	characteristic
75	#6867	—	37	2	no	—
76	C₃×C₃₇	—	3	1	no	—
77	C₃×C₃₇	—	3	2	no	—
78	C₃×C₃₇	—	9	3	no	—
79	C₃×C₃₇	C₉×H(3,𝔽₃)	1	1	yes	characteristic
80	C₃×C₃₇	C₃×#731	1	1	yes	characteristic
81	C₃×C₃₇	#6908	1	1	yes	characteristic
82	C₃×C₃₇	#6908	1	1	yes	characteristic
83	C₃×3₋¹⁺²	—	37	2	no	—
84	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
85	C₃²×C₉	—	37	1	no	—
86	C₃²×C₉	—	37	1	no	—
87	C₃²×C₉	—	37	1	no	—
88	C₃²×C₉	—	111	1	no	—
89	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
90	#731	—	37	9	no	—
91	C₉⋊C₉	—	111	4	no	—
92	C₃²⋊C₉	—	37	2	no	—
93	C₃²⋊C₉	—	37	2	no	—
94	C₃²⋊C₉	—	111	2	no	—
95	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	2	yes	—
96	C₉²	—	111	1	no	—
97	C₉²	—	111	2	no	—
98	C₉²	—	111	2	no	—
99	C₃₇	C₉×#731	1	1	yes	characteristic, complemented, Sylow
100	C₃³	—	3	1	no	—
101	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
102	3₋¹⁺²	—	37	6	no	—
103	3₋¹⁺²	—	111	4	no	—
104	H(3,𝔽₃)	—	3	2	no	—
105	H(3,𝔽₃)	C₉×C₃₇⋊C₃	1	3	yes	—
106	C₃×C₉	—	3	1	no	—
107	C₃×C₉	—	3	3	no	—
108	C₃×C₉	—	3	2	no	—
109	C₃×C₉	—	37	1	no	—
110	C₃×C₉	—	37	3	no	—
111	C₃×C₉	—	37	2	no	—
112	C₃×C₉	—	333	3	no	—
113	C₃×C₉	—	111	3	no	—
114	C₃×C₉	—	111	2	no	—
115	C₃×C₉	—	111	2	no	—
116	C₃×C₉	—	111	1	no	—
117	C₃×C₉	—	111	1	no	—
118	C₃×C₉	—	111	2	no	—
119	C₃×C₉	—	111	2	no	—
120	C₃×C₉	—	111	3	no	—
121	C₃²	—	3	1	no	—
122	C₃²	—	3	1	no	—
123	C₃²	—	3	1	no	—
124	C₃²	—	3	3	no	—
125	C₃²	—	9	1	no	—
126	C₃²	—	9	1	no	—
127	C₃²	—	9	1	no	—
128	C₃²	C₃×C₉×C₃₇⋊C₃	1	1	yes	characteristic
129	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
130	C₃²	#15688532	1	2	yes	—
131	C₉	—	3	3	no	—
132	C₉	—	9	2	no	—
133	C₉	—	37	3	no	—
134	C₉	—	333	2	no	—
135	C₉	—	333	2	no	—
136	C₉	—	111	3	no	—
137	C₉	—	111	6	no	—
138	C₉	—	111	2	no	—
139	C₉	—	111	3	no	—
140	C₃	—	3	1	no	—
141	C₃	—	3	2	no	—
142	C₃	—	9	3	no	—
143	C₃	#22582539	1	1	yes	characteristic
144	C₃	#731×C₃₇⋊C₃	1	1	yes	characteristic
145	C₃	#22582629	1	1	yes	characteristic
146	C₃	#22582631	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×C₉×#66524

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

—

C₉×#66524

—

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

—

—

—

—

—

—

—

yes

—

C₉×3₋¹⁺²

—

C₉×H(3,𝔽₃)

—

C₃×C₉²

—

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

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

C₉²

—

111

—

C₉²

—

111

—

C₉²

—

111

—

C₃₇

C₉×#731

yes

characteristic, complemented, Sylow

100

C₃³

—

101

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

102

3₋¹⁺²

—

103

3₋¹⁺²

—

111

—

104

H(3,𝔽₃)

—

105

H(3,𝔽₃)

C₉×C₃₇⋊C₃

yes

—

106

C₃×C₉

—

107

C₃×C₉

—

108

C₃×C₉

—

109

C₃×C₉

—

110

C₃×C₉

—

111

C₃×C₉

—

112

C₃×C₉

—

333

—

113

C₃×C₉

—

111

—

114

C₃×C₉

—

111

—

115

C₃×C₉

—

111

—

116

C₃×C₉

—

111

—

117

C₃×C₉

—

111

—

118

C₃×C₉

—

111

—

119

C₃×C₉

—

111

—

120

C₃×C₉

—

111

—

121

C₃²

—

122

C₃²

—

123

C₃²

—

124

C₃²

—

125

C₃²

—

126

C₃²

—

127

C₃²

—

128

C₃²

C₃×C₉×C₃₇⋊C₃

yes

characteristic

129

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

130

C₃²

#15688532

yes

—

131

C₉

—

132

C₉

—

133

C₉

—

134

C₉

—

333

—

135

C₉

—

333

—

136

C₉

—

111

—

137

C₉

—

111

—

138

C₉

—

111

—

139

C₉

—

111

—

140

C₃

—

141

C₃

—

142

C₃

—

143

C₃

#22582539

yes

characteristic

144

C₃

#731×C₃₇⋊C₃

yes

characteristic

145

C₃

#22582629

yes

characteristic

146

C₃

#22582631

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	C₉×#731
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

C₉×#731

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×C₉×#66524	C₃	semidirect
5. #22582480	C₃	semidirect
6. #22582478	C₃	semidirect
8. #22582469	C₃	semidirect
15. #15688515	C₉	semidirect
34. #11758608	C₃×C₉	semidirect
44. C₃×#66524	C₃×C₉	semidirect
66. #66524	#731	semidirect
99. C₃₇	C₉×#731	semidirect

H ≅ G/N

type

2. #22582548

C₃

semidirect

4. C₃×C₉×#66524

C₃

semidirect

5. #22582480

C₃

semidirect

6. #22582478

C₃

semidirect

8. #22582469

C₃

semidirect

15. #15688515

C₉

semidirect

34. #11758608

C₃×C₉

semidirect

44. C₃×#66524

C₃×C₉

semidirect

66. #66524

#731

semidirect

99. C₃₇

C₉×#731

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×C₉×#66524	3
#22582480	3
#22582478	3
H(3,𝔽₃)⋊#66524	3
#22582469	3
C₃₇×H(3,𝔽₃)⋊C₉	3
C₉×#731	37

subgroup

index

C₃×C₉×#66524

C₃₇×H(3,𝔽₃)⋊C₉

Subgroups — #41006381

Conjugacy classes of subgroups

Complements

Maximal subgroups