Group #41006924 — subgroups

Subgroups — #41006924

Order 26973.

240 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 64 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	#22582568	C₃	1	2	yes	maximal, complemented
2	H(3,𝔽₃)×#66524	C₃	1	1	yes	maximal, characteristic
3	C₃×#15688504	C₃	1	1	yes	maximal, characteristic, complemented
4	3₋¹⁺²⋊#66524	C₃	1	2	yes	maximal, complemented
5	3₋¹⁺²⋊#66524	C₃	1	2	yes	maximal, complemented
6	H(3,𝔽₃)⋊#66524	C₃	1	1	yes	maximal, characteristic
7	H(3,𝔽₃)⋊#66524	C₃	1	1	yes	maximal, characteristic
8	#22582470	C₃	1	2	yes	maximal, complemented
9	C₃×C₃₇×#731	C₃	1	1	yes	maximal, characteristic
10	C₃²×#66524	—	3	1	no	—
11	C₃²×#66524	C₃²	1	1	yes	characteristic
12	C₃²×#66524	C₃²	1	1	yes	characteristic
13	C₃²×#66524	C₃²	1	1	yes	characteristic
14	#15688504	—	3	2	no	—
15	#15688504	—	3	1	no	—
16	#15688504	—	3	4	no	—
17	#15688504	—	3	2	no	—
18	C₃²⋊#66524	—	3	2	no	—
19	C₃²⋊#66524	C₃²	1	2	yes	—
20	C₃²⋊#66524	C₃²	1	2	yes	—
21	C₃²⋊#66524	C₃²	1	2	yes	—
22	C₃×3₋¹⁺²×C₃₇	C₃²	1	2	yes	—
23	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
24	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
25	C₃₇×#731	—	3	3	no	—
26	C₃³×C₃₇	—	3	1	no	—
27	C₃³×C₃₇	C₃³	1	1	yes	characteristic
28	3₋¹⁺²×C₃₇	—	3	2	no	—
29	3₋¹⁺²×C₃₇	—	3	4	no	—
30	H(3,𝔽₃)×C₃₇	—	3	2	no	—
31	H(3,𝔽₃)×C₃₇	3₋¹⁺²	1	3	yes	—
32	C₃×C₉×C₃₇	—	3	1	no	—
33	C₃×C₉×C₃₇	—	3	2	no	—
34	C₃×C₉×C₃₇	—	3	1	no	—
35	C₃×C₉×C₃₇	—	3	2	no	—
36	C₃×#66524	—	3	3	no	—
37	C₃×#66524	—	3	2	no	—
38	C₃×#66524	—	3	2	no	—
39	C₃×#66524	—	3	2	no	—
40	C₃×#66524	—	3	1	no	—
41	C₃×#66524	—	3	1	no	—
42	C₃×#66524	—	9	3	no	—
43	C₃×#66524	—	9	1	no	—
44	C₃×#66524	—	9	1	no	—
45	C₃×#66524	H(3,𝔽₃)	1	2	yes	—
46	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
47	#10624793	—	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	C₃×H(3,𝔽₃)	1	1	yes	characteristic
56	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
57	C₃²×C₃₇	#737	1	2	yes	—
58	C₉×C₃₇	—	9	1	no	—
59	C₉×C₃₇	—	9	2	no	—
60	#66524	—	3	2	no	—
61	#66524	—	9	2	no	—
62	#66524	—	9	2	no	—
63	#66524	—	9	1	no	—
64	#66524	—	9	1	no	—
65	#66524	—	9	2	no	—
66	#66524	#731	1	3	yes	complemented
67	C₃×#731	C₃₇⋊C₃	1	1	yes	characteristic
68	#6894	—	37	2	no	—
69	C₉×H(3,𝔽₃)	—	37	1	no	—
70	C₃×C₉⋊C₉	—	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	—	37	1	no	—
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	#6913	1	1	yes	characteristic
82	C₃×C₃₇	#6913	1	1	yes	characteristic
83	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	2	yes	—
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	—	3	3	no	—
91	C₉⋊C₉	—	111	1	no	—
92	C₉⋊C₉	—	111	2	no	—
93	C₉⋊C₉	—	111	2	no	—
94	C₉⋊C₉	—	111	4	no	—
95	C₃²⋊C₉	—	37	2	no	—
96	C₃²⋊C₉	—	37	2	no	—
97	C₃²⋊C₉	—	37	2	no	—
98	C₃²⋊C₉	—	111	2	no	—
99	C₃₇	#10624793	1	1	yes	characteristic, complemented, Sylow
100	C₃³	—	3	1	no	—
101	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
102	3₋¹⁺²	—	3	2	no	—
103	3₋¹⁺²	—	3	4	no	—
104	H(3,𝔽₃)	—	3	2	no	—
105	H(3,𝔽₃)	#11758608	1	3	yes	—
106	C₃×C₉	—	3	1	no	—
107	C₃×C₉	—	3	2	no	—
108	C₃×C₉	—	3	1	no	—
109	C₃×C₉	—	3	2	no	—
110	C₃×C₉	—	37	1	no	—
111	C₃×C₉	—	37	2	no	—
112	C₃×C₉	—	333	3	no	—
113	C₃×C₉	—	333	1	no	—
114	C₃×C₉	—	333	1	no	—
115	C₃×C₉	—	111	3	no	—
116	C₃×C₉	—	111	2	no	—
117	C₃×C₉	—	111	2	no	—
118	C₃×C₉	—	111	2	no	—
119	C₃×C₉	—	111	1	no	—
120	C₃×C₉	—	111	1	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	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
129	C₃²	C₃×#11758608	1	1	yes	characteristic
130	C₃²	#15688530	1	2	yes	—
131	C₉	—	9	1	no	—
132	C₉	—	9	2	no	—
133	C₉	—	37	3	no	—
134	C₉	—	333	2	no	—
135	C₉	—	333	1	no	—
136	C₉	—	333	2	no	—
137	C₉	—	333	1	no	—
138	C₉	—	333	2	no	—
139	C₉	—	111	2	no	—
140	C₃	—	3	1	no	—
141	C₃	—	3	2	no	—
142	C₃	—	9	3	no	—
143	C₃	#22582556	1	1	yes	characteristic
144	C₃	#731×C₃₇⋊C₃	1	1	yes	characteristic
145	C₃	#22582659	1	1	yes	characteristic
146	C₃	#22582660	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22582568

C₃

yes

maximal, complemented

H(3,𝔽₃)×#66524

C₃

yes

maximal, characteristic

C₃×#15688504

C₃

yes

maximal, characteristic, complemented

3₋¹⁺²⋊#66524

C₃

yes

maximal, complemented

3₋¹⁺²⋊#66524

C₃

yes

maximal, complemented

H(3,𝔽₃)⋊#66524

C₃

yes

maximal, characteristic

H(3,𝔽₃)⋊#66524

C₃

yes

maximal, characteristic

#22582470

C₃

yes

maximal, complemented

C₃×C₃₇×#731

C₃

yes

maximal, characteristic

C₃²×#66524

—

C₃²×#66524

C₃²

yes

characteristic

C₃²×#66524

C₃²

yes

characteristic

C₃²×#66524

C₃²

yes

—

—

—

—

—

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

—

C₃×3₋¹⁺²×C₃₇

C₃²

yes

—

C₃×H(3,𝔽₃)×C₃₇

C₃²

yes

characteristic

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×#731

—

C₃³×C₃₇

—

C₃³×C₃₇

C₃³

yes

characteristic

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

—

H(3,𝔽₃)×C₃₇

—

H(3,𝔽₃)×C₃₇

3₋¹⁺²

yes

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

—

—

—

—

—

—

—

—

—

yes

—

yes

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₃₇

#737

yes

—

C₉×C₃₇

—

C₉×C₃₇

—

—

—

—

—

—

—

yes

C₃₇⋊C₃

yes

characteristic

#6894

—

C₉×H(3,𝔽₃)

—

C₃×C₉⋊C₉

—

3₋¹⁺²⋊C₉

—

3₋¹⁺²⋊C₉

—

H(3,𝔽₃)⋊C₉

—

H(3,𝔽₃)⋊C₉

—

#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×C₃₇⋊C₃

yes

—

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

—

C₃²⋊C₉

—

111

—

C₃₇

#10624793

yes

characteristic, complemented, Sylow

100

C₃³

—

101

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

102

3₋¹⁺²

—

103

3₋¹⁺²

—

104

H(3,𝔽₃)

—

105

H(3,𝔽₃)

#11758608

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

—

333

—

114

C₃×C₉

—

333

—

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

H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

129

C₃²

C₃×#11758608

yes

characteristic

130

C₃²

#15688530

yes

—

131

C₉

—

132

C₉

—

133

C₉

—

134

C₉

—

333

—

135

C₉

—

333

—

136

C₉

—

333

—

137

C₉

—

333

—

138

C₉

—

333

—

139

C₉

—

111

—

140

C₃

—

141

C₃

—

142

C₃

—

143

C₃

#22582556

yes

characteristic

144

C₃

#731×C₃₇⋊C₃

yes

characteristic

145

C₃

#22582659

yes

characteristic

146

C₃

#22582660

yes

characteristic

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₃₇⋊C₃
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃×C₃₇×#731	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃×H(3,𝔽₃)×C₃₇	C₃²
hypercenter	C₃×#731	C₃₇⋊C₃
hypocenter	C₃₇	#10624793
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×#731

subgroup

G/H

center

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃×C₃₇×#731

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

prime radical

C₃×H(3,𝔽₃)×C₃₇

C₃²

C₃₇⋊C₃

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ C₃×#731

N	H ≅ G/N	type
1. #22582568	C₃	semidirect
3. C₃×#15688504	C₃	semidirect
4. 3₋¹⁺²⋊#66524	C₃	semidirect
5. 3₋¹⁺²⋊#66524	C₃	semidirect
8. #22582470	C₃	semidirect
66. #66524	#731	semidirect
99. C₃₇	#10624793	semidirect

H ≅ G/N

type

1. #22582568

C₃

semidirect

3. C₃×#15688504

C₃

semidirect

4. 3₋¹⁺²⋊#66524

C₃

semidirect

5. 3₋¹⁺²⋊#66524

C₃

semidirect

8. #22582470

C₃

semidirect

66. #66524

#731

semidirect

99. C₃₇

#10624793

semidirect

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

subgroup	index
#22582568	3
H(3,𝔽₃)×#66524	3
C₃×#15688504	3
3₋¹⁺²⋊#66524	3
H(3,𝔽₃)⋊#66524	3
#22582470	3
C₃×C₃₇×#731	3
#10624793	37

subgroup

index

C₃×C₃₇×#731

Subgroups — #41006924

Conjugacy classes of subgroups

Complements

Maximal subgroups