Group #41006939 — subgroups

Subgroups — #41006939

Order 26973.

372 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 70 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×#11758606	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×#15688512	C₃	1	1	yes	maximal, characteristic
3	#22582572	C₃	1	2	yes	maximal, complemented
4	#22582478	C₃	1	4	yes	maximal, complemented
5	H(3,𝔽₃)⋊#66524	C₃	1	2	yes	maximal, complemented
6	#22582467	C₃	1	2	yes	maximal, complemented
7	C₉×H(3,𝔽₃)×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
9	C₃×#11758608	C₃²	1	2	yes	—
10	C₃×#11758606	—	3	2	no	—
11	C₃×#11758606	—	3	1	no	—
12	C₃×#11758606	—	3	9	no	—
13	C₃²×#66524	C₃²	1	2	yes	—
14	#15688512	—	3	3	no	—
15	#15688506	—	3	2	no	—
16	#15688504	—	3	4	no	—
17	C₃²⋊#66524	C₃²	1	4	yes	—
18	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
19	C₃²×C₉×C₃₇	—	3	1	no	—
20	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
21	C₃₇×C₃²⋊C₉	—	3	2	no	—
22	C₃₇×C₃²⋊C₉	C₃²	1	2	yes	—
23	C₃³×C₃₇	—	3	1	no	—
24	C₃³×C₃₇	C₃³	1	1	yes	characteristic
25	C₃²×C₃₇⋊C₃	—	3	1	no	—
26	C₃²×C₃₇⋊C₃	—	3	3	no	—
27	C₃²×C₃₇⋊C₃	—	3	1	no	—
28	C₃²×C₃₇⋊C₃	—	3	2	no	—
29	C₃²×C₃₇⋊C₃	—	9	2	no	—
30	H(3,𝔽₃)×C₃₇	—	3	2	no	—
31	H(3,𝔽₃)×C₃₇	3₋¹⁺²	1	3	yes	complemented
32	C₃×C₉×C₃₇	—	3	3	no	—
33	C₃×C₉×C₃₇	—	3	2	no	—
34	C₃×C₉×C₃₇	—	9	3	no	—
35	C₃×C₉×C₃₇	H(3,𝔽₃)	1	2	yes	—
36	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
37	#11758608	—	3	2	no	—
38	#11758608	—	3	4	no	—
39	C₃×#66524	—	3	2	no	—
40	C₃×#66524	—	3	6	no	—
41	C₃×#66524	—	3	4	no	—
42	C₃×#66524	—	3	2	no	—
43	#11758606	—	3	18	no	—
44	#11758606	—	3	9	no	—
45	#10624794	—	37	1	no	maximal, Sylow
46	C₃²×C₃₇	—	3	2	no	—
47	C₃²×C₃₇	—	3	1	no	—
48	C₃²×C₃₇	—	3	3	no	—
49	C₃²×C₃₇	—	9	2	no	—
50	C₃²×C₃₇	—	9	1	no	—
51	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
52	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
53	C₃²×C₃₇	#737	1	2	yes	—
54	C₃×C₃₇⋊C₃	—	3	3	no	—
55	C₃×C₃₇⋊C₃	—	9	2	no	—
56	C₃×C₃₇⋊C₃	—	9	1	no	—
57	C₃×C₃₇⋊C₃	—	9	3	no	—
58	C₃×C₃₇⋊C₃	—	9	6	no	—
59	C₉×C₃₇	—	3	2	no	—
60	C₉×C₃₇	—	9	2	no	—
61	C₉×C₃₇	AΣL(1,𝔽₂₇)	1	3	yes	complemented
62	#66524	—	3	6	no	—
63	#66524	—	9	4	no	—
64	#66524	—	9	2	no	—
65	C₃²×3₋¹⁺²	—	37	1	no	—
66	C₃×AΣL(1,𝔽₂₇)	—	37	1	no	—
67	#6894	—	37	2	no	—
68	C₉×H(3,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic, complemented
69	3₋¹⁺²⋊C₉	—	37	4	no	—
70	H(3,𝔽₃)⋊C₉	—	37	2	no	—
71	#6866	—	37	2	no	—
72	C₃×C₃₇	—	3	1	no	—
73	C₃×C₃₇	—	3	2	no	—
74	C₃×C₃₇	—	9	3	no	—
75	C₃×C₃₇	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
76	C₃×C₃₇	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
77	C₃×C₃₇	#6913	1	2	yes	—
78	C₃₇⋊C₃	—	9	3	no	—
79	C₃⁴	—	37	1	no	—
80	C₃×3₋¹⁺²	—	37	2	no	—
81	C₃×3₋¹⁺²	—	111	9	no	—
82	C₃×3₋¹⁺²	—	111	2	no	—
83	C₃×3₋¹⁺²	—	111	1	no	—
84	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
85	C₃²×C₉	—	3	1	no	—
86	C₃²×C₉	—	37	2	no	—
87	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
88	AΣL(1,𝔽₂₇)	—	111	3	no	—
89	C₉⋊C₉	—	111	4	no	—
90	C₉⋊C₉	—	111	2	no	—
91	C₃²⋊C₉	—	3	2	no	—
92	C₃²⋊C₉	—	37	4	no	—
93	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	2	yes	—
94	C₃₇	#10624794	1	1	yes	characteristic, complemented, Sylow
95	C₃³	—	3	1	no	—
96	C₃³	—	333	2	no	—
97	C₃³	—	111	3	no	—
98	C₃³	—	111	1	no	—
99	C₃³	—	111	2	no	—
100	C₃³	—	111	1	no	—
101	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
102	3₋¹⁺²	—	111	18	no	—
103	3₋¹⁺²	—	111	2	no	—
104	3₋¹⁺²	—	111	4	no	—
105	3₋¹⁺²	—	111	9	no	—
106	H(3,𝔽₃)	—	3	2	no	—
107	H(3,𝔽₃)	#11758606	1	3	yes	complemented
108	C₃×C₉	—	3	3	no	—
109	C₃×C₉	—	3	2	no	—
110	C₃×C₉	—	9	3	no	—
111	C₃×C₉	—	111	4	no	—
112	C₃×C₉	—	111	2	no	—
113	C₃×C₉	—	111	2	no	—
114	C₃×C₉	—	111	6	no	—
115	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
116	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
117	C₃²	—	3	1	no	—
118	C₃²	—	3	2	no	—
119	C₃²	—	3	3	no	—
120	C₃²	—	9	1	no	—
121	C₃²	—	9	2	no	—
122	C₃²	—	333	3	no	—
123	C₃²	—	333	6	no	—
124	C₃²	—	333	1	no	—
125	C₃²	—	333	2	no	—
126	C₃²	—	111	3	no	—
127	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
128	C₃²	C₃×#11758606	1	1	yes	characteristic
129	C₃²	#15688531	1	2	yes	—
130	C₉	—	3	2	no	—
131	C₉	—	9	2	no	—
132	C₉	—	333	2	no	—
133	C₉	—	333	4	no	—
134	C₉	—	111	6	no	—
135	C₉	#15688512	1	3	yes	complemented
136	C₃	—	3	1	no	—
137	C₃	—	3	2	no	—
138	C₃	—	9	3	no	—
139	C₃	—	333	3	no	—
140	C₃	#22582559	1	1	yes	characteristic
141	C₃	C₃×#15688512	1	1	yes	characteristic
142	C₃	#22582656	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#11758606

C₃

yes

maximal, characteristic, complemented

C₃×#15688512

C₃

yes

maximal, characteristic

#22582572

C₃

yes

maximal, complemented

#22582478

C₃

yes

maximal, complemented

H(3,𝔽₃)⋊#66524

C₃

yes

maximal, complemented

#22582467

C₃

yes

maximal, complemented

C₉×H(3,𝔽₃)×C₃₇

C₃

yes

maximal, characteristic, complemented

C₃³×C₃₇⋊C₃

C₃²

yes

characteristic

C₃×#11758608

C₃²

yes

—

C₃×#11758606

—

C₃×#11758606

—

C₃×#11758606

—

C₃²×#66524

C₃²

yes

—

—

—

—

C₃²

yes

—

C₃×H(3,𝔽₃)×C₃₇

C₃²

yes

characteristic

C₃²×C₉×C₃₇

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×C₃²⋊C₉

—

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₃³×C₃₇

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

H(3,𝔽₃)×C₃₇

—

H(3,𝔽₃)×C₃₇

3₋¹⁺²

yes

complemented

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

H(3,𝔽₃)

yes

—

C₃×C₉×C₃₇

yes

—

—

—

—

—

—

—

—

—

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

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

AΣL(1,𝔽₂₇)

yes

—

—

—

C₃²×3₋¹⁺²

—

C₃×AΣL(1,𝔽₂₇)

—

#6894

—

C₉×H(3,𝔽₃)

C₃₇⋊C₃

yes

characteristic, complemented

3₋¹⁺²⋊C₉

—

H(3,𝔽₃)⋊C₉

—

#6866

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₉⋊H(3,𝔽₃)

yes

characteristic

C₃×C₃₇

C₃×AΣL(1,𝔽₂₇)

yes

characteristic

C₃×C₃₇

#6913

yes

—

C₃₇⋊C₃

—

C₃⁴

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×H(3,𝔽₃)

C₃×C₃₇⋊C₃

yes

characteristic

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

AΣL(1,𝔽₂₇)

—

111

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

C₃₇

#10624794

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

333

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

100

C₃³

—

111

—

101

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

102

3₋¹⁺²

—

111

—

103

3₋¹⁺²

—

111

—

104

3₋¹⁺²

—

111

—

105

3₋¹⁺²

—

111

—

106

H(3,𝔽₃)

—

107

H(3,𝔽₃)

#11758606

yes

complemented

108

C₃×C₉

—

109

C₃×C₉

—

110

C₃×C₉

—

111

C₃×C₉

—

111

—

112

C₃×C₉

—

111

—

113

C₃×C₉

—

111

—

114

C₃×C₉

—

111

—

115

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

—

116

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

—

122

C₃²

—

333

—

123

C₃²

—

333

—

124

C₃²

—

333

—

125

C₃²

—

333

—

126

C₃²

—

111

—

127

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

128

C₃²

C₃×#11758606

yes

characteristic

129

C₃²

#15688531

yes

—

130

C₉

—

131

C₉

—

132

C₉

—

333

—

133

C₉

—

333

—

134

C₉

—

111

—

135

C₉

#15688512

yes

complemented

136

C₃

—

137

C₃

—

138

C₃

—

139

C₃

—

333

—

140

C₃

#22582559

yes

characteristic

141

C₃

C₃×#15688512

yes

characteristic

142

C₃

#22582656

yes

—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₉×H(3,𝔽₃)×C₃₇	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃×#15688512	C₃
hypercenter	C₉×H(3,𝔽₃)	C₃₇⋊C₃
hypocenter	C₃₇	#10624794
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃²×C₉ ◁ C₉×H(3,𝔽₃)

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₉×H(3,𝔽₃)×C₃₇

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

prime radical

C₃×#15688512

C₃

hypercenter

C₉×H(3,𝔽₃)

C₃₇⋊C₃

hypocenter

C₃₇

#10624794

LCS

G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃²×C₉ ◁ C₉×H(3,𝔽₃)

N	H ≅ G/N	type
1. C₃²×#11758606	C₃	semidirect
3. #22582572	C₃	semidirect
4. #22582478	C₃	semidirect
5. H(3,𝔽₃)⋊#66524	C₃	semidirect
6. #22582467	C₃	semidirect
7. C₉×H(3,𝔽₃)×C₃₇	C₃	semidirect
31. H(3,𝔽₃)×C₃₇	3₋¹⁺²	semidirect
61. C₉×C₃₇	AΣL(1,𝔽₂₇)	semidirect
68. C₉×H(3,𝔽₃)	C₃₇⋊C₃	semidirect
94. C₃₇	#10624794	semidirect
107. H(3,𝔽₃)	#11758606	semidirect
135. C₉	#15688512	semidirect

H ≅ G/N

type

1. C₃²×#11758606

C₃

semidirect

3. #22582572

C₃

semidirect

4. #22582478

C₃

semidirect

5. H(3,𝔽₃)⋊#66524

C₃

semidirect

6. #22582467

C₃

semidirect

7. C₉×H(3,𝔽₃)×C₃₇

C₃

semidirect

31. H(3,𝔽₃)×C₃₇

3₋¹⁺²

semidirect

61. C₉×C₃₇

AΣL(1,𝔽₂₇)

semidirect

68. C₉×H(3,𝔽₃)

C₃₇⋊C₃

semidirect

94. C₃₇

#10624794

semidirect

107. H(3,𝔽₃)

#11758606

semidirect

135. C₉

#15688512

semidirect

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

subgroup	index
C₃²×#11758606	3
C₃×#15688512	3
#22582572	3
#22582478	3
H(3,𝔽₃)⋊#66524	3
#22582467	3
C₉×H(3,𝔽₃)×C₃₇	3
#10624794	37

subgroup

index

C₉×H(3,𝔽₃)×C₃₇

Subgroups — #41006939

Conjugacy classes of subgroups

Complements

Maximal subgroups