Group #41006293 — subgroups

Subgroups — #41006293

Order 26973.

344 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 112 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×#15688531	C₃	1	1	yes	maximal, characteristic
2	#22582538	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582538	C₃	1	1	yes	maximal, characteristic, complemented
4	#22582480	C₃	1	6	yes	maximal, complemented
5	#22582477	C₃	1	3	yes	maximal
6	C₃×C₃₇×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
7	#15688531	—	3	3	no	—
8	#15688531	—	3	3	no	—
9	#15688531	C₉	1	9	yes	complemented
10	C₃×#11758608	C₃²	1	2	yes	—
11	C₃×#11758605	—	3	1	no	—
12	C₃×#11758606	—	3	1	no	—
13	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	1	yes	characteristic
14	C₃×C₉×C₃₇⋊C₃	—	3	1	no	—
15	#15688506	—	3	1	no	—
16	#15688506	—	3	1	no	—
17	#15688505	—	3	1	no	—
18	#15688505	—	3	1	no	—
19	C₃²⋊#66524	C₃²	1	3	yes	—
20	C₃²⋊#66524	C₃²	1	3	yes	—
21	C₉×#66524	—	3	1	no	—
22	C₉×#66524	—	3	1	no	—
23	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
24	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
25	C₃₇×C₉⋊C₉	—	3	1	no	—
26	C₃₇×C₉⋊C₉	—	3	1	no	—
27	C₃₇×C₉⋊C₉	—	3	1	no	—
28	C₃³×C₃₇	C₃³	1	1	yes	characteristic
29	C₃²×C₃₇⋊C₃	—	3	1	no	—
30	C₃×C₉×C₃₇	—	3	3	no	—
31	C₃×C₉×C₃₇	—	9	3	no	—
32	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	—
33	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
34	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
35	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
36	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	—
37	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	—
38	C₃²⋊(C₃₇⋊C₃)	—	3	1	no	—
39	C₃²⋊(C₃₇⋊C₃)	—	3	1	no	—
40	C₃²⋊(C₃₇⋊C₃)	C₃×C₉	1	3	yes	—
41	#11758608	—	3	2	no	—
42	#11758608	—	3	2	no	—
43	#11758608	C₃×C₉	1	6	yes	—
44	C₃×#66524	—	3	2	no	—
45	C₃×#66524	—	3	3	no	—
46	C₃×#66524	—	3	3	no	—
47	#11758606	—	3	9	no	—
48	#11758605	—	3	9	no	—
49	C₉×C₃₇⋊C₃	—	3	9	no	—
50	#10624700	—	37	1	no	maximal, Sylow
51	C₃²×C₃₇	—	3	3	no	—
52	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
53	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
54	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
55	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
56	C₃×C₃₇⋊C₃	—	3	3	no	—
57	C₃×C₃₇⋊C₃	—	9	3	no	—
58	C₉×C₃₇	—	9	3	no	—
59	C₉×C₃₇	C₃²⋊C₉	1	3	yes	—
60	C₉×C₃₇	C₃²⋊C₉	1	3	yes	—
61	C₉×C₃₇	C₃²⋊C₉	1	3	yes	—
62	#66524	—	3	6	no	—
63	#66524	—	9	3	no	—
64	#66524	—	9	3	no	—
65	C₃×#737	—	37	1	no	—
66	#6887	—	37	1	no	—
67	#6887	—	37	1	no	—
68	C₃×C₉⋊C₉	C₃₇⋊C₃	1	1	yes	characteristic, complemented
69	3₋¹⁺²⋊C₉	—	37	6	no	—
70	H(3,𝔽₃)⋊C₉	—	37	3	no	—
71	C₃×C₃₇	—	3	1	no	—
72	C₃×C₃₇	—	3	1	no	—
73	C₃×C₃₇	—	3	1	no	—
74	C₃×C₃₇	C₃×C₃²⋊C₉	1	1	yes	characteristic
75	C₃×C₃₇	#6913	1	3	yes	—
76	C₃₇⋊C₃	—	9	3	no	—
77	#737	—	37	9	no	—
78	#737	—	111	3	no	—
79	#737	—	111	3	no	—
80	C₃×3₋¹⁺²	—	37	2	no	—
81	C₃×3₋¹⁺²	—	111	1	no	—
82	C₃×3₋¹⁺²	—	111	1	no	—
83	C₃×H(3,𝔽₃)	—	37	1	no	—
84	C₃²×C₉	—	111	1	no	—
85	C₃²×C₉	C₃×C₃₇⋊C₃	1	3	yes	—
86	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
87	C₉⋊C₉	—	3	1	no	—
88	C₉⋊C₉	—	3	1	no	—
89	C₉⋊C₉	—	3	1	no	—
90	C₉⋊C₉	—	111	1	no	—
91	C₉⋊C₉	—	111	1	no	—
92	C₉⋊C₉	—	111	1	no	—
93	C₉⋊C₉	—	111	1	no	—
94	C₃²⋊C₉	—	37	3	no	—
95	C₃²⋊C₉	—	37	3	no	—
96	C₉²	—	111	1	no	—
97	C₉²	—	111	1	no	—
98	C₃₇	#10624700	1	1	yes	characteristic, complemented, Sylow
99	C₃³	—	111	1	no	—
100	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
101	3₋¹⁺²	—	37	6	no	—
102	3₋¹⁺²	—	111	2	no	—
103	3₋¹⁺²	—	111	2	no	—
104	3₋¹⁺²	—	111	9	no	—
105	3₋¹⁺²	—	111	9	no	—
106	H(3,𝔽₃)	—	37	3	no	—
107	H(3,𝔽₃)	—	111	1	no	—
108	H(3,𝔽₃)	—	111	1	no	—
109	C₃×C₉	—	3	3	no	—
110	C₃×C₉	—	9	3	no	—
111	C₃×C₉	—	111	3	no	—
112	C₃×C₉	—	111	3	no	—
113	C₃×C₉	—	111	2	no	—
114	C₃×C₉	—	111	9	no	—
115	C₃×C₉	C₉×C₃₇⋊C₃	1	3	yes	—
116	C₃×C₉	#11758605	1	3	yes	—
117	C₃×C₉	#11758606	1	3	yes	—
118	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
119	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
120	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
121	C₃²	—	3	3	no	—
122	C₃²	—	333	3	no	—
123	C₃²	—	111	3	no	—
124	C₃²	C₃×C₉×C₃₇⋊C₃	1	1	yes	characteristic
125	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
126	C₃²	C₃×#11758606	1	1	yes	characteristic
127	C₃²	C₃×#11758605	1	1	yes	characteristic
128	C₉	—	9	3	no	—
129	C₉	—	333	3	no	—
130	C₉	—	333	3	no	—
131	C₉	—	111	6	no	—
132	C₉	#15688502	1	3	yes	—
133	C₉	#15688502	1	3	yes	—
134	C₉	#15688502	1	3	yes	—
135	C₃	—	3	1	no	—
136	C₃	—	3	1	no	—
137	C₃	—	3	1	no	—
138	C₃	—	333	3	no	—
139	C₃	C₃×#15688502	1	1	yes	characteristic
140	C₃	#22582661	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688531

C₃

yes

maximal, characteristic

#22582538

C₃

yes

maximal, characteristic, complemented

#22582538

C₃

yes

maximal, characteristic, complemented

#22582480

C₃

yes

maximal, complemented

#22582477

C₃

yes

maximal

C₃×C₃₇×C₉⋊C₉

C₃

yes

maximal, characteristic, complemented

#15688531

—

#15688531

—

#15688531

C₉

yes

complemented

C₃×#11758608

C₃²

yes

—

C₃×#11758605

—

C₃×#11758606

—

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

yes

characteristic

C₃×C₉×C₃₇⋊C₃

—

—

—

—

—

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

—

C₉×#66524

—

C₉×#66524

—

C₃²×C₉×C₃₇

C₃²

yes

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×C₉⋊C₉

—

C₃₇×C₉⋊C₉

—

C₃₇×C₉⋊C₉

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

C₃×C₉

yes

—

C₃×C₉×C₃₇

yes

C₃×C₉×C₃₇

yes

C₃×C₉×C₃₇

yes

C₃×C₉×C₃₇

3₋¹⁺²

yes

—

C₃×C₉×C₃₇

3₋¹⁺²

yes

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

C₃×C₉

yes

—

#11758608

—

#11758608

—

#11758608

C₃×C₉

yes

—

—

—

—

—

—

C₉×C₃₇⋊C₃

—

#10624700

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

C₃²×C₉

yes

characteristic

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

C₉×C₃₇

C₃²⋊C₉

yes

—

C₉×C₃₇

C₃²⋊C₉

yes

—

C₉×C₃₇

C₃²⋊C₉

yes

—

—

—

—

—

—

—

C₃×C₉⋊C₉

C₃₇⋊C₃

yes

characteristic, complemented

3₋¹⁺²⋊C₉

—

H(3,𝔽₃)⋊C₉

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃×C₃²⋊C₉

yes

characteristic

C₃×C₃₇

#6913

yes

—

C₃₇⋊C₃

—

#737

—

#737

—

111

—

#737

—

111

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

111

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₉²

—

111

—

C₉²

—

111

—

C₃₇

#10624700

yes

characteristic, complemented, Sylow

C₃³

—

111

—

100

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

101

3₋¹⁺²

—

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)

—

111

—

108

H(3,𝔽₃)

—

111

—

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

C₉×C₃₇⋊C₃

yes

—

116

C₃×C₉

#11758605

yes

—

117

C₃×C₉

#11758606

yes

—

118

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

119

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

120

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

121

C₃²

—

122

C₃²

—

333

—

123

C₃²

—

111

—

124

C₃²

C₃×C₉×C₃₇⋊C₃

yes

characteristic

125

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

126

C₃²

C₃×#11758606

yes

characteristic

127

C₃²

C₃×#11758605

yes

characteristic

128

C₉

—

129

C₉

—

333

—

130

C₉

—

333

—

131

C₉

—

111

—

132

C₉

#15688502

yes

—

133

C₉

#15688502

yes

—

134

C₉

#15688502

yes

—

135

C₃

—

136

C₃

—

137

C₃

—

138

C₃

—

333

—

139

C₃

C₃×#15688502

yes

characteristic

140

C₃

#22582661

yes

—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator	C₃²×C₃₇	C₃²×C₉
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃×C₃₇×C₉⋊C₉	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²
cosocle	C₃³×C₃₇	C₃³
hypercenter	C₃×C₉⋊C₉	C₃₇⋊C₃
hypocenter	C₃₇	#10624700
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃²×C₉ ◁ C₃×C₉⋊C₉

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

commutator

C₃²×C₃₇

C₃²×C₉

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃×C₃₇×C₉⋊C₉

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

prime radical

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

cosocle

C₃³×C₃₇

C₃³

hypercenter

C₃×C₉⋊C₉

C₃₇⋊C₃

hypocenter

C₃₇

#10624700

LCS

G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃²×C₉ ◁ C₃×C₉⋊C₉

N	H ≅ G/N	type
2. #22582538	C₃	semidirect
3. #22582538	C₃	semidirect
4. #22582480	C₃	semidirect
6. C₃×C₃₇×C₉⋊C₉	C₃	semidirect
9. #15688531	C₉	semidirect
68. C₃×C₉⋊C₉	C₃₇⋊C₃	semidirect
98. C₃₇	#10624700	semidirect

H ≅ G/N

type

2. #22582538

C₃

semidirect

3. #22582538

C₃

semidirect

4. #22582480

C₃

semidirect

6. C₃×C₃₇×C₉⋊C₉

C₃

semidirect

9. #15688531

C₉

semidirect

68. C₃×C₉⋊C₉

C₃₇⋊C₃

semidirect

98. C₃₇

#10624700

semidirect

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

subgroup	index
C₃×#15688531	3
#22582538	3
#22582480	3
#22582477	3
C₃×C₃₇×C₉⋊C₉	3
#10624700	37

subgroup

index

C₃×C₃₇×C₉⋊C₉

Subgroups — #41006293

Conjugacy classes of subgroups

Complements

Maximal subgroups