Group #41006502 — subgroups

Subgroups — #41006502

Order 26973.

530 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 67 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×C₉×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×#15688521	C₃	1	2	yes	maximal, complemented
3	C₃×#15688515	C₃	1	3	yes	maximal, complemented
4	C₃×#15688512	C₃	1	3	yes	maximal, complemented
5	#22582562	C₃	1	3	yes	maximal, complemented
6	C₃×C₃₇×#732	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
8	C₃×#11758608	C₃²	1	6	yes	—
9	C₃²×#66524	C₃²	1	1	yes	characteristic
10	C₃²×#66524	C₃²	1	1	yes	characteristic
11	C₃×C₉×C₃₇⋊C₃	—	3	3	no	—
12	C₃×C₉×C₃₇⋊C₃	—	3	3	no	—
13	C₃×C₉×C₃₇⋊C₃	—	3	3	no	—
14	C₃×C₉×C₃₇⋊C₃	—	3	3	no	—
15	#15688521	—	3	6	no	—
16	#15688515	—	3	9	no	—
17	#15688512	—	3	9	no	—
18	C₃²⋊#66524	—	3	9	no	—
19	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	3	yes	—
20	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
21	C₃₇×#732	—	3	3	no	—
22	C₃³×C₃₇	—	3	3	no	—
23	C₃³×C₃₇	C₃³	1	1	yes	characteristic
24	C₃²×C₃₇⋊C₃	—	3	1	no	—
25	C₃²×C₃₇⋊C₃	—	3	1	no	—
26	C₃²×C₃₇⋊C₃	—	3	1	no	—
27	C₃²×C₃₇⋊C₃	—	3	9	no	—
28	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	3	yes	—
29	H(3,𝔽₃)×C₃₇	—	3	3	no	—
30	H(3,𝔽₃)×C₃₇	—	3	3	no	—
31	H(3,𝔽₃)×C₃₇	—	3	3	no	—
32	C₃×C₉×C₃₇	—	3	1	no	—
33	C₃×C₉×C₃₇	—	3	1	no	—
34	C₃×C₉×C₃₇	—	3	1	no	—
35	C₃×C₉×C₃₇	H(3,𝔽₃)	1	3	yes	—
36	#11758608	—	3	6	no	—
37	#11758608	—	3	6	no	—
38	#11758608	—	3	6	no	—
39	C₃×#66524	—	3	1	no	—
40	C₃×#66524	—	3	1	no	—
41	C₃×#66524	—	3	1	no	—
42	C₃×#66524	—	3	1	no	—
43	C₃×#66524	—	3	1	no	—
44	C₃×#66524	—	3	1	no	—
45	C₃×#66524	—	3	6	no	—
46	C₃×#66524	H(3,𝔽₃)	1	3	yes	—
47	C₃×#66524	H(3,𝔽₃)	1	3	yes	—
48	C₉×C₃₇⋊C₃	—	3	27	no	—
49	#10624734	—	37	1	no	maximal, Sylow
50	C₃²×C₃₇	—	3	3	no	—
51	C₃²×C₃₇	—	9	3	no	—
52	C₃²×C₃₇	—	9	9	no	—
53	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
54	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
55	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
56	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
57	C₃×C₃₇⋊C₃	—	3	9	no	—
58	C₃×C₃₇⋊C₃	—	3	9	no	—
59	C₃×C₃₇⋊C₃	—	3	3	no	—
60	C₃×C₃₇⋊C₃	—	3	9	no	—
61	C₃×C₃₇⋊C₃	—	3	9	no	—
62	C₉×C₃₇	—	3	3	no	—
63	#66524	—	3	3	no	—
64	#66524	—	3	3	no	—
65	#66524	—	9	6	no	—
66	C₃³×C₉	—	37	1	no	—
67	C₃×#733	—	37	2	no	—
68	C₃×#732	C₃₇⋊C₃	1	1	yes	characteristic, complemented
69	C₃×#731	—	37	3	no	—
70	C₃×AΣL(1,𝔽₂₇)	—	37	3	no	—
71	#6893	—	37	3	no	—
72	C₃×C₃₇	—	3	1	no	—
73	C₃×C₃₇	—	3	1	no	—
74	C₃×C₃₇	—	3	1	no	—
75	C₃×C₃₇	—	27	3	no	—
76	C₃×C₃₇	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
77	C₃×C₃₇	#6908	1	3	yes	—
78	C₃₇⋊C₃	—	3	9	no	—
79	C₃⁴	—	37	1	no	—
80	C₃×3₋¹⁺²	—	37	6	no	—
81	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	3	yes	—
82	C₃²×C₉	—	37	1	no	—
83	C₃²×C₉	—	37	1	no	—
84	C₃²×C₉	—	111	3	no	—
85	C₃²×C₉	—	111	3	no	—
86	C₃²×C₉	—	111	3	no	—
87	C₃²×C₉	—	111	3	no	—
88	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
89	#733	—	111	6	no	—
90	#732	—	3	3	no	—
91	#731	—	111	9	no	—
92	AΣL(1,𝔽₂₇)	—	111	9	no	—
93	C₃²⋊C₉	—	111	9	no	—
94	C₃₇	#10624734	1	1	yes	characteristic, complemented, Sylow
95	C₃³	—	3	3	no	—
96	C₃³	—	37	3	no	—
97	C₃³	—	111	9	no	—
98	C₃³	—	111	1	no	—
99	C₃³	—	111	1	no	—
100	C₃³	—	111	1	no	—
101	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
102	3₋¹⁺²	—	111	6	no	—
103	3₋¹⁺²	—	111	6	no	—
104	3₋¹⁺²	—	111	6	no	—
105	H(3,𝔽₃)	—	3	3	no	—
106	H(3,𝔽₃)	—	3	3	no	—
107	H(3,𝔽₃)	—	3	3	no	—
108	C₃×C₉	—	3	1	no	—
109	C₃×C₉	—	3	1	no	—
110	C₃×C₉	—	3	1	no	—
111	C₃×C₉	—	37	3	no	—
112	C₃×C₉	—	37	3	no	—
113	C₃×C₉	—	111	27	no	—
114	C₃×C₉	—	111	6	no	—
115	C₃×C₉	—	111	1	no	—
116	C₃×C₉	—	111	1	no	—
117	C₃×C₉	—	111	1	no	—
118	C₃×C₉	—	111	1	no	—
119	C₃×C₉	—	111	1	no	—
120	C₃×C₉	—	111	1	no	—
121	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
122	C₃²	—	3	3	no	—
123	C₃²	—	9	3	no	—
124	C₃²	—	9	9	no	—
125	C₃²	—	111	9	no	—
126	C₃²	—	111	9	no	—
127	C₃²	—	111	9	no	—
128	C₃²	—	111	3	no	—
129	C₃²	—	111	9	no	—
130	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
131	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
132	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
133	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
134	C₉	—	3	3	no	—
135	C₉	—	333	6	no	—
136	C₉	—	111	3	no	—
137	C₉	—	111	3	no	—
138	C₃	—	3	1	no	—
139	C₃	—	3	1	no	—
140	C₃	—	3	1	no	—
141	C₃	—	27	3	no	—
142	C₃	—	111	9	no	—
143	C₃	(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)	1	1	yes	characteristic
144	C₃	#22582631	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₉×C₃₇⋊C₃

C₃

yes

maximal, characteristic, complemented

C₃×#15688521

C₃

yes

maximal, complemented

C₃×#15688515

C₃

yes

maximal, complemented

C₃×#15688512

C₃

yes

maximal, complemented

#22582562

C₃

yes

maximal, complemented

C₃×C₃₇×#732

C₃

yes

maximal, characteristic, complemented

C₃³×C₃₇⋊C₃

C₃²

yes

characteristic

C₃×#11758608

C₃²

yes

—

C₃²×#66524

C₃²

yes

characteristic

C₃²×#66524

C₃²

yes

characteristic

C₃×C₉×C₃₇⋊C₃

—

C₃×C₉×C₃₇⋊C₃

—

C₃×C₉×C₃₇⋊C₃

—

C₃×C₉×C₃₇⋊C₃

—

—

—

—

—

C₃×H(3,𝔽₃)×C₃₇

C₃²

yes

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×#732

—

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)

yes

—

H(3,𝔽₃)×C₃₇

—

H(3,𝔽₃)×C₃₇

—

H(3,𝔽₃)×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

H(3,𝔽₃)

yes

—

—

—

—

—

—

—

—

—

—

—

yes

—

C₃×#66524

H(3,𝔽₃)

yes

—

C₉×C₃₇⋊C₃

—

#10624734

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

#66524

—

#66524

—

#66524

—

C₃³×C₉

—

C₃×#733

—

C₃×#732

C₃₇⋊C₃

yes

characteristic, complemented

C₃×#731

—

C₃×AΣL(1,𝔽₂₇)

—

#6893

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃²⋊H(3,𝔽₃)

yes

characteristic

C₃×C₃₇

#6908

yes

—

C₃₇⋊C₃

—

C₃⁴

—

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

C₃×C₃₇⋊C₃

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

111

—

C₃²×C₉

—

111

—

C₃²×C₉

—

111

—

C₃²×C₉

—

111

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

#733

—

111

—

#732

—

#731

—

111

—

AΣL(1,𝔽₂₇)

—

111

—

C₃²⋊C₉

—

111

—

C₃₇

#10624734

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

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

H(3,𝔽₃)

—

106

H(3,𝔽₃)

—

107

H(3,𝔽₃)

—

108

C₃×C₉

—

109

C₃×C₉

—

110

C₃×C₉

—

111

C₃×C₉

—

112

C₃×C₉

—

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

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

—

122

C₃²

—

123

C₃²

—

124

C₃²

—

125

C₃²

—

111

—

126

C₃²

—

111

—

127

C₃²

—

111

—

128

C₃²

—

111

—

129

C₃²

—

111

—

130

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

131

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

132

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

133

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

134

C₉

—

135

C₉

—

333

—

136

C₉

—

111

—

137

C₉

—

111

—

138

C₃

—

139

C₃

—

140

C₃

—

141

C₃

—

142

C₃

—

111

—

143

C₃

(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)

yes

characteristic

144

C₃

#22582631

yes

—

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₃₇⋊C₃
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃×C₃₇×#732	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
hypercenter	C₃×#732	C₃₇⋊C₃
hypocenter	C₃₇	#10624734
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×#732

subgroup

G/H

center

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃×C₃₇×#732

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

C₃₇⋊C₃

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ C₃×#732

N	H ≅ G/N	type
1. C₃²×C₉×C₃₇⋊C₃	C₃	semidirect
2. C₃×#15688521	C₃	semidirect
3. C₃×#15688515	C₃	semidirect
4. C₃×#15688512	C₃	semidirect
5. #22582562	C₃	semidirect
6. C₃×C₃₇×#732	C₃	semidirect
68. C₃×#732	C₃₇⋊C₃	semidirect
94. C₃₇	#10624734	semidirect

H ≅ G/N

type

1. C₃²×C₉×C₃₇⋊C₃

C₃

semidirect

2. C₃×#15688521

C₃

semidirect

3. C₃×#15688515

C₃

semidirect

4. C₃×#15688512

C₃

semidirect

5. #22582562

C₃

semidirect

6. C₃×C₃₇×#732

C₃

semidirect

68. C₃×#732

C₃₇⋊C₃

semidirect

94. C₃₇

#10624734

semidirect

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

subgroup	index
C₃²×C₉×C₃₇⋊C₃	3
C₃×#15688521	3
C₃×#15688515	3
C₃×#15688512	3
#22582562	3
C₃×C₃₇×#732	3
#10624734	37

subgroup

index

C₃²×C₉×C₃₇⋊C₃

C₃×C₃₇×#732

Subgroups — #41006502

Conjugacy classes of subgroups

Complements

Maximal subgroups