Group #41005339 — subgroups

Subgroups — #41005339

Order 26973.

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

normal subgroups only

This group has 356 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	C₃²×#11758608	C₃	1	1	yes	maximal, characteristic
2	#22582554	C₃	1	1	yes	maximal, characteristic
3	#22582554	C₃	1	1	yes	maximal, characteristic
4	#22582448	C₃	1	9	yes	maximal, complemented
5	C₃×C₃₇×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
6	C₃×#11758608	—	3	1	no	—
7	C₃×#11758608	—	3	1	no	—
8	C₃×#11758608	—	3	9	no	—
9	C₃×#11758608	—	3	1	no	—
10	C₃×#11758608	—	3	1	no	—
11	C₃×#11758608	C₃²	1	3	yes	—
12	C₃²×#66524	—	3	1	no	—
13	C₃²⋊#66524	—	3	1	no	—
14	C₃²⋊#66524	—	3	1	no	—
15	C₃²⋊#66524	—	3	1	no	—
16	C₃²⋊#66524	—	3	1	no	—
17	C₃²⋊#66524	C₃²	1	3	yes	—
18	C₃²⋊#66524	C₃²	1	3	yes	—
19	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
20	C₃²×C₉×C₃₇	—	3	1	no	—
21	C₃₇×C₃²⋊C₉	—	3	1	no	—
22	C₃₇×C₃²⋊C₉	—	3	1	no	—
23	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
24	C₃³×C₃₇	—	3	3	no	—
25	C₃³×C₃₇	—	3	3	no	—
26	C₃³×C₃₇	—	9	1	no	—
27	C₃³×C₃₇	—	9	1	no	—
28	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
29	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
30	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
31	C₃³×C₃₇	C₃³	1	1	yes	characteristic
32	C₃×C₉×C₃₇	—	3	3	no	—
33	C₃×C₉×C₃₇	—	9	3	no	—
34	#11758608	—	3	9	no	—
35	#11758608	—	3	9	no	—
36	#11758608	—	3	9	no	—
37	#11758608	—	9	3	no	—
38	#11758608	—	9	3	no	—
39	C₃×#66524	—	3	9	no	—
40	C₃×#66524	—	3	3	no	—
41	C₃×#66524	—	3	3	no	—
42	C₃×#66524	—	3	3	no	—
43	#10624592	—	37	1	no	maximal, Sylow
44	C₃²×C₃₇	—	3	1	no	—
45	C₃²×C₃₇	—	3	1	no	—
46	C₃²×C₃₇	—	3	1	no	—
47	C₃²×C₃₇	—	3	1	no	—
48	C₃²×C₃₇	—	3	1	no	—
49	C₃²×C₃₇	—	3	1	no	—
50	C₃²×C₃₇	—	3	1	no	—
51	C₃²×C₃₇	—	3	1	no	—
52	C₃²×C₃₇	—	3	1	no	—
53	C₃²×C₃₇	—	3	1	no	—
54	C₃²×C₃₇	—	3	1	no	—
55	C₃²×C₃₇	—	3	1	no	—
56	C₃²×C₃₇	—	3	1	no	—
57	C₃²×C₃₇	—	3	1	no	—
58	C₃²×C₃₇	—	9	9	no	—
59	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	3	yes	—
60	C₃²×C₃₇	#731	1	3	yes	—
61	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
62	C₉×C₃₇	—	9	3	no	—
63	#66524	—	3	9	no	—
64	#66524	—	9	3	no	—
65	#66524	—	9	3	no	—
66	C₃²×3₋¹⁺²	—	37	1	no	—
67	#6891	—	37	1	no	—
68	#6891	—	37	1	no	—
69	C₃×C₃²⋊C₉	C₃₇⋊C₃	1	1	yes	characteristic
70	#6858	—	37	9	no	—
71	C₃×C₃₇	—	3	3	no	—
72	C₃×C₃₇	—	3	3	no	—
73	C₃×C₃₇	—	9	1	no	—
74	C₃×C₃₇	—	9	1	no	—
75	C₃×C₃₇	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
76	C₃×C₃₇	C₃×#731	1	1	yes	characteristic
77	C₃×C₃₇	#6910	1	1	yes	characteristic
78	C₃×C₃₇	#6912	1	1	yes	characteristic
79	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
80	C₃×3₋¹⁺²	—	37	3	no	—
81	C₃×3₋¹⁺²	—	111	1	no	—
82	C₃×3₋¹⁺²	—	111	1	no	—
83	C₃×3₋¹⁺²	—	111	1	no	—
84	C₃×3₋¹⁺²	—	111	1	no	—
85	C₃×3₋¹⁺²	—	111	9	no	—
86	C₃²×C₉	—	3	1	no	—
87	C₃²×C₉	—	111	1	no	—
88	C₃²⋊C₉	—	3	1	no	—
89	C₃²⋊C₉	—	3	1	no	—
90	C₃²⋊C₉	—	37	3	no	—
91	C₃²⋊C₉	—	37	3	no	—
92	C₃²⋊C₉	—	111	1	no	—
93	C₃²⋊C₉	—	111	1	no	—
94	C₃²⋊C₉	—	111	1	no	—
95	C₃²⋊C₉	—	111	1	no	—
96	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
97	C₃₇	#10624592	1	1	yes	characteristic, complemented, Sylow
98	C₃³	—	3	3	no	—
99	C₃³	—	3	3	no	—
100	C₃³	—	9	1	no	—
101	C₃³	—	9	1	no	—
102	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
103	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
104	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
105	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
106	3₋¹⁺²	—	333	3	no	—
107	3₋¹⁺²	—	333	3	no	—
108	3₋¹⁺²	—	111	9	no	—
109	3₋¹⁺²	—	111	9	no	—
110	3₋¹⁺²	—	111	9	no	—
111	C₃×C₉	—	3	3	no	—
112	C₃×C₉	—	9	3	no	—
113	C₃×C₉	—	111	3	no	—
114	C₃×C₉	—	111	3	no	—
115	C₃×C₉	—	111	9	no	—
116	C₃×C₉	—	111	3	no	—
117	C₃²	—	3	1	no	—
118	C₃²	—	3	1	no	—
119	C₃²	—	3	1	no	—
120	C₃²	—	3	1	no	—
121	C₃²	—	3	1	no	—
122	C₃²	—	3	1	no	—
123	C₃²	—	3	1	no	—
124	C₃²	—	3	1	no	—
125	C₃²	—	3	1	no	—
126	C₃²	—	3	1	no	—
127	C₃²	—	3	1	no	—
128	C₃²	—	3	1	no	—
129	C₃²	—	3	1	no	—
130	C₃²	—	3	1	no	—
131	C₃²	—	9	9	no	—
132	C₃²	#15688512	1	3	yes	—
133	C₃²	#15688516	1	3	yes	—
134	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
135	C₉	—	9	3	no	—
136	C₉	—	333	3	no	—
137	C₉	—	333	3	no	—
138	C₉	—	111	9	no	—
139	C₃	—	3	3	no	—
140	C₃	—	3	3	no	—
141	C₃	—	9	1	no	—
142	C₃	—	9	1	no	—
143	C₃	C₃×#15688512	1	1	yes	characteristic
144	C₃	C₃×#15688516	1	1	yes	characteristic
145	C₃	#22582644	1	1	yes	characteristic
146	C₃	#22582651	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#11758608

C₃

yes

maximal, characteristic

#22582554

C₃

yes

maximal, characteristic

#22582554

C₃

yes

maximal, characteristic

#22582448

C₃

yes

maximal, complemented

C₃×C₃₇×C₃²⋊C₉

C₃

yes

maximal, characteristic

—

—

—

—

—

C₃²

yes

—

C₃²×#66524

—

—

—

—

—

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

—

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃²×C₉×C₃₇

—

C₃₇×C₃²⋊C₉

—

C₃₇×C₃²⋊C₉

—

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

yes

C₃³×C₃₇

yes

C₃³×C₃₇

yes

C₃³×C₃₇

C₃³

yes

characteristic

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

—

—

—

—

—

—

—

—

—

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

AΣL(1,𝔽₂₇)

yes

—

C₃²×C₃₇

#731

yes

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₃₇

—

#66524

—

#66524

—

#66524

—

C₃²×3₋¹⁺²

—

#6891

—

#6891

—

C₃×C₃²⋊C₉

C₃₇⋊C₃

yes

characteristic

#6858

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃×AΣL(1,𝔽₂₇)

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃⁴

C₃×C₃₇⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃²×C₉

—

C₃²×C₉

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

C₃₇

#10624592

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

100

C₃³

—

101

C₃³

—

102

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

103

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

104

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

105

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

106

3₋¹⁺²

—

333

—

107

3₋¹⁺²

—

333

—

108

3₋¹⁺²

—

111

—

109

3₋¹⁺²

—

111

—

110

3₋¹⁺²

—

111

—

111

C₃×C₉

—

112

C₃×C₉

—

113

C₃×C₉

—

111

—

114

C₃×C₉

—

111

—

115

C₃×C₉

—

111

—

116

C₃×C₉

—

111

—

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

—

122

C₃²

—

123

C₃²

—

124

C₃²

—

125

C₃²

—

126

C₃²

—

127

C₃²

—

128

C₃²

—

129

C₃²

—

130

C₃²

—

131

C₃²

—

132

C₃²

#15688512

yes

—

133

C₃²

#15688516

yes

—

134

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

135

C₉

—

136

C₉

—

333

—

137

C₉

—

333

—

138

C₉

—

111

—

139

C₃

—

140

C₃

—

141

C₃

—

142

C₃

—

143

C₃

C₃×#15688512

yes

characteristic

144

C₃

C₃×#15688516

yes

characteristic

145

C₃

#22582644

yes

characteristic

146

C₃

#22582651

yes

characteristic

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃×C₃₇×C₃²⋊C₉	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃⁴×C₃₇	C₃²
hypercenter	C₃×C₃²⋊C₉	C₃₇⋊C₃
hypocenter	C₃₇	#10624592
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃×C₃²⋊C₉

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃×C₃₇×C₃²⋊C₉

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

prime radical

C₃⁴×C₃₇

C₃²

hypercenter

C₃×C₃²⋊C₉

C₃₇⋊C₃

hypocenter

C₃₇

#10624592

LCS

G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃×C₃²⋊C₉

N	H ≅ G/N	type
4. #22582448	C₃	semidirect
97. C₃₇	#10624592	semidirect

H ≅ G/N

type

4. #22582448

C₃

semidirect

97. C₃₇

#10624592

semidirect

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

subgroup	index
C₃²×#11758608	3
#22582554	3
#22582448	3
C₃×C₃₇×C₃²⋊C₉	3
#10624592	37

subgroup

index

C₃×C₃₇×C₃²⋊C₉

Subgroups — #41005339

Conjugacy classes of subgroups

Complements

Maximal subgroups