Group #41004754 — subgroups

Subgroups — #41004754

Order 26973.

254 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 73 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×C₃²⋊#66524	C₃	1	2	yes	maximal
2	C₃×C₃²⋊#66524	C₃	1	1	yes	maximal, characteristic
3	C₃×C₃₇×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
4	C₃²×#66524	—	3	1	no	—
5	C₃²×#66524	—	3	2	no	—
6	C₃²⋊#66524	—	3	1	no	—
7	C₃²⋊#66524	—	3	1	no	—
8	C₃²⋊#66524	—	3	6	no	—
9	C₃²⋊#66524	C₉	1	3	yes	complemented
10	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
11	C₃²×C₉×C₃₇	—	3	1	no	—
12	C₃₇×C₃²⋊C₉	—	3	3	no	—
13	C₃³×C₃₇	—	3	1	no	—
14	C₃³×C₃₇	—	3	2	no	—
15	C₃³×C₃₇	—	3	1	no	—
16	C₃³×C₃₇	—	3	1	no	—
17	C₃³×C₃₇	—	3	2	no	—
18	C₃³×C₃₇	—	3	2	no	—
19	C₃³×C₃₇	—	3	1	no	—
20	C₃³×C₃₇	—	3	2	no	—
21	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
22	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
23	C₃³×C₃₇	3₋¹⁺²	1	1	yes	characteristic
24	C₃³×C₃₇	3₋¹⁺²	1	1	yes	characteristic
25	C₃×C₉×C₃₇	—	3	3	no	—
26	C₃×C₉×C₃₇	—	9	2	no	—
27	C₃×C₉×C₃₇	—	9	1	no	—
28	C₃×#66524	—	3	3	no	—
29	C₃×#66524	—	3	6	no	—
30	C₃×#66524	—	9	1	no	—
31	C₃×#66524	—	9	1	no	—
32	C₃×#66524	—	9	2	no	—
33	C₃×#66524	—	9	2	no	—
34	C₃×#66524	—	9	2	no	—
35	C₃×#66524	—	9	1	no	—
36	#10624450	—	37	1	no	maximal, Sylow
37	C₃²×C₃₇	—	3	1	no	—
38	C₃²×C₃₇	—	3	1	no	—
39	C₃²×C₃₇	—	3	1	no	—
40	C₃²×C₃₇	—	3	1	no	—
41	C₃²×C₃₇	—	3	1	no	—
42	C₃²×C₃₇	—	3	1	no	—
43	C₃²×C₃₇	—	3	2	no	—
44	C₃²×C₃₇	—	3	2	no	—
45	C₃²×C₃₇	—	3	2	no	—
46	C₃²×C₃₇	—	9	1	no	—
47	C₃²×C₃₇	—	9	2	no	—
48	C₃²×C₃₇	—	9	2	no	—
49	C₃²×C₃₇	—	9	2	no	—
50	C₃²×C₃₇	—	9	1	no	—
51	C₃²×C₃₇	—	9	1	no	—
52	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
53	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	2	yes	—
54	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
55	C₃²×C₃₇	#731	1	1	yes	characteristic
56	C₃²×C₃₇	#731	1	1	yes	characteristic
57	C₃²×C₃₇	#731	1	2	yes	—
58	C₃²×C₃₇	#731	1	2	yes	—
59	C₃²×C₃₇	#732	1	1	yes	characteristic
60	C₃²×C₃₇	#733	1	2	yes	—
61	C₉×C₃₇	—	9	3	no	—
62	#66524	—	9	6	no	—
63	#66524	—	9	3	no	—
64	C₃×C₃²⋊C₉	—	37	1	no	—
65	C₃×C₃²⋊C₉	—	37	2	no	—
66	C₃×C₃²⋊C₉	C₃₇⋊C₃	1	1	yes	characteristic
67	C₃×C₃₇	—	9	1	no	—
68	C₃×C₃₇	—	9	1	no	—
69	C₃×C₃₇	—	9	1	no	—
70	C₃×C₃₇	#6857	1	1	yes	characteristic
71	C₃×C₃₇	#6858	1	2	yes	—
72	C₃×C₃₇	#6858	1	2	yes	—
73	C₃×C₃₇	#6859	1	2	yes	—
74	C₃×C₃₇	#6861	1	1	yes	characteristic
75	C₃×C₃₇	#6862	1	1	yes	characteristic
76	C₃×C₃₇	#6867	1	1	yes	characteristic
77	C₃×C₃₇	H(3,𝔽₃)⋊C₉	1	1	yes	characteristic
78	C₃×C₃₇	3₋¹⁺²⋊C₉	1	2	yes	—
79	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
80	C₃²×C₉	—	3	1	no	—
81	C₃²×C₉	—	111	1	no	—
82	C₃²×C₉	—	111	2	no	—
83	C₃²⋊C₉	—	3	3	no	—
84	C₃²⋊C₉	—	37	3	no	—
85	C₃²⋊C₉	—	111	1	no	—
86	C₃²⋊C₉	—	111	1	no	—
87	C₃²⋊C₉	—	111	6	no	—
88	C₃₇	#10624450	1	1	yes	characteristic, complemented, Sylow
89	C₃³	—	3	2	no	—
90	C₃³	—	3	1	no	—
91	C₃³	—	3	1	no	—
92	C₃³	—	3	1	no	—
93	C₃³	—	3	2	no	—
94	C₃³	—	3	2	no	—
95	C₃³	—	3	1	no	—
96	C₃³	—	3	2	no	—
97	C₃³	C₉×C₃₇⋊C₃	1	1	yes	characteristic
98	C₃³	#11758605	1	1	yes	characteristic
99	C₃³	#11758606	1	1	yes	characteristic
100	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
101	C₃×C₉	—	3	3	no	—
102	C₃×C₉	—	9	2	no	—
103	C₃×C₉	—	9	1	no	—
104	C₃×C₉	—	333	1	no	—
105	C₃×C₉	—	333	1	no	—
106	C₃×C₉	—	333	2	no	—
107	C₃×C₉	—	333	2	no	—
108	C₃×C₉	—	333	2	no	—
109	C₃×C₉	—	333	1	no	—
110	C₃×C₉	—	111	3	no	—
111	C₃×C₉	—	111	6	no	—
112	C₃²	—	3	1	no	—
113	C₃²	—	3	1	no	—
114	C₃²	—	3	1	no	—
115	C₃²	—	3	1	no	—
116	C₃²	—	3	1	no	—
117	C₃²	—	3	1	no	—
118	C₃²	—	3	2	no	—
119	C₃²	—	3	2	no	—
120	C₃²	—	3	2	no	—
121	C₃²	—	9	1	no	—
122	C₃²	—	9	2	no	—
123	C₃²	—	9	2	no	—
124	C₃²	—	9	2	no	—
125	C₃²	—	9	1	no	—
126	C₃²	—	9	1	no	—
127	C₃²	#15688502	1	1	yes	characteristic
128	C₃²	#15688512	1	2	yes	—
129	C₃²	#15688513	1	1	yes	characteristic
130	C₃²	#15688515	1	1	yes	characteristic
131	C₃²	#15688516	1	1	yes	characteristic
132	C₃²	#15688517	1	2	yes	—
133	C₃²	#15688517	1	2	yes	—
134	C₃²	#15688519	1	1	yes	characteristic
135	C₃²	#15688520	1	2	yes	—
136	C₉	—	9	3	no	—
137	C₉	—	333	6	no	—
138	C₉	—	333	3	no	—
139	C₃	—	9	1	no	—
140	C₃	—	9	1	no	—
141	C₃	—	9	1	no	—
142	C₃	#22582445	1	1	yes	characteristic
143	C₃	#22582447	1	2	yes	—
144	C₃	#22582448	1	2	yes	—
145	C₃	#22582452	1	2	yes	—
146	C₃	#22582456	1	1	yes	characteristic
147	C₃	C₃₇⋊#6862	1	1	yes	characteristic
148	C₃	#22582469	1	1	yes	characteristic
149	C₃	#22582477	1	1	yes	characteristic
150	C₃	#22582478	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₃²⋊#66524

C₃

yes

maximal

C₃×C₃²⋊#66524

C₃

yes

maximal, characteristic

C₃×C₃₇×C₃²⋊C₉

C₃

yes

maximal, characteristic

C₃²×#66524

—

C₃²×#66524

—

—

—

—

C₉

yes

complemented

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃²×C₉×C₃₇

—

C₃₇×C₃²⋊C₉

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

C₃×C₉

yes

characteristic

C₃³×C₃₇

H(3,𝔽₃)

yes

characteristic

C₃³×C₃₇

3₋¹⁺²

yes

characteristic

C₃³×C₃₇

3₋¹⁺²

yes

characteristic

C₃×C₉×C₃₇

—

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

C₃²⋊C₉

yes

characteristic

C₃²×C₃₇

AΣL(1,𝔽₂₇)

yes

—

C₃²×C₃₇

AΣL(1,𝔽₂₇)

yes

C₃²×C₃₇

yes

C₃²×C₃₇

yes

C₃²×C₃₇

yes

—

C₃²×C₃₇

#731

yes

—

C₃²×C₃₇

#732

yes

characteristic

C₃²×C₃₇

#733

yes

—

C₉×C₃₇

—

#66524

—

#66524

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

C₃₇⋊C₃

yes

characteristic

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

#6857

yes

characteristic

C₃×C₃₇

#6858

yes

—

C₃×C₃₇

#6858

yes

—

C₃×C₃₇

#6859

yes

—

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃×C₃₇

H(3,𝔽₃)⋊C₉

yes

characteristic

C₃×C₃₇

3₋¹⁺²⋊C₉

yes

—

C₃⁴

C₃×C₃₇⋊C₃

yes

characteristic

C₃²×C₉

—

C₃²×C₉

—

111

—

C₃²×C₉

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃₇

#10624450

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₉×C₃₇⋊C₃

yes

C₃³

yes

C₃³

yes

100

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

101

C₃×C₉

—

102

C₃×C₉

—

103

C₃×C₉

—

104

C₃×C₉

—

333

—

105

C₃×C₉

—

333

—

106

C₃×C₉

—

333

—

107

C₃×C₉

—

333

—

108

C₃×C₉

—

333

—

109

C₃×C₉

—

333

—

110

C₃×C₉

—

111

—

111

C₃×C₉

—

111

—

112

C₃²

—

113

C₃²

—

114

C₃²

—

115

C₃²

—

116

C₃²

—

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

—

122

C₃²

—

123

C₃²

—

124

C₃²

—

125

C₃²

—

126

C₃²

—

127

C₃²

#15688502

yes

characteristic

128

C₃²

#15688512

yes

—

129

C₃²

#15688513

yes

characteristic

130

C₃²

#15688515

yes

characteristic

131

C₃²

#15688516

yes

characteristic

132

C₃²

#15688517

yes

—

133

C₃²

#15688517

yes

—

134

C₃²

#15688519

yes

characteristic

135

C₃²

#15688520

yes

—

136

C₉

—

137

C₉

—

333

—

138

C₉

—

333

—

139

C₃

—

140

C₃

—

141

C₃

—

142

C₃

#22582445

yes

characteristic

143

C₃

#22582447

yes

—

144

C₃

#22582448

yes

—

145

C₃

#22582452

yes

—

146

C₃

#22582456

yes

characteristic

147

C₃

C₃₇⋊#6862

yes

characteristic

148

C₃

#22582469

yes

characteristic

149

C₃

#22582477

yes

characteristic

150

C₃

#22582478

yes

—

	subgroup	G/H
center	C₃³	C₃²⋊(C₃₇⋊C₃)
commutator	C₃³×C₃₇	C₃×C₉
Frattini	C₃⁴	C₃×C₃₇⋊C₃
Fitting	C₃×C₃₇×C₃²⋊C₉	C₃
socle	C₃³×C₃₇	H(3,𝔽₃)
prime radical, cosocle	C₃⁴×C₃₇	C₃²
hypercenter	C₃×C₃²⋊C₉	C₃₇⋊C₃
hypocenter	C₃₇	#10624450
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃³ ◁ C₃⁴ ◁ C₃×C₃²⋊C₉

subgroup

G/H

center

C₃³

C₃²⋊(C₃₇⋊C₃)

commutator

C₃³×C₃₇

C₃×C₉

Frattini

C₃⁴

C₃×C₃₇⋊C₃

Fitting

C₃×C₃₇×C₃²⋊C₉

C₃

socle

C₃³×C₃₇

H(3,𝔽₃)

prime radical, cosocle

C₃⁴×C₃₇

C₃²

hypercenter

C₃×C₃²⋊C₉

C₃₇⋊C₃

hypocenter

C₃₇

#10624450

LCS

G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃³ ◁ C₃⁴ ◁ C₃×C₃²⋊C₉

N	H ≅ G/N	type
9. C₃²⋊#66524	C₉	semidirect
88. C₃₇	#10624450	semidirect

H ≅ G/N

type

9. C₃²⋊#66524

C₉

semidirect

88. C₃₇

#10624450

semidirect

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

subgroup	index
C₃×C₃²⋊#66524	3
C₃×C₃₇×C₃²⋊C₉	3
#10624450	37

subgroup

index

C₃×C₃²⋊#66524

C₃×C₃₇×C₃²⋊C₉

#10624450

Subgroups — #41004754

Conjugacy classes of subgroups

Complements

Maximal subgroups