Group #41005162 — subgroups

Subgroups — #41005162

Order 26973.

506 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 157 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
2	C₃×#15688505	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×#15688502	C₃	1	3	yes	maximal
4	C₃×C₉×#66524	C₃	1	1	yes	maximal, characteristic, complemented
5	#22582441	C₃	1	6	yes	maximal, complemented
6	C₃×C₃₇×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
8	C₃×#11758606	—	3	3	no	—
9	C₃×#11758606	—	3	3	no	—
10	C₃×#11758606	—	3	3	no	—
11	C₃×#11758606	C₉	1	9	yes	complemented
12	C₃²×#66524	C₃²	1	3	yes	—
13	C₃²×#66524	C₃²	1	2	yes	—
14	C₃²×#66524	C₃²	1	3	yes	—
15	#15688505	—	3	3	no	—
16	#15688502	—	3	9	no	—
17	C₉×#66524	—	3	3	no	—
18	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
19	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
20	C₃₇×C₉⋊C₉	—	3	3	no	—
21	C₃³×C₃₇	C₃³	1	1	yes	characteristic
22	C₃²×C₃₇⋊C₃	—	3	1	no	—
23	C₃²×C₃₇⋊C₃	—	3	3	no	—
24	C₃²×C₃₇⋊C₃	—	3	1	no	—
25	C₃²×C₃₇⋊C₃	—	3	1	no	—
26	C₃²×C₃₇⋊C₃	—	3	6	no	—
27	C₃²×C₃₇⋊C₃	C₃×C₉	1	3	yes	—
28	C₃×C₉×C₃₇	—	3	3	no	—
29	C₃×C₉×C₃₇	—	3	3	no	—
30	C₃×C₉×C₃₇	—	3	6	no	—
31	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	—
32	C₃×C₉×C₃₇	H(3,𝔽₃)	1	3	yes	—
33	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	—
34	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	—
35	C₃×#66524	—	3	3	no	—
36	C₃×#66524	—	3	2	no	—
37	C₃×#66524	—	3	2	no	—
38	C₃×#66524	—	3	2	no	—
39	C₃×#66524	—	3	6	no	—
40	C₃×#66524	—	3	3	no	—
41	C₃×#66524	—	3	3	no	—
42	C₃×#66524	—	3	6	no	—
43	C₃×#66524	C₃×C₉	1	6	yes	—
44	C₃×#66524	3₋¹⁺²	1	9	yes	complemented
45	#11758606	—	3	27	no	—
46	#10624560	—	37	1	no	maximal, Sylow
47	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
48	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
49	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
50	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
51	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
52	C₃²×C₃₇	#737	1	6	yes	—
53	C₃×C₃₇⋊C₃	—	3	3	no	—
54	C₃×C₃₇⋊C₃	—	3	9	no	—
55	C₃×C₃₇⋊C₃	—	9	3	no	—
56	C₃×C₃₇⋊C₃	—	9	3	no	—
57	C₃×C₃₇⋊C₃	—	9	2	no	—
58	C₃×C₃₇⋊C₃	—	9	1	no	—
59	C₉×C₃₇	—	9	3	no	—
60	C₉×C₃₇	C₃²⋊C₉	1	9	yes	complemented
61	#66524	—	3	6	no	—
62	#66524	—	3	9	no	—
63	#66524	—	9	3	no	—
64	C₃²×3₋¹⁺²	—	37	1	no	—
65	C₃×C₉⋊C₉	—	37	1	no	—
66	C₃×C₉⋊C₉	C₃₇⋊C₃	1	1	yes	characteristic, complemented
67	C₃×C₃²⋊C₉	—	37	3	no	—
68	C₃×C₉²	—	37	1	no	—
69	#6855	—	37	6	no	—
70	C₃×C₃₇	C₃×C₃²⋊C₉	1	1	yes	characteristic
71	C₃×C₃₇	C₉×3₋¹⁺²	1	3	yes	—
72	C₃×C₃₇	C₉⋊H(3,𝔽₃)	1	3	yes	—
73	C₃×C₃₇	#6896	1	3	yes	—
74	C₃×C₃₇	#6899	1	2	yes	—
75	C₃×C₃₇	#6900	1	1	yes	characteristic
76	C₃₇⋊C₃	—	9	3	no	—
77	C₃⁴	—	37	1	no	—
78	C₃×3₋¹⁺²	—	37	9	no	—
79	C₃×3₋¹⁺²	—	111	3	no	—
80	C₃×3₋¹⁺²	—	111	3	no	—
81	C₃×3₋¹⁺²	—	111	3	no	—
82	C₃²×C₉	—	37	2	no	—
83	C₃²×C₉	—	37	3	no	—
84	C₃²×C₉	—	37	3	no	—
85	C₃²×C₉	C₃×C₃₇⋊C₃	1	3	yes	—
86	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
87	C₉⋊C₉	—	3	3	no	—
88	C₉⋊C₉	—	111	3	no	—
89	C₃²⋊C₉	—	111	9	no	—
90	C₉²	—	111	3	no	—
91	C₃₇	#10624560	1	1	yes	characteristic, complemented, Sylow
92	C₃³	—	37	3	no	—
93	C₃³	—	111	1	no	—
94	C₃³	—	111	3	no	—
95	C₃³	—	111	1	no	—
96	C₃³	—	111	1	no	—
97	C₃³	—	111	6	no	—
98	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
99	3₋¹⁺²	—	111	27	no	—
100	C₃×C₉	—	3	3	no	—
101	C₃×C₉	—	3	3	no	—
102	C₃×C₉	—	3	6	no	—
103	C₃×C₉	—	37	6	no	—
104	C₃×C₉	—	37	9	no	—
105	C₃×C₉	—	111	3	no	—
106	C₃×C₉	—	111	3	no	—
107	C₃×C₉	—	111	6	no	—
108	C₃×C₉	—	111	3	no	—
109	C₃×C₉	—	111	2	no	—
110	C₃×C₉	—	111	2	no	—
111	C₃×C₉	—	111	2	no	—
112	C₃×C₉	—	111	6	no	—
113	C₃×C₉	C₉×C₃₇⋊C₃	1	3	yes	—
114	C₃×C₉	#11758605	1	3	yes	—
115	C₃×C₉	#11758606	1	3	yes	—
116	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
117	C₃²	—	333	3	no	—
118	C₃²	—	333	3	no	—
119	C₃²	—	333	2	no	—
120	C₃²	—	333	1	no	—
121	C₃²	—	111	3	no	—
122	C₃²	—	111	9	no	—
123	C₃²	C₃×C₉×C₃₇⋊C₃	1	1	yes	characteristic
124	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
125	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	3	yes	—
126	C₃²	C₃×#11758606	1	1	yes	characteristic
127	C₃²	C₃×#11758605	1	1	yes	characteristic
128	C₃²	#15688532	1	6	yes	—
129	C₉	—	9	3	no	—
130	C₉	—	333	3	no	—
131	C₉	—	111	6	no	—
132	C₉	—	111	9	no	—
133	C₉	#15688502	1	9	yes	complemented
134	C₃	—	333	3	no	—
135	C₃	C₃×#15688502	1	1	yes	characteristic
136	C₃	#22582542	1	3	yes	—
137	C₃	#22582555	1	3	yes	—
138	C₃	#22582583	1	3	yes	—
139	C₃	#22582600	1	2	yes	—
140	C₃	#22582604	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#11758606

C₃

yes

maximal, characteristic

C₃×#15688505

C₃

yes

maximal, characteristic, complemented

C₃×#15688502

C₃

yes

maximal

C₃×C₉×#66524

C₃

yes

maximal, characteristic, complemented

#22582441

C₃

yes

maximal, complemented

C₃×C₃₇×C₉⋊C₉

C₃

yes

maximal, characteristic, complemented

C₃³×C₃₇⋊C₃

C₃²

yes

—

—

—

C₉

yes

complemented

C₃²×#66524

C₃²

yes

—

C₃²×#66524

C₃²

yes

—

C₃²×#66524

C₃²

yes

—

#15688505

—

#15688502

—

C₉×#66524

—

C₃²×C₉×C₃₇

C₃²

yes

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×C₉⋊C₉

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

C₃×C₉

yes

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

C₃×C₉

yes

—

C₃×C₉×C₃₇

H(3,𝔽₃)

yes

—

C₃×C₉×C₃₇

3₋¹⁺²

yes

—

C₃×C₉×C₃₇

3₋¹⁺²

yes

—

—

—

—

—

—

—

—

—

C₃×C₉

yes

—

C₃×#66524

3₋¹⁺²

yes

—

—

C₃²×C₃₇

C₃²×C₉

yes

characteristic

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

—

C₃²×C₃₇

C₃×3₋¹⁺²

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

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

C₉×C₃₇

C₃²⋊C₉

yes

—

—

—

C₃²×3₋¹⁺²

—

C₃×C₉⋊C₉

—

C₃×C₉⋊C₉

C₃₇⋊C₃

yes

characteristic, complemented

C₃×C₃²⋊C₉

—

C₃×C₉²

—

#6855

—

C₃×C₃₇

C₃×C₃²⋊C₉

yes

characteristic

C₃×C₃₇

C₉×3₋¹⁺²

yes

—

C₃×C₃₇

C₉⋊H(3,𝔽₃)

yes

—

C₃×C₃₇

#6896

yes

—

C₃×C₃₇

#6899

yes

—

C₃×C₃₇

#6900

yes

characteristic

C₃₇⋊C₃

—

C₃⁴

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₉²

—

111

—

C₃₇

#10624560

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

111

—

100

C₃×C₉

—

101

C₃×C₉

—

102

C₃×C₉

—

103

C₃×C₉

—

104

C₃×C₉

—

105

C₃×C₉

—

111

—

106

C₃×C₉

—

111

—

107

C₃×C₉

—

111

—

108

C₃×C₉

—

111

—

109

C₃×C₉

—

111

—

110

C₃×C₉

—

111

—

111

C₃×C₉

—

111

—

112

C₃×C₉

—

111

—

113

C₃×C₉

C₉×C₃₇⋊C₃

yes

—

114

C₃×C₉

#11758605

yes

—

115

C₃×C₉

#11758606

yes

—

116

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

—

117

C₃²

—

333

—

118

C₃²

—

333

—

119

C₃²

—

333

—

120

C₃²

—

333

—

121

C₃²

—

111

—

122

C₃²

—

111

—

123

C₃²

C₃×C₉×C₃₇⋊C₃

yes

characteristic

124

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

125

C₃²

3₋¹⁺²×C₃₇⋊C₃

yes

—

126

C₃²

C₃×#11758606

yes

characteristic

127

C₃²

C₃×#11758605

yes

characteristic

128

C₃²

#15688532

yes

—

129

C₉

—

130

C₉

—

333

—

131

C₉

—

111

—

132

C₉

—

111

—

133

C₉

#15688502

yes

complemented

134

C₃

—

333

—

135

C₃

C₃×#15688502

yes

characteristic

136

C₃

#22582542

yes

—

137

C₃

#22582555

yes

—

138

C₃

#22582583

yes

—

139

C₃

#22582600

yes

—

140

C₃

#22582604

yes

characteristic

	subgroup	G/H
center, Frattini	C₃³	C₃²×C₃₇⋊C₃
commutator	C₃²×C₃₇	C₃²×C₉
Fitting	C₃×C₃₇×C₉⋊C₉	C₃
socle, cosocle	C₃³×C₃₇	C₃³
prime radical	C₃³×C₃₇⋊C₃	C₃²
hypercenter	C₃×C₉⋊C₉	C₃₇⋊C₃
hypocenter	C₃₇	#10624560
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃³ ◁ C₃×C₉⋊C₉

subgroup

G/H

center, Frattini

C₃³

C₃²×C₃₇⋊C₃

commutator

C₃²×C₃₇

C₃²×C₉

Fitting

C₃×C₃₇×C₉⋊C₉

C₃

socle, cosocle

C₃³×C₃₇

C₃³

prime radical

C₃³×C₃₇⋊C₃

C₃²

hypercenter

C₃×C₉⋊C₉

C₃₇⋊C₃

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃₇

C₁ ◁ C₃³ ◁ C₃×C₉⋊C₉

N	H ≅ G/N	type
2. C₃×#15688505	C₃	semidirect
4. C₃×C₉×#66524	C₃	semidirect
5. #22582441	C₃	semidirect
6. C₃×C₃₇×C₉⋊C₉	C₃	semidirect
11. C₃×#11758606	C₉	semidirect
44. C₃×#66524	3₋¹⁺²	semidirect
60. C₉×C₃₇	C₃²⋊C₉	semidirect
66. C₃×C₉⋊C₉	C₃₇⋊C₃	semidirect
91. C₃₇	#10624560	semidirect
133. C₉	#15688502	semidirect

H ≅ G/N

type

2. C₃×#15688505

C₃

semidirect

4. C₃×C₉×#66524

C₃

semidirect

5. #22582441

C₃

semidirect

6. C₃×C₃₇×C₉⋊C₉

C₃

semidirect

11. C₃×#11758606

C₉

semidirect

44. C₃×#66524

3₋¹⁺²

semidirect

60. C₉×C₃₇

C₃²⋊C₉

semidirect

66. C₃×C₉⋊C₉

C₃₇⋊C₃

semidirect

91. C₃₇

#10624560

semidirect

133. C₉

#15688502

semidirect

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

subgroup	index
C₃²×#11758606	3
C₃×#15688505	3
C₃×#15688502	3
C₃×C₉×#66524	3
#22582441	3
C₃×C₃₇×C₉⋊C₉	3
#10624560	37

subgroup

index

C₃×C₉×#66524

C₃×C₃₇×C₉⋊C₉

Subgroups — #41005162

Conjugacy classes of subgroups

Complements

Maximal subgroups