Group #41007035 — subgroups

Subgroups — #41007035

Order 26973.

236 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 70 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×#15688517	C₃	1	1	yes	maximal, characteristic
2	#22582572	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582559	C₃	1	1	yes	maximal, characteristic
4	C₃×#15688506	C₃	1	1	yes	maximal, characteristic, complemented
5	3₋¹⁺²⋊#66524	C₃	1	2	yes	maximal, complemented
6	#22582478	C₃	1	2	yes	maximal, complemented
7	#22582475	C₃	1	2	yes	maximal
8	#22582469	C₃	1	2	yes	maximal, complemented
9	C₉×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
10	C₃×#11758608	C₃²	1	1	yes	characteristic
11	C₃×#11758606	—	3	1	no	—
12	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	1	yes	characteristic
13	C₃²×#66524	C₃²	1	1	yes	characteristic
14	C₃²×#66524	C₃²	1	2	yes	—
15	#15688517	—	3	3	no	—
16	#15688506	—	3	2	no	—
17	#15688506	—	3	1	no	—
18	#15688506	—	3	1	no	—
19	#15688504	—	3	2	no	—
20	C₃²⋊#66524	C₃²	1	2	yes	—
21	C₃²⋊#66524	C₃²	1	2	yes	—
22	#15688502	—	3	2	no	—
23	C₃×3₋¹⁺²×C₃₇	C₃²	1	1	yes	characteristic
24	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
25	C₃₇×C₉⋊C₉	—	3	2	no	—
26	C₃₇×C₃²⋊C₉	C₃²	1	2	yes	—
27	C₉²×C₃₇	—	3	1	no	—
28	C₃³×C₃₇	C₃³	1	1	yes	characteristic
29	C₃²×C₃₇⋊C₃	—	3	1	no	—
30	3₋¹⁺²×C₃₇	—	3	2	no	—
31	3₋¹⁺²×C₃₇	3₋¹⁺²	1	3	yes	complemented
32	C₃×C₉×C₃₇	—	3	1	no	—
33	C₃×C₉×C₃₇	—	3	3	no	—
34	C₃×C₉×C₃₇	—	3	2	no	—
35	C₃×C₉×C₃₇	H(3,𝔽₃)	1	2	yes	—
36	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
37	C₃²⋊(C₃₇⋊C₃)	—	3	1	no	—
38	C₃²⋊(C₃₇⋊C₃)	—	3	2	no	—
39	#11758608	—	3	1	no	—
40	#11758608	—	3	2	no	—
41	C₃×#66524	—	3	2	no	—
42	C₃×#66524	—	3	2	no	—
43	C₃×#66524	—	3	1	no	—
44	C₃×#66524	—	3	2	no	—
45	C₃×#66524	—	3	1	no	—
46	C₃×#66524	—	3	1	no	—
47	C₃×#66524	—	3	2	no	—
48	C₃×#66524	—	9	2	no	—
49	#11758606	—	9	3	no	—
50	#10624800	—	37	1	no	maximal, Sylow
51	C₃²×C₃₇	—	3	2	no	—
52	C₃²×C₃₇	—	3	1	no	—
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	#737	1	2	yes	—
56	C₃×C₃₇⋊C₃	—	9	2	no	—
57	C₃×C₃₇⋊C₃	—	9	1	no	—
58	C₃×C₃₇⋊C₃	—	9	1	no	—
59	C₉×C₃₇	—	3	3	no	—
60	C₉×C₃₇	—	3	2	no	—
61	C₉×C₃₇	—	9	2	no	—
62	C₉×C₃₇	#731	1	3	yes	complemented
63	#66524	—	9	2	no	—
64	#66524	—	9	2	no	—
65	#66524	—	9	1	no	—
66	#66524	—	9	1	no	—
67	#66524	—	9	2	no	—
68	C₃×#731	—	37	1	no	—
69	#6894	—	37	1	no	—
70	C₉⋊H(3,𝔽₃)	—	37	1	no	—
71	C₉×3₋¹⁺²	C₃₇⋊C₃	1	1	yes	characteristic, complemented
72	C₃×C₉⋊C₉	—	37	1	no	—
73	3₋¹⁺²⋊C₉	—	37	2	no	—
74	3₋¹⁺²⋊C₉	—	37	2	no	—
75	H(3,𝔽₃)⋊C₉	—	37	2	no	—
76	#6867	—	37	2	no	—
77	C₃×C₃₇	—	3	1	no	—
78	C₃×C₃₇	—	3	2	no	—
79	C₃×C₃₇	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
80	C₃×C₃₇	C₃×#731	1	1	yes	characteristic
81	C₃×C₃₇	#6913	1	2	yes	—
82	C₃₇⋊C₃	—	27	1	no	—
83	C₃×3₋¹⁺²	—	37	1	no	—
84	C₃×3₋¹⁺²	—	111	1	no	—
85	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
86	C₃×H(3,𝔽₃)	—	37	1	no	—
87	C₃²×C₉	—	37	2	no	—
88	C₃²×C₉	—	37	1	no	—
89	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
90	#731	—	111	3	no	—
91	C₉⋊C₉	—	3	2	no	—
92	C₉⋊C₉	—	111	2	no	—
93	C₉⋊C₉	—	111	2	no	—
94	C₉⋊C₉	—	111	1	no	—
95	C₉⋊C₉	—	111	1	no	—
96	C₃²⋊C₉	—	37	2	no	—
97	C₃²⋊C₉	—	37	2	no	—
98	C₃²⋊C₉	—	111	2	no	—
99	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	2	yes	—
100	C₉²	—	3	1	no	—
101	C₃₇	#10624800	1	1	yes	characteristic, complemented, Sylow
102	C₃³	—	111	1	no	—
103	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
104	3₋¹⁺²	—	3	2	no	—
105	3₋¹⁺²	—	333	3	no	—
106	3₋¹⁺²	—	111	1	no	—
107	3₋¹⁺²	—	111	2	no	—
108	3₋¹⁺²	#11758606	1	3	yes	complemented
109	H(3,𝔽₃)	—	111	1	no	—
110	H(3,𝔽₃)	—	111	2	no	—
111	C₃×C₉	—	3	1	no	—
112	C₃×C₉	—	3	3	no	—
113	C₃×C₉	—	3	2	no	—
114	C₃×C₉	—	333	2	no	—
115	C₃×C₉	—	111	2	no	—
116	C₃×C₉	—	111	2	no	—
117	C₃×C₉	—	111	1	no	—
118	C₃×C₉	—	111	2	no	—
119	C₃×C₉	—	111	1	no	—
120	C₃×C₉	—	111	1	no	—
121	C₃×C₉	—	111	2	no	—
122	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
123	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
124	C₃²	—	3	1	no	—
125	C₃²	—	3	2	no	—
126	C₃²	—	333	1	no	—
127	C₃²	—	333	1	no	—
128	C₃²	—	333	2	no	—
129	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
130	C₃²	C₃×#11758606	1	1	yes	characteristic
131	C₃²	#15688531	1	2	yes	—
132	C₉	—	3	3	no	—
133	C₉	—	3	2	no	—
134	C₉	—	9	2	no	—
135	C₉	—	333	1	no	—
136	C₉	—	333	1	no	—
137	C₉	—	333	2	no	—
138	C₉	—	333	2	no	—
139	C₉	—	333	2	no	—
140	C₉	#15688517	1	3	yes	complemented
141	C₃	—	3	1	no	—
142	C₃	—	3	2	no	—
143	C₃	—	999	1	no	—
144	C₃	#22582559	1	1	yes	characteristic
145	C₃	C₃×#15688517	1	1	yes	characteristic
146	C₃	#22582656	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688517

C₃

yes

maximal, characteristic

#22582572

C₃

yes

maximal, characteristic, complemented

#22582559

C₃

yes

maximal, characteristic

C₃×#15688506

C₃

yes

maximal, characteristic, complemented

3₋¹⁺²⋊#66524

C₃

yes

maximal, complemented

#22582478

C₃

yes

maximal, complemented

#22582475

C₃

yes

maximal

#22582469

C₃

yes

maximal, complemented

C₉×3₋¹⁺²×C₃₇

C₃

yes

maximal, characteristic, complemented

C₃×#11758608

C₃²

yes

characteristic

C₃×#11758606

—

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

yes

characteristic

C₃²×#66524

C₃²

yes

characteristic

C₃²×#66524

C₃²

yes

—

—

—

—

—

—

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

—

#15688502

—

C₃×3₋¹⁺²×C₃₇

C₃²

yes

characteristic

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×C₉⋊C₉

—

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₉²×C₃₇

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

3₋¹⁺²

yes

complemented

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

H(3,𝔽₃)

yes

—

C₃×C₉×C₃₇

H(3,𝔽₃)

yes

characteristic

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

—

—

—

—

—

—

—

—

—

—

—

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₃₇

#737

yes

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

yes

—

—

—

—

—

—

—

C₉⋊H(3,𝔽₃)

—

C₉×3₋¹⁺²

C₃₇⋊C₃

yes

characteristic, complemented

C₃×C₉⋊C₉

—

3₋¹⁺²⋊C₉

—

3₋¹⁺²⋊C₉

—

H(3,𝔽₃)⋊C₉

—

#6867

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₉⋊H(3,𝔽₃)

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

—

C₃₇⋊C₃

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

characteristic

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

#731

—

111

—

C₉⋊C₉

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

100

C₉²

—

101

C₃₇

#10624800

yes

characteristic, complemented, Sylow

102

C₃³

—

111

—

103

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

104

3₋¹⁺²

—

105

3₋¹⁺²

—

333

—

106

3₋¹⁺²

—

111

—

107

3₋¹⁺²

—

111

—

108

3₋¹⁺²

#11758606

yes

complemented

109

H(3,𝔽₃)

—

111

—

110

H(3,𝔽₃)

—

111

—

111

C₃×C₉

—

112

C₃×C₉

—

113

C₃×C₉

—

114

C₃×C₉

—

333

—

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

—

111

—

122

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

—

123

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

124

C₃²

—

125

C₃²

—

126

C₃²

—

333

—

127

C₃²

—

333

—

128

C₃²

—

333

—

129

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

130

C₃²

C₃×#11758606

yes

characteristic

131

C₃²

#15688531

yes

—

132

C₉

—

133

C₉

—

134

C₉

—

135

C₉

—

333

—

136

C₉

—

333

—

137

C₉

—

333

—

138

C₉

—

333

—

139

C₉

—

333

—

140

C₉

#15688517

yes

complemented

141

C₃

—

142

C₃

—

143

C₃

—

999

—

144

C₃

#22582559

yes

characteristic

145

C₃

C₃×#15688517

yes

characteristic

146

C₃

#22582656

yes

—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₉×3₋¹⁺²×C₃₇	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²
hypercenter	C₉×3₋¹⁺²	C₃₇⋊C₃
hypocenter	C₃₇	#10624800
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃²×C₉ ◁ C₉×3₋¹⁺²

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₉×3₋¹⁺²×C₃₇

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

prime radical

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

hypercenter

C₉×3₋¹⁺²

C₃₇⋊C₃

hypocenter

C₃₇

#10624800

LCS

G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃²×C₉ ◁ C₉×3₋¹⁺²

N	H ≅ G/N	type
2. #22582572	C₃	semidirect
4. C₃×#15688506	C₃	semidirect
5. 3₋¹⁺²⋊#66524	C₃	semidirect
6. #22582478	C₃	semidirect
8. #22582469	C₃	semidirect
9. C₉×3₋¹⁺²×C₃₇	C₃	semidirect
31. 3₋¹⁺²×C₃₇	3₋¹⁺²	semidirect
62. C₉×C₃₇	#731	semidirect
71. C₉×3₋¹⁺²	C₃₇⋊C₃	semidirect
101. C₃₇	#10624800	semidirect
108. 3₋¹⁺²	#11758606	semidirect
140. C₉	#15688517	semidirect

H ≅ G/N

type

2. #22582572

C₃

semidirect

4. C₃×#15688506

C₃

semidirect

5. 3₋¹⁺²⋊#66524

C₃

semidirect

6. #22582478

C₃

semidirect

8. #22582469

C₃

semidirect

9. C₉×3₋¹⁺²×C₃₇

C₃

semidirect

31. 3₋¹⁺²×C₃₇

3₋¹⁺²

semidirect

62. C₉×C₃₇

#731

semidirect

71. C₉×3₋¹⁺²

C₃₇⋊C₃

semidirect

101. C₃₇

#10624800

semidirect

108. 3₋¹⁺²

#11758606

semidirect

140. C₉

#15688517

semidirect

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

subgroup	index
C₃×#15688517	3
#22582572	3
#22582559	3
C₃×#15688506	3
3₋¹⁺²⋊#66524	3
#22582478	3
#22582475	3
#22582469	3
C₉×3₋¹⁺²×C₃₇	3
#10624800	37

subgroup

index

C₉×3₋¹⁺²×C₃₇

Subgroups — #41007035

Conjugacy classes of subgroups

Complements

Maximal subgroups