Group #41006785 — subgroups

Subgroups — #41006785

Order 26973.

316 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 76 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	#22582578	C₃	1	2	yes	maximal, complemented
2	C₃×#15688504	C₃	1	1	yes	maximal, characteristic, complemented
3	3₋¹⁺²⋊#66524	C₃	1	2	yes	maximal, complemented
4	3₋¹⁺²⋊#66524	C₃	1	2	yes	maximal, complemented
5	3₋¹⁺²⋊#66524	C₃	1	2	yes	maximal, complemented
6	H(3,𝔽₃)⋊#66524	C₃	1	1	yes	maximal, characteristic
7	H(3,𝔽₃)⋊#66524	C₃	1	1	yes	maximal, characteristic
8	H(3,𝔽₃)⋊#66524	C₃	1	1	yes	maximal, characteristic
9	C₃×C₃₇×#737	C₃	1	1	yes	maximal, characteristic
10	C₃²×#66524	C₃²	1	1	yes	characteristic
11	C₃²×#66524	C₃²	1	1	yes	characteristic
12	C₃²×#66524	C₃²	1	1	yes	characteristic
13	#15688504	—	3	1	no	—
14	#15688504	—	3	2	no	—
15	#15688504	—	3	2	no	—
16	#15688504	—	3	4	no	—
17	C₃²⋊#66524	C₃²	1	2	yes	—
18	C₃²⋊#66524	C₃²	1	2	yes	—
19	C₃²⋊#66524	C₃²	1	2	yes	—
20	C₃₇×#737	—	3	6	no	—
21	C₃₇×#737	—	3	3	no	—
22	C₃×3₋¹⁺²×C₃₇	—	3	2	no	—
23	C₃×3₋¹⁺²×C₃₇	C₃²	1	2	yes	—
24	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
25	C₃²×C₉×C₃₇	—	3	1	no	—
26	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
27	C₃³×C₃₇	—	3	1	no	—
28	C₃³×C₃₇	C₃³	1	1	yes	characteristic
29	3₋¹⁺²×C₃₇	—	3	18	no	—
30	3₋¹⁺²×C₃₇	—	3	2	no	—
31	3₋¹⁺²×C₃₇	—	3	2	no	—
32	3₋¹⁺²×C₃₇	3₋¹⁺²	1	6	yes	—
33	H(3,𝔽₃)×C₃₇	—	3	2	no	—
34	H(3,𝔽₃)×C₃₇	3₋¹⁺²	1	3	yes	—
35	C₃×C₉×C₃₇	—	3	9	no	—
36	C₃×C₉×C₃₇	—	3	2	no	—
37	C₃×C₉×C₃₇	—	3	2	no	—
38	C₃×C₉×C₃₇	—	3	1	no	—
39	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
40	C₃×C₉×C₃₇	H(3,𝔽₃)	1	2	yes	—
41	C₃×#66524	—	3	1	no	—
42	C₃×#66524	—	3	3	no	—
43	C₃×#66524	—	3	2	no	—
44	C₃×#66524	—	3	2	no	—
45	C₃×#66524	—	3	2	no	—
46	C₃×#66524	—	3	1	no	—
47	C₃×#66524	—	3	1	no	—
48	C₃×#66524	—	9	1	no	—
49	C₃×#66524	—	9	1	no	—
50	#10624780	—	37	1	no	maximal, Sylow
51	C₃²×C₃₇	—	3	1	no	—
52	C₃²×C₃₇	—	3	1	no	—
53	C₃²×C₃₇	—	3	1	no	—
54	C₃²×C₃₇	—	3	3	no	—
55	C₃²×C₃₇	—	9	1	no	—
56	C₃²×C₃₇	—	9	1	no	—
57	C₃²×C₃₇	—	9	1	no	—
58	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
59	C₃²×C₃₇	C₃×3₋¹⁺²	1	2	yes	—
60	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
61	C₉×C₃₇	—	3	1	no	—
62	C₉×C₃₇	—	3	6	no	—
63	C₉×C₃₇	—	3	2	no	—
64	#66524	—	3	3	no	—
65	#66524	—	9	2	no	—
66	#66524	—	9	2	no	—
67	#66524	—	9	1	no	—
68	#66524	—	9	1	no	—
69	#66524	—	9	2	no	—
70	C₃×#737	C₃₇⋊C₃	1	1	yes	characteristic
71	#6895	—	37	2	no	—
72	C₃×C₉⋊C₉	—	37	1	no	—
73	3₋¹⁺²⋊C₉	—	37	2	no	—
74	3₋¹⁺²⋊C₉	—	37	2	no	—
75	3₋¹⁺²⋊C₉	—	37	2	no	—
76	H(3,𝔽₃)⋊C₉	—	37	1	no	—
77	H(3,𝔽₃)⋊C₉	—	37	1	no	—
78	H(3,𝔽₃)⋊C₉	—	37	1	no	—
79	C₃×C₃₇	—	3	2	no	—
80	C₃×C₃₇	—	3	1	no	—
81	C₃×C₃₇	—	9	3	no	—
82	C₃×C₃₇	#6891	1	1	yes	characteristic
83	C₃×C₃₇	#6908	1	1	yes	characteristic
84	C₃×C₃₇	#6913	1	1	yes	characteristic
85	C₃×C₃₇	#6913	1	1	yes	characteristic
86	#737	—	3	6	no	—
87	#737	—	3	3	no	—
88	C₃×3₋¹⁺²	—	3	2	no	—
89	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	2	yes	—
90	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
91	C₃²×C₉	—	3	1	no	—
92	C₃²×C₉	—	37	1	no	—
93	C₃²×C₉	—	37	1	no	—
94	C₃²×C₉	—	37	1	no	—
95	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
96	C₉⋊C₉	—	111	1	no	—
97	C₉⋊C₉	—	111	2	no	—
98	C₉⋊C₉	—	111	2	no	—
99	C₉⋊C₉	—	111	4	no	—
100	C₃²⋊C₉	—	37	2	no	—
101	C₃²⋊C₉	—	37	2	no	—
102	C₃²⋊C₉	—	37	2	no	—
103	C₃₇	#10624780	1	1	yes	characteristic, complemented, Sylow
104	C₃³	—	3	1	no	—
105	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
106	3₋¹⁺²	—	3	18	no	—
107	3₋¹⁺²	—	3	2	no	—
108	3₋¹⁺²	—	3	2	no	—
109	3₋¹⁺²	#11758608	1	6	yes	—
110	H(3,𝔽₃)	—	3	2	no	—
111	H(3,𝔽₃)	#11758608	1	3	yes	—
112	C₃×C₉	—	3	9	no	—
113	C₃×C₉	—	3	2	no	—
114	C₃×C₉	—	3	2	no	—
115	C₃×C₉	—	3	1	no	—
116	C₃×C₉	—	333	1	no	—
117	C₃×C₉	—	333	1	no	—
118	C₃×C₉	—	111	2	no	—
119	C₃×C₉	—	111	2	no	—
120	C₃×C₉	—	111	2	no	—
121	C₃×C₉	—	111	1	no	—
122	C₃×C₉	—	111	1	no	—
123	C₃×C₉	—	111	1	no	—
124	C₃×C₉	—	111	3	no	—
125	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
126	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
127	C₃²	—	3	1	no	—
128	C₃²	—	3	1	no	—
129	C₃²	—	3	1	no	—
130	C₃²	—	3	3	no	—
131	C₃²	—	9	1	no	—
132	C₃²	—	9	1	no	—
133	C₃²	—	9	1	no	—
134	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
135	C₃²	C₃×#11758608	1	2	yes	—
136	C₃²	C₃×#11758608	1	1	yes	characteristic
137	C₉	—	3	1	no	—
138	C₉	—	3	6	no	—
139	C₉	—	3	2	no	—
140	C₉	—	333	2	no	—
141	C₉	—	333	1	no	—
142	C₉	—	333	2	no	—
143	C₉	—	333	1	no	—
144	C₉	—	333	2	no	—
145	C₉	—	111	3	no	—
146	C₃	—	3	2	no	—
147	C₃	—	3	1	no	—
148	C₃	—	9	3	no	—
149	C₃	#22582554	1	1	yes	characteristic
150	C₃	#22582630	1	1	yes	characteristic
151	C₃	#22582655	1	1	yes	characteristic
152	C₃	#22582656	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22582578

C₃

yes

maximal, complemented

C₃×#15688504

C₃

yes

maximal, characteristic, complemented

3₋¹⁺²⋊#66524

C₃

yes

maximal, complemented

3₋¹⁺²⋊#66524

C₃

yes

maximal, complemented

3₋¹⁺²⋊#66524

C₃

yes

maximal, complemented

H(3,𝔽₃)⋊#66524

C₃

yes

maximal, characteristic

H(3,𝔽₃)⋊#66524

C₃

yes

maximal, characteristic

H(3,𝔽₃)⋊#66524

C₃

yes

maximal, characteristic

C₃×C₃₇×#737

C₃

yes

maximal, characteristic

C₃²×#66524

C₃²

yes

characteristic

C₃²×#66524

C₃²

yes

characteristic

C₃²×#66524

C₃²

yes

—

—

—

—

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

—

C₃₇×#737

—

C₃₇×#737

—

C₃×3₋¹⁺²×C₃₇

—

C₃×3₋¹⁺²×C₃₇

C₃²

yes

—

C₃×H(3,𝔽₃)×C₃₇

C₃²

yes

characteristic

C₃²×C₉×C₃₇

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃³×C₃₇

—

C₃³×C₃₇

C₃³

yes

characteristic

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

3₋¹⁺²

yes

—

H(3,𝔽₃)×C₃₇

—

H(3,𝔽₃)×C₃₇

3₋¹⁺²

yes

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

H(3,𝔽₃)

yes

characteristic

C₃×C₉×C₃₇

H(3,𝔽₃)

yes

—

—

—

—

—

—

—

—

—

—

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

—

C₃²×C₃₇

C₃×3₋¹⁺²

yes

characteristic

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

—

—

—

—

—

—

C₃₇⋊C₃

yes

characteristic

#6895

—

C₃×C₉⋊C₉

—

3₋¹⁺²⋊C₉

—

3₋¹⁺²⋊C₉

—

3₋¹⁺²⋊C₉

—

H(3,𝔽₃)⋊C₉

—

H(3,𝔽₃)⋊C₉

—

H(3,𝔽₃)⋊C₉

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

—

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

—

C₃×H(3,𝔽₃)

C₃×C₃₇⋊C₃

yes

characteristic

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

100

C₃²⋊C₉

—

101

C₃²⋊C₉

—

102

C₃²⋊C₉

—

103

C₃₇

#10624780

yes

characteristic, complemented, Sylow

104

C₃³

—

105

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

106

3₋¹⁺²

—

107

3₋¹⁺²

—

108

3₋¹⁺²

—

109

3₋¹⁺²

#11758608

yes

—

110

H(3,𝔽₃)

—

111

H(3,𝔽₃)

#11758608

yes

—

112

C₃×C₉

—

113

C₃×C₉

—

114

C₃×C₉

—

115

C₃×C₉

—

116

C₃×C₉

—

333

—

117

C₃×C₉

—

333

—

118

C₃×C₉

—

111

—

119

C₃×C₉

—

111

—

120

C₃×C₉

—

111

—

121

C₃×C₉

—

111

—

122

C₃×C₉

—

111

—

123

C₃×C₉

—

111

—

124

C₃×C₉

—

111

—

125

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

126

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

—

127

C₃²

—

128

C₃²

—

129

C₃²

—

130

C₃²

—

131

C₃²

—

132

C₃²

—

133

C₃²

—

134

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

135

C₃²

C₃×#11758608

yes

—

136

C₃²

C₃×#11758608

yes

characteristic

137

C₉

—

138

C₉

—

139

C₉

—

140

C₉

—

333

—

141

C₉

—

333

—

142

C₉

—

333

—

143

C₉

—

333

—

144

C₉

—

333

—

145

C₉

—

111

—

146

C₃

—

147

C₃

—

148

C₃

—

149

C₃

#22582554

yes

characteristic

150

C₃

#22582630

yes

characteristic

151

C₃

#22582655

yes

characteristic

152

C₃

#22582656

yes

characteristic

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃×C₃₇×#737	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃×H(3,𝔽₃)×C₃₇	C₃²
hypercenter	C₃×#737	C₃₇⋊C₃
hypocenter	C₃₇	#10624780
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃²×C₉ ◁ C₃×#737

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃×C₃₇×#737

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

prime radical

C₃×H(3,𝔽₃)×C₃₇

C₃²

C₃₇⋊C₃

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃²×C₉ ◁ C₃×#737

N	H ≅ G/N	type
1. #22582578	C₃	semidirect
2. C₃×#15688504	C₃	semidirect
3. 3₋¹⁺²⋊#66524	C₃	semidirect
4. 3₋¹⁺²⋊#66524	C₃	semidirect
5. 3₋¹⁺²⋊#66524	C₃	semidirect
103. C₃₇	#10624780	semidirect

H ≅ G/N

type

1. #22582578

C₃

semidirect

2. C₃×#15688504

C₃

semidirect

3. 3₋¹⁺²⋊#66524

C₃

semidirect

4. 3₋¹⁺²⋊#66524

C₃

semidirect

5. 3₋¹⁺²⋊#66524

C₃

semidirect

103. C₃₇

#10624780

semidirect

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

subgroup	index
#22582578	3
C₃×#15688504	3
3₋¹⁺²⋊#66524	3
H(3,𝔽₃)⋊#66524	3
C₃×C₃₇×#737	3
#10624780	37

subgroup

index

C₃×C₃₇×#737

Subgroups — #41006785

Conjugacy classes of subgroups

Complements

Maximal subgroups