Group #41005198 — subgroups

Subgroups — #41005198

Order 26973.

470 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 112 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×#11758605	C₃	1	1	yes	maximal, characteristic
2	C₃×#15688504	C₃	1	3	yes	maximal, complemented
3	C₃×#15688502	C₃	1	3	yes	maximal
4	#22582441	C₃	1	3	yes	maximal, complemented
5	#22582441	C₃	1	1	yes	maximal, characteristic, complemented
6	#22582441	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃₇×#6855	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
9	C₃×#11758605	—	3	3	no	—
10	C₃×#11758605	—	3	3	no	—
11	C₃×#11758605	—	3	3	no	—
12	C₃×#11758605	—	3	3	no	—
13	C₃²×#66524	C₃²	1	3	yes	—
14	C₃²×#66524	C₃²	1	1	yes	characteristic
15	C₃²×#66524	C₃²	1	1	yes	characteristic
16	C₃²×#66524	C₃²	1	3	yes	—
17	#15688504	—	3	9	no	—
18	#15688502	—	3	9	no	—
19	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
20	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
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	3	no	—
27	C₃²×C₃₇⋊C₃	—	3	3	no	—
28	C₃²×C₃₇⋊C₃	3₋¹⁺²	1	3	yes	—
29	C₃×C₉×C₃₇	—	3	1	no	—
30	C₃×C₉×C₃₇	—	3	1	no	—
31	C₃×C₉×C₃₇	—	3	1	no	—
32	C₃×C₉×C₃₇	—	3	3	no	—
33	C₃×C₉×C₃₇	—	3	3	no	—
34	C₃×C₉×C₃₇	—	3	3	no	—
35	C₃×C₉×C₃₇	—	3	3	no	—
36	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	—
37	C₃×#66524	—	3	3	no	—
38	C₃×#66524	—	3	1	no	—
39	C₃×#66524	—	3	1	no	—
40	C₃×#66524	—	3	1	no	—
41	C₃×#66524	—	3	1	no	—
42	C₃×#66524	—	3	1	no	—
43	C₃×#66524	—	3	1	no	—
44	C₃×#66524	—	3	3	no	—
45	C₃×#66524	—	3	3	no	—
46	C₃×#66524	—	3	3	no	—
47	C₃×#66524	—	3	3	no	—
48	C₃×#66524	—	3	3	no	—
49	C₃×#66524	—	3	3	no	—
50	C₃×#66524	H(3,𝔽₃)	1	3	yes	—
51	C₃×#66524	3₋¹⁺²	1	3	yes	—
52	C₃×#66524	3₋¹⁺²	1	9	yes	complemented
53	#11758605	—	3	27	no	—
54	#10624565	—	37	1	no	maximal, Sylow
55	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
56	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
57	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
58	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
59	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
60	C₃²×C₃₇	#737	1	3	yes	—
61	C₃²×C₃₇	#737	1	3	yes	—
62	C₃×C₃₇⋊C₃	—	3	3	no	—
63	C₃×C₃₇⋊C₃	—	3	9	no	—
64	C₃×C₃₇⋊C₃	—	9	3	no	—
65	C₃×C₃₇⋊C₃	—	9	3	no	—
66	C₃×C₃₇⋊C₃	—	9	1	no	—
67	C₃×C₃₇⋊C₃	—	9	1	no	—
68	C₃×C₃₇⋊C₃	—	9	1	no	—
69	C₉×C₃₇	—	3	3	no	—
70	C₉×C₃₇	—	9	3	no	—
71	#66524	—	3	3	no	—
72	#66524	—	3	3	no	—
73	#66524	—	3	9	no	—
74	#66524	—	9	3	no	—
75	C₃²×3₋¹⁺²	—	37	1	no	—
76	C₃×C₉⋊C₉	—	37	3	no	—
77	C₃×C₃²⋊C₉	—	37	3	no	—
78	#6855	—	37	3	no	—
79	#6855	—	37	1	no	—
80	#6855	—	37	1	no	—
81	#6855	C₃₇⋊C₃	1	1	yes	characteristic, complemented
82	C₃×C₃₇	#6891	1	1	yes	characteristic
83	C₃×C₃₇	C₉⋊H(3,𝔽₃)	1	3	yes	—
84	C₃×C₃₇	#6894	1	3	yes	—
85	C₃×C₃₇	#6896	1	3	yes	—
86	C₃×C₃₇	#6899	1	1	yes	characteristic
87	C₃×C₃₇	#6899	1	1	yes	characteristic
88	C₃×C₃₇	#6900	1	1	yes	characteristic
89	C₃₇⋊C₃	—	9	3	no	—
90	C₃⁴	—	37	1	no	—
91	C₃×3₋¹⁺²	—	111	3	no	—
92	C₃×3₋¹⁺²	—	111	3	no	—
93	C₃×3₋¹⁺²	—	111	3	no	—
94	C₃×3₋¹⁺²	—	111	3	no	—
95	C₃²×C₉	—	37	1	no	—
96	C₃²×C₉	—	37	1	no	—
97	C₃²×C₉	—	37	3	no	—
98	C₃²×C₉	—	37	3	no	—
99	C₃²×C₉	C₃×C₃₇⋊C₃	1	3	yes	—
100	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
101	C₉⋊C₉	—	111	9	no	—
102	C₃²⋊C₉	—	111	9	no	—
103	C₃₇	#10624565	1	1	yes	characteristic, complemented, Sylow
104	C₃³	—	37	3	no	—
105	C₃³	—	111	1	no	—
106	C₃³	—	111	3	no	—
107	C₃³	—	111	1	no	—
108	C₃³	—	111	1	no	—
109	C₃³	—	111	3	no	—
110	C₃³	—	111	3	no	—
111	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
112	3₋¹⁺²	—	111	27	no	—
113	C₃×C₉	—	3	1	no	—
114	C₃×C₉	—	3	1	no	—
115	C₃×C₉	—	3	1	no	—
116	C₃×C₉	—	3	3	no	—
117	C₃×C₉	—	3	3	no	—
118	C₃×C₉	—	3	3	no	—
119	C₃×C₉	—	3	3	no	—
120	C₃×C₉	—	37	3	no	—
121	C₃×C₉	—	37	3	no	—
122	C₃×C₉	—	37	9	no	—
123	C₃×C₉	—	111	3	no	—
124	C₃×C₉	—	111	3	no	—
125	C₃×C₉	—	111	3	no	—
126	C₃×C₉	—	111	3	no	—
127	C₃×C₉	—	111	3	no	—
128	C₃×C₉	—	111	1	no	—
129	C₃×C₉	—	111	1	no	—
130	C₃×C₉	—	111	1	no	—
131	C₃×C₉	—	111	1	no	—
132	C₃×C₉	—	111	1	no	—
133	C₃×C₉	—	111	1	no	—
134	C₃×C₉	—	111	3	no	—
135	C₃×C₉	—	111	3	no	—
136	C₃×C₉	#11758605	1	3	yes	—
137	C₃²	—	333	3	no	—
138	C₃²	—	333	3	no	—
139	C₃²	—	333	1	no	—
140	C₃²	—	333	1	no	—
141	C₃²	—	333	1	no	—
142	C₃²	—	111	3	no	—
143	C₃²	—	111	9	no	—
144	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
145	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	1	yes	characteristic
146	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	1	yes	characteristic
147	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	3	yes	—
148	C₃²	C₃×#11758605	1	1	yes	characteristic
149	C₃²	#15688531	1	3	yes	—
150	C₃²	#15688532	1	3	yes	—
151	C₉	—	3	3	no	—
152	C₉	—	9	3	no	—
153	C₉	—	333	3	no	—
154	C₉	—	111	3	no	—
155	C₉	—	111	3	no	—
156	C₉	—	111	9	no	—
157	C₃	—	333	3	no	—
158	C₃	#22582553	1	1	yes	characteristic
159	C₃	#22582557	1	3	yes	—
160	C₃	#22582570	1	3	yes	—
161	C₃	#22582582	1	3	yes	—
162	C₃	#22582596	1	1	yes	characteristic
163	C₃	#22582599	1	1	yes	characteristic
164	C₃	#22582605	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#11758605

C₃

yes

maximal, characteristic

C₃×#15688504

C₃

yes

maximal, complemented

C₃×#15688502

C₃

yes

maximal

#22582441

C₃

yes

maximal, complemented

#22582441

C₃

yes

maximal, characteristic, complemented

#22582441

C₃

yes

maximal, characteristic, complemented

C₃₇×#6855

C₃

yes

maximal, characteristic, complemented

C₃³×C₃₇⋊C₃

C₃²

yes

—

—

—

—

C₃²×#66524

C₃²

yes

—

C₃²×#66524

C₃²

yes

characteristic

C₃²×#66524

C₃²

yes

characteristic

C₃²×#66524

C₃²

yes

—

#15688504

—

#15688502

—

C₃²×C₉×C₃₇

C₃²

yes

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

3₋¹⁺²

yes

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

3₋¹⁺²

yes

—

—

—

—

—

—

—

—

—

—

—

—

—

—

yes

—

C₃×#66524

3₋¹⁺²

yes

—

C₃×#66524

3₋¹⁺²

yes

—

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

—

C₃²×C₃₇

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₃₇

#737

yes

—

C₃²×C₃₇

#737

yes

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

C₉×C₃₇

—

—

—

—

—

C₃²×3₋¹⁺²

—

C₃×C₉⋊C₉

—

C₃×C₃²⋊C₉

—

—

—

—

C₃₇⋊C₃

yes

characteristic, complemented

C₃×C₃₇

#6891

yes

characteristic

C₃×C₃₇

C₉⋊H(3,𝔽₃)

yes

—

C₃×C₃₇

#6894

yes

—

C₃×C₃₇

#6896

yes

—

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃₇⋊C₃

—

C₃⁴

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

—

100

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

101

C₉⋊C₉

—

111

—

102

C₃²⋊C₉

—

111

—

103

C₃₇

#10624565

yes

characteristic, complemented, Sylow

104

C₃³

—

105

C₃³

—

111

—

106

C₃³

—

111

—

107

C₃³

—

111

—

108

C₃³

—

111

—

109

C₃³

—

111

—

110

C₃³

—

111

—

111

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

112

3₋¹⁺²

—

111

—

113

C₃×C₉

—

114

C₃×C₉

—

115

C₃×C₉

—

116

C₃×C₉

—

117

C₃×C₉

—

118

C₃×C₉

—

119

C₃×C₉

—

120

C₃×C₉

—

121

C₃×C₉

—

122

C₃×C₉

—

123

C₃×C₉

—

111

—

124

C₃×C₉

—

111

—

125

C₃×C₉

—

111

—

126

C₃×C₉

—

111

—

127

C₃×C₉

—

111

—

128

C₃×C₉

—

111

—

129

C₃×C₉

—

111

—

130

C₃×C₉

—

111

—

131

C₃×C₉

—

111

—

132

C₃×C₉

—

111

—

133

C₃×C₉

—

111

—

134

C₃×C₉

—

111

—

135

C₃×C₉

—

111

—

136

C₃×C₉

#11758605

yes

—

137

C₃²

—

333

—

138

C₃²

—

333

—

139

C₃²

—

333

—

140

C₃²

—

333

—

141

C₃²

—

333

—

142

C₃²

—

111

—

143

C₃²

—

111

—

144

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

145

C₃²

3₋¹⁺²×C₃₇⋊C₃

yes

characteristic

146

C₃²

3₋¹⁺²×C₃₇⋊C₃

yes

characteristic

147

C₃²

3₋¹⁺²×C₃₇⋊C₃

yes

—

148

C₃²

C₃×#11758605

yes

characteristic

149

C₃²

#15688531

yes

—

150

C₃²

#15688532

yes

—

151

C₉

—

152

C₉

—

153

C₉

—

333

—

154

C₉

—

111

—

155

C₉

—

111

—

156

C₉

—

111

—

157

C₃

—

333

—

158

C₃

#22582553

yes

characteristic

159

C₃

#22582557

yes

—

160

C₃

#22582570

yes

—

161

C₃

#22582582

yes

—

162

C₃

#22582596

yes

characteristic

163

C₃

#22582599

yes

characteristic

164

C₃

#22582605

yes

characteristic

	subgroup	G/H
center, Frattini	C₃³	C₃²×C₃₇⋊C₃
commutator, socle, cosocle	C₃³×C₃₇	C₃³
Fitting	C₃₇×#6855	C₃
prime radical	C₃³×C₃₇⋊C₃	C₃²
hypercenter	#6855	C₃₇⋊C₃
hypocenter	C₃₇	#10624565
LCS	G ▷ C₃³×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃³ ◁ #6855

subgroup

G/H

center, Frattini

C₃³

C₃²×C₃₇⋊C₃

commutator, socle, cosocle

C₃³×C₃₇

C₃³

Fitting

C₃₇×#6855

C₃

prime radical

C₃³×C₃₇⋊C₃

C₃²

C₃₇⋊C₃

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃₇

C₁ ◁ C₃³ ◁ #6855

N	H ≅ G/N	type
2. C₃×#15688504	C₃	semidirect
4. #22582441	C₃	semidirect
5. #22582441	C₃	semidirect
6. #22582441	C₃	semidirect
7. C₃₇×#6855	C₃	semidirect
52. C₃×#66524	3₋¹⁺²	semidirect
81. #6855	C₃₇⋊C₃	semidirect
103. C₃₇	#10624565	semidirect

H ≅ G/N

type

2. C₃×#15688504

C₃

semidirect

4. #22582441

C₃

semidirect

5. #22582441

C₃

semidirect

6. #22582441

C₃

semidirect

7. C₃₇×#6855

C₃

semidirect

52. C₃×#66524

3₋¹⁺²

semidirect

81. #6855

C₃₇⋊C₃

semidirect

103. C₃₇

#10624565

semidirect

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

subgroup	index
C₃²×#11758605	3
C₃×#15688504	3
C₃×#15688502	3
#22582441	3
C₃₇×#6855	3
#10624565	37

subgroup

index

C₃₇×#6855

Subgroups — #41005198

Conjugacy classes of subgroups

Complements

Maximal subgroups