Group #41005142 — subgroups

Subgroups — #41005142

Order 26973.

608 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 142 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	C₃²×C₉×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
3	C₃×#15688502	C₃	1	2	yes	maximal
4	#22582441	C₃	1	4	yes	maximal, complemented
5	#22582441	C₃	1	2	yes	maximal, complemented
6	#22582441	C₃	1	2	yes	maximal, 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	6	no	—
11	C₃×#11758605	C₉	1	9	yes	complemented
12	C₃²×#66524	C₃²	1	4	yes	—
13	C₃²×#66524	C₃²	1	2	yes	—
14	C₃²×#66524	C₃²	1	2	yes	—
15	C₃×C₉×C₃₇⋊C₃	—	3	3	no	—
16	C₃×C₉×C₃₇⋊C₃	—	3	3	no	—
17	C₃×C₉×C₃₇⋊C₃	—	3	6	no	—
18	#15688502	—	3	6	no	—
19	C₃²×C₉×C₃₇	C₃²	1	2	yes	—
20	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
21	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
22	C₃³×C₃₇	C₃³	1	1	yes	characteristic
23	C₃²×C₃₇⋊C₃	—	3	2	no	—
24	C₃²×C₃₇⋊C₃	—	3	1	no	—
25	C₃²×C₃₇⋊C₃	—	3	2	no	—
26	C₃²×C₃₇⋊C₃	—	3	4	no	—
27	C₃²×C₃₇⋊C₃	C₃×C₉	1	3	yes	—
28	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	3	yes	—
29	C₃²×C₃₇⋊C₃	3₋¹⁺²	1	6	yes	—
30	C₃×C₉×C₃₇	—	3	1	no	—
31	C₃×C₉×C₃₇	—	3	1	no	—
32	C₃×C₉×C₃₇	—	3	2	no	—
33	C₃×C₉×C₃₇	—	3	2	no	—
34	C₃×C₉×C₃₇	—	3	2	no	—
35	C₃×C₉×C₃₇	—	3	2	no	—
36	C₃×C₉×C₃₇	—	3	4	no	—
37	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	complemented
38	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	complemented
39	C₃×#66524	—	3	2	no	—
40	C₃×#66524	—	3	2	no	—
41	C₃×#66524	—	3	2	no	—
42	C₃×#66524	—	3	2	no	—
43	C₃×#66524	—	3	4	no	—
44	C₃×#66524	—	3	4	no	—
45	C₃×#66524	—	3	4	no	—
46	C₃×#66524	—	3	8	no	—
47	C₃×#66524	C₃×C₉	1	6	yes	complemented
48	C₃×#66524	3₋¹⁺²	1	6	yes	complemented
49	#11758605	—	3	27	no	—
50	C₉×C₃₇⋊C₃	—	3	27	no	—
51	#10624558	—	37	1	no	maximal, Sylow
52	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
53	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
54	C₃²×C₃₇	C₃×3₋¹⁺²	1	2	yes	—
55	C₃²×C₃₇	C₃×3₋¹⁺²	1	2	yes	—
56	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
57	C₃²×C₃₇	#737	1	2	yes	—
58	C₃²×C₃₇	#737	1	4	yes	—
59	C₃×C₃₇⋊C₃	—	3	6	no	—
60	C₃×C₃₇⋊C₃	—	3	3	no	—
61	C₃×C₃₇⋊C₃	—	3	3	no	—
62	C₃×C₃₇⋊C₃	—	3	6	no	—
63	C₃×C₃₇⋊C₃	—	3	6	no	—
64	C₃×C₃₇⋊C₃	—	3	12	no	—
65	C₃×C₃₇⋊C₃	C₃²⋊C₉	1	9	yes	—
66	C₉×C₃₇	—	3	3	no	—
67	C₉×C₃₇	—	3	3	no	—
68	C₉×C₃₇	—	9	2	no	—
69	#66524	—	3	6	no	—
70	#66524	—	3	6	no	—
71	#66524	—	9	4	no	—
72	C₃²×3₋¹⁺²	—	37	1	no	—
73	C₃³×C₉	—	37	1	no	—
74	C₃×C₃²⋊C₉	—	37	2	no	—
75	#6855	—	37	4	no	—
76	#6855	—	37	2	no	—
77	#6855	—	37	2	no	—
78	#6855	C₃₇⋊C₃	1	1	yes	characteristic, complemented
79	C₃×C₃₇	C₃×C₃²⋊C₉	1	1	yes	characteristic
80	C₃×C₃₇	C₉×3₋¹⁺²	1	2	yes	—
81	C₃×C₃₇	C₉×3₋¹⁺²	1	1	yes	characteristic
82	C₃×C₃₇	#6891	1	1	yes	characteristic
83	C₃×C₃₇	#6893	1	2	yes	—
84	C₃×C₃₇	#6894	1	2	yes	—
85	C₃×C₃₇	#6894	1	4	yes	—
86	C₃₇⋊C₃	—	3	9	no	—
87	C₃⁴	—	37	1	no	—
88	C₃×3₋¹⁺²	—	37	9	no	—
89	C₃×3₋¹⁺²	—	111	3	no	—
90	C₃×3₋¹⁺²	—	111	6	no	—
91	C₃²×C₉	—	37	2	no	—
92	C₃²×C₉	—	37	2	no	—
93	C₃²×C₉	—	37	4	no	—
94	C₃²×C₉	—	111	6	no	—
95	C₃²×C₉	—	111	3	no	—
96	C₃²×C₉	—	111	3	no	—
97	C₃²×C₉	C₃×C₃₇⋊C₃	1	2	yes	—
98	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
99	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
100	C₃²⋊C₉	—	111	6	no	—
101	C₃₇	#10624558	1	1	yes	characteristic, complemented, Sylow
102	C₃³	—	37	3	no	—
103	C₃³	—	37	3	no	—
104	C₃³	—	37	6	no	—
105	C₃³	—	111	2	no	—
106	C₃³	—	111	1	no	—
107	C₃³	—	111	2	no	—
108	C₃³	—	111	4	no	—
109	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
110	3₋¹⁺²	—	111	27	no	—
111	C₃×C₉	—	3	1	no	—
112	C₃×C₉	—	3	1	no	—
113	C₃×C₉	—	3	2	no	—
114	C₃×C₉	—	3	2	no	—
115	C₃×C₉	—	3	2	no	—
116	C₃×C₉	—	3	2	no	—
117	C₃×C₉	—	3	4	no	—
118	C₃×C₉	—	37	6	no	—
119	C₃×C₉	—	37	6	no	—
120	C₃×C₉	—	111	27	no	—
121	C₃×C₉	—	111	4	no	—
122	C₃×C₉	—	111	4	no	—
123	C₃×C₉	—	111	8	no	—
124	C₃×C₉	—	111	2	no	—
125	C₃×C₉	—	111	2	no	—
126	C₃×C₉	—	111	2	no	—
127	C₃×C₉	—	111	4	no	—
128	C₃×C₉	—	111	2	no	—
129	C₃×C₉	C₉×C₃₇⋊C₃	1	3	yes	complemented
130	C₃×C₉	#11758605	1	3	yes	complemented
131	C₃²	—	37	9	no	—
132	C₃²	—	111	6	no	—
133	C₃²	—	111	3	no	—
134	C₃²	—	111	3	no	—
135	C₃²	—	111	6	no	—
136	C₃²	—	111	6	no	—
137	C₃²	—	111	12	no	—
138	C₃²	C₃×C₉×C₃₇⋊C₃	1	1	yes	characteristic
139	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
140	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	2	yes	—
141	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	2	yes	—
142	C₃²	C₃×#11758605	1	1	yes	characteristic
143	C₃²	#15688531	1	2	yes	—
144	C₃²	#15688532	1	4	yes	—
145	C₉	—	3	3	no	—
146	C₉	—	3	3	no	—
147	C₉	—	9	2	no	—
148	C₉	—	333	4	no	—
149	C₉	—	111	6	no	—
150	C₉	—	111	6	no	—
151	C₃	—	111	9	no	—
152	C₃	C₃²⋊C₉×C₃₇⋊C₃	1	1	yes	characteristic
153	C₃	#22582543	1	2	yes	—
154	C₃	C₉×#11758605	1	1	yes	characteristic
155	C₃	#22582553	1	1	yes	characteristic
156	C₃	#22582563	1	2	yes	—
157	C₃	#22582567	1	2	yes	—
158	C₃	#22582570	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#11758605

C₃

yes

maximal, characteristic

C₃²×C₉×C₃₇⋊C₃

C₃

yes

maximal, characteristic

C₃×#15688502

C₃

yes

maximal

#22582441

C₃

yes

maximal, complemented

#22582441

C₃

yes

maximal, complemented

#22582441

C₃

yes

maximal, complemented

C₃₇×#6855

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

—

C₃×C₉×C₃₇⋊C₃

—

C₃×C₉×C₃₇⋊C₃

—

C₃×C₉×C₃₇⋊C₃

—

#15688502

—

C₃²×C₉×C₃₇

C₃²

yes

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

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

C₃×C₉

yes

—

C₃²×C₃₇⋊C₃

H(3,𝔽₃)

yes

—

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

C₃×C₉

yes

complemented

C₃×C₉×C₃₇

3₋¹⁺²

yes

—

—

—

—

—

—

—

—

C₃×C₉

yes

complemented

C₃×#66524

3₋¹⁺²

yes

complemented

#11758605

—

C₉×C₃₇⋊C₃

—

#10624558

—

maximal, Sylow

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

—

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

C₃²⋊C₉

yes

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

#66524

—

#66524

—

#66524

—

C₃²×3₋¹⁺²

—

C₃³×C₉

—

C₃×C₃²⋊C₉

—

—

—

—

C₃₇⋊C₃

yes

characteristic, complemented

C₃×C₃₇

C₃×C₃²⋊C₉

yes

characteristic

C₃×C₃₇

C₉×3₋¹⁺²

yes

—

C₃×C₃₇

C₉×3₋¹⁺²

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

—

C₃×C₃₇

#6894

yes

—

C₃×C₃₇

#6894

yes

—

C₃₇⋊C₃

—

C₃⁴

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

111

—

C₃²×C₉

—

111

—

C₃²×C₉

—

111

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

100

C₃²⋊C₉

—

111

—

101

C₃₇

#10624558

yes

characteristic, complemented, Sylow

102

C₃³

—

103

C₃³

—

104

C₃³

—

105

C₃³

—

111

—

106

C₃³

—

111

—

107

C₃³

—

111

—

108

C₃³

—

111

—

109

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

110

3₋¹⁺²

—

111

—

111

C₃×C₉

—

112

C₃×C₉

—

113

C₃×C₉

—

114

C₃×C₉

—

115

C₃×C₉

—

116

C₃×C₉

—

117

C₃×C₉

—

118

C₃×C₉

—

119

C₃×C₉

—

120

C₃×C₉

—

111

—

121

C₃×C₉

—

111

—

122

C₃×C₉

—

111

—

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

C₉×C₃₇⋊C₃

yes

complemented

130

C₃×C₉

#11758605

yes

complemented

131

C₃²

—

132

C₃²

—

111

—

133

C₃²

—

111

—

134

C₃²

—

111

—

135

C₃²

—

111

—

136

C₃²

—

111

—

137

C₃²

—

111

—

138

C₃²

C₃×C₉×C₃₇⋊C₃

yes

characteristic

139

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

140

C₃²

3₋¹⁺²×C₃₇⋊C₃

yes

—

141

C₃²

3₋¹⁺²×C₃₇⋊C₃

yes

—

142

C₃²

C₃×#11758605

yes

characteristic

143

C₃²

#15688531

yes

—

144

C₃²

#15688532

yes

—

145

C₉

—

146

C₉

—

147

C₉

—

148

C₉

—

333

—

149

C₉

—

111

—

150

C₉

—

111

—

151

C₃

—

111

—

152

C₃

C₃²⋊C₉×C₃₇⋊C₃

yes

characteristic

153

C₃

#22582543

yes

—

154

C₃

C₉×#11758605

yes

characteristic

155

C₃

#22582553

yes

characteristic

156

C₃

#22582563

yes

—

157

C₃

#22582567

yes

—

158

C₃

#22582570

yes

—

	subgroup	G/H
center, Frattini	C₃³	C₃²×C₃₇⋊C₃
commutator	C₃²×C₃₇	C₃²×C₉
Fitting	C₃₇×#6855	C₃
socle, cosocle	C₃³×C₃₇	C₃³
prime radical	C₃³×C₃₇⋊C₃	C₃²
hypercenter	#6855	C₃₇⋊C₃
hypocenter	C₃₇	#10624558
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃³ ◁ #6855

subgroup

G/H

center, Frattini

C₃³

C₃²×C₃₇⋊C₃

commutator

C₃²×C₃₇

C₃²×C₉

Fitting

C₃₇×#6855

C₃

socle, cosocle

C₃³×C₃₇

C₃³

prime radical

C₃³×C₃₇⋊C₃

C₃²

C₃₇⋊C₃

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃₇

C₁ ◁ C₃³ ◁ #6855

N	H ≅ G/N	type
4. #22582441	C₃	semidirect
5. #22582441	C₃	semidirect
6. #22582441	C₃	semidirect
7. C₃₇×#6855	C₃	semidirect
11. C₃×#11758605	C₉	semidirect
37. C₃×C₉×C₃₇	C₃×C₉	semidirect
38. C₃×C₉×C₃₇	3₋¹⁺²	semidirect
47. C₃×#66524	C₃×C₉	semidirect
48. C₃×#66524	3₋¹⁺²	semidirect
78. #6855	C₃₇⋊C₃	semidirect
101. C₃₇	#10624558	semidirect
129. C₃×C₉	C₉×C₃₇⋊C₃	semidirect
130. C₃×C₉	#11758605	semidirect

H ≅ G/N

type

4. #22582441

C₃

semidirect

5. #22582441

C₃

semidirect

6. #22582441

C₃

semidirect

7. C₃₇×#6855

C₃

semidirect

11. C₃×#11758605

C₉

semidirect

37. C₃×C₉×C₃₇

C₃×C₉

semidirect

38. C₃×C₉×C₃₇

3₋¹⁺²

semidirect

47. C₃×#66524

C₃×C₉

semidirect

48. C₃×#66524

3₋¹⁺²

semidirect

78. #6855

C₃₇⋊C₃

semidirect

101. C₃₇

#10624558

semidirect

129. C₃×C₉

C₉×C₃₇⋊C₃

semidirect

130. C₃×C₉

#11758605

semidirect

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

subgroup	index
C₃²×#11758605	3
C₃²×C₉×C₃₇⋊C₃	3
C₃×#15688502	3
#22582441	3
C₃₇×#6855	3
#10624558	37

subgroup

index

C₃²×C₉×C₃₇⋊C₃

C₃₇×#6855

Subgroups — #41005142

Conjugacy classes of subgroups

Complements

Maximal subgroups