Group #36381360 — subgroups

Subgroups — #36381360

Order 22599.

414 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 61 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	#21419438	C₃	1	1	yes	maximal, characteristic, complemented
2	#21419409	C₃	1	2	yes	maximal, complemented
3	#21419405	C₃	1	1	yes	maximal, characteristic
4	#21419402	C₃	1	2	yes	maximal, complemented
5	#21419401	C₃	1	2	yes	maximal, complemented
6	#21419400	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃×#14578902	C₃	1	1	yes	maximal, characteristic, complemented
8	#21419322	C₃	1	2	yes	maximal, complemented
9	C₃×C₃₁×#731	C₃	1	1	yes	maximal, characteristic, complemented
10	#14578911	—	3	3	no	—
11	#14578911	—	3	6	no	—
12	C₃×#11719689	—	3	3	no	—
13	C₃×#11719689	C₃²	1	2	yes	—
14	C₃×#11719689	C₃²	1	1	yes	characteristic
15	C₃×#11719689	C₃²	1	1	yes	characteristic
16	C₃×#11719689	C₃²	1	1	yes	characteristic
17	C₃×#11719689	C₃²	1	2	yes	—
18	C₃×C₃²⋊(C₃₁⋊C₃)	C₃²	1	2	yes	—
19	#14578902	—	3	3	no	—
20	#14578901	—	3	6	no	—
21	#14578899	—	3	3	no	—
22	#14578898	—	3	6	no	—
23	#14578897	—	3	6	no	—
24	#14578896	—	3	3	no	—
25	(C₃×C₃₁⋊C₃)⋊C₉	—	3	6	no	—
26	C₃×3₋¹⁺²×C₃₁	C₃²	1	2	yes	—
27	C₃×H(3,𝔽₃)×C₃₁	C₃²	1	1	yes	characteristic
28	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
29	C₃₁×#731	—	3	3	no	—
30	C₃³×C₃₁	—	3	1	no	—
31	C₃³×C₃₁	C₃³	1	1	yes	characteristic
32	C₃²×C₃₁⋊C₃	—	3	2	no	—
33	3₋¹⁺²×C₃₁	—	3	2	no	—
34	3₋¹⁺²×C₃₁	—	3	2	no	—
35	3₋¹⁺²×C₃₁	—	3	2	no	—
36	H(3,𝔽₃)×C₃₁	—	3	2	no	—
37	H(3,𝔽₃)×C₃₁	H(3,𝔽₃)	1	3	yes	—
38	C₃×C₉×C₃₁	—	3	1	no	—
39	C₃×C₉×C₃₁	—	3	2	no	—
40	C₃×C₉×C₃₁	—	3	1	no	—
41	C₃×C₉×C₃₁	—	3	2	no	—
42	C₃²⋊(C₃₁⋊C₃)	—	3	2	no	—
43	C₃²⋊(C₃₁⋊C₃)	—	3	2	no	—
44	C₃²⋊(C₃₁⋊C₃)	—	3	2	no	—
45	#11719689	—	3	18	no	—
46	#11719689	—	3	2	no	—
47	#11719689	—	3	2	no	—
48	#11719689	—	3	2	no	—
49	#11719689	—	3	2	no	—
50	#11719689	—	3	2	no	—
51	#11719689	—	3	2	no	—
52	#11719689	—	3	2	no	—
53	#11719689	—	3	2	no	—
54	#11719689	—	3	2	no	—
55	#11719689	—	3	1	no	—
56	#11719689	—	3	1	no	—
57	#11719689	H(3,𝔽₃)	1	3	yes	—
58	C₃×C₃₁⋊C₉	—	3	9	no	—
59	C₃×C₃₁⋊C₉	—	3	1	no	—
60	C₃×C₃₁⋊C₉	—	3	2	no	—
61	C₃×C₃₁⋊C₉	—	3	1	no	—
62	C₃×C₃₁⋊C₉	—	3	1	no	—
63	C₃×C₃₁⋊C₉	H(3,𝔽₃)	1	6	yes	—
64	#10624751	—	31	1	no	maximal, Sylow
65	C₃²×C₃₁	—	3	3	no	—
66	C₃²×C₃₁	—	9	1	no	—
67	C₃²×C₃₁	—	27	1	no	—
68	C₃²×C₃₁	C₃×H(3,𝔽₃)	1	2	yes	—
69	C₃²×C₃₁	C₃×H(3,𝔽₃)	1	1	yes	characteristic
70	C₃²×C₃₁	C₃×H(3,𝔽₃)	1	1	yes	characteristic
71	C₃×C₃₁⋊C₃	—	9	6	no	—
72	C₃×C₃₁⋊C₃	—	9	2	no	—
73	C₉×C₃₁	—	9	1	no	—
74	C₉×C₃₁	—	9	2	no	—
75	C₃₁⋊C₉	—	3	3	no	—
76	C₃₁⋊C₉	—	3	6	no	—
77	C₃₁⋊C₉	—	9	2	no	—
78	C₃₁⋊C₉	—	9	1	no	—
79	C₃₁⋊C₉	—	9	1	no	—
80	3₋¹⁺⁴	—	31	1	no	—
81	#6912	—	31	2	no	—
82	#6911	—	31	1	no	—
83	#6910	—	31	2	no	—
84	#6910	—	31	2	no	—
85	#6910	—	31	1	no	—
86	C₃×#733	—	31	1	no	—
87	C₃×#731	C₃₁⋊C₃	1	1	yes	characteristic, complemented
88	#6893	—	31	2	no	—
89	C₃×C₃₁	—	3	2	no	—
90	C₃×C₃₁	—	3	1	no	—
91	C₃×C₃₁	—	3	1	no	—
92	C₃×C₃₁	—	9	3	no	—
93	C₃×C₃₁	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
94	C₃₁⋊C₃	—	27	2	no	—
95	#737	—	93	6	no	—
96	#737	—	93	3	no	—
97	C₃×3₋¹⁺²	—	31	2	no	—
98	C₃×3₋¹⁺²	—	31	1	no	—
99	C₃×3₋¹⁺²	—	31	1	no	—
100	C₃×3₋¹⁺²	—	31	2	no	—
101	C₃×3₋¹⁺²	—	31	1	no	—
102	C₃×3₋¹⁺²	—	93	3	no	—
103	C₃×3₋¹⁺²	C₃×C₃₁⋊C₃	1	2	yes	—
104	C₃×H(3,𝔽₃)	—	31	2	no	—
105	C₃×H(3,𝔽₃)	C₃×C₃₁⋊C₃	1	1	yes	characteristic
106	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
107	#733	—	93	3	no	—
108	#733	—	93	6	no	—
109	#732	—	93	3	no	—
110	#731	—	3	3	no	—
111	#731	—	93	6	no	—
112	#731	—	93	6	no	—
113	#731	—	93	3	no	—
114	C₃²⋊C₉	—	93	6	no	—
115	C₃₁	#10624751	1	1	yes	characteristic, complemented, Sylow
116	C₃³	—	3	1	no	—
117	C₃³	—	93	2	no	—
118	C₃³	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
119	3₋¹⁺²	—	3	2	no	—
120	3₋¹⁺²	—	3	2	no	—
121	3₋¹⁺²	—	3	2	no	—
122	3₋¹⁺²	—	31	3	no	—
123	3₋¹⁺²	—	93	2	no	—
124	3₋¹⁺²	—	93	2	no	—
125	3₋¹⁺²	—	93	1	no	—
126	3₋¹⁺²	—	93	1	no	—
127	3₋¹⁺²	—	93	2	no	—
128	3₋¹⁺²	—	93	2	no	—
129	3₋¹⁺²	—	93	2	no	—
130	3₋¹⁺²	—	93	18	no	—
131	3₋¹⁺²	—	93	2	no	—
132	3₋¹⁺²	—	93	2	no	—
133	3₋¹⁺²	—	93	2	no	—
134	3₋¹⁺²	—	93	2	no	—
135	H(3,𝔽₃)	—	3	2	no	—
136	H(3,𝔽₃)	—	93	2	no	—
137	H(3,𝔽₃)	—	93	2	no	—
138	H(3,𝔽₃)	—	93	2	no	—
139	H(3,𝔽₃)	C₃²⋊(C₃₁⋊C₃)	1	3	yes	—
140	C₃×C₉	—	3	2	no	—
141	C₃×C₉	—	3	2	no	—
142	C₃×C₉	—	3	1	no	—
143	C₃×C₉	—	3	1	no	—
144	C₃×C₉	—	31	6	no	—
145	C₃×C₉	—	93	2	no	—
146	C₃×C₉	—	93	1	no	—
147	C₃×C₉	—	93	1	no	—
148	C₃×C₉	—	93	9	no	—
149	C₃×C₉	—	93	1	no	—
150	C₃²	—	3	3	no	—
151	C₃²	—	9	1	no	—
152	C₃²	—	279	6	no	—
153	C₃²	—	279	2	no	—
154	C₃²	—	27	1	no	—
155	C₃²	H(3,𝔽₃)×C₃₁⋊C₃	1	1	yes	characteristic
156	C₃²	H(3,𝔽₃)×C₃₁⋊C₃	1	2	yes	—
157	C₃²	C₃×C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
158	C₉	—	9	1	no	—
159	C₉	—	9	2	no	—
160	C₉	—	279	2	no	—
161	C₉	—	279	1	no	—
162	C₉	—	279	1	no	—
163	C₉	—	93	6	no	—
164	C₉	—	93	3	no	—
165	C₃	—	3	2	no	—
166	C₃	—	3	1	no	—
167	C₃	—	3	1	no	—
168	C₃	—	9	3	no	—
169	C₃	—	837	2	no	—
170	C₃	(C₃×C₃₁⋊C₃)⋊H(3,𝔽₃)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21419438

C₃

yes

maximal, characteristic, complemented

#21419409

C₃

yes

maximal, complemented

#21419405

C₃

yes

maximal, characteristic

#21419402

C₃

yes

maximal, complemented

#21419401

C₃

yes

maximal, complemented

#21419400

C₃

yes

maximal, characteristic, complemented

C₃×#14578902

C₃

yes

maximal, characteristic, complemented

#21419322

C₃

yes

maximal, complemented

C₃×C₃₁×#731

C₃

yes

maximal, characteristic, complemented

—

—

—

C₃²

yes

—

C₃×#11719689

C₃²

yes

characteristic

C₃×#11719689

C₃²

yes

characteristic

C₃×#11719689

C₃²

yes

characteristic

C₃×#11719689

C₃²

yes

—

C₃×C₃²⋊(C₃₁⋊C₃)

C₃²

yes

—

—

—

—

—

—

—

(C₃×C₃₁⋊C₃)⋊C₉

—

C₃×3₋¹⁺²×C₃₁

C₃²

yes

—

C₃×H(3,𝔽₃)×C₃₁

C₃²

yes

characteristic

C₃²×C₉×C₃₁

C₃²

yes

characteristic

C₃₁×#731

—

C₃³×C₃₁

—

C₃³×C₃₁

C₃³

yes

characteristic

C₃²×C₃₁⋊C₃

—

3₋¹⁺²×C₃₁

—

3₋¹⁺²×C₃₁

—

3₋¹⁺²×C₃₁

—

H(3,𝔽₃)×C₃₁

—

H(3,𝔽₃)×C₃₁

H(3,𝔽₃)

yes

—

C₃×C₉×C₃₁

—

C₃×C₉×C₃₁

—

C₃×C₉×C₃₁

—

C₃×C₉×C₃₁

—

C₃²⋊(C₃₁⋊C₃)

—

C₃²⋊(C₃₁⋊C₃)

—

C₃²⋊(C₃₁⋊C₃)

—

—

—

—

—

—

—

—

—

—

—

—

—

yes

—

C₃×C₃₁⋊C₉

—

C₃×C₃₁⋊C₉

—

C₃×C₃₁⋊C₉

—

C₃×C₃₁⋊C₉

—

C₃×C₃₁⋊C₉

—

C₃×C₃₁⋊C₉

H(3,𝔽₃)

yes

—

#10624751

—

maximal, Sylow

C₃²×C₃₁

—

C₃²×C₃₁

—

C₃²×C₃₁

—

C₃²×C₃₁

C₃×H(3,𝔽₃)

yes

—

C₃²×C₃₁

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₁

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₃₁⋊C₃

—

C₃×C₃₁⋊C₃

—

C₉×C₃₁

—

C₉×C₃₁

—

C₃₁⋊C₉

—

C₃₁⋊C₉

—

C₃₁⋊C₉

—

C₃₁⋊C₉

—

C₃₁⋊C₉

—

3₋¹⁺⁴

—

—

—

—

—

—

—

C₃₁⋊C₃

yes

characteristic, complemented

#6893

—

C₃×C₃₁

—

C₃×C₃₁

—

C₃×C₃₁

—

C₃×C₃₁

—

C₃×C₃₁

C₃²⋊H(3,𝔽₃)

yes

characteristic

C₃₁⋊C₃

—

#737

—

#737

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

100

C₃×3₋¹⁺²

—

101

C₃×3₋¹⁺²

—

102

C₃×3₋¹⁺²

—

103

C₃×3₋¹⁺²

C₃×C₃₁⋊C₃

yes

—

104

C₃×H(3,𝔽₃)

—

105

C₃×H(3,𝔽₃)

C₃×C₃₁⋊C₃

yes

characteristic

106

C₃²×C₉

C₃×C₃₁⋊C₃

yes

characteristic

107

#733

—

108

#733

—

109

#732

—

110

#731

—

111

#731

—

112

#731

—

113

#731

—

114

C₃²⋊C₉

—

115

C₃₁

#10624751

yes

characteristic, complemented, Sylow

116

C₃³

—

117

C₃³

—

118

C₃³

C₃²×C₃₁⋊C₃

yes

characteristic

119

3₋¹⁺²

—

120

3₋¹⁺²

—

121

3₋¹⁺²

—

122

3₋¹⁺²

—

123

3₋¹⁺²

—

124

3₋¹⁺²

—

125

3₋¹⁺²

—

126

3₋¹⁺²

—

127

3₋¹⁺²

—

128

3₋¹⁺²

—

129

3₋¹⁺²

—

130

3₋¹⁺²

—

131

3₋¹⁺²

—

132

3₋¹⁺²

—

133

3₋¹⁺²

—

134

3₋¹⁺²

—

135

H(3,𝔽₃)

—

136

H(3,𝔽₃)

—

137

H(3,𝔽₃)

—

138

H(3,𝔽₃)

—

139

H(3,𝔽₃)

C₃²⋊(C₃₁⋊C₃)

yes

—

140

C₃×C₉

—

141

C₃×C₉

—

142

C₃×C₉

—

143

C₃×C₉

—

144

C₃×C₉

—

145

C₃×C₉

—

146

C₃×C₉

—

147

C₃×C₉

—

148

C₃×C₉

—

149

C₃×C₉

—

150

C₃²

—

151

C₃²

—

152

C₃²

—

279

—

153

C₃²

—

279

—

154

C₃²

—

155

C₃²

H(3,𝔽₃)×C₃₁⋊C₃

yes

characteristic

156

C₃²

H(3,𝔽₃)×C₃₁⋊C₃

yes

—

157

C₃²

C₃×C₃²⋊(C₃₁⋊C₃)

yes

characteristic

158

C₉

—

159

C₉

—

160

C₉

—

279

—

161

C₉

—

279

—

162

C₉

—

279

—

163

C₉

—

164

C₉

—

165

C₃

—

166

C₃

—

167

C₃

—

168

C₃

—

169

C₃

—

837

—

170

C₃

(C₃×C₃₁⋊C₃)⋊H(3,𝔽₃)

yes

characteristic

	subgroup	G/H
center	C₃	(C₃×C₃₁⋊C₃)⋊H(3,𝔽₃)
commutator, cosocle	C₃³×C₃₁	C₃³
Frattini	C₃³	C₃²×C₃₁⋊C₃
Fitting	C₃×C₃₁×#731	C₃
socle	C₃×C₃₁	C₃²⋊H(3,𝔽₃)
prime radical	#21419405	C₃
hypercenter	C₃×#731	C₃₁⋊C₃
hypocenter	C₃₁	#10624751
LCS	G ▷ C₃³×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃×#731

subgroup

G/H

center

C₃

(C₃×C₃₁⋊C₃)⋊H(3,𝔽₃)

commutator, cosocle

C₃³×C₃₁

C₃³

Frattini

C₃³

C₃²×C₃₁⋊C₃

Fitting

C₃×C₃₁×#731

C₃

socle

C₃×C₃₁

C₃²⋊H(3,𝔽₃)

C₃

C₃₁⋊C₃

C₃₁

G ▷ C₃³×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₃ ◁ C₃³ ◁ C₃×#731

N	H ≅ G/N	type
1. #21419438	C₃	semidirect
2. #21419409	C₃	semidirect
4. #21419402	C₃	semidirect
5. #21419401	C₃	semidirect
6. #21419400	C₃	semidirect
7. C₃×#14578902	C₃	semidirect
8. #21419322	C₃	semidirect
9. C₃×C₃₁×#731	C₃	semidirect
87. C₃×#731	C₃₁⋊C₃	semidirect
115. C₃₁	#10624751	semidirect

H ≅ G/N

type

1. #21419438

C₃

semidirect

2. #21419409

C₃

semidirect

4. #21419402

C₃

semidirect

5. #21419401

C₃

semidirect

6. #21419400

C₃

semidirect

7. C₃×#14578902

C₃

semidirect

8. #21419322

C₃

semidirect

9. C₃×C₃₁×#731

C₃

semidirect

87. C₃×#731

C₃₁⋊C₃

semidirect

115. C₃₁

#10624751

semidirect

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

subgroup	index
#21419438	3
#21419409	3
#21419405	3
#21419402	3
#21419401	3
#21419400	3
C₃×#14578902	3
#21419322	3
C₃×C₃₁×#731	3
#10624751	31

subgroup

index

C₃×C₃₁×#731

Subgroups — #36381360

Conjugacy classes of subgroups

Complements

Maximal subgroups