Group #23744741 — subgroups

Subgroups — C₁₃×#10624590 #23744741

Order 9477.

560 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 86 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×C₁₃×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃²×C₁₃×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	C₁₃×#6891	C₃	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#6891	C₃	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#6857	C₃	1	9	yes	maximal, complemented
6	C₃⁴×C₁₃	—	3	1	no	—
7	C₃⁴×C₁₃	C₃²	1	1	yes	characteristic
8	C₃×C₁₃×3₋¹⁺²	—	3	1	no	—
9	C₃×C₁₃×3₋¹⁺²	—	3	1	no	—
10	C₃×C₁₃×3₋¹⁺²	—	3	9	no	—
11	C₃×C₁₃×3₋¹⁺²	—	3	1	no	—
12	C₃×C₁₃×3₋¹⁺²	C₃²	1	3	yes	complemented
13	C₃×C₁₃×3₋¹⁺²	C₃²	1	3	yes	complemented
14	C₃×C₁₃×H(3,𝔽₃)	—	3	1	no	—
15	C₃×C₁₃×H(3,𝔽₃)	—	3	1	no	—
16	C₃×C₁₃×H(3,𝔽₃)	—	3	9	no	—
17	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	3	yes	—
18	C₃²×C₉×C₁₃	—	3	1	no	—
19	C₁₃×C₃²⋊C₉	—	3	1	no	—
20	C₁₃×C₃²⋊C₉	—	3	1	no	—
21	C₁₃×C₃²⋊C₉	—	3	1	no	—
22	C₁₃×C₃²⋊C₉	—	3	1	no	—
23	C₁₃×C₃²⋊C₉	—	3	1	no	—
24	C₁₃×C₃²⋊C₉	C₃²	1	3	yes	complemented
25	#10624590	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₃³×C₁₃	—	3	9	no	—
27	C₃³×C₁₃	—	3	3	no	—
28	C₃³×C₁₃	—	3	3	no	—
29	C₃³×C₁₃	—	3	3	no	—
30	C₃³×C₁₃	—	9	1	no	—
31	C₃³×C₁₃	—	9	1	no	—
32	C₃³×C₁₃	—	9	3	no	—
33	C₃³×C₁₃	—	9	3	no	—
34	C₃³×C₁₃	—	9	3	no	—
35	C₃³×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
36	C₃³×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
37	C₃³×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
38	C₃³×C₁₃	C₃³	1	1	yes	characteristic
39	C₁₃×3₋¹⁺²	—	3	9	no	—
40	C₁₃×3₋¹⁺²	—	3	9	no	—
41	C₁₃×3₋¹⁺²	—	3	9	no	—
42	C₁₃×3₋¹⁺²	—	9	3	no	—
43	C₁₃×3₋¹⁺²	—	9	3	no	—
44	C₁₃×H(3,𝔽₃)	—	3	9	no	—
45	C₁₃×H(3,𝔽₃)	—	3	9	no	—
46	C₁₃×H(3,𝔽₃)	—	3	9	no	—
47	C₃×C₉×C₁₃	—	3	9	no	—
48	C₃×C₉×C₁₃	—	3	3	no	—
49	C₃×C₉×C₁₃	—	3	3	no	—
50	C₃×C₉×C₁₃	—	3	3	no	—
51	C₃²×3₋¹⁺²	C₃×C₁₃	1	1	yes	characteristic, complemented
52	C₃²×H(3,𝔽₃)	C₃×C₁₃	1	1	yes	characteristic
53	#6891	C₃×C₁₃	1	1	yes	characteristic, complemented
54	#6891	C₃×C₁₃	1	1	yes	characteristic, complemented
55	#6857	C₃×C₁₃	1	9	yes	complemented
56	C₃²×C₁₃	—	3	1	no	—
57	C₃²×C₁₃	—	3	1	no	—
58	C₃²×C₁₃	—	3	1	no	—
59	C₃²×C₁₃	—	3	1	no	—
60	C₃²×C₁₃	—	3	1	no	—
61	C₃²×C₁₃	—	3	1	no	—
62	C₃²×C₁₃	—	3	1	no	—
63	C₃²×C₁₃	—	3	1	no	—
64	C₃²×C₁₃	—	3	9	no	—
65	C₃²×C₁₃	—	3	1	no	—
66	C₃²×C₁₃	—	3	1	no	—
67	C₃²×C₁₃	—	3	1	no	—
68	C₃²×C₁₃	—	3	1	no	—
69	C₃²×C₁₃	—	3	1	no	—
70	C₃²×C₁₃	—	3	1	no	—
71	C₃²×C₁₃	—	9	9	no	—
72	C₃²×C₁₃	—	9	27	no	—
73	C₃²×C₁₃	—	9	3	no	—
74	C₃²×C₁₃	—	9	3	no	—
75	C₃²×C₁₃	—	9	3	no	—
76	C₃²×C₁₃	AΣL(1,𝔽₂₇)	1	3	yes	—
77	C₃²×C₁₃	AΣL(1,𝔽₂₇)	1	3	yes	—
78	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
79	C₉×C₁₃	—	3	9	no	—
80	C₉×C₁₃	—	9	3	no	—
81	C₉×C₁₃	—	9	3	no	—
82	C₃⁴	—	3	1	no	—
83	C₃⁴	C₃²×C₁₃	1	1	yes	characteristic
84	C₃×3₋¹⁺²	—	3	1	no	—
85	C₃×3₋¹⁺²	—	3	1	no	—
86	C₃×3₋¹⁺²	—	3	9	no	—
87	C₃×3₋¹⁺²	—	3	1	no	—
88	C₃×3₋¹⁺²	C₃²×C₁₃	1	3	yes	complemented
89	C₃×3₋¹⁺²	C₃²×C₁₃	1	3	yes	complemented
90	C₃×H(3,𝔽₃)	—	3	1	no	—
91	C₃×H(3,𝔽₃)	—	3	1	no	—
92	C₃×H(3,𝔽₃)	—	3	9	no	—
93	C₃×H(3,𝔽₃)	C₃²×C₁₃	1	3	yes	—
94	C₃²×C₉	—	3	1	no	—
95	C₃²⋊C₉	—	3	1	no	—
96	C₃²⋊C₉	—	3	1	no	—
97	C₃²⋊C₉	—	3	1	no	—
98	C₃²⋊C₉	—	3	1	no	—
99	C₃²⋊C₉	—	3	1	no	—
100	C₃²⋊C₉	C₃²×C₁₃	1	3	yes	complemented
101	C₃×C₁₃	—	3	3	no	—
102	C₃×C₁₃	—	3	3	no	—
103	C₃×C₁₃	—	9	1	no	—
104	C₃×C₁₃	—	9	1	no	—
105	C₃×C₁₃	—	9	9	no	—
106	C₃×C₁₃	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
107	C₃×C₁₃	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
108	C₃×C₁₃	#6909	1	1	yes	characteristic
109	C₃×C₁₃	#6910	1	1	yes	characteristic
110	C₃³	—	3	9	no	—
111	C₃³	—	3	3	no	—
112	C₃³	—	3	3	no	—
113	C₃³	—	3	3	no	—
114	C₃³	—	9	1	no	—
115	C₃³	—	9	1	no	—
116	C₃³	—	9	3	no	—
117	C₃³	—	9	3	no	—
118	C₃³	—	9	3	no	—
119	C₃³	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
120	C₃³	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
121	C₃³	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
122	C₃³	C₃³×C₁₃	1	1	yes	characteristic
123	3₋¹⁺²	—	3	9	no	—
124	3₋¹⁺²	—	3	9	no	—
125	3₋¹⁺²	—	3	9	no	—
126	3₋¹⁺²	—	9	3	no	—
127	3₋¹⁺²	—	9	3	no	—
128	H(3,𝔽₃)	—	3	9	no	—
129	H(3,𝔽₃)	—	3	9	no	—
130	H(3,𝔽₃)	—	3	9	no	—
131	C₃×C₉	—	3	9	no	—
132	C₃×C₉	—	3	3	no	—
133	C₃×C₉	—	3	3	no	—
134	C₃×C₉	—	3	3	no	—
135	C₁₃	#10624590	1	1	yes	characteristic, direct product factor, Sylow
136	C₃²	—	3	1	no	—
137	C₃²	—	3	1	no	—
138	C₃²	—	3	1	no	—
139	C₃²	—	3	1	no	—
140	C₃²	—	3	1	no	—
141	C₃²	—	3	1	no	—
142	C₃²	—	3	1	no	—
143	C₃²	—	3	1	no	—
144	C₃²	—	3	9	no	—
145	C₃²	—	3	1	no	—
146	C₃²	—	3	1	no	—
147	C₃²	—	3	1	no	—
148	C₃²	—	3	1	no	—
149	C₃²	—	3	1	no	—
150	C₃²	—	3	1	no	—
151	C₃²	—	9	9	no	—
152	C₃²	—	9	27	no	—
153	C₃²	—	9	3	no	—
154	C₃²	—	9	3	no	—
155	C₃²	—	9	3	no	—
156	C₃²	C₁₃×AΣL(1,𝔽₂₇)	1	3	yes	—
157	C₃²	C₁₃×AΣL(1,𝔽₂₇)	1	3	yes	—
158	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
159	C₉	—	3	9	no	—
160	C₉	—	9	3	no	—
161	C₉	—	9	3	no	—
162	C₃	—	3	3	no	—
163	C₃	—	3	3	no	—
164	C₃	—	9	1	no	—
165	C₃	—	9	1	no	—
166	C₃	—	9	9	no	—
167	C₃	C₃×C₁₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
168	C₃	C₃×C₁₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
169	C₃	C₁₃×#6909	1	1	yes	characteristic
170	C₃	C₁₃×#6910	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₁₃×3₋¹⁺²

C₃

yes

maximal, characteristic, complemented

C₃²×C₁₃×H(3,𝔽₃)

C₃

yes

maximal, characteristic

C₁₃×#6891

C₃

yes

maximal, characteristic, complemented

C₁₃×#6891

C₃

yes

maximal, characteristic, complemented

C₁₃×#6857

C₃

yes

maximal, complemented

C₃⁴×C₁₃

—

C₃⁴×C₁₃

C₃²

yes

characteristic

C₃×C₁₃×3₋¹⁺²

—

C₃×C₁₃×3₋¹⁺²

—

C₃×C₁₃×3₋¹⁺²

—

C₃×C₁₃×3₋¹⁺²

—

C₃×C₁₃×3₋¹⁺²

C₃²

yes

complemented

C₃×C₁₃×3₋¹⁺²

C₃²

yes

complemented

C₃×C₁₃×H(3,𝔽₃)

—

C₃×C₁₃×H(3,𝔽₃)

—

C₃×C₁₃×H(3,𝔽₃)

—

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

—

C₃²×C₉×C₁₃

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

C₃²

yes

complemented

#10624590

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

yes

C₃³×C₁₃

yes

C₃³×C₁₃

yes

C₃³×C₁₃

C₃³

yes

characteristic

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃²×3₋¹⁺²

C₃×C₁₃

yes

characteristic, complemented

C₃²×H(3,𝔽₃)

C₃×C₁₃

yes

characteristic

#6891

C₃×C₁₃

yes

characteristic, complemented

#6891

C₃×C₁₃

yes

characteristic, complemented

#6857

C₃×C₁₃

yes

complemented

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

AΣL(1,𝔽₂₇)

yes

—

C₃²×C₁₃

AΣL(1,𝔽₂₇)

yes

—

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₃⁴

—

C₃⁴

C₃²×C₁₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃²×C₁₃

yes

complemented

C₃×3₋¹⁺²

C₃²×C₁₃

yes

complemented

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃²×C₁₃

yes

—

C₃²×C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

100

C₃²⋊C₉

C₃²×C₁₃

yes

complemented

101

C₃×C₁₃

—

102

C₃×C₁₃

—

103

C₃×C₁₃

—

104

C₃×C₁₃

—

105

C₃×C₁₃

—

106

C₃×C₁₃

C₃×AΣL(1,𝔽₂₇)

yes

characteristic

107

C₃×C₁₃

C₃×AΣL(1,𝔽₂₇)

yes

characteristic

108

C₃×C₁₃

#6909

yes

characteristic

109

C₃×C₁₃

#6910

yes

characteristic

110

C₃³

—

111

C₃³

—

112

C₃³

—

113

C₃³

—

114

C₃³

—

115

C₃³

—

116

C₃³

—

117

C₃³

—

118

C₃³

—

119

C₃³

C₁₃×H(3,𝔽₃)

yes

characteristic

120

C₃³

C₁₃×H(3,𝔽₃)

yes

characteristic

121

C₃³

C₁₃×H(3,𝔽₃)

yes

characteristic

122

C₃³

C₃³×C₁₃

yes

characteristic

123

3₋¹⁺²

—

124

3₋¹⁺²

—

125

3₋¹⁺²

—

126

3₋¹⁺²

—

127

3₋¹⁺²

—

128

H(3,𝔽₃)

—

129

H(3,𝔽₃)

—

130

H(3,𝔽₃)

—

131

C₃×C₉

—

132

C₃×C₉

—

133

C₃×C₉

—

134

C₃×C₉

—

135

C₁₃

#10624590

yes

characteristic, direct product factor, Sylow

136

C₃²

—

137

C₃²

—

138

C₃²

—

139

C₃²

—

140

C₃²

—

141

C₃²

—

142

C₃²

—

143

C₃²

—

144

C₃²

—

145

C₃²

—

146

C₃²

—

147

C₃²

—

148

C₃²

—

149

C₃²

—

150

C₃²

—

151

C₃²

—

152

C₃²

—

153

C₃²

—

154

C₃²

—

155

C₃²

—

156

C₃²

C₁₃×AΣL(1,𝔽₂₇)

yes

—

157

C₃²

C₁₃×AΣL(1,𝔽₂₇)

yes

—

158

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

159

C₉

—

160

C₉

—

161

C₉

—

162

C₃

—

163

C₃

—

164

C₃

—

165

C₃

—

166

C₃

—

167

C₃

C₃×C₁₃×AΣL(1,𝔽₂₇)

yes

characteristic

168

C₃

C₃×C₁₃×AΣL(1,𝔽₂₇)

yes

characteristic

169

C₃

C₁₃×#6909

yes

characteristic

170

C₃

C₁₃×#6910

yes

characteristic

	subgroup	G/H
center, socle	C₃²×C₁₃	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₁₃
prime radical	C₃²×C₁₃×H(3,𝔽₃)	C₃
LCS	G ▷ C₃³ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×C₁₃ ◁ C₃⁴×C₁₃ ◁ G

subgroup

G/H

center, socle

C₃²×C₁₃

C₃×H(3,𝔽₃)

commutator, Frattini, cosocle

C₃³

C₃³×C₁₃

prime radical

C₃²×C₁₃×H(3,𝔽₃)

C₃

LCS

G ▷ C₃³ ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₃²×C₁₃ ◁ C₃⁴×C₁₃ ◁ G

N	H ≅ G/N	type
1. C₃²×C₁₃×3₋¹⁺²	C₃	semidirect
3. C₁₃×#6891	C₃	semidirect
4. C₁₃×#6891	C₃	semidirect
5. C₁₃×#6857	C₃	semidirect
12. C₃×C₁₃×3₋¹⁺²	C₃²	semidirect
13. C₃×C₁₃×3₋¹⁺²	C₃²	semidirect
24. C₁₃×C₃²⋊C₉	C₃²	semidirect
25. #10624590	C₁₃	direct (N × H)
51. C₃²×3₋¹⁺²	C₃×C₁₃	semidirect
53. #6891	C₃×C₁₃	semidirect
54. #6891	C₃×C₁₃	semidirect
55. #6857	C₃×C₁₃	semidirect
88. C₃×3₋¹⁺²	C₃²×C₁₃	semidirect
89. C₃×3₋¹⁺²	C₃²×C₁₃	semidirect
100. C₃²⋊C₉	C₃²×C₁₃	semidirect
135. C₁₃	#10624590	direct (N × H)

H ≅ G/N

type

1. C₃²×C₁₃×3₋¹⁺²

C₃

semidirect

3. C₁₃×#6891

C₃

semidirect

4. C₁₃×#6891

C₃

semidirect

5. C₁₃×#6857

C₃

semidirect

12. C₃×C₁₃×3₋¹⁺²

C₃²

semidirect

13. C₃×C₁₃×3₋¹⁺²

C₃²

semidirect

24. C₁₃×C₃²⋊C₉

C₃²

semidirect

25. #10624590

C₁₃

direct (N × H)

51. C₃²×3₋¹⁺²

C₃×C₁₃

semidirect

53. #6891

C₃×C₁₃

semidirect

54. #6891

C₃×C₁₃

semidirect

55. #6857

C₃×C₁₃

semidirect

88. C₃×3₋¹⁺²

C₃²×C₁₃

semidirect

89. C₃×3₋¹⁺²

C₃²×C₁₃

semidirect

100. C₃²⋊C₉

C₃²×C₁₃

semidirect

135. C₁₃

#10624590

direct (N × H)

Subgroups — C₁₃×#10624590 #23744741

Conjugacy classes of subgroups

Complements

Subgroups — C13×#10624590 #23744741

Conjugacy classes of subgroups

Complements

Subgroups — C₁₃×#10624590 #23744741