Group #23744949 — subgroups

Subgroups — C₁₃×#10624798 #23744949

Order 9477.

368 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 92 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	C₃×C₁₃×AΣL(1,𝔽₂₇)	C₃	1	1	yes	maximal, characteristic
3	C₁₃×#6894	C₃	1	1	yes	maximal, characteristic, complemented
4	C₁₃×C₉⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
5	C₉×C₁₃×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
6	C₁₃×3₋¹⁺²⋊C₉	C₃	1	2	yes	maximal, complemented
7	C₁₃×3₋¹⁺²⋊C₉	C₃	1	2	yes	maximal, complemented
8	C₁₃×H(3,𝔽₃)⋊C₉	C₃	1	2	yes	maximal, complemented
9	C₁₃×#6866	C₃	1	2	yes	maximal, complemented
10	C₃⁴×C₁₃	C₃²	1	1	yes	characteristic
11	C₃×C₁₃×3₋¹⁺²	—	3	9	no	—
12	C₃×C₁₃×3₋¹⁺²	—	3	2	no	—
13	C₃×C₁₃×3₋¹⁺²	—	3	1	no	—
14	C₃×C₁₃×3₋¹⁺²	—	3	1	no	—
15	C₃×C₁₃×3₋¹⁺²	C₃²	1	1	yes	characteristic
16	C₃×C₁₃×3₋¹⁺²	C₃²	1	1	yes	characteristic
17	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
18	C₃²×C₉×C₁₃	C₃²	1	2	yes	—
19	C₃²×C₉×C₁₃	C₃²	1	1	yes	characteristic
20	C₁₃×AΣL(1,𝔽₂₇)	—	3	3	no	—
21	C₁₃×C₉⋊C₉	—	3	2	no	—
22	C₁₃×C₉⋊C₉	—	3	2	no	—
23	C₁₃×C₉⋊C₉	—	3	1	no	—
24	C₁₃×C₃²⋊C₉	—	3	2	no	—
25	C₁₃×C₃²⋊C₉	C₃²	1	2	yes	—
26	C₁₃×C₃²⋊C₉	C₃²	1	2	yes	—
27	C₁₃×C₃²⋊C₉	C₃²	1	2	yes	—
28	C₉²×C₁₃	—	3	1	no	—
29	#10624798	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₃³×C₁₃	—	3	3	no	—
31	C₃³×C₁₃	—	3	1	no	—
32	C₃³×C₁₃	—	3	1	no	—
33	C₃³×C₁₃	—	3	1	no	—
34	C₃³×C₁₃	—	3	2	no	—
35	C₃³×C₁₃	—	9	2	no	—
36	C₃³×C₁₃	C₃³	1	1	yes	characteristic
37	C₁₃×3₋¹⁺²	—	3	18	no	—
38	C₁₃×3₋¹⁺²	—	3	2	no	—
39	C₁₃×3₋¹⁺²	—	3	9	no	—
40	C₁₃×3₋¹⁺²	—	3	1	no	—
41	C₁₃×3₋¹⁺²	—	3	2	no	—
42	C₁₃×3₋¹⁺²	—	9	3	no	—
43	C₁₃×3₋¹⁺²	3₋¹⁺²	1	3	yes	complemented
44	C₁₃×H(3,𝔽₃)	—	3	1	no	—
45	C₁₃×H(3,𝔽₃)	—	3	2	no	—
46	C₃×C₉×C₁₃	—	3	1	no	—
47	C₃×C₉×C₁₃	—	3	2	no	—
48	C₃×C₉×C₁₃	—	3	3	no	—
49	C₃×C₉×C₁₃	—	3	6	no	—
50	C₃×C₉×C₁₃	—	3	2	no	—
51	C₃×C₉×C₁₃	—	3	2	no	—
52	C₃×C₉×C₁₃	—	3	2	no	—
53	C₃×C₉×C₁₃	—	3	1	no	—
54	C₃×C₉×C₁₃	H(3,𝔽₃)	1	2	yes	—
55	C₃×C₉×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
56	C₃²×3₋¹⁺²	C₃×C₁₃	1	1	yes	characteristic, complemented
57	C₃×AΣL(1,𝔽₂₇)	C₃×C₁₃	1	1	yes	characteristic
58	#6894	C₃×C₁₃	1	1	yes	characteristic, complemented
59	C₉⋊H(3,𝔽₃)	C₃×C₁₃	1	1	yes	characteristic, complemented
60	C₉×3₋¹⁺²	C₃×C₁₃	1	1	yes	characteristic, complemented
61	3₋¹⁺²⋊C₉	C₃×C₁₃	1	2	yes	complemented
62	3₋¹⁺²⋊C₉	C₃×C₁₃	1	2	yes	complemented
63	H(3,𝔽₃)⋊C₉	C₃×C₁₃	1	2	yes	complemented
64	#6866	C₃×C₁₃	1	2	yes	complemented
65	C₃²×C₁₃	—	3	2	no	—
66	C₃²×C₁₃	—	3	1	no	—
67	C₃²×C₁₃	—	3	3	no	—
68	C₃²×C₁₃	—	9	3	no	—
69	C₃²×C₁₃	—	9	6	no	—
70	C₃²×C₁₃	—	9	2	no	—
71	C₃²×C₁₃	—	9	1	no	—
72	C₃²×C₁₃	—	9	2	no	—
73	C₃²×C₁₃	—	9	1	no	—
74	C₃²×C₁₃	—	9	1	no	—
75	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
76	C₃²×C₁₃	C₃×3₋¹⁺²	1	1	yes	characteristic
77	C₃²×C₁₃	#737	1	2	yes	—
78	C₉×C₁₃	—	3	3	no	—
79	C₉×C₁₃	—	3	2	no	—
80	C₉×C₁₃	—	3	6	no	—
81	C₉×C₁₃	—	9	2	no	—
82	C₉×C₁₃	—	9	2	no	—
83	C₉×C₁₃	—	9	1	no	—
84	C₉×C₁₃	—	9	2	no	—
85	C₉×C₁₃	AΣL(1,𝔽₂₇)	1	3	yes	complemented
86	C₃⁴	C₃²×C₁₃	1	1	yes	characteristic
87	C₃×3₋¹⁺²	—	3	9	no	—
88	C₃×3₋¹⁺²	—	3	2	no	—
89	C₃×3₋¹⁺²	—	3	1	no	—
90	C₃×3₋¹⁺²	—	3	1	no	—
91	C₃×3₋¹⁺²	C₃²×C₁₃	1	1	yes	characteristic
92	C₃×3₋¹⁺²	C₃²×C₁₃	1	1	yes	characteristic
93	C₃×H(3,𝔽₃)	C₃²×C₁₃	1	1	yes	characteristic
94	C₃²×C₉	C₃²×C₁₃	1	2	yes	—
95	C₃²×C₉	C₃²×C₁₃	1	1	yes	characteristic
96	AΣL(1,𝔽₂₇)	—	3	3	no	—
97	C₉⋊C₉	—	3	2	no	—
98	C₉⋊C₉	—	3	2	no	—
99	C₉⋊C₉	—	3	1	no	—
100	C₃²⋊C₉	—	3	2	no	—
101	C₃²⋊C₉	C₃²×C₁₃	1	2	yes	—
102	C₃²⋊C₉	C₃²×C₁₃	1	2	yes	—
103	C₃²⋊C₉	C₃²×C₁₃	1	2	yes	—
104	C₉²	—	3	1	no	—
105	C₃×C₁₃	—	3	1	no	—
106	C₃×C₁₃	—	3	2	no	—
107	C₃×C₁₃	—	9	3	no	—
108	C₃×C₁₃	—	27	1	no	—
109	C₃×C₁₃	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
110	C₃×C₁₃	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
111	C₃×C₁₃	#6913	1	2	yes	—
112	C₃³	—	3	3	no	—
113	C₃³	—	3	1	no	—
114	C₃³	—	3	1	no	—
115	C₃³	—	3	1	no	—
116	C₃³	—	3	2	no	—
117	C₃³	—	9	2	no	—
118	C₃³	C₃³×C₁₃	1	1	yes	characteristic
119	3₋¹⁺²	—	3	18	no	—
120	3₋¹⁺²	—	3	2	no	—
121	3₋¹⁺²	—	3	9	no	—
122	3₋¹⁺²	—	3	1	no	—
123	3₋¹⁺²	—	3	2	no	—
124	3₋¹⁺²	—	9	3	no	—
125	3₋¹⁺²	C₁₃×3₋¹⁺²	1	3	yes	complemented
126	H(3,𝔽₃)	—	3	1	no	—
127	H(3,𝔽₃)	—	3	2	no	—
128	C₃×C₉	—	3	1	no	—
129	C₃×C₉	—	3	2	no	—
130	C₃×C₉	—	3	3	no	—
131	C₃×C₉	—	3	6	no	—
132	C₃×C₉	—	3	2	no	—
133	C₃×C₉	—	3	2	no	—
134	C₃×C₉	—	3	2	no	—
135	C₃×C₉	—	3	1	no	—
136	C₃×C₉	C₁₃×H(3,𝔽₃)	1	2	yes	—
137	C₃×C₉	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
138	C₁₃	#10624798	1	1	yes	characteristic, direct product factor, Sylow
139	C₃²	—	3	1	no	—
140	C₃²	—	3	2	no	—
141	C₃²	—	3	3	no	—
142	C₃²	—	9	3	no	—
143	C₃²	—	9	6	no	—
144	C₃²	—	9	1	no	—
145	C₃²	—	9	2	no	—
146	C₃²	—	9	1	no	—
147	C₃²	—	9	1	no	—
148	C₃²	—	9	2	no	—
149	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
150	C₃²	C₃×C₁₃×3₋¹⁺²	1	1	yes	characteristic
151	C₃²	C₁₃×#737	1	2	yes	—
152	C₉	—	3	3	no	—
153	C₉	—	3	2	no	—
154	C₉	—	3	6	no	—
155	C₉	—	9	1	no	—
156	C₉	—	9	2	no	—
157	C₉	—	9	2	no	—
158	C₉	—	9	2	no	—
159	C₉	C₁₃×AΣL(1,𝔽₂₇)	1	3	yes	complemented
160	C₃	—	3	1	no	—
161	C₃	—	3	2	no	—
162	C₃	—	9	3	no	—
163	C₃	—	27	1	no	—
164	C₃	C₁₃×C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
165	C₃	C₃×C₁₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
166	C₃	C₁₃×#6913	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₁₃×3₋¹⁺²

C₃

yes

maximal, characteristic, complemented

C₃×C₁₃×AΣL(1,𝔽₂₇)

C₃

yes

maximal, characteristic

C₁₃×#6894

C₃

yes

maximal, characteristic, complemented

C₁₃×C₉⋊H(3,𝔽₃)

C₃

yes

maximal, characteristic, complemented

C₉×C₁₃×3₋¹⁺²

C₃

yes

maximal, characteristic, complemented

C₁₃×3₋¹⁺²⋊C₉

C₃

yes

maximal, complemented

C₁₃×3₋¹⁺²⋊C₉

C₃

yes

maximal, complemented

C₁₃×H(3,𝔽₃)⋊C₉

C₃

yes

maximal, complemented

C₁₃×#6866

C₃

yes

maximal, complemented

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

characteristic

C₃×C₁₃×3₋¹⁺²

C₃²

yes

characteristic

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

characteristic

C₃²×C₉×C₁₃

C₃²

yes

—

C₃²×C₉×C₁₃

C₃²

yes

characteristic

C₁₃×AΣL(1,𝔽₂₇)

—

C₁₃×C₉⋊C₉

—

C₁₃×C₉⋊C₉

—

C₁₃×C₉⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

C₃²

yes

—

C₁₃×C₃²⋊C₉

C₃²

yes

—

C₁₃×C₃²⋊C₉

C₃²

yes

—

C₉²×C₁₃

—

#10624798

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

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

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

3₋¹⁺²

yes

complemented

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

H(3,𝔽₃)

yes

—

C₃×C₉×C₁₃

H(3,𝔽₃)

yes

characteristic

C₃²×3₋¹⁺²

C₃×C₁₃

yes

characteristic, complemented

C₃×AΣL(1,𝔽₂₇)

C₃×C₁₃

yes

characteristic

#6894

C₃×C₁₃

yes

characteristic, complemented

C₉⋊H(3,𝔽₃)

C₃×C₁₃

yes

characteristic, complemented

C₉×3₋¹⁺²

C₃×C₁₃

yes

characteristic, complemented

3₋¹⁺²⋊C₉

C₃×C₁₃

yes

complemented

3₋¹⁺²⋊C₉

C₃×C₁₃

yes

complemented

H(3,𝔽₃)⋊C₉

C₃×C₁₃

yes

complemented

#6866

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

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₁₃

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₁₃

#737

yes

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

AΣL(1,𝔽₂₇)

yes

complemented

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

characteristic

C₃×3₋¹⁺²

C₃²×C₁₃

yes

characteristic

C₃×H(3,𝔽₃)

C₃²×C₁₃

yes

characteristic

C₃²×C₉

C₃²×C₁₃

yes

—

C₃²×C₉

C₃²×C₁₃

yes

characteristic

AΣL(1,𝔽₂₇)

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

100

C₃²⋊C₉

—

101

C₃²⋊C₉

C₃²×C₁₃

yes

—

102

C₃²⋊C₉

C₃²×C₁₃

yes

—

103

C₃²⋊C₉

C₃²×C₁₃

yes

—

104

C₉²

—

105

C₃×C₁₃

—

106

C₃×C₁₃

—

107

C₃×C₁₃

—

108

C₃×C₁₃

—

109

C₃×C₁₃

C₉⋊H(3,𝔽₃)

yes

characteristic

110

C₃×C₁₃

C₃×AΣL(1,𝔽₂₇)

yes

characteristic

111

C₃×C₁₃

#6913

yes

—

112

C₃³

—

113

C₃³

—

114

C₃³

—

115

C₃³

—

116

C₃³

—

117

C₃³

—

118

C₃³

C₃³×C₁₃

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

C₁₃×3₋¹⁺²

yes

complemented

126

H(3,𝔽₃)

—

127

H(3,𝔽₃)

—

128

C₃×C₉

—

129

C₃×C₉

—

130

C₃×C₉

—

131

C₃×C₉

—

132

C₃×C₉

—

133

C₃×C₉

—

134

C₃×C₉

—

135

C₃×C₉

—

136

C₃×C₉

C₁₃×H(3,𝔽₃)

yes

—

137

C₃×C₉

C₁₃×H(3,𝔽₃)

yes

characteristic

138

C₁₃

#10624798

yes

characteristic, direct product factor, Sylow

139

C₃²

—

140

C₃²

—

141

C₃²

—

142

C₃²

—

143

C₃²

—

144

C₃²

—

145

C₃²

—

146

C₃²

—

147

C₃²

—

148

C₃²

—

149

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

150

C₃²

C₃×C₁₃×3₋¹⁺²

yes

characteristic

151

C₃²

C₁₃×#737

yes

—

152

C₉

—

153

C₉

—

154

C₉

—

155

C₉

—

156

C₉

—

157

C₉

—

158

C₉

—

159

C₉

C₁₃×AΣL(1,𝔽₂₇)

yes

complemented

160

C₃

—

161

C₃

—

162

C₃

—

163

C₃

—

164

C₃

C₁₃×C₉⋊H(3,𝔽₃)

yes

characteristic

165

C₃

C₃×C₁₃×AΣL(1,𝔽₂₇)

yes

characteristic

166

C₃

C₁₃×#6913

yes

—

	subgroup	G/H
center, socle	C₃²×C₁₃	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₁₃
prime radical	C₃×C₁₃×AΣL(1,𝔽₂₇)	C₃
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₁₃ ◁ C₃²×C₉×C₁₃ ◁ G

subgroup

G/H

center, socle

C₃²×C₁₃

C₃×H(3,𝔽₃)

commutator, Frattini, cosocle

C₃³

C₃³×C₁₃

prime radical

C₃×C₁₃×AΣL(1,𝔽₂₇)

C₃

LCS

G ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃²×C₁₃ ◁ C₃²×C₉×C₁₃ ◁ G

N	H ≅ G/N	type
1. C₃²×C₁₃×3₋¹⁺²	C₃	semidirect
3. C₁₃×#6894	C₃	semidirect
4. C₁₃×C₉⋊H(3,𝔽₃)	C₃	semidirect
5. C₉×C₁₃×3₋¹⁺²	C₃	semidirect
6. C₁₃×3₋¹⁺²⋊C₉	C₃	semidirect
7. C₁₃×3₋¹⁺²⋊C₉	C₃	semidirect
8. C₁₃×H(3,𝔽₃)⋊C₉	C₃	semidirect
9. C₁₃×#6866	C₃	semidirect
29. #10624798	C₁₃	direct (N × H)
43. C₁₃×3₋¹⁺²	3₋¹⁺²	semidirect
56. C₃²×3₋¹⁺²	C₃×C₁₃	semidirect
58. #6894	C₃×C₁₃	semidirect
59. C₉⋊H(3,𝔽₃)	C₃×C₁₃	semidirect
60. C₉×3₋¹⁺²	C₃×C₁₃	semidirect
61. 3₋¹⁺²⋊C₉	C₃×C₁₃	semidirect
62. 3₋¹⁺²⋊C₉	C₃×C₁₃	semidirect
63. H(3,𝔽₃)⋊C₉	C₃×C₁₃	semidirect
64. #6866	C₃×C₁₃	semidirect
85. C₉×C₁₃	AΣL(1,𝔽₂₇)	semidirect
125. 3₋¹⁺²	C₁₃×3₋¹⁺²	semidirect
138. C₁₃	#10624798	direct (N × H)
159. C₉	C₁₃×AΣL(1,𝔽₂₇)	semidirect

H ≅ G/N

type

1. C₃²×C₁₃×3₋¹⁺²

C₃

semidirect

3. C₁₃×#6894

C₃

semidirect

4. C₁₃×C₉⋊H(3,𝔽₃)

C₃

semidirect

5. C₉×C₁₃×3₋¹⁺²

C₃

semidirect

6. C₁₃×3₋¹⁺²⋊C₉

C₃

semidirect

7. C₁₃×3₋¹⁺²⋊C₉

C₃

semidirect

8. C₁₃×H(3,𝔽₃)⋊C₉

C₃

semidirect

9. C₁₃×#6866

C₃

semidirect

29. #10624798

C₁₃

direct (N × H)

43. C₁₃×3₋¹⁺²

3₋¹⁺²

semidirect

56. C₃²×3₋¹⁺²

C₃×C₁₃

semidirect

58. #6894

C₃×C₁₃

semidirect

59. C₉⋊H(3,𝔽₃)

C₃×C₁₃

semidirect

60. C₉×3₋¹⁺²

C₃×C₁₃

semidirect

61. 3₋¹⁺²⋊C₉

C₃×C₁₃

semidirect

62. 3₋¹⁺²⋊C₉

C₃×C₁₃

semidirect

63. H(3,𝔽₃)⋊C₉

C₃×C₁₃

semidirect

64. #6866

C₃×C₁₃

semidirect

85. C₉×C₁₃

AΣL(1,𝔽₂₇)

semidirect

125. 3₋¹⁺²

C₁₃×3₋¹⁺²

semidirect

138. C₁₃

#10624798

direct (N × H)

159. C₉

C₁₃×AΣL(1,𝔽₂₇)

semidirect

Subgroups — C₁₃×#10624798 #23744949

Conjugacy classes of subgroups

Complements

Subgroups — C13×#10624798 #23744949

Conjugacy classes of subgroups

Complements

Subgroups — C₁₃×#10624798 #23744949