Group #23744760 — subgroups

Subgroups — C₁₃×#10624609 #23744760

Order 9477.

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

normal subgroups only

This group has 398 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	C₁₃×#6891	C₃	1	1	yes	maximal, characteristic, complemented
3	C₁₃×C₃²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
4	C₃×C₁₃×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#6857	C₃	1	9	yes	maximal, complemented
6	C₃⁴×C₁₃	C₃²	1	1	yes	characteristic
7	C₃×C₁₃×3₋¹⁺²	—	3	1	no	—
8	C₃×C₁₃×3₋¹⁺²	—	3	1	no	—
9	C₃×C₁₃×3₋¹⁺²	—	3	9	no	—
10	C₃×C₁₃×3₋¹⁺²	C₃²	1	3	yes	—
11	C₃×C₁₃×3₋¹⁺²	C₃²	1	3	yes	—
12	C₃×C₁₃×H(3,𝔽₃)	—	3	1	no	—
13	C₃×C₁₃×H(3,𝔽₃)	—	3	1	no	—
14	C₃×C₁₃×H(3,𝔽₃)	—	3	1	no	—
15	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	3	yes	—
16	C₃²×C₉×C₁₃	—	3	1	no	—
17	C₃²×C₉×C₁₃	—	3	1	no	—
18	C₁₃×C₃²⋊C₉	—	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	C₃²	1	3	yes	—
24	#10624609	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₃³×C₁₃	—	3	3	no	—
26	C₃³×C₁₃	—	3	3	no	—
27	C₃³×C₁₃	—	3	3	no	—
28	C₃³×C₁₃	—	9	1	no	—
29	C₃³×C₁₃	—	9	1	no	—
30	C₃³×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
31	C₃³×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
32	C₃³×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
33	C₃³×C₁₃	C₃³	1	1	yes	characteristic
34	C₁₃×3₋¹⁺²	—	3	9	no	—
35	C₁₃×3₋¹⁺²	—	3	9	no	—
36	C₁₃×3₋¹⁺²	—	3	9	no	—
37	C₁₃×3₋¹⁺²	—	9	3	no	—
38	C₁₃×H(3,𝔽₃)	—	9	3	no	—
39	C₁₃×H(3,𝔽₃)	—	9	3	no	—
40	C₁₃×H(3,𝔽₃)	—	9	3	no	—
41	C₁₃×H(3,𝔽₃)	—	9	3	no	—
42	C₃×C₉×C₁₃	—	3	9	no	—
43	C₃×C₉×C₁₃	—	3	3	no	—
44	C₃×C₉×C₁₃	—	3	3	no	—
45	C₃×C₉×C₁₃	—	3	3	no	—
46	C₃×C₉×C₁₃	—	9	3	no	—
47	C₃²×3₋¹⁺²	C₃×C₁₃	1	1	yes	characteristic, complemented
48	#6891	C₃×C₁₃	1	1	yes	characteristic, complemented
49	C₃²⋊H(3,𝔽₃)	C₃×C₁₃	1	1	yes	characteristic
50	C₃×C₃²⋊C₉	C₃×C₁₃	1	1	yes	characteristic, complemented
51	#6857	C₃×C₁₃	1	9	yes	complemented
52	C₃²×C₁₃	—	3	1	no	—
53	C₃²×C₁₃	—	3	1	no	—
54	C₃²×C₁₃	—	3	1	no	—
55	C₃²×C₁₃	—	3	1	no	—
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	1	no	—
65	C₃²×C₁₃	—	3	1	no	—
66	C₃²×C₁₃	—	9	9	no	—
67	C₃²×C₁₃	—	9	3	no	—
68	C₃²×C₁₃	—	9	3	no	—
69	C₃²×C₁₃	—	9	3	no	—
70	C₃²×C₁₃	—	9	3	no	—
71	C₃²×C₁₃	AΣL(1,𝔽₂₇)	1	3	yes	—
72	C₃²×C₁₃	#731	1	3	yes	—
73	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
74	C₉×C₁₃	—	3	9	no	—
75	C₉×C₁₃	—	9	3	no	—
76	C₉×C₁₃	—	9	3	no	—
77	C₃⁴	C₃²×C₁₃	1	1	yes	characteristic
78	C₃×3₋¹⁺²	—	3	1	no	—
79	C₃×3₋¹⁺²	—	3	1	no	—
80	C₃×3₋¹⁺²	—	3	9	no	—
81	C₃×3₋¹⁺²	C₃²×C₁₃	1	3	yes	—
82	C₃×3₋¹⁺²	C₃²×C₁₃	1	3	yes	—
83	C₃×H(3,𝔽₃)	—	3	1	no	—
84	C₃×H(3,𝔽₃)	—	3	1	no	—
85	C₃×H(3,𝔽₃)	—	3	1	no	—
86	C₃×H(3,𝔽₃)	C₃²×C₁₃	1	3	yes	—
87	C₃²×C₉	—	3	1	no	—
88	C₃²×C₉	—	3	1	no	—
89	C₃²⋊C₉	—	3	1	no	—
90	C₃²⋊C₉	—	3	1	no	—
91	C₃²⋊C₉	—	3	1	no	—
92	C₃²⋊C₉	—	3	1	no	—
93	C₃²⋊C₉	—	3	1	no	—
94	C₃²⋊C₉	C₃²×C₁₃	1	3	yes	—
95	C₃×C₁₃	—	3	3	no	—
96	C₃×C₁₃	—	3	3	no	—
97	C₃×C₁₃	—	9	1	no	—
98	C₃×C₁₃	—	9	1	no	—
99	C₃×C₁₃	—	27	3	no	—
100	C₃×C₁₃	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
101	C₃×C₁₃	C₃×#731	1	1	yes	characteristic
102	C₃×C₁₃	#6909	1	1	yes	characteristic
103	C₃×C₁₃	#6909	1	1	yes	characteristic
104	C₃³	—	3	3	no	—
105	C₃³	—	3	3	no	—
106	C₃³	—	3	3	no	—
107	C₃³	—	9	1	no	—
108	C₃³	—	9	1	no	—
109	C₃³	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
110	C₃³	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
111	C₃³	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
112	C₃³	C₃³×C₁₃	1	1	yes	characteristic
113	3₋¹⁺²	—	3	9	no	—
114	3₋¹⁺²	—	3	9	no	—
115	3₋¹⁺²	—	3	9	no	—
116	3₋¹⁺²	—	9	3	no	—
117	H(3,𝔽₃)	—	9	3	no	—
118	H(3,𝔽₃)	—	9	3	no	—
119	H(3,𝔽₃)	—	9	3	no	—
120	H(3,𝔽₃)	—	9	3	no	—
121	C₃×C₉	—	3	9	no	—
122	C₃×C₉	—	3	3	no	—
123	C₃×C₉	—	3	3	no	—
124	C₃×C₉	—	3	3	no	—
125	C₃×C₉	—	9	3	no	—
126	C₁₃	#10624609	1	1	yes	characteristic, direct product factor, Sylow
127	C₃²	—	3	1	no	—
128	C₃²	—	3	1	no	—
129	C₃²	—	3	1	no	—
130	C₃²	—	3	1	no	—
131	C₃²	—	3	1	no	—
132	C₃²	—	3	1	no	—
133	C₃²	—	3	1	no	—
134	C₃²	—	3	1	no	—
135	C₃²	—	3	1	no	—
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	—	9	9	no	—
142	C₃²	—	9	3	no	—
143	C₃²	—	9	3	no	—
144	C₃²	—	9	3	no	—
145	C₃²	—	9	3	no	—
146	C₃²	C₁₃×AΣL(1,𝔽₂₇)	1	3	yes	—
147	C₃²	C₁₃×#731	1	3	yes	—
148	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
149	C₉	—	3	9	no	—
150	C₉	—	9	3	no	—
151	C₉	—	9	3	no	—
152	C₃	—	3	3	no	—
153	C₃	—	3	3	no	—
154	C₃	—	9	1	no	—
155	C₃	—	9	1	no	—
156	C₃	—	27	3	no	—
157	C₃	C₃×C₁₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
158	C₃	C₃×C₁₃×#731	1	1	yes	characteristic
159	C₃	C₁₃×#6909	1	1	yes	characteristic
160	C₃	C₁₃×#6909	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₁₃×3₋¹⁺²

C₃

yes

maximal, characteristic, complemented

C₁₃×#6891

C₃

yes

maximal, characteristic, complemented

C₁₃×C₃²⋊H(3,𝔽₃)

C₃

yes

maximal, characteristic

C₃×C₁₃×C₃²⋊C₉

C₃

yes

maximal, characteristic, complemented

C₁₃×#6857

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

C₃²

yes

—

C₃×C₁₃×3₋¹⁺²

C₃²

yes

—

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

—

C₃²×C₉×C₁₃

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

C₃²

yes

—

#10624609

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

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×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃²×3₋¹⁺²

C₃×C₁₃

yes

characteristic, complemented

#6891

C₃×C₁₃

yes

characteristic, complemented

C₃²⋊H(3,𝔽₃)

C₃×C₁₃

yes

characteristic

C₃×C₃²⋊C₉

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

AΣL(1,𝔽₂₇)

yes

—

C₃²×C₁₃

#731

yes

—

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₃⁴

C₃²×C₁₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃²×C₁₃

yes

—

C₃×3₋¹⁺²

C₃²×C₁₃

yes

—

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

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃²×C₁₃

yes

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

100

C₃×C₁₃

C₃×AΣL(1,𝔽₂₇)

yes

characteristic

101

C₃×C₁₃

C₃×#731

yes

characteristic

102

C₃×C₁₃

#6909

yes

characteristic

103

C₃×C₁₃

#6909

yes

characteristic

104

C₃³

—

105

C₃³

—

106

C₃³

—

107

C₃³

—

108

C₃³

—

109

C₃³

C₁₃×H(3,𝔽₃)

yes

characteristic

110

C₃³

C₁₃×H(3,𝔽₃)

yes

characteristic

111

C₃³

C₁₃×H(3,𝔽₃)

yes

characteristic

112

C₃³

C₃³×C₁₃

yes

characteristic

113

3₋¹⁺²

—

114

3₋¹⁺²

—

115

3₋¹⁺²

—

116

3₋¹⁺²

—

117

H(3,𝔽₃)

—

118

H(3,𝔽₃)

—

119

H(3,𝔽₃)

—

120

H(3,𝔽₃)

—

121

C₃×C₉

—

122

C₃×C₉

—

123

C₃×C₉

—

124

C₃×C₉

—

125

C₃×C₉

—

126

C₁₃

#10624609

yes

characteristic, direct product factor, Sylow

127

C₃²

—

128

C₃²

—

129

C₃²

—

130

C₃²

—

131

C₃²

—

132

C₃²

—

133

C₃²

—

134

C₃²

—

135

C₃²

—

136

C₃²

—

137

C₃²

—

138

C₃²

—

139

C₃²

—

140

C₃²

—

141

C₃²

—

142

C₃²

—

143

C₃²

—

144

C₃²

—

145

C₃²

—

146

C₃²

C₁₃×AΣL(1,𝔽₂₇)

yes

—

147

C₃²

C₁₃×#731

yes

—

148

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

149

C₉

—

150

C₉

—

151

C₉

—

152

C₃

—

153

C₃

—

154

C₃

—

155

C₃

—

156

C₃

—

157

C₃

C₃×C₁₃×AΣL(1,𝔽₂₇)

yes

characteristic

158

C₃

C₃×C₁₃×#731

yes

characteristic

159

C₃

C₁₃×#6909

yes

characteristic

160

C₃

C₁₃×#6909

yes

characteristic

	subgroup	G/H
center, socle	C₃²×C₁₃	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₁₃
prime radical	C₁₃×C₃²⋊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

C₁₃×C₃²⋊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
2. C₁₃×#6891	C₃	semidirect
4. C₃×C₁₃×C₃²⋊C₉	C₃	semidirect
5. C₁₃×#6857	C₃	semidirect
24. #10624609	C₁₃	direct (N × H)
47. C₃²×3₋¹⁺²	C₃×C₁₃	semidirect
48. #6891	C₃×C₁₃	semidirect
50. C₃×C₃²⋊C₉	C₃×C₁₃	semidirect
51. #6857	C₃×C₁₃	semidirect
126. C₁₃	#10624609	direct (N × H)

H ≅ G/N

type

1. C₃²×C₁₃×3₋¹⁺²

C₃

semidirect

2. C₁₃×#6891

C₃

semidirect

4. C₃×C₁₃×C₃²⋊C₉

C₃

semidirect

5. C₁₃×#6857

C₃

semidirect

24. #10624609

C₁₃

direct (N × H)

47. C₃²×3₋¹⁺²

C₃×C₁₃

semidirect

48. #6891

C₃×C₁₃

semidirect

50. C₃×C₃²⋊C₉

C₃×C₁₃

semidirect

51. #6857

C₃×C₁₃

semidirect

126. C₁₃

#10624609

direct (N × H)

Subgroups — C₁₃×#10624609 #23744760

Conjugacy classes of subgroups

Complements

Subgroups — C13×#10624609 #23744760

Conjugacy classes of subgroups

Complements

Subgroups — C₁₃×#10624609 #23744760