Group #23744749 — subgroups

Subgroups — C₁₃×#10624598 #23744749

Order 9477.

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

normal subgroups only

This group has 302 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	C₁₃×#6891	C₃	1	1	yes	maximal, characteristic, complemented
2	C₁₃×C₃²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	C₃×C₁₃×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₁₃×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#6859	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×H(3,𝔽₃)	—	3	1	no	—
9	C₃×C₁₃×H(3,𝔽₃)	—	3	1	no	—
10	C₃×C₁₃×H(3,𝔽₃)	—	3	1	no	—
11	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	3	yes	—
12	C₃²×C₉×C₁₃	—	3	1	no	—
13	C₃²×C₉×C₁₃	—	3	1	no	—
14	C₁₃×C₃²⋊C₉	—	3	1	no	—
15	C₁₃×C₃²⋊C₉	—	3	1	no	—
16	C₁₃×C₃²⋊C₉	—	3	1	no	—
17	C₁₃×C₃²⋊C₉	—	3	1	no	—
18	C₁₃×C₃²⋊C₉	—	3	1	no	—
19	C₁₃×C₃²⋊C₉	—	3	1	no	—
20	C₁₃×C₃²⋊C₉	C₃²	1	3	yes	—
21	C₁₃×C₃²⋊C₉	C₃²	1	3	yes	—
22	C₁₃×C₃²⋊C₉	C₃²	1	3	yes	—
23	#10624598	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₃³×C₁₃	—	3	3	no	—
25	C₃³×C₁₃	—	3	3	no	—
26	C₃³×C₁₃	—	3	3	no	—
27	C₃³×C₁₃	—	9	1	no	—
28	C₃³×C₁₃	—	9	1	no	—
29	C₃³×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
30	C₃³×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
31	C₃³×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
32	C₃³×C₁₃	C₃³	1	1	yes	characteristic
33	C₁₃×3₋¹⁺²	—	9	3	no	—
34	C₁₃×H(3,𝔽₃)	—	9	3	no	—
35	C₁₃×H(3,𝔽₃)	—	9	3	no	—
36	C₁₃×H(3,𝔽₃)	—	9	3	no	—
37	C₁₃×H(3,𝔽₃)	—	9	3	no	—
38	C₃×C₉×C₁₃	—	3	3	no	—
39	C₃×C₉×C₁₃	—	3	3	no	—
40	C₃×C₉×C₁₃	—	3	3	no	—
41	C₃×C₉×C₁₃	—	9	3	no	—
42	C₃×C₉×C₁₃	—	9	3	no	—
43	#6891	C₃×C₁₃	1	1	yes	characteristic, complemented
44	C₃²⋊H(3,𝔽₃)	C₃×C₁₃	1	1	yes	characteristic
45	C₃×C₃²⋊C₉	C₃×C₁₃	1	1	yes	characteristic, complemented
46	C₃×C₃²⋊C₉	C₃×C₁₃	1	1	yes	characteristic, complemented
47	#6859	C₃×C₁₃	1	9	yes	complemented
48	C₃²×C₁₃	—	3	1	no	—
49	C₃²×C₁₃	—	3	1	no	—
50	C₃²×C₁₃	—	3	1	no	—
51	C₃²×C₁₃	—	3	1	no	—
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	—	9	9	no	—
63	C₃²×C₁₃	—	9	3	no	—
64	C₃²×C₁₃	—	9	3	no	—
65	C₃²×C₁₃	—	9	3	no	—
66	C₃²×C₁₃	—	9	3	no	—
67	C₃²×C₁₃	#731	1	3	yes	—
68	C₃²×C₁₃	#732	1	3	yes	—
69	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
70	C₉×C₁₃	—	9	3	no	—
71	C₉×C₁₃	—	9	3	no	—
72	C₉×C₁₃	—	9	3	no	—
73	C₃⁴	C₃²×C₁₃	1	1	yes	characteristic
74	C₃×3₋¹⁺²	—	3	1	no	—
75	C₃×H(3,𝔽₃)	—	3	1	no	—
76	C₃×H(3,𝔽₃)	—	3	1	no	—
77	C₃×H(3,𝔽₃)	—	3	1	no	—
78	C₃×H(3,𝔽₃)	C₃²×C₁₃	1	3	yes	—
79	C₃²×C₉	—	3	1	no	—
80	C₃²×C₉	—	3	1	no	—
81	C₃²⋊C₉	—	3	1	no	—
82	C₃²⋊C₉	—	3	1	no	—
83	C₃²⋊C₉	—	3	1	no	—
84	C₃²⋊C₉	—	3	1	no	—
85	C₃²⋊C₉	—	3	1	no	—
86	C₃²⋊C₉	—	3	1	no	—
87	C₃²⋊C₉	C₃²×C₁₃	1	3	yes	—
88	C₃²⋊C₉	C₃²×C₁₃	1	3	yes	—
89	C₃²⋊C₉	C₃²×C₁₃	1	3	yes	—
90	C₃×C₁₃	—	3	3	no	—
91	C₃×C₁₃	—	3	3	no	—
92	C₃×C₁₃	—	9	1	no	—
93	C₃×C₁₃	—	9	1	no	—
94	C₃×C₁₃	—	27	3	no	—
95	C₃×C₁₃	C₃×#731	1	1	yes	characteristic
96	C₃×C₁₃	C₃×#732	1	1	yes	characteristic
97	C₃×C₁₃	#6909	1	1	yes	characteristic
98	C₃×C₁₃	#6910	1	1	yes	characteristic
99	C₃³	—	3	3	no	—
100	C₃³	—	3	3	no	—
101	C₃³	—	3	3	no	—
102	C₃³	—	9	1	no	—
103	C₃³	—	9	1	no	—
104	C₃³	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
105	C₃³	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
106	C₃³	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
107	C₃³	C₃³×C₁₃	1	1	yes	characteristic
108	3₋¹⁺²	—	9	3	no	—
109	H(3,𝔽₃)	—	9	3	no	—
110	H(3,𝔽₃)	—	9	3	no	—
111	H(3,𝔽₃)	—	9	3	no	—
112	H(3,𝔽₃)	—	9	3	no	—
113	C₃×C₉	—	3	3	no	—
114	C₃×C₉	—	3	3	no	—
115	C₃×C₉	—	3	3	no	—
116	C₃×C₉	—	9	3	no	—
117	C₃×C₉	—	9	3	no	—
118	C₁₃	#10624598	1	1	yes	characteristic, direct product factor, Sylow
119	C₃²	—	3	1	no	—
120	C₃²	—	3	1	no	—
121	C₃²	—	3	1	no	—
122	C₃²	—	3	1	no	—
123	C₃²	—	3	1	no	—
124	C₃²	—	3	1	no	—
125	C₃²	—	3	1	no	—
126	C₃²	—	3	1	no	—
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	—	9	9	no	—
134	C₃²	—	9	3	no	—
135	C₃²	—	9	3	no	—
136	C₃²	—	9	3	no	—
137	C₃²	—	9	3	no	—
138	C₃²	C₁₃×#731	1	3	yes	—
139	C₃²	C₁₃×#732	1	3	yes	—
140	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
141	C₉	—	9	3	no	—
142	C₉	—	9	3	no	—
143	C₉	—	9	3	no	—
144	C₃	—	3	3	no	—
145	C₃	—	3	3	no	—
146	C₃	—	9	1	no	—
147	C₃	—	9	1	no	—
148	C₃	—	27	3	no	—
149	C₃	C₃×C₁₃×#731	1	1	yes	characteristic
150	C₃	C₃×C₁₃×#732	1	1	yes	characteristic
151	C₃	C₁₃×#6909	1	1	yes	characteristic
152	C₃	C₁₃×#6910	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

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

C₃×C₁₃×C₃²⋊C₉

C₃

yes

maximal, characteristic, complemented

C₁₃×#6859

C₃

yes

maximal, complemented

C₃⁴×C₁₃

C₃²

yes

characteristic

C₃×C₁₃×3₋¹⁺²

—

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

—

C₁₃×C₃²⋊C₉

C₃²

yes

—

C₁₃×C₃²⋊C₉

C₃²

yes

—

C₁₃×C₃²⋊C₉

C₃²

yes

—

#10624598

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×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

—

#6891

C₃×C₁₃

yes

characteristic, complemented

C₃²⋊H(3,𝔽₃)

C₃×C₁₃

yes

characteristic

C₃×C₃²⋊C₉

C₃×C₁₃

yes

characteristic, complemented

C₃×C₃²⋊C₉

C₃×C₁₃

yes

characteristic, complemented

#6859

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

#731

yes

—

C₃²×C₁₃

#732

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×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⋊C₉

C₃²×C₁₃

yes

—

C₃²⋊C₉

C₃²×C₁₃

yes

—

C₃²⋊C₉

C₃²×C₁₃

yes

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

yes

C₃×C₁₃

yes

C₃×C₁₃

yes

C₃×C₁₃

yes

C₃³

—

100

C₃³

—

101

C₃³

—

102

C₃³

—

103

C₃³

—

104

C₃³

C₁₃×H(3,𝔽₃)

yes

characteristic

105

C₃³

C₁₃×H(3,𝔽₃)

yes

characteristic

106

C₃³

C₁₃×H(3,𝔽₃)

yes

characteristic

107

C₃³

C₃³×C₁₃

yes

characteristic

108

3₋¹⁺²

—

109

H(3,𝔽₃)

—

110

H(3,𝔽₃)

—

111

H(3,𝔽₃)

—

112

H(3,𝔽₃)

—

113

C₃×C₉

—

114

C₃×C₉

—

115

C₃×C₉

—

116

C₃×C₉

—

117

C₃×C₉

—

118

C₁₃

#10624598

yes

characteristic, direct product factor, Sylow

119

C₃²

—

120

C₃²

—

121

C₃²

—

122

C₃²

—

123

C₃²

—

124

C₃²

—

125

C₃²

—

126

C₃²

—

127

C₃²

—

128

C₃²

—

129

C₃²

—

130

C₃²

—

131

C₃²

—

132

C₃²

—

133

C₃²

—

134

C₃²

—

135

C₃²

—

136

C₃²

—

137

C₃²

—

138

C₃²

C₁₃×#731

yes

—

139

C₃²

C₁₃×#732

yes

—

140

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

141

C₉

—

142

C₉

—

143

C₉

—

144

C₃

—

145

C₃

—

146

C₃

—

147

C₃

—

148

C₃

—

149

C₃

C₃×C₁₃×#731

yes

characteristic

150

C₃

C₃×C₁₃×#732

yes

characteristic

151

C₃

C₁₃×#6909

yes

characteristic

152

C₃

C₁₃×#6910

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. C₁₃×#6891	C₃	semidirect
3. C₃×C₁₃×C₃²⋊C₉	C₃	semidirect
4. C₃×C₁₃×C₃²⋊C₉	C₃	semidirect
5. C₁₃×#6859	C₃	semidirect
23. #10624598	C₁₃	direct (N × H)
43. #6891	C₃×C₁₃	semidirect
45. C₃×C₃²⋊C₉	C₃×C₁₃	semidirect
46. C₃×C₃²⋊C₉	C₃×C₁₃	semidirect
47. #6859	C₃×C₁₃	semidirect
118. C₁₃	#10624598	direct (N × H)

H ≅ G/N

type

1. C₁₃×#6891

C₃

semidirect

3. C₃×C₁₃×C₃²⋊C₉

C₃

semidirect

4. C₃×C₁₃×C₃²⋊C₉

C₃

semidirect

5. C₁₃×#6859

C₃

semidirect

23. #10624598

C₁₃

direct (N × H)

43. #6891

C₃×C₁₃

semidirect

45. C₃×C₃²⋊C₉

C₃×C₁₃

semidirect

46. C₃×C₃²⋊C₉

C₃×C₁₃

semidirect

47. #6859

C₃×C₁₃

semidirect

118. C₁₃

#10624598

direct (N × H)

Subgroups — C₁₃×#10624598 #23744749

Conjugacy classes of subgroups

Complements

Subgroups — C13×#10624598 #23744749

Conjugacy classes of subgroups

Complements

Subgroups — C₁₃×#10624598 #23744749