Group #23744951 — subgroups

Subgroups — C₁₃×#10624800 #23744951

Order 9477.

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

normal subgroups only

This group has 236 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	C₃×C₁₃×#731	C₃	1	1	yes	maximal, characteristic
2	C₁₃×#6894	C₃	1	1	yes	maximal, characteristic, complemented
3	C₁₃×C₉⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
4	C₉×C₁₃×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃×C₁₃×C₉⋊C₉	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
9	C₁₃×#6867	C₃	1	2	yes	maximal, complemented
10	C₃×C₁₃×3₋¹⁺²	—	3	1	no	—
11	C₃×C₁₃×3₋¹⁺²	C₃²	1	1	yes	characteristic
12	C₃×C₁₃×3₋¹⁺²	C₃²	1	1	yes	characteristic
13	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
14	C₃²×C₉×C₁₃	C₃²	1	1	yes	characteristic
15	C₃²×C₉×C₁₃	C₃²	1	2	yes	—
16	C₃²×C₉×C₁₃	C₃²	1	1	yes	characteristic
17	C₁₃×#731	—	3	3	no	—
18	C₁₃×C₉⋊C₉	—	3	2	no	—
19	C₁₃×C₉⋊C₉	—	3	1	no	—
20	C₁₃×C₉⋊C₉	—	3	2	no	—
21	C₁₃×C₉⋊C₉	—	3	2	no	—
22	C₁₃×C₉⋊C₉	—	3	1	no	—
23	C₁₃×C₃²⋊C₉	—	3	2	no	—
24	C₁₃×C₃²⋊C₉	C₃²	1	2	yes	—
25	C₁₃×C₃²⋊C₉	C₃²	1	2	yes	—
26	C₁₃×C₃²⋊C₉	C₃²	1	2	yes	—
27	C₉²×C₁₃	—	3	1	no	—
28	#10624800	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₃³×C₁₃	—	3	1	no	—
30	C₃³×C₁₃	C₃³	1	1	yes	characteristic
31	C₁₃×3₋¹⁺²	—	3	2	no	—
32	C₁₃×3₋¹⁺²	—	3	1	no	—
33	C₁₃×3₋¹⁺²	—	3	2	no	—
34	C₁₃×3₋¹⁺²	—	9	3	no	—
35	C₁₃×3₋¹⁺²	3₋¹⁺²	1	3	yes	complemented
36	C₁₃×H(3,𝔽₃)	—	3	1	no	—
37	C₁₃×H(3,𝔽₃)	—	3	2	no	—
38	C₃×C₉×C₁₃	—	3	1	no	—
39	C₃×C₉×C₁₃	—	3	3	no	—
40	C₃×C₉×C₁₃	—	3	2	no	—
41	C₃×C₉×C₁₃	—	3	2	no	—
42	C₃×C₉×C₁₃	—	3	2	no	—
43	C₃×C₉×C₁₃	—	3	2	no	—
44	C₃×C₉×C₁₃	—	3	1	no	—
45	C₃×C₉×C₁₃	—	3	1	no	—
46	C₃×C₉×C₁₃	—	3	1	no	—
47	C₃×C₉×C₁₃	—	3	2	no	—
48	C₃×C₉×C₁₃	—	9	2	no	—
49	C₃×C₉×C₁₃	H(3,𝔽₃)	1	2	yes	—
50	C₃×C₉×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
51	C₃×#731	C₃×C₁₃	1	1	yes	characteristic
52	#6894	C₃×C₁₃	1	1	yes	characteristic, complemented
53	C₉⋊H(3,𝔽₃)	C₃×C₁₃	1	1	yes	characteristic
54	C₉×3₋¹⁺²	C₃×C₁₃	1	1	yes	characteristic, complemented
55	C₃×C₉⋊C₉	C₃×C₁₃	1	1	yes	characteristic, complemented
56	3₋¹⁺²⋊C₉	C₃×C₁₃	1	2	yes	complemented
57	3₋¹⁺²⋊C₉	C₃×C₁₃	1	2	yes	complemented
58	H(3,𝔽₃)⋊C₉	C₃×C₁₃	1	2	yes	—
59	#6867	C₃×C₁₃	1	2	yes	complemented
60	C₃²×C₁₃	—	3	2	no	—
61	C₃²×C₁₃	—	3	1	no	—
62	C₃²×C₁₃	—	9	2	no	—
63	C₃²×C₁₃	—	9	1	no	—
64	C₃²×C₁₃	—	9	1	no	—
65	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
66	C₃²×C₁₃	C₃×3₋¹⁺²	1	1	yes	characteristic
67	C₃²×C₁₃	#737	1	2	yes	—
68	C₉×C₁₃	—	3	3	no	—
69	C₉×C₁₃	—	3	2	no	—
70	C₉×C₁₃	—	9	2	no	—
71	C₉×C₁₃	—	9	2	no	—
72	C₉×C₁₃	—	9	2	no	—
73	C₉×C₁₃	—	9	1	no	—
74	C₉×C₁₃	—	9	1	no	—
75	C₉×C₁₃	—	9	2	no	—
76	C₉×C₁₃	#731	1	3	yes	complemented
77	C₃×3₋¹⁺²	—	3	1	no	—
78	C₃×3₋¹⁺²	C₃²×C₁₃	1	1	yes	characteristic
79	C₃×3₋¹⁺²	C₃²×C₁₃	1	1	yes	characteristic
80	C₃×H(3,𝔽₃)	C₃²×C₁₃	1	1	yes	characteristic
81	C₃²×C₉	C₃²×C₁₃	1	1	yes	characteristic
82	C₃²×C₉	C₃²×C₁₃	1	1	yes	characteristic
83	C₃²×C₉	C₃²×C₁₃	1	2	yes	—
84	#731	—	3	3	no	—
85	C₉⋊C₉	—	3	2	no	—
86	C₉⋊C₉	—	3	1	no	—
87	C₉⋊C₉	—	3	2	no	—
88	C₉⋊C₉	—	3	2	no	—
89	C₉⋊C₉	—	3	1	no	—
90	C₃²⋊C₉	—	3	2	no	—
91	C₃²⋊C₉	C₃²×C₁₃	1	2	yes	—
92	C₃²⋊C₉	C₃²×C₁₃	1	2	yes	—
93	C₃²⋊C₉	C₃²×C₁₃	1	2	yes	—
94	C₉²	—	3	1	no	—
95	C₃×C₁₃	—	3	1	no	—
96	C₃×C₁₃	—	3	2	no	—
97	C₃×C₁₃	—	27	1	no	—
98	C₃×C₁₃	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
99	C₃×C₁₃	C₃×#731	1	1	yes	characteristic
100	C₃×C₁₃	#6913	1	2	yes	—
101	C₃³	—	3	1	no	—
102	C₃³	C₃³×C₁₃	1	1	yes	characteristic
103	3₋¹⁺²	—	3	2	no	—
104	3₋¹⁺²	—	3	1	no	—
105	3₋¹⁺²	—	3	2	no	—
106	3₋¹⁺²	—	9	3	no	—
107	3₋¹⁺²	C₁₃×3₋¹⁺²	1	3	yes	complemented
108	H(3,𝔽₃)	—	3	1	no	—
109	H(3,𝔽₃)	—	3	2	no	—
110	C₃×C₉	—	3	1	no	—
111	C₃×C₉	—	3	3	no	—
112	C₃×C₉	—	3	2	no	—
113	C₃×C₉	—	3	2	no	—
114	C₃×C₉	—	3	2	no	—
115	C₃×C₉	—	3	1	no	—
116	C₃×C₉	—	3	2	no	—
117	C₃×C₉	—	3	1	no	—
118	C₃×C₉	—	3	1	no	—
119	C₃×C₉	—	3	2	no	—
120	C₃×C₉	—	9	2	no	—
121	C₃×C₉	C₁₃×H(3,𝔽₃)	1	2	yes	—
122	C₃×C₉	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
123	C₁₃	#10624800	1	1	yes	characteristic, direct product factor, Sylow
124	C₃²	—	3	1	no	—
125	C₃²	—	3	2	no	—
126	C₃²	—	9	1	no	—
127	C₃²	—	9	1	no	—
128	C₃²	—	9	2	no	—
129	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
130	C₃²	C₃×C₁₃×3₋¹⁺²	1	1	yes	characteristic
131	C₃²	C₁₃×#737	1	2	yes	—
132	C₉	—	3	3	no	—
133	C₉	—	3	2	no	—
134	C₉	—	9	1	no	—
135	C₉	—	9	1	no	—
136	C₉	—	9	2	no	—
137	C₉	—	9	2	no	—
138	C₉	—	9	2	no	—
139	C₉	—	9	2	no	—
140	C₉	C₁₃×#731	1	3	yes	complemented
141	C₃	—	3	1	no	—
142	C₃	—	3	2	no	—
143	C₃	—	27	1	no	—
144	C₃	C₁₃×C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
145	C₃	C₃×C₁₃×#731	1	1	yes	characteristic
146	C₃	C₁₃×#6913	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₁₃×#731

C₃

yes

maximal, characteristic

C₁₃×#6894

C₃

yes

maximal, characteristic, complemented

C₁₃×C₉⋊H(3,𝔽₃)

C₃

yes

maximal, characteristic

C₉×C₁₃×3₋¹⁺²

C₃

yes

maximal, characteristic, complemented

C₃×C₁₃×C₉⋊C₉

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

C₁₃×#6867

C₃

yes

maximal, complemented

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

characteristic

C₃²×C₉×C₁₃

C₃²

yes

—

C₃²×C₉×C₁₃

C₃²

yes

characteristic

C₁₃×#731

—

C₁₃×C₉⋊C₉

—

C₁₃×C₉⋊C₉

—

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

—

#10624800

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

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

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

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

H(3,𝔽₃)

yes

—

C₃×C₉×C₁₃

H(3,𝔽₃)

yes

characteristic

C₃×#731

C₃×C₁₃

yes

characteristic

#6894

C₃×C₁₃

yes

characteristic, complemented

C₉⋊H(3,𝔽₃)

C₃×C₁₃

yes

characteristic

C₉×3₋¹⁺²

C₃×C₁₃

yes

characteristic, complemented

C₃×C₉⋊C₉

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

—

#6867

C₃×C₁₃

yes

complemented

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

—

C₉×C₁₃

#731

yes

complemented

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

characteristic

C₃²×C₉

C₃²×C₁₃

yes

characteristic

C₃²×C₉

C₃²×C₁₃

yes

—

#731

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃²×C₁₃

yes

—

C₃²⋊C₉

C₃²×C₁₃

yes

—

C₃²⋊C₉

C₃²×C₁₃

yes

—

C₉²

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

C₉⋊H(3,𝔽₃)

yes

characteristic

C₃×C₁₃

C₃×#731

yes

characteristic

100

C₃×C₁₃

#6913

yes

—

101

C₃³

—

102

C₃³

C₃³×C₁₃

yes

characteristic

103

3₋¹⁺²

—

104

3₋¹⁺²

—

105

3₋¹⁺²

—

106

3₋¹⁺²

—

107

3₋¹⁺²

C₁₃×3₋¹⁺²

yes

complemented

108

H(3,𝔽₃)

—

109

H(3,𝔽₃)

—

110

C₃×C₉

—

111

C₃×C₉

—

112

C₃×C₉

—

113

C₃×C₉

—

114

C₃×C₉

—

115

C₃×C₉

—

116

C₃×C₉

—

117

C₃×C₉

—

118

C₃×C₉

—

119

C₃×C₉

—

120

C₃×C₉

—

121

C₃×C₉

C₁₃×H(3,𝔽₃)

yes

—

122

C₃×C₉

C₁₃×H(3,𝔽₃)

yes

characteristic

123

C₁₃

#10624800

yes

characteristic, direct product factor, Sylow

124

C₃²

—

125

C₃²

—

126

C₃²

—

127

C₃²

—

128

C₃²

—

129

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

130

C₃²

C₃×C₁₃×3₋¹⁺²

yes

characteristic

131

C₃²

C₁₃×#737

yes

—

132

C₉

—

133

C₉

—

134

C₉

—

135

C₉

—

136

C₉

—

137

C₉

—

138

C₉

—

139

C₉

—

140

C₉

C₁₃×#731

yes

complemented

141

C₃

—

142

C₃

—

143

C₃

—

144

C₃

C₁₃×C₉⋊H(3,𝔽₃)

yes

characteristic

145

C₃

C₃×C₁₃×#731

yes

characteristic

146

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×H(3,𝔽₃)	C₃²
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×H(3,𝔽₃)

C₃²

LCS

G ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃²×C₁₃ ◁ C₃²×C₉×C₁₃ ◁ G

N	H ≅ G/N	type
2. C₁₃×#6894	C₃	semidirect
4. C₉×C₁₃×3₋¹⁺²	C₃	semidirect
5. C₃×C₁₃×C₉⋊C₉	C₃	semidirect
6. C₁₃×3₋¹⁺²⋊C₉	C₃	semidirect
7. C₁₃×3₋¹⁺²⋊C₉	C₃	semidirect
9. C₁₃×#6867	C₃	semidirect
28. #10624800	C₁₃	direct (N × H)
35. C₁₃×3₋¹⁺²	3₋¹⁺²	semidirect
52. #6894	C₃×C₁₃	semidirect
54. C₉×3₋¹⁺²	C₃×C₁₃	semidirect
55. C₃×C₉⋊C₉	C₃×C₁₃	semidirect
56. 3₋¹⁺²⋊C₉	C₃×C₁₃	semidirect
57. 3₋¹⁺²⋊C₉	C₃×C₁₃	semidirect
59. #6867	C₃×C₁₃	semidirect
76. C₉×C₁₃	#731	semidirect
107. 3₋¹⁺²	C₁₃×3₋¹⁺²	semidirect
123. C₁₃	#10624800	direct (N × H)
140. C₉	C₁₃×#731	semidirect

H ≅ G/N

type

2. C₁₃×#6894

C₃

semidirect

4. C₉×C₁₃×3₋¹⁺²

C₃

semidirect

5. C₃×C₁₃×C₉⋊C₉

C₃

semidirect

6. C₁₃×3₋¹⁺²⋊C₉

C₃

semidirect

7. C₁₃×3₋¹⁺²⋊C₉

C₃

semidirect

9. C₁₃×#6867

C₃

semidirect

28. #10624800

C₁₃

direct (N × H)

35. C₁₃×3₋¹⁺²

3₋¹⁺²

semidirect

52. #6894

C₃×C₁₃

semidirect

54. C₉×3₋¹⁺²

C₃×C₁₃

semidirect

55. C₃×C₉⋊C₉

C₃×C₁₃

semidirect

56. 3₋¹⁺²⋊C₉

C₃×C₁₃

semidirect

57. 3₋¹⁺²⋊C₉

C₃×C₁₃

semidirect

59. #6867

C₃×C₁₃

semidirect

76. C₉×C₁₃

#731

semidirect

107. 3₋¹⁺²

C₁₃×3₋¹⁺²

semidirect

123. C₁₃

#10624800

direct (N × H)

140. C₉

C₁₃×#731

semidirect

Subgroups — C₁₃×#10624800 #23744951

Conjugacy classes of subgroups

Complements

Subgroups — C13×#10624800 #23744951

Conjugacy classes of subgroups

Complements

Subgroups — C₁₃×#10624800 #23744951