Group #23745087 — subgroups

Subgroups — C₁₃×#10624936 #23745087

Order 9477.

948 conjugacy classes of subgroups; only the 470 normal subgroups are shown.

normal subgroups only

This group has 948 conjugacy classes of subgroups — too many to draw. Only its 470 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	C₁₃×3₋¹⁺⁴	C₃	1	3	yes	maximal, complemented
2	C₃×C₁₃×#737	C₃	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#6913	C₃	1	18	yes	maximal, complemented
4	C₁₃×#6910	C₃	1	3	yes	maximal, complemented
5	C₁₃×#6909	C₃	1	3	yes	maximal, complemented
6	C₁₃×#6908	C₃	1	9	yes	maximal, complemented
7	C₃×C₁₃×#733	C₃	1	2	yes	maximal, complemented
8	C₃×C₁₃×#732	C₃	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#737	—	3	3	no	—
10	C₁₃×#737	—	3	6	no	—
11	C₁₃×#737	—	3	18	no	—
12	C₁₃×#737	C₃²	1	27	yes	complemented
13	C₃×C₁₃×3₋¹⁺²	—	3	9	no	—
14	C₃×C₁₃×3₋¹⁺²	C₃²	1	6	yes	—
15	C₃×C₁₃×3₋¹⁺²	C₃²	1	1	yes	characteristic
16	C₃×C₁₃×3₋¹⁺²	C₃²	1	1	yes	characteristic
17	C₃×C₁₃×3₋¹⁺²	C₃²	1	6	yes	complemented
18	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
19	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	3	yes	—
20	C₃²×C₉×C₁₃	C₃²	1	1	yes	characteristic
21	C₃²×C₉×C₁₃	C₃²	1	3	yes	complemented
22	C₁₃×#733	C₃²	1	6	yes	complemented
23	C₁₃×#733	C₃²	1	12	yes	complemented
24	C₁₃×#732	C₃²	1	3	yes	—
25	C₁₃×#732	C₃²	1	6	yes	—
26	C₁₃×#731	C₃²	1	27	yes	complemented
27	C₁₃×AΣL(1,𝔽₂₇)	C₃²	1	27	yes	complemented
28	#10624936	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₃³×C₁₃	—	3	3	no	—
30	C₃³×C₁₃	C₃³	1	3	yes	—
31	C₃³×C₁₃	C₃³	1	1	yes	characteristic
32	C₁₃×3₋¹⁺²	—	3	6	no	—
33	C₁₃×3₋¹⁺²	—	3	6	no	—
34	C₁₃×3₋¹⁺²	—	3	2	no	—
35	C₁₃×3₋¹⁺²	—	3	1	no	—
36	C₁₃×3₋¹⁺²	—	3	54	no	—
37	C₁₃×3₋¹⁺²	—	3	18	no	—
38	C₁₃×3₋¹⁺²	H(3,𝔽₃)	1	3	yes	—
39	C₁₃×3₋¹⁺²	H(3,𝔽₃)	1	6	yes	—
40	C₁₃×3₋¹⁺²	C₃³	1	18	yes	—
41	C₁₃×H(3,𝔽₃)	—	3	2	no	—
42	C₁₃×H(3,𝔽₃)	—	3	1	no	—
43	C₁₃×H(3,𝔽₃)	—	3	6	no	—
44	C₁₃×H(3,𝔽₃)	C₃³	1	9	yes	—
45	C₃×C₉×C₁₃	—	3	2	no	—
46	C₃×C₉×C₁₃	—	3	1	no	—
47	C₃×C₉×C₁₃	—	3	6	no	—
48	C₃×C₉×C₁₃	—	3	9	no	—
49	C₃×C₉×C₁₃	—	3	27	no	—
50	C₃×C₉×C₁₃	C₃³	1	3	yes	—
51	C₃×C₉×C₁₃	C₃³	1	1	yes	characteristic
52	C₃×C₉×C₁₃	C₃³	1	2	yes	—
53	C₃×C₉×C₁₃	C₃³	1	3	yes	—
54	3₋¹⁺⁴	C₃×C₁₃	1	3	yes	complemented
55	C₃×#737	C₃×C₁₃	1	1	yes	characteristic, complemented
56	#6913	C₃×C₁₃	1	18	yes	complemented
57	#6910	C₃×C₁₃	1	3	yes	complemented
58	#6909	C₃×C₁₃	1	3	yes	complemented
59	#6908	C₃×C₁₃	1	9	yes	complemented
60	C₃×#733	C₃×C₁₃	1	2	yes	complemented
61	C₃×#732	C₃×C₁₃	1	1	yes	characteristic, complemented
62	C₃²×C₁₃	—	3	3	no	—
63	C₃²×C₁₃	—	3	9	no	—
64	C₃²×C₁₃	—	9	1	no	—
65	C₃²×C₁₃	—	9	3	no	—
66	C₃²×C₁₃	—	27	3	no	—
67	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	2	yes	—
68	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
69	C₃²×C₁₃	C₃⁴	1	1	yes	characteristic
70	C₉×C₁₃	—	3	1	no	—
71	C₉×C₁₃	—	3	18	no	—
72	C₉×C₁₃	—	3	2	no	—
73	C₉×C₁₃	—	3	3	no	—
74	C₉×C₁₃	C₃×H(3,𝔽₃)	1	9	yes	—
75	#737	—	3	3	no	—
76	#737	—	3	6	no	—
77	#737	—	3	18	no	—
78	#737	C₃²×C₁₃	1	27	yes	complemented
79	C₃×3₋¹⁺²	—	3	9	no	—
80	C₃×3₋¹⁺²	C₃²×C₁₃	1	6	yes	—
81	C₃×3₋¹⁺²	C₃²×C₁₃	1	1	yes	characteristic
82	C₃×3₋¹⁺²	C₃²×C₁₃	1	1	yes	characteristic
83	C₃×3₋¹⁺²	C₃²×C₁₃	1	6	yes	complemented
84	C₃×H(3,𝔽₃)	C₃²×C₁₃	1	1	yes	characteristic
85	C₃×H(3,𝔽₃)	C₃²×C₁₃	1	3	yes	—
86	C₃²×C₉	C₃²×C₁₃	1	1	yes	characteristic
87	C₃²×C₉	C₃²×C₁₃	1	3	yes	complemented
88	#733	C₃²×C₁₃	1	12	yes	complemented
89	#733	C₃²×C₁₃	1	6	yes	complemented
90	#732	C₃²×C₁₃	1	6	yes	—
91	#732	C₃²×C₁₃	1	3	yes	—
92	#731	C₃²×C₁₃	1	27	yes	complemented
93	AΣL(1,𝔽₂₇)	C₃²×C₁₃	1	27	yes	complemented
94	C₃×C₁₃	—	3	1	no	—
95	C₃×C₁₃	—	3	1	no	—
96	C₃×C₁₃	—	3	2	no	—
97	C₃×C₁₃	—	9	3	no	—
98	C₃×C₁₃	—	9	9	no	—
99	C₃×C₁₃	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
100	C₃³	—	3	3	no	—
101	C₃³	C₃³×C₁₃	1	3	yes	—
102	C₃³	C₃³×C₁₃	1	1	yes	characteristic
103	3₋¹⁺²	—	3	6	no	—
104	3₋¹⁺²	—	3	6	no	—
105	3₋¹⁺²	—	3	2	no	—
106	3₋¹⁺²	—	3	1	no	—
107	3₋¹⁺²	—	3	54	no	—
108	3₋¹⁺²	—	3	18	no	—
109	3₋¹⁺²	C₁₃×H(3,𝔽₃)	1	3	yes	—
110	3₋¹⁺²	C₁₃×H(3,𝔽₃)	1	6	yes	—
111	3₋¹⁺²	C₃³×C₁₃	1	18	yes	—
112	H(3,𝔽₃)	—	3	2	no	—
113	H(3,𝔽₃)	—	3	1	no	—
114	H(3,𝔽₃)	—	3	6	no	—
115	H(3,𝔽₃)	C₃³×C₁₃	1	9	yes	—
116	C₃×C₉	—	3	2	no	—
117	C₃×C₉	—	3	1	no	—
118	C₃×C₉	—	3	6	no	—
119	C₃×C₉	—	3	9	no	—
120	C₃×C₉	—	3	27	no	—
121	C₃×C₉	C₃³×C₁₃	1	3	yes	—
122	C₃×C₉	C₃³×C₁₃	1	1	yes	characteristic
123	C₃×C₉	C₃³×C₁₃	1	2	yes	—
124	C₃×C₉	C₃³×C₁₃	1	3	yes	—
125	C₁₃	#10624936	1	1	yes	characteristic, direct product factor, Sylow
126	C₃²	—	3	3	no	—
127	C₃²	—	3	9	no	—
128	C₃²	—	9	1	no	—
129	C₃²	—	9	3	no	—
130	C₃²	—	27	3	no	—
131	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	2	yes	—
132	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
133	C₃²	C₃⁴×C₁₃	1	1	yes	characteristic
134	C₉	—	3	1	no	—
135	C₉	—	3	18	no	—
136	C₉	—	3	2	no	—
137	C₉	—	3	3	no	—
138	C₉	C₃×C₁₃×H(3,𝔽₃)	1	9	yes	—
139	C₃	—	3	1	no	—
140	C₃	—	3	1	no	—
141	C₃	—	3	2	no	—
142	C₃	—	9	3	no	—
143	C₃	—	9	9	no	—
144	C₃	C₃²×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×3₋¹⁺⁴

C₃

yes

maximal, complemented

C₃×C₁₃×#737

C₃

yes

maximal, characteristic, complemented

C₁₃×#6913

C₃

yes

maximal, complemented

C₁₃×#6910

C₃

yes

maximal, complemented

C₁₃×#6909

C₃

yes

maximal, complemented

C₁₃×#6908

C₃

yes

maximal, complemented

C₃×C₁₃×#733

C₃

yes

maximal, complemented

C₃×C₁₃×#732

C₃

yes

maximal, characteristic, complemented

C₁₃×#737

—

C₁₃×#737

—

C₁₃×#737

—

C₁₃×#737

C₃²

yes

complemented

C₃×C₁₃×3₋¹⁺²

—

C₃×C₁₃×3₋¹⁺²

C₃²

yes

—

C₃×C₁₃×3₋¹⁺²

C₃²

yes

characteristic

C₃×C₁₃×3₋¹⁺²

C₃²

yes

characteristic

C₃×C₁₃×3₋¹⁺²

C₃²

yes

complemented

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

characteristic

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

—

C₃²×C₉×C₁₃

C₃²

yes

characteristic

C₃²×C₉×C₁₃

C₃²

yes

complemented

C₁₃×#733

C₃²

yes

complemented

C₁₃×#733

C₃²

yes

complemented

C₁₃×#732

C₃²

yes

—

C₁₃×#732

C₃²

yes

—

C₁₃×#731

C₃²

yes

complemented

C₁₃×AΣL(1,𝔽₂₇)

C₃²

yes

complemented

#10624936

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₃³×C₁₃

—

C₃³×C₁₃

C₃³

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

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

H(3,𝔽₃)

yes

—

C₁₃×3₋¹⁺²

H(3,𝔽₃)

yes

—

C₁₃×3₋¹⁺²

C₃³

yes

—

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

C₃³

yes

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

C₃³

yes

—

C₃×C₉×C₁₃

C₃³

yes

characteristic

C₃×C₉×C₁₃

C₃³

yes

—

C₃×C₉×C₁₃

C₃³

yes

—

3₋¹⁺⁴

C₃×C₁₃

yes

complemented

C₃×#737

C₃×C₁₃

yes

characteristic, complemented

#6913

C₃×C₁₃

yes

complemented

#6910

C₃×C₁₃

yes

complemented

#6909

C₃×C₁₃

yes

complemented

#6908

C₃×C₁₃

yes

complemented

C₃×#733

C₃×C₁₃

yes

complemented

C₃×#732

C₃×C₁₃

yes

characteristic, complemented

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

—

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₁₃

C₃⁴

yes

characteristic

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

C₃×H(3,𝔽₃)

yes

—

—

—

—

C₃²×C₁₃

yes

complemented

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃²×C₁₃

yes

—

C₃×3₋¹⁺²

C₃²×C₁₃

yes

characteristic

C₃×3₋¹⁺²

C₃²×C₁₃

yes

characteristic

C₃×3₋¹⁺²

C₃²×C₁₃

yes

complemented

C₃×H(3,𝔽₃)

C₃²×C₁₃

yes

characteristic

C₃×H(3,𝔽₃)

C₃²×C₁₃

yes

—

C₃²×C₉

C₃²×C₁₃

yes

characteristic

C₃²×C₉

C₃²×C₁₃

yes

complemented

#733

C₃²×C₁₃

yes

complemented

#733

C₃²×C₁₃

yes

complemented

#732

C₃²×C₁₃

yes

—

#732

C₃²×C₁₃

yes

—

#731

C₃²×C₁₃

yes

complemented

AΣL(1,𝔽₂₇)

C₃²×C₁₃

yes

complemented

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

C₃²×H(3,𝔽₃)

yes

characteristic

100

C₃³

—

101

C₃³

C₃³×C₁₃

yes

—

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

—

108

3₋¹⁺²

—

109

3₋¹⁺²

C₁₃×H(3,𝔽₃)

yes

—

110

3₋¹⁺²

C₁₃×H(3,𝔽₃)

yes

—

111

3₋¹⁺²

C₃³×C₁₃

yes

—

112

H(3,𝔽₃)

—

113

H(3,𝔽₃)

—

114

H(3,𝔽₃)

—

115

H(3,𝔽₃)

C₃³×C₁₃

yes

—

116

C₃×C₉

—

117

C₃×C₉

—

118

C₃×C₉

—

119

C₃×C₉

—

120

C₃×C₉

—

121

C₃×C₉

C₃³×C₁₃

yes

—

122

C₃×C₉

C₃³×C₁₃

yes

characteristic

123

C₃×C₉

C₃³×C₁₃

yes

—

124

C₃×C₉

C₃³×C₁₃

yes

—

125

C₁₃

#10624936

yes

characteristic, direct product factor, Sylow

126

C₃²

—

127

C₃²

—

128

C₃²

—

129

C₃²

—

130

C₃²

—

131

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

—

132

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

133

C₃²

C₃⁴×C₁₃

yes

characteristic

134

C₉

—

135

C₉

—

136

C₉

—

137

C₉

—

138

C₉

C₃×C₁₃×H(3,𝔽₃)

yes

—

139

C₃

—

140

C₃

—

141

C₃

—

142

C₃

—

143

C₃

—

144

C₃

C₃²×C₁₃×H(3,𝔽₃)

yes

characteristic

	subgroup	G/H
center, socle	C₃×C₁₃	C₃²×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃²	C₃⁴×C₁₃
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₁₃ ◁ C₃×C₁₃×3₋¹⁺² ◁ G

subgroup

G/H

center, socle

C₃×C₁₃

C₃²×H(3,𝔽₃)

commutator, Frattini, cosocle

C₃²

C₃⁴×C₁₃

LCS

G ▷ C₃² ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃×C₁₃ ◁ C₃×C₁₃×3₋¹⁺² ◁ G

N	H ≅ G/N	type
1. C₁₃×3₋¹⁺⁴	C₃	semidirect
2. C₃×C₁₃×#737	C₃	semidirect
3. C₁₃×#6913	C₃	semidirect
4. C₁₃×#6910	C₃	semidirect
5. C₁₃×#6909	C₃	semidirect
6. C₁₃×#6908	C₃	semidirect
7. C₃×C₁₃×#733	C₃	semidirect
8. C₃×C₁₃×#732	C₃	semidirect
12. C₁₃×#737	C₃²	semidirect
17. C₃×C₁₃×3₋¹⁺²	C₃²	semidirect
21. C₃²×C₉×C₁₃	C₃²	semidirect
22. C₁₃×#733	C₃²	semidirect
23. C₁₃×#733	C₃²	semidirect
26. C₁₃×#731	C₃²	semidirect
27. C₁₃×AΣL(1,𝔽₂₇)	C₃²	semidirect
28. #10624936	C₁₃	direct (N × H)
54. 3₋¹⁺⁴	C₃×C₁₃	semidirect
55. C₃×#737	C₃×C₁₃	semidirect
56. #6913	C₃×C₁₃	semidirect
57. #6910	C₃×C₁₃	semidirect
58. #6909	C₃×C₁₃	semidirect
59. #6908	C₃×C₁₃	semidirect
60. C₃×#733	C₃×C₁₃	semidirect
61. C₃×#732	C₃×C₁₃	semidirect
78. #737	C₃²×C₁₃	semidirect
83. C₃×3₋¹⁺²	C₃²×C₁₃	semidirect
87. C₃²×C₉	C₃²×C₁₃	semidirect
88. #733	C₃²×C₁₃	semidirect
89. #733	C₃²×C₁₃	semidirect
92. #731	C₃²×C₁₃	semidirect
93. AΣL(1,𝔽₂₇)	C₃²×C₁₃	semidirect
125. C₁₃	#10624936	direct (N × H)

H ≅ G/N

type

1. C₁₃×3₋¹⁺⁴

C₃

semidirect

2. C₃×C₁₃×#737

C₃

semidirect

3. C₁₃×#6913

C₃

semidirect

4. C₁₃×#6910

C₃

semidirect

5. C₁₃×#6909

C₃

semidirect

6. C₁₃×#6908

C₃

semidirect

7. C₃×C₁₃×#733

C₃

semidirect

8. C₃×C₁₃×#732

C₃

semidirect

12. C₁₃×#737

C₃²

semidirect

17. C₃×C₁₃×3₋¹⁺²

C₃²

semidirect

21. C₃²×C₉×C₁₃

C₃²

semidirect

22. C₁₃×#733

C₃²

semidirect

23. C₁₃×#733

C₃²

semidirect

26. C₁₃×#731

C₃²

semidirect

27. C₁₃×AΣL(1,𝔽₂₇)

C₃²

semidirect

28. #10624936

C₁₃

direct (N × H)

54. 3₋¹⁺⁴

C₃×C₁₃

semidirect

55. C₃×#737

C₃×C₁₃

semidirect

56. #6913

C₃×C₁₃

semidirect

57. #6910

C₃×C₁₃

semidirect

58. #6909

C₃×C₁₃

semidirect

59. #6908

C₃×C₁₃

semidirect

60. C₃×#733

C₃×C₁₃

semidirect

61. C₃×#732

C₃×C₁₃

semidirect

78. #737

C₃²×C₁₃

semidirect

83. C₃×3₋¹⁺²

C₃²×C₁₃

semidirect

87. C₃²×C₉

C₃²×C₁₃

semidirect

88. #733

C₃²×C₁₃

semidirect

89. #733

C₃²×C₁₃

semidirect

92. #731

C₃²×C₁₃

semidirect

93. AΣL(1,𝔽₂₇)

C₃²×C₁₃

semidirect

125. C₁₃

#10624936

direct (N × H)

Subgroups — C₁₃×#10624936 #23745087

Conjugacy classes of subgroups

Complements

Subgroups — C13×#10624936 #23745087

Conjugacy classes of subgroups

Complements

Subgroups — C₁₃×#10624936 #23745087