Group #23744746 — subgroups

Subgroups — C₁₃×#10624595 #23744746

Order 9477.

318 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 80 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×#737	C₃	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#6894	C₃	1	2	yes	maximal, complemented
3	C₉×C₁₃×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#6860	C₃	1	2	yes	maximal, complemented
5	C₁₃×#6859	C₃	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#6858	C₃	1	2	yes	maximal, complemented
7	C₁₃×#6857	C₃	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#6857	C₃	1	2	yes	maximal, complemented
9	C₁₃×#6856	C₃	1	1	yes	maximal, characteristic
10	C₁₃×#737	—	3	9	no	—
11	C₃×C₁₃×3₋¹⁺²	—	3	1	no	—
12	C₃×C₁₃×3₋¹⁺²	—	3	1	no	—
13	C₃×C₁₃×3₋¹⁺²	C₃²	1	2	yes	—
14	C₃×C₁₃×3₋¹⁺²	C₃²	1	2	yes	—
15	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
16	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
17	C₃²×C₉×C₁₃	—	3	1	no	—
18	C₃²×C₉×C₁₃	—	3	1	no	—
19	C₃²×C₉×C₁₃	C₃²	1	1	yes	characteristic
20	C₁₃×C₉⋊C₉	—	3	2	no	—
21	C₁₃×C₉⋊C₉	—	3	2	no	—
22	C₁₃×C₉⋊C₉	—	3	2	no	—
23	C₁₃×C₃²⋊C₉	—	3	1	no	—
24	C₁₃×C₃²⋊C₉	—	3	1	no	—
25	C₁₃×C₃²⋊C₉	C₃²	1	2	yes	—
26	C₁₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
27	C₁₃×C₃²⋊C₉	C₃²	1	2	yes	—
28	C₁₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
29	#10624595	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₃³×C₁₃	—	3	1	no	—
31	C₃³×C₁₃	—	3	1	no	—
32	C₃³×C₁₃	C₃³	1	1	yes	characteristic
33	C₁₃×3₋¹⁺²	—	3	9	no	—
34	C₁₃×3₋¹⁺²	—	3	9	no	—
35	C₁₃×3₋¹⁺²	—	3	6	no	—
36	C₁₃×3₋¹⁺²	—	9	2	no	—
37	C₁₃×H(3,𝔽₃)	—	3	3	no	—
38	C₁₃×H(3,𝔽₃)	—	3	3	no	—
39	C₃×C₉×C₁₃	—	3	9	no	—
40	C₃×C₉×C₁₃	—	3	2	no	—
41	C₃×C₉×C₁₃	—	3	3	no	—
42	C₃×C₉×C₁₃	—	3	1	no	—
43	C₃×C₉×C₁₃	—	3	1	no	—
44	C₃×C₉×C₁₃	—	3	2	no	—
45	C₃×C₉×C₁₃	—	3	2	no	—
46	C₃×C₉×C₁₃	—	3	2	no	—
47	C₃×C₉×C₁₃	—	9	3	no	—
48	C₃×C₉×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
49	C₃×C₉×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
50	C₃×C₉×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
51	C₃×#737	C₃×C₁₃	1	1	yes	characteristic, complemented
52	#6894	C₃×C₁₃	1	2	yes	complemented
53	C₉×H(3,𝔽₃)	C₃×C₁₃	1	1	yes	characteristic, complemented
54	#6860	C₃×C₁₃	1	2	yes	complemented
55	#6859	C₃×C₁₃	1	1	yes	characteristic, complemented
56	#6858	C₃×C₁₃	1	2	yes	complemented
57	#6857	C₃×C₁₃	1	1	yes	characteristic, complemented
58	#6857	C₃×C₁₃	1	2	yes	complemented
59	#6856	C₃×C₁₃	1	1	yes	characteristic
60	C₃²×C₁₃	—	3	2	no	—
61	C₃²×C₁₃	—	3	1	no	—
62	C₃²×C₁₃	—	3	3	no	—
63	C₃²×C₁₃	—	3	1	no	—
64	C₃²×C₁₃	—	3	3	no	—
65	C₃²×C₁₃	—	9	2	no	—
66	C₃²×C₁₃	—	9	2	no	—
67	C₃²×C₁₃	—	9	1	no	—
68	C₃²×C₁₃	—	9	1	no	—
69	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
70	C₉×C₁₃	—	3	1	no	—
71	C₉×C₁₃	—	3	6	no	—
72	C₉×C₁₃	—	3	1	no	—
73	C₉×C₁₃	—	9	2	no	—
74	C₉×C₁₃	—	9	2	no	—
75	C₉×C₁₃	—	9	1	no	—
76	C₉×C₁₃	—	9	1	no	—
77	C₉×C₁₃	—	9	2	no	—
78	C₉×C₁₃	AΣL(1,𝔽₂₇)	1	3	yes	—
79	#737	—	3	9	no	—
80	C₃×3₋¹⁺²	—	3	1	no	—
81	C₃×3₋¹⁺²	—	3	1	no	—
82	C₃×3₋¹⁺²	C₃²×C₁₃	1	2	yes	—
83	C₃×3₋¹⁺²	C₃²×C₁₃	1	2	yes	—
84	C₃×H(3,𝔽₃)	C₃²×C₁₃	1	1	yes	characteristic
85	C₃×H(3,𝔽₃)	C₃²×C₁₃	1	1	yes	characteristic
86	C₃²×C₉	—	3	1	no	—
87	C₃²×C₉	—	3	1	no	—
88	C₃²×C₉	C₃²×C₁₃	1	1	yes	characteristic
89	C₉⋊C₉	—	3	2	no	—
90	C₉⋊C₉	—	3	2	no	—
91	C₉⋊C₉	—	3	2	no	—
92	C₃²⋊C₉	—	3	1	no	—
93	C₃²⋊C₉	—	3	1	no	—
94	C₃²⋊C₉	C₃²×C₁₃	1	2	yes	—
95	C₃²⋊C₉	C₃²×C₁₃	1	1	yes	characteristic
96	C₃²⋊C₉	C₃²×C₁₃	1	2	yes	—
97	C₃²⋊C₉	C₃²×C₁₃	1	1	yes	characteristic
98	C₃×C₁₃	—	9	1	no	—
99	C₃×C₁₃	—	9	3	no	—
100	C₃×C₁₃	—	9	3	no	—
101	C₃×C₁₃	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
102	C₃×C₁₃	#6908	1	1	yes	characteristic
103	C₃×C₁₃	#6913	1	2	yes	—
104	C₃³	—	3	1	no	—
105	C₃³	—	3	1	no	—
106	C₃³	C₃³×C₁₃	1	1	yes	characteristic
107	3₋¹⁺²	—	3	9	no	—
108	3₋¹⁺²	—	3	9	no	—
109	3₋¹⁺²	—	3	6	no	—
110	3₋¹⁺²	—	9	2	no	—
111	H(3,𝔽₃)	—	3	3	no	—
112	H(3,𝔽₃)	—	3	3	no	—
113	C₃×C₉	—	3	9	no	—
114	C₃×C₉	—	3	2	no	—
115	C₃×C₉	—	3	3	no	—
116	C₃×C₉	—	3	1	no	—
117	C₃×C₉	—	3	1	no	—
118	C₃×C₉	—	3	2	no	—
119	C₃×C₉	—	3	2	no	—
120	C₃×C₉	—	3	2	no	—
121	C₃×C₉	—	9	3	no	—
122	C₃×C₉	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
123	C₃×C₉	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
124	C₃×C₉	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
125	C₁₃	#10624595	1	1	yes	characteristic, direct product factor, Sylow
126	C₃²	—	3	1	no	—
127	C₃²	—	3	3	no	—
128	C₃²	—	3	1	no	—
129	C₃²	—	3	3	no	—
130	C₃²	—	3	2	no	—
131	C₃²	—	9	1	no	—
132	C₃²	—	9	1	no	—
133	C₃²	—	9	2	no	—
134	C₃²	—	9	2	no	—
135	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
136	C₉	—	3	1	no	—
137	C₉	—	3	6	no	—
138	C₉	—	3	1	no	—
139	C₉	—	9	1	no	—
140	C₉	—	9	1	no	—
141	C₉	—	9	2	no	—
142	C₉	—	9	2	no	—
143	C₉	—	9	2	no	—
144	C₉	C₁₃×AΣL(1,𝔽₂₇)	1	3	yes	—
145	C₃	—	9	1	no	—
146	C₃	—	9	3	no	—
147	C₃	—	9	3	no	—
148	C₃	C₃×C₁₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
149	C₃	C₁₃×#6908	1	1	yes	characteristic
150	C₃	C₁₃×#6913	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₁₃×#737

C₃

yes

maximal, characteristic, complemented

C₁₃×#6894

C₃

yes

maximal, complemented

C₉×C₁₃×H(3,𝔽₃)

C₃

yes

maximal, characteristic, complemented

C₁₃×#6860

C₃

yes

maximal, complemented

C₁₃×#6859

C₃

yes

maximal, characteristic, complemented

C₁₃×#6858

C₃

yes

maximal, complemented

C₁₃×#6857

C₃

yes

maximal, characteristic, complemented

C₁₃×#6857

C₃

yes

maximal, complemented

C₁₃×#6856

C₃

yes

maximal, characteristic

C₁₃×#737

—

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)

C₃²

yes

characteristic

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

characteristic

C₃²×C₉×C₁₃

—

C₃²×C₉×C₁₃

—

C₃²×C₉×C₁₃

C₃²

yes

characteristic

C₁₃×C₉⋊C₉

—

C₁₃×C₉⋊C₉

—

C₁₃×C₉⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

C₃²

yes

—

C₁₃×C₃²⋊C₉

C₃²

yes

characteristic

C₁₃×C₃²⋊C₉

C₃²

yes

—

C₁₃×C₃²⋊C₉

C₃²

yes

characteristic

#10624595

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

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×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

yes

C₃×C₉×C₁₃

yes

C₃×C₉×C₁₃

yes

C₃×C₁₃

yes

characteristic, complemented

#6894

C₃×C₁₃

yes

complemented

C₉×H(3,𝔽₃)

C₃×C₁₃

yes

characteristic, complemented

#6860

C₃×C₁₃

yes

complemented

#6859

C₃×C₁₃

yes

characteristic, complemented

#6858

C₃×C₁₃

yes

complemented

#6857

C₃×C₁₃

yes

characteristic, complemented

#6857

C₃×C₁₃

yes

complemented

#6856

C₃×C₁₃

yes

characteristic

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

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

AΣL(1,𝔽₂₇)

yes

—

#737

—

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)

C₃²×C₁₃

yes

characteristic

C₃×H(3,𝔽₃)

C₃²×C₁₃

yes

characteristic

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃²×C₁₃

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃²×C₁₃

yes

—

C₃²⋊C₉

C₃²×C₁₃

yes

characteristic

C₃²⋊C₉

C₃²×C₁₃

yes

—

C₃²⋊C₉

C₃²×C₁₃

yes

characteristic

C₃×C₁₃

—

C₃×C₁₃

—

100

C₃×C₁₃

—

101

C₃×C₁₃

C₃×AΣL(1,𝔽₂₇)

yes

characteristic

102

C₃×C₁₃

#6908

yes

characteristic

103

C₃×C₁₃

#6913

yes

—

104

C₃³

—

105

C₃³

—

106

C₃³

C₃³×C₁₃

yes

characteristic

107

3₋¹⁺²

—

108

3₋¹⁺²

—

109

3₋¹⁺²

—

110

3₋¹⁺²

—

111

H(3,𝔽₃)

—

112

H(3,𝔽₃)

—

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

—

122

C₃×C₉

C₁₃×H(3,𝔽₃)

yes

characteristic

123

C₃×C₉

C₁₃×H(3,𝔽₃)

yes

characteristic

124

C₃×C₉

C₁₃×H(3,𝔽₃)

yes

characteristic

125

C₁₃

#10624595

yes

characteristic, direct product factor, Sylow

126

C₃²

—

127

C₃²

—

128

C₃²

—

129

C₃²

—

130

C₃²

—

131

C₃²

—

132

C₃²

—

133

C₃²

—

134

C₃²

—

135

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

136

C₉

—

137

C₉

—

138

C₉

—

139

C₉

—

140

C₉

—

141

C₉

—

142

C₉

—

143

C₉

—

144

C₉

C₁₃×AΣL(1,𝔽₂₇)

yes

—

145

C₃

—

146

C₃

—

147

C₃

—

148

C₃

C₃×C₁₃×AΣL(1,𝔽₂₇)

yes

characteristic

149

C₃

C₁₃×#6908

yes

characteristic

150

C₃

C₁₃×#6913

yes

—

	subgroup	G/H
center, socle	C₃²×C₁₃	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₁₃
prime radical	C₁₃×#6856	C₃
LCS	G ▷ C₃³ ▷ C₃² ▷ 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

C₁₃×#6856

C₃

LCS

G ▷ C₃³ ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₃²×C₁₃ ◁ C₃²×C₉×C₁₃ ◁ G

N	H ≅ G/N	type
1. C₃×C₁₃×#737	C₃	semidirect
2. C₁₃×#6894	C₃	semidirect
3. C₉×C₁₃×H(3,𝔽₃)	C₃	semidirect
4. C₁₃×#6860	C₃	semidirect
5. C₁₃×#6859	C₃	semidirect
6. C₁₃×#6858	C₃	semidirect
7. C₁₃×#6857	C₃	semidirect
8. C₁₃×#6857	C₃	semidirect
29. #10624595	C₁₃	direct (N × H)
51. C₃×#737	C₃×C₁₃	semidirect
52. #6894	C₃×C₁₃	semidirect
53. C₉×H(3,𝔽₃)	C₃×C₁₃	semidirect
54. #6860	C₃×C₁₃	semidirect
55. #6859	C₃×C₁₃	semidirect
56. #6858	C₃×C₁₃	semidirect
57. #6857	C₃×C₁₃	semidirect
58. #6857	C₃×C₁₃	semidirect
125. C₁₃	#10624595	direct (N × H)

H ≅ G/N

type

1. C₃×C₁₃×#737

C₃

semidirect

2. C₁₃×#6894

C₃

semidirect

3. C₉×C₁₃×H(3,𝔽₃)

C₃

semidirect

4. C₁₃×#6860

C₃

semidirect

5. C₁₃×#6859

C₃

semidirect

6. C₁₃×#6858

C₃

semidirect

7. C₁₃×#6857

C₃

semidirect

8. C₁₃×#6857

C₃

semidirect

29. #10624595

C₁₃

direct (N × H)

51. C₃×#737

C₃×C₁₃

semidirect

52. #6894

C₃×C₁₃

semidirect

53. C₉×H(3,𝔽₃)

C₃×C₁₃

semidirect

54. #6860

C₃×C₁₃

semidirect

55. #6859

C₃×C₁₃

semidirect

56. #6858

C₃×C₁₃

semidirect

57. #6857

C₃×C₁₃

semidirect

58. #6857

C₃×C₁₃

semidirect

125. C₁₃

#10624595

direct (N × H)

Subgroups — C₁₃×#10624595 #23744746

Conjugacy classes of subgroups

Complements

Subgroups — C13×#10624595 #23744746

Conjugacy classes of subgroups

Complements

Subgroups — C₁₃×#10624595 #23744746