Group #23744752 — subgroups

Subgroups — C₁₃×#10624601 #23744752

Order 9477.

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

normal subgroups only

This group has 232 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	C₁₃×C₉⋊H(3,𝔽₃)	C₃	1	2	yes	maximal, complemented
2	C₉×C₁₃×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#6887	C₃	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#6861	C₃	1	2	yes	maximal, complemented
5	C₁₃×#6861	C₃	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#6859	C₃	1	2	yes	maximal, complemented
7	C₁₃×#6859	C₃	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#6856	C₃	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#6856	C₃	1	2	yes	maximal, complemented
10	C₃×C₁₃×3₋¹⁺²	—	3	2	no	—
11	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
12	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	2	yes	—
13	C₃²×C₉×C₁₃	—	3	1	no	—
14	C₃²×C₉×C₁₃	C₃²	1	1	yes	characteristic
15	C₁₃×C₉⋊C₉	—	3	1	no	—
16	C₁₃×C₉⋊C₉	—	3	1	no	—
17	C₁₃×C₃²⋊C₉	—	3	1	no	—
18	C₁₃×C₃²⋊C₉	—	3	1	no	—
19	C₁₃×C₃²⋊C₉	—	3	2	no	—
20	C₁₃×C₃²⋊C₉	—	3	2	no	—
21	C₁₃×C₃²⋊C₉	C₃²	1	2	yes	—
22	C₁₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
23	C₁₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
24	C₁₃×C₃²⋊C₉	C₃²	1	2	yes	—
25	C₁₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
26	C₁₃×C₃²⋊C₉	C₃²	1	2	yes	—
27	C₉²×C₁₃	—	3	1	no	—
28	#10624601	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₃³×C₁₃	—	3	1	no	—
30	C₃³×C₁₃	—	3	2	no	—
31	C₃³×C₁₃	C₃³	1	1	yes	characteristic
32	C₁₃×3₋¹⁺²	—	9	6	no	—
33	C₁₃×H(3,𝔽₃)	—	3	3	no	—
34	C₁₃×H(3,𝔽₃)	—	9	2	no	—
35	C₃×C₉×C₁₃	—	3	3	no	—
36	C₃×C₉×C₁₃	—	3	2	no	—
37	C₃×C₉×C₁₃	—	3	1	no	—
38	C₃×C₉×C₁₃	—	3	1	no	—
39	C₃×C₉×C₁₃	—	3	1	no	—
40	C₃×C₉×C₁₃	—	3	2	no	—
41	C₃×C₉×C₁₃	—	3	2	no	—
42	C₃×C₉×C₁₃	—	9	3	no	—
43	C₃×C₉×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
44	C₃×C₉×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
45	C₃×C₉×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
46	C₉⋊H(3,𝔽₃)	C₃×C₁₃	1	2	yes	complemented
47	C₉×H(3,𝔽₃)	C₃×C₁₃	1	1	yes	characteristic, complemented
48	#6887	C₃×C₁₃	1	1	yes	characteristic, complemented
49	#6861	C₃×C₁₃	1	2	yes	complemented
50	#6861	C₃×C₁₃	1	1	yes	characteristic, complemented
51	#6859	C₃×C₁₃	1	2	yes	complemented
52	#6859	C₃×C₁₃	1	1	yes	characteristic, complemented
53	#6856	C₃×C₁₃	1	1	yes	characteristic, complemented
54	#6856	C₃×C₁₃	1	2	yes	complemented
55	C₃²×C₁₃	—	3	2	no	—
56	C₃²×C₁₃	—	3	1	no	—
57	C₃²×C₁₃	—	3	1	no	—
58	C₃²×C₁₃	—	3	3	no	—
59	C₃²×C₁₃	—	9	2	no	—
60	C₃²×C₁₃	—	9	1	no	—
61	C₃²×C₁₃	—	9	2	no	—
62	C₃²×C₁₃	—	9	2	no	—
63	C₃²×C₁₃	—	9	2	no	—
64	C₃²×C₁₃	—	9	2	no	—
65	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
66	C₉×C₁₃	—	3	1	no	—
67	C₉×C₁₃	—	3	1	no	—
68	C₉×C₁₃	—	9	2	no	—
69	C₉×C₁₃	—	9	1	no	—
70	C₉×C₁₃	—	9	1	no	—
71	C₉×C₁₃	—	9	2	no	—
72	C₉×C₁₃	—	9	2	no	—
73	C₉×C₁₃	—	9	1	no	—
74	C₉×C₁₃	#732	1	3	yes	—
75	C₃×3₋¹⁺²	—	3	2	no	—
76	C₃×H(3,𝔽₃)	C₃²×C₁₃	1	1	yes	characteristic
77	C₃×H(3,𝔽₃)	C₃²×C₁₃	1	2	yes	—
78	C₃²×C₉	—	3	1	no	—
79	C₃²×C₉	C₃²×C₁₃	1	1	yes	characteristic
80	C₉⋊C₉	—	3	1	no	—
81	C₉⋊C₉	—	3	1	no	—
82	C₃²⋊C₉	—	3	1	no	—
83	C₃²⋊C₉	—	3	1	no	—
84	C₃²⋊C₉	—	3	2	no	—
85	C₃²⋊C₉	—	3	2	no	—
86	C₃²⋊C₉	C₃²×C₁₃	1	1	yes	characteristic
87	C₃²⋊C₉	C₃²×C₁₃	1	2	yes	—
88	C₃²⋊C₉	C₃²×C₁₃	1	1	yes	characteristic
89	C₃²⋊C₉	C₃²×C₁₃	1	2	yes	—
90	C₃²⋊C₉	C₃²×C₁₃	1	1	yes	characteristic
91	C₃²⋊C₉	C₃²×C₁₃	1	2	yes	—
92	C₉²	—	3	1	no	—
93	C₃×C₁₃	—	9	1	no	—
94	C₃×C₁₃	—	9	3	no	—
95	C₃×C₁₃	—	27	2	no	—
96	C₃×C₁₃	C₃×#732	1	1	yes	characteristic
97	C₃×C₁₃	#6908	1	1	yes	characteristic
98	C₃×C₁₃	#6913	1	2	yes	—
99	C₃³	—	3	1	no	—
100	C₃³	—	3	2	no	—
101	C₃³	C₃³×C₁₃	1	1	yes	characteristic
102	3₋¹⁺²	—	9	6	no	—
103	H(3,𝔽₃)	—	3	3	no	—
104	H(3,𝔽₃)	—	9	2	no	—
105	C₃×C₉	—	3	3	no	—
106	C₃×C₉	—	3	1	no	—
107	C₃×C₉	—	3	2	no	—
108	C₃×C₉	—	3	1	no	—
109	C₃×C₉	—	3	1	no	—
110	C₃×C₉	—	3	2	no	—
111	C₃×C₉	—	3	2	no	—
112	C₃×C₉	—	9	3	no	—
113	C₃×C₉	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
114	C₃×C₉	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
115	C₃×C₉	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
116	C₁₃	#10624601	1	1	yes	characteristic, direct product factor, Sylow
117	C₃²	—	3	1	no	—
118	C₃²	—	3	1	no	—
119	C₃²	—	3	3	no	—
120	C₃²	—	3	2	no	—
121	C₃²	—	9	1	no	—
122	C₃²	—	9	2	no	—
123	C₃²	—	9	2	no	—
124	C₃²	—	9	2	no	—
125	C₃²	—	9	2	no	—
126	C₃²	—	9	2	no	—
127	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
128	C₉	—	3	1	no	—
129	C₉	—	3	1	no	—
130	C₉	—	9	1	no	—
131	C₉	—	9	1	no	—
132	C₉	—	9	2	no	—
133	C₉	—	9	2	no	—
134	C₉	—	9	1	no	—
135	C₉	—	9	2	no	—
136	C₉	C₁₃×#732	1	3	yes	—
137	C₃	—	9	1	no	—
138	C₃	—	9	3	no	—
139	C₃	—	27	2	no	—
140	C₃	C₃×C₁₃×#732	1	1	yes	characteristic
141	C₃	C₁₃×#6908	1	1	yes	characteristic
142	C₃	C₁₃×#6913	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×C₉⋊H(3,𝔽₃)

C₃

yes

maximal, complemented

C₉×C₁₃×H(3,𝔽₃)

C₃

yes

maximal, characteristic, complemented

C₁₃×#6887

C₃

yes

maximal, characteristic, complemented

C₁₃×#6861

C₃

yes

maximal, complemented

C₁₃×#6861

C₃

yes

maximal, characteristic, complemented

C₁₃×#6859

C₃

yes

maximal, complemented

C₁₃×#6859

C₃

yes

maximal, characteristic, complemented

C₁₃×#6856

C₃

yes

maximal, characteristic, complemented

C₁₃×#6856

C₃

yes

maximal, complemented

C₃×C₁₃×3₋¹⁺²

—

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

characteristic

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

—

C₃²×C₉×C₁₃

—

C₃²×C₉×C₁₃

C₃²

yes

characteristic

C₁₃×C₉⋊C₉

—

C₁₃×C₉⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

C₃²

yes

—

C₁₃×C₃²⋊C₉

C₃²

yes

characteristic

C₁₃×C₃²⋊C₉

C₃²

yes

characteristic

C₁₃×C₃²⋊C₉

C₃²

yes

—

C₁₃×C₃²⋊C₉

C₃²

yes

characteristic

C₁₃×C₃²⋊C₉

C₃²

yes

—

C₉²×C₁₃

—

#10624601

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

C₃³

yes

characteristic

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

yes

C₃×C₉×C₁₃

yes

C₃×C₉×C₁₃

yes

C₉⋊H(3,𝔽₃)

C₃×C₁₃

yes

complemented

C₉×H(3,𝔽₃)

C₃×C₁₃

yes

characteristic, complemented

#6887

C₃×C₁₃

yes

characteristic, complemented

#6861

C₃×C₁₃

yes

complemented

#6861

C₃×C₁₃

yes

characteristic, complemented

#6859

C₃×C₁₃

yes

complemented

#6859

C₃×C₁₃

yes

characteristic, complemented

#6856

C₃×C₁₃

yes

characteristic, complemented

#6856

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

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

#732

yes

—

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

C₃²×C₁₃

yes

characteristic

C₃×H(3,𝔽₃)

C₃²×C₁₃

yes

—

C₃²×C₉

—

C₃²×C₉

C₃²×C₁₃

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃²×C₁₃

yes

characteristic

C₃²⋊C₉

C₃²×C₁₃

yes

—

C₃²⋊C₉

C₃²×C₁₃

yes

characteristic

C₃²⋊C₉

C₃²×C₁₃

yes

—

C₃²⋊C₉

C₃²×C₁₃

yes

characteristic

C₃²⋊C₉

C₃²×C₁₃

yes

—

C₉²

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

yes

C₃×C₁₃

yes

C₃×C₁₃

yes

—

C₃³

—

100

C₃³

—

101

C₃³

C₃³×C₁₃

yes

characteristic

102

3₋¹⁺²

—

103

H(3,𝔽₃)

—

104

H(3,𝔽₃)

—

105

C₃×C₉

—

106

C₃×C₉

—

107

C₃×C₉

—

108

C₃×C₉

—

109

C₃×C₉

—

110

C₃×C₉

—

111

C₃×C₉

—

112

C₃×C₉

—

113

C₃×C₉

C₁₃×H(3,𝔽₃)

yes

characteristic

114

C₃×C₉

C₁₃×H(3,𝔽₃)

yes

characteristic

115

C₃×C₉

C₁₃×H(3,𝔽₃)

yes

characteristic

116

C₁₃

#10624601

yes

characteristic, direct product factor, Sylow

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

—

122

C₃²

—

123

C₃²

—

124

C₃²

—

125

C₃²

—

126

C₃²

—

127

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

128

C₉

—

129

C₉

—

130

C₉

—

131

C₉

—

132

C₉

—

133

C₉

—

134

C₉

—

135

C₉

—

136

C₉

C₁₃×#732

yes

—

137

C₃

—

138

C₃

—

139

C₃

—

140

C₃

C₃×C₁₃×#732

yes

characteristic

141

C₃

C₁₃×#6908

yes

characteristic

142

C₃

C₁₃×#6913

yes

—

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

LCS

G ▷ C₃³ ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₃²×C₁₃ ◁ C₃²×C₉×C₁₃ ◁ G

N	H ≅ G/N	type
1. C₁₃×C₉⋊H(3,𝔽₃)	C₃	semidirect
2. C₉×C₁₃×H(3,𝔽₃)	C₃	semidirect
3. C₁₃×#6887	C₃	semidirect
4. C₁₃×#6861	C₃	semidirect
5. C₁₃×#6861	C₃	semidirect
6. C₁₃×#6859	C₃	semidirect
7. C₁₃×#6859	C₃	semidirect
8. C₁₃×#6856	C₃	semidirect
9. C₁₃×#6856	C₃	semidirect
28. #10624601	C₁₃	direct (N × H)
46. C₉⋊H(3,𝔽₃)	C₃×C₁₃	semidirect
47. C₉×H(3,𝔽₃)	C₃×C₁₃	semidirect
48. #6887	C₃×C₁₃	semidirect
49. #6861	C₃×C₁₃	semidirect
50. #6861	C₃×C₁₃	semidirect
51. #6859	C₃×C₁₃	semidirect
52. #6859	C₃×C₁₃	semidirect
53. #6856	C₃×C₁₃	semidirect
54. #6856	C₃×C₁₃	semidirect
116. C₁₃	#10624601	direct (N × H)

H ≅ G/N

type

1. C₁₃×C₉⋊H(3,𝔽₃)

C₃

semidirect

2. C₉×C₁₃×H(3,𝔽₃)

C₃

semidirect

3. C₁₃×#6887

C₃

semidirect

4. C₁₃×#6861

C₃

semidirect

5. C₁₃×#6861

C₃

semidirect

6. C₁₃×#6859

C₃

semidirect

7. C₁₃×#6859

C₃

semidirect

8. C₁₃×#6856

C₃

semidirect

9. C₁₃×#6856

C₃

semidirect

28. #10624601

C₁₃

direct (N × H)

46. C₉⋊H(3,𝔽₃)

C₃×C₁₃

semidirect

47. C₉×H(3,𝔽₃)

C₃×C₁₃

semidirect

48. #6887

C₃×C₁₃

semidirect

49. #6861

C₃×C₁₃

semidirect

50. #6861

C₃×C₁₃

semidirect

51. #6859

C₃×C₁₃

semidirect

52. #6859

C₃×C₁₃

semidirect

53. #6856

C₃×C₁₃

semidirect

54. #6856

C₃×C₁₃

semidirect

116. C₁₃

#10624601

direct (N × H)

Subgroups — C₁₃×#10624601 #23744752

Conjugacy classes of subgroups

Complements

Subgroups — C13×#10624601 #23744752

Conjugacy classes of subgroups

Complements

Subgroups — C₁₃×#10624601 #23744752