Group #23744734 — subgroups

Subgroups — C₁₃×#10624583 #23744734

Order 9477.

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

normal subgroups only

This group has 334 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	C₉×C₁₃×3₋¹⁺²	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	2	no	—
12	C₃×C₁₃×3₋¹⁺²	C₃²	1	2	yes	—
13	C₃×C₁₃×3₋¹⁺²	C₃²	1	2	yes	—
14	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
15	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
16	C₃²×C₉×C₁₃	—	3	1	no	—
17	C₃²×C₉×C₁₃	—	3	1	no	—
18	C₃²×C₉×C₁₃	C₃²	1	1	yes	characteristic
19	C₁₃×C₉⋊C₉	—	3	4	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	2	yes	—
24	C₁₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
25	C₉²×C₁₃	—	3	2	no	—
26	#10624583	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₃³×C₁₃	—	3	1	no	—
28	C₃³×C₁₃	—	3	1	no	—
29	C₃³×C₁₃	C₃³	1	1	yes	characteristic
30	C₁₃×3₋¹⁺²	—	3	18	no	—
31	C₁₃×3₋¹⁺²	—	3	6	no	—
32	C₁₃×3₋¹⁺²	—	3	6	no	—
33	C₁₃×H(3,𝔽₃)	—	3	3	no	—
34	C₁₃×H(3,𝔽₃)	—	3	3	no	—
35	C₃×C₉×C₁₃	—	3	2	no	—
36	C₃×C₉×C₁₃	—	3	2	no	—
37	C₃×C₉×C₁₃	—	3	9	no	—
38	C₃×C₉×C₁₃	—	3	3	no	—
39	C₃×C₉×C₁₃	—	3	2	no	—
40	C₃×C₉×C₁₃	—	3	2	no	—
41	C₃×C₉×C₁₃	—	3	1	no	—
42	C₃×C₉×C₁₃	—	3	1	no	—
43	C₃×C₉×C₁₃	—	9	3	no	—
44	C₃×C₉×C₁₃	H(3,𝔽₃)	1	2	yes	—
45	C₃×C₉×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
46	C₃×#737	C₃×C₁₃	1	1	yes	characteristic, complemented
47	C₉×3₋¹⁺²	C₃×C₁₃	1	2	yes	complemented
48	C₉×H(3,𝔽₃)	C₃×C₁₃	1	1	yes	characteristic, complemented
49	#6860	C₃×C₁₃	1	2	yes	complemented
50	#6859	C₃×C₁₃	1	1	yes	characteristic, complemented
51	#6858	C₃×C₁₃	1	2	yes	complemented
52	#6857	C₃×C₁₃	1	1	yes	characteristic, complemented
53	#6857	C₃×C₁₃	1	2	yes	complemented
54	#6856	C₃×C₁₃	1	1	yes	characteristic
55	C₃²×C₁₃	—	3	1	no	—
56	C₃²×C₁₃	—	3	3	no	—
57	C₃²×C₁₃	—	3	1	no	—
58	C₃²×C₁₃	—	3	3	no	—
59	C₃²×C₁₃	—	3	2	no	—
60	C₃²×C₁₃	—	9	1	no	—
61	C₃²×C₁₃	—	9	2	no	—
62	C₃²×C₁₃	—	9	2	no	—
63	C₃²×C₁₃	—	9	1	no	—
64	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
65	C₉×C₁₃	—	3	6	no	—
66	C₉×C₁₃	—	3	2	no	—
67	C₉×C₁₃	—	3	6	no	—
68	C₉×C₁₃	—	9	1	no	—
69	C₉×C₁₃	—	9	1	no	—
70	C₉×C₁₃	—	9	2	no	—
71	C₉×C₁₃	—	9	2	no	—
72	C₉×C₁₃	AΣL(1,𝔽₂₇)	1	3	yes	—
73	#737	—	3	9	no	—
74	C₃×3₋¹⁺²	—	3	2	no	—
75	C₃×3₋¹⁺²	C₃²×C₁₃	1	2	yes	—
76	C₃×3₋¹⁺²	C₃²×C₁₃	1	2	yes	—
77	C₃×H(3,𝔽₃)	C₃²×C₁₃	1	1	yes	characteristic
78	C₃×H(3,𝔽₃)	C₃²×C₁₃	1	1	yes	characteristic
79	C₃²×C₉	—	3	1	no	—
80	C₃²×C₉	—	3	1	no	—
81	C₃²×C₉	C₃²×C₁₃	1	1	yes	characteristic
82	C₉⋊C₉	—	3	4	no	—
83	C₃²⋊C₉	—	3	2	no	—
84	C₃²⋊C₉	C₃²×C₁₃	1	2	yes	—
85	C₃²⋊C₉	C₃²×C₁₃	1	1	yes	characteristic
86	C₃²⋊C₉	C₃²×C₁₃	1	2	yes	—
87	C₃²⋊C₉	C₃²×C₁₃	1	1	yes	characteristic
88	C₉²	—	3	2	no	—
89	C₃×C₁₃	—	9	1	no	—
90	C₃×C₁₃	—	9	3	no	—
91	C₃×C₁₃	—	9	3	no	—
92	C₃×C₁₃	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
93	C₃×C₁₃	#6908	1	2	yes	—
94	C₃×C₁₃	#6908	1	1	yes	characteristic
95	C₃³	—	3	1	no	—
96	C₃³	—	3	1	no	—
97	C₃³	C₃³×C₁₃	1	1	yes	characteristic
98	3₋¹⁺²	—	3	18	no	—
99	3₋¹⁺²	—	3	6	no	—
100	3₋¹⁺²	—	3	6	no	—
101	H(3,𝔽₃)	—	3	3	no	—
102	H(3,𝔽₃)	—	3	3	no	—
103	C₃×C₉	—	3	2	no	—
104	C₃×C₉	—	3	2	no	—
105	C₃×C₉	—	3	9	no	—
106	C₃×C₉	—	3	3	no	—
107	C₃×C₉	—	3	2	no	—
108	C₃×C₉	—	3	2	no	—
109	C₃×C₉	—	3	1	no	—
110	C₃×C₉	—	3	1	no	—
111	C₃×C₉	—	9	3	no	—
112	C₃×C₉	C₁₃×H(3,𝔽₃)	1	2	yes	—
113	C₃×C₉	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
114	C₁₃	#10624583	1	1	yes	characteristic, direct product factor, Sylow
115	C₃²	—	3	1	no	—
116	C₃²	—	3	3	no	—
117	C₃²	—	3	1	no	—
118	C₃²	—	3	3	no	—
119	C₃²	—	3	2	no	—
120	C₃²	—	9	1	no	—
121	C₃²	—	9	2	no	—
122	C₃²	—	9	2	no	—
123	C₃²	—	9	1	no	—
124	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
125	C₉	—	3	6	no	—
126	C₉	—	3	2	no	—
127	C₉	—	3	6	no	—
128	C₉	—	9	1	no	—
129	C₉	—	9	1	no	—
130	C₉	—	9	2	no	—
131	C₉	—	9	2	no	—
132	C₉	C₁₃×AΣL(1,𝔽₂₇)	1	3	yes	—
133	C₃	—	9	1	no	—
134	C₃	—	9	3	no	—
135	C₃	—	9	3	no	—
136	C₃	C₃×C₁₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
137	C₃	C₁₃×#6908	1	2	yes	—
138	C₃	C₁₃×#6908	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₁₃×#737

C₃

yes

maximal, characteristic, complemented

C₉×C₁₃×3₋¹⁺²

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

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×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

C₃²

yes

—

C₁₃×C₃²⋊C₉

C₃²

yes

characteristic

C₁₃×C₃²⋊C₉

C₃²

yes

—

C₁₃×C₃²⋊C₉

C₃²

yes

characteristic

C₉²×C₁₃

—

#10624583

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×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

H(3,𝔽₃)

yes

—

C₃×C₉×C₁₃

H(3,𝔽₃)

yes

characteristic

C₃×#737

C₃×C₁₃

yes

characteristic, complemented

C₉×3₋¹⁺²

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

AΣL(1,𝔽₂₇)

yes

—

#737

—

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

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃²×C₁₃

yes

—

C₃²⋊C₉

C₃²×C₁₃

yes

characteristic

C₃²⋊C₉

C₃²×C₁₃

yes

—

C₃²⋊C₉

C₃²×C₁₃

yes

characteristic

C₉²

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

C₃×AΣL(1,𝔽₂₇)

yes

characteristic

C₃×C₁₃

#6908

yes

—

C₃×C₁₃

#6908

yes

characteristic

C₃³

—

C₃³

—

C₃³

C₃³×C₁₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

100

3₋¹⁺²

—

101

H(3,𝔽₃)

—

102

H(3,𝔽₃)

—

103

C₃×C₉

—

104

C₃×C₉

—

105

C₃×C₉

—

106

C₃×C₉

—

107

C₃×C₉

—

108

C₃×C₉

—

109

C₃×C₉

—

110

C₃×C₉

—

111

C₃×C₉

—

112

C₃×C₉

C₁₃×H(3,𝔽₃)

yes

—

113

C₃×C₉

C₁₃×H(3,𝔽₃)

yes

characteristic

114

C₁₃

#10624583

yes

characteristic, direct product factor, Sylow

115

C₃²

—

116

C₃²

—

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

—

122

C₃²

—

123

C₃²

—

124

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

125

C₉

—

126

C₉

—

127

C₉

—

128

C₉

—

129

C₉

—

130

C₉

—

131

C₉

—

132

C₉

C₁₃×AΣL(1,𝔽₂₇)

yes

—

133

C₃

—

134

C₃

—

135

C₃

—

136

C₃

C₃×C₁₃×AΣL(1,𝔽₂₇)

yes

characteristic

137

C₃

C₁₃×#6908

yes

—

138

C₃

C₁₃×#6908

yes

characteristic

	subgroup	G/H
center	C₃×C₉×C₁₃	H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₁₃
socle	C₃²×C₁₃	C₃×H(3,𝔽₃)
prime radical	C₁₃×#6856	C₃
LCS	G ▷ C₃³ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃×C₉×C₁₃ ◁ C₃²×C₉×C₁₃ ◁ G

subgroup

G/H

center

C₃×C₉×C₁₃

H(3,𝔽₃)

commutator, Frattini, cosocle

C₃³

C₃³×C₁₃

socle

C₃²×C₁₃

C₃×H(3,𝔽₃)

prime radical

C₁₃×#6856

C₃

LCS

G ▷ C₃³ ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₃×C₉×C₁₃ ◁ C₃²×C₉×C₁₃ ◁ G

N	H ≅ G/N	type
1. C₃×C₁₃×#737	C₃	semidirect
2. C₉×C₁₃×3₋¹⁺²	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
26. #10624583	C₁₃	direct (N × H)
46. C₃×#737	C₃×C₁₃	semidirect
47. C₉×3₋¹⁺²	C₃×C₁₃	semidirect
48. C₉×H(3,𝔽₃)	C₃×C₁₃	semidirect
49. #6860	C₃×C₁₃	semidirect
50. #6859	C₃×C₁₃	semidirect
51. #6858	C₃×C₁₃	semidirect
52. #6857	C₃×C₁₃	semidirect
53. #6857	C₃×C₁₃	semidirect
114. C₁₃	#10624583	direct (N × H)

H ≅ G/N

type

1. C₃×C₁₃×#737

C₃

semidirect

2. C₉×C₁₃×3₋¹⁺²

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

26. #10624583

C₁₃

direct (N × H)

46. C₃×#737

C₃×C₁₃

semidirect

47. C₉×3₋¹⁺²

C₃×C₁₃

semidirect

48. C₉×H(3,𝔽₃)

C₃×C₁₃

semidirect

49. #6860

C₃×C₁₃

semidirect

50. #6859

C₃×C₁₃

semidirect

51. #6858

C₃×C₁₃

semidirect

52. #6857

C₃×C₁₃

semidirect

53. #6857

C₃×C₁₃

semidirect

114. C₁₃

#10624583

direct (N × H)

Subgroups — C₁₃×#10624583 #23744734

Conjugacy classes of subgroups

Complements

Subgroups — C13×#10624583 #23744734

Conjugacy classes of subgroups

Complements

Subgroups — C₁₃×#10624583 #23744734