Group #23745085 — subgroups

Subgroups — C₁₃×#10624934 #23745085

Order 9477.

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

normal subgroups only

This group has 980 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	C₃²×C₁₃×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#6913	C₃	1	18	yes	maximal, complemented
4	C₁₃×#6912	C₃	1	2	yes	maximal, complemented
5	C₁₃×#6911	C₃	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#6910	C₃	1	3	yes	maximal, complemented
7	C₁₃×#6908	C₃	1	9	yes	maximal, complemented
8	C₃×C₁₃×AΣL(1,𝔽₂₇)	C₃	1	3	yes	maximal, complemented
9	C₃⁴×C₁₃	C₃²	1	1	yes	characteristic
10	C₁₃×#737	—	3	18	no	—
11	C₁₃×#737	C₃²	1	27	yes	complemented
12	C₃×C₁₃×3₋¹⁺²	—	3	9	no	—
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	1	yes	characteristic
18	C₃×C₁₃×3₋¹⁺²	C₃²	1	6	yes	complemented
19	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	3	yes	—
20	C₃²×C₉×C₁₃	C₃²	1	3	yes	complemented
21	C₁₃×#733	C₃²	1	18	yes	complemented
22	C₁₃×#732	C₃²	1	9	yes	complemented
23	C₁₃×#731	C₃²	1	27	yes	complemented
24	C₁₃×AΣL(1,𝔽₂₇)	C₃²	1	9	yes	complemented
25	C₁₃×AΣL(1,𝔽₂₇)	C₃²	1	18	yes	complemented
26	#10624934	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₃³×C₁₃	—	3	3	no	—
28	C₃³×C₁₃	—	3	3	no	—
29	C₃³×C₁₃	—	9	3	no	—
30	C₃³×C₁₃	C₃³	1	1	yes	characteristic
31	C₃³×C₁₃	C₃³	1	2	yes	—
32	C₃³×C₁₃	C₃³	1	1	yes	characteristic
33	C₁₃×3₋¹⁺²	—	3	6	no	—
34	C₁₃×3₋¹⁺²	—	3	3	no	—
35	C₁₃×3₋¹⁺²	—	3	3	no	—
36	C₁₃×3₋¹⁺²	—	3	6	no	—
37	C₁₃×3₋¹⁺²	—	3	18	no	—
38	C₁₃×3₋¹⁺²	—	3	54	no	—
39	C₁₃×3₋¹⁺²	H(3,𝔽₃)	1	9	yes	—
40	C₁₃×3₋¹⁺²	C₃³	1	18	yes	—
41	C₁₃×H(3,𝔽₃)	—	3	6	no	—
42	C₁₃×H(3,𝔽₃)	C₃³	1	9	yes	—
43	C₃×C₉×C₁₃	—	3	6	no	—
44	C₃×C₉×C₁₃	—	3	9	no	—
45	C₃×C₉×C₁₃	—	3	27	no	—
46	C₃×C₉×C₁₃	C₃³	1	3	yes	—
47	C₃×C₉×C₁₃	C₃³	1	3	yes	—
48	C₃×C₉×C₁₃	C₃³	1	3	yes	—
49	3₋¹⁺⁴	C₃×C₁₃	1	3	yes	complemented
50	C₃²×3₋¹⁺²	C₃×C₁₃	1	1	yes	characteristic, complemented
51	#6913	C₃×C₁₃	1	18	yes	complemented
52	#6912	C₃×C₁₃	1	2	yes	complemented
53	#6911	C₃×C₁₃	1	1	yes	characteristic, complemented
54	#6910	C₃×C₁₃	1	3	yes	complemented
55	#6908	C₃×C₁₃	1	9	yes	complemented
56	C₃×AΣL(1,𝔽₂₇)	C₃×C₁₃	1	3	yes	complemented
57	C₃²×C₁₃	—	3	3	no	—
58	C₃²×C₁₃	—	3	1	no	—
59	C₃²×C₁₃	—	3	9	no	—
60	C₃²×C₁₃	—	3	2	no	—
61	C₃²×C₁₃	—	9	1	no	—
62	C₃²×C₁₃	—	9	9	no	—
63	C₃²×C₁₃	—	9	2	no	—
64	C₃²×C₁₃	—	27	3	no	—
65	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	3	yes	—
66	C₃²×C₁₃	C₃⁴	1	1	yes	characteristic
67	C₉×C₁₃	—	3	18	no	—
68	C₉×C₁₃	—	3	3	no	—
69	C₉×C₁₃	—	3	3	no	—
70	C₉×C₁₃	C₃×H(3,𝔽₃)	1	9	yes	—
71	C₃⁴	C₃²×C₁₃	1	1	yes	characteristic
72	#737	—	3	18	no	—
73	#737	C₃²×C₁₃	1	27	yes	complemented
74	C₃×3₋¹⁺²	—	3	9	no	—
75	C₃×3₋¹⁺²	—	3	9	no	—
76	C₃×3₋¹⁺²	C₃²×C₁₃	1	6	yes	—
77	C₃×3₋¹⁺²	C₃²×C₁₃	1	1	yes	characteristic
78	C₃×3₋¹⁺²	C₃²×C₁₃	1	1	yes	characteristic
79	C₃×3₋¹⁺²	C₃²×C₁₃	1	1	yes	characteristic
80	C₃×3₋¹⁺²	C₃²×C₁₃	1	6	yes	complemented
81	C₃×H(3,𝔽₃)	C₃²×C₁₃	1	3	yes	—
82	C₃²×C₉	C₃²×C₁₃	1	3	yes	complemented
83	#733	C₃²×C₁₃	1	18	yes	complemented
84	#732	C₃²×C₁₃	1	9	yes	complemented
85	#731	C₃²×C₁₃	1	27	yes	complemented
86	AΣL(1,𝔽₂₇)	C₃²×C₁₃	1	18	yes	complemented
87	AΣL(1,𝔽₂₇)	C₃²×C₁₃	1	9	yes	complemented
88	C₃×C₁₃	—	3	3	no	—
89	C₃×C₁₃	—	3	1	no	—
90	C₃×C₁₃	—	3	3	no	—
91	C₃×C₁₃	—	9	2	no	—
92	C₃×C₁₃	—	9	9	no	—
93	C₃×C₁₃	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
94	C₃³	—	3	3	no	—
95	C₃³	—	3	3	no	—
96	C₃³	—	9	3	no	—
97	C₃³	C₃³×C₁₃	1	1	yes	characteristic
98	C₃³	C₃³×C₁₃	1	2	yes	—
99	C₃³	C₃³×C₁₃	1	1	yes	characteristic
100	3₋¹⁺²	—	3	6	no	—
101	3₋¹⁺²	—	3	3	no	—
102	3₋¹⁺²	—	3	3	no	—
103	3₋¹⁺²	—	3	6	no	—
104	3₋¹⁺²	—	3	18	no	—
105	3₋¹⁺²	—	3	54	no	—
106	3₋¹⁺²	C₁₃×H(3,𝔽₃)	1	9	yes	—
107	3₋¹⁺²	C₃³×C₁₃	1	18	yes	—
108	H(3,𝔽₃)	—	3	6	no	—
109	H(3,𝔽₃)	C₃³×C₁₃	1	9	yes	—
110	C₃×C₉	—	3	6	no	—
111	C₃×C₉	—	3	9	no	—
112	C₃×C₉	—	3	27	no	—
113	C₃×C₉	C₃³×C₁₃	1	3	yes	—
114	C₃×C₉	C₃³×C₁₃	1	3	yes	—
115	C₃×C₉	C₃³×C₁₃	1	3	yes	—
116	C₁₃	#10624934	1	1	yes	characteristic, direct product factor, Sylow
117	C₃²	—	3	3	no	—
118	C₃²	—	3	1	no	—
119	C₃²	—	3	9	no	—
120	C₃²	—	3	2	no	—
121	C₃²	—	9	1	no	—
122	C₃²	—	9	9	no	—
123	C₃²	—	9	2	no	—
124	C₃²	—	27	3	no	—
125	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	3	yes	—
126	C₃²	C₃⁴×C₁₃	1	1	yes	characteristic
127	C₉	—	3	18	no	—
128	C₉	—	3	3	no	—
129	C₉	—	3	3	no	—
130	C₉	C₃×C₁₃×H(3,𝔽₃)	1	9	yes	—
131	C₃	—	3	1	no	—
132	C₃	—	3	3	no	—
133	C₃	—	3	3	no	—
134	C₃	—	9	2	no	—
135	C₃	—	9	9	no	—
136	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

C₃²×C₁₃×3₋¹⁺²

C₃

yes

maximal, characteristic, complemented

C₁₃×#6913

C₃

yes

maximal, complemented

C₁₃×#6912

C₃

yes

maximal, complemented

C₁₃×#6911

C₃

yes

maximal, characteristic, complemented

C₁₃×#6910

C₃

yes

maximal, complemented

C₁₃×#6908

C₃

yes

maximal, complemented

C₃×C₁₃×AΣL(1,𝔽₂₇)

C₃

yes

maximal, complemented

C₃⁴×C₁₃

C₃²

yes

characteristic

C₁₃×#737

—

C₁₃×#737

C₃²

yes

complemented

C₃×C₁₃×3₋¹⁺²

—

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

characteristic

C₃×C₁₃×3₋¹⁺²

C₃²

yes

complemented

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

—

C₃²×C₉×C₁₃

C₃²

yes

complemented

C₁₃×#733

C₃²

yes

complemented

C₁₃×#732

C₃²

yes

complemented

C₁₃×#731

C₃²

yes

complemented

C₁₃×AΣL(1,𝔽₂₇)

C₃²

yes

complemented

C₁₃×AΣL(1,𝔽₂₇)

C₃²

yes

complemented

#10624934

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

C₃³

yes

characteristic

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

C₃³

yes

—

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

C₃³

yes

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

C₃³

yes

—

C₃×C₉×C₁₃

C₃³

yes

—

C₃×C₉×C₁₃

C₃³

yes

—

3₋¹⁺⁴

C₃×C₁₃

yes

complemented

C₃²×3₋¹⁺²

C₃×C₁₃

yes

characteristic, complemented

#6913

C₃×C₁₃

yes

complemented

#6912

C₃×C₁₃

yes

complemented

#6911

C₃×C₁₃

yes

characteristic, complemented

#6910

C₃×C₁₃

yes

complemented

#6908

C₃×C₁₃

yes

complemented

C₃×AΣL(1,𝔽₂₇)

C₃×C₁₃

yes

complemented

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

—

C₃²×C₁₃

C₃⁴

yes

characteristic

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

C₃×H(3,𝔽₃)

yes

—

C₃⁴

C₃²×C₁₃

yes

characteristic

#737

—

#737

C₃²×C₁₃

yes

complemented

C₃×3₋¹⁺²

—

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

characteristic

C₃×3₋¹⁺²

C₃²×C₁₃

yes

complemented

C₃×H(3,𝔽₃)

C₃²×C₁₃

yes

—

C₃²×C₉

C₃²×C₁₃

yes

complemented

#733

C₃²×C₁₃

yes

complemented

#732

C₃²×C₁₃

yes

complemented

#731

C₃²×C₁₃

yes

complemented

AΣL(1,𝔽₂₇)

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

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃³×C₁₃

yes

characteristic

C₃³

C₃³×C₁₃

yes

—

C₃³

C₃³×C₁₃

yes

characteristic

100

3₋¹⁺²

—

101

3₋¹⁺²

—

102

3₋¹⁺²

—

103

3₋¹⁺²

—

104

3₋¹⁺²

—

105

3₋¹⁺²

—

106

3₋¹⁺²

C₁₃×H(3,𝔽₃)

yes

—

107

3₋¹⁺²

C₃³×C₁₃

yes

—

108

H(3,𝔽₃)

—

109

H(3,𝔽₃)

C₃³×C₁₃

yes

—

110

C₃×C₉

—

111

C₃×C₉

—

112

C₃×C₉

—

113

C₃×C₉

C₃³×C₁₃

yes

—

114

C₃×C₉

C₃³×C₁₃

yes

—

115

C₃×C₉

C₃³×C₁₃

yes

—

116

C₁₃

#10624934

yes

characteristic, direct product factor, Sylow

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

—

122

C₃²

—

123

C₃²

—

124

C₃²

—

125

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

—

126

C₃²

C₃⁴×C₁₃

yes

characteristic

127

C₉

—

128

C₉

—

129

C₉

—

130

C₉

C₃×C₁₃×H(3,𝔽₃)

yes

—

131

C₃

—

132

C₃

—

133

C₃

—

134

C₃

—

135

C₃

—

136

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. C₃²×C₁₃×3₋¹⁺²	C₃	semidirect
3. C₁₃×#6913	C₃	semidirect
4. C₁₃×#6912	C₃	semidirect
5. C₁₃×#6911	C₃	semidirect
6. C₁₃×#6910	C₃	semidirect
7. C₁₃×#6908	C₃	semidirect
8. C₃×C₁₃×AΣL(1,𝔽₂₇)	C₃	semidirect
11. C₁₃×#737	C₃²	semidirect
18. C₃×C₁₃×3₋¹⁺²	C₃²	semidirect
20. C₃²×C₉×C₁₃	C₃²	semidirect
21. C₁₃×#733	C₃²	semidirect
22. C₁₃×#732	C₃²	semidirect
23. C₁₃×#731	C₃²	semidirect
24. C₁₃×AΣL(1,𝔽₂₇)	C₃²	semidirect
25. C₁₃×AΣL(1,𝔽₂₇)	C₃²	semidirect
26. #10624934	C₁₃	direct (N × H)
49. 3₋¹⁺⁴	C₃×C₁₃	semidirect
50. C₃²×3₋¹⁺²	C₃×C₁₃	semidirect
51. #6913	C₃×C₁₃	semidirect
52. #6912	C₃×C₁₃	semidirect
53. #6911	C₃×C₁₃	semidirect
54. #6910	C₃×C₁₃	semidirect
55. #6908	C₃×C₁₃	semidirect
56. C₃×AΣL(1,𝔽₂₇)	C₃×C₁₃	semidirect
73. #737	C₃²×C₁₃	semidirect
80. C₃×3₋¹⁺²	C₃²×C₁₃	semidirect
82. C₃²×C₉	C₃²×C₁₃	semidirect
83. #733	C₃²×C₁₃	semidirect
84. #732	C₃²×C₁₃	semidirect
85. #731	C₃²×C₁₃	semidirect
86. AΣL(1,𝔽₂₇)	C₃²×C₁₃	semidirect
87. AΣL(1,𝔽₂₇)	C₃²×C₁₃	semidirect
116. C₁₃	#10624934	direct (N × H)

H ≅ G/N

type

1. C₁₃×3₋¹⁺⁴

C₃

semidirect

2. C₃²×C₁₃×3₋¹⁺²

C₃

semidirect

3. C₁₃×#6913

C₃

semidirect

4. C₁₃×#6912

C₃

semidirect

5. C₁₃×#6911

C₃

semidirect

6. C₁₃×#6910

C₃

semidirect

7. C₁₃×#6908

C₃

semidirect

8. C₃×C₁₃×AΣL(1,𝔽₂₇)

C₃

semidirect

11. C₁₃×#737

C₃²

semidirect

18. C₃×C₁₃×3₋¹⁺²

C₃²

semidirect

20. C₃²×C₉×C₁₃

C₃²

semidirect

21. C₁₃×#733

C₃²

semidirect

22. C₁₃×#732

C₃²

semidirect

23. C₁₃×#731

C₃²

semidirect

24. C₁₃×AΣL(1,𝔽₂₇)

C₃²

semidirect

25. C₁₃×AΣL(1,𝔽₂₇)

C₃²

semidirect

26. #10624934

C₁₃

direct (N × H)

49. 3₋¹⁺⁴

C₃×C₁₃

semidirect

50. C₃²×3₋¹⁺²

C₃×C₁₃

semidirect

51. #6913

C₃×C₁₃

semidirect

52. #6912

C₃×C₁₃

semidirect

53. #6911

C₃×C₁₃

semidirect

54. #6910

C₃×C₁₃

semidirect

55. #6908

C₃×C₁₃

semidirect

56. C₃×AΣL(1,𝔽₂₇)

C₃×C₁₃

semidirect

73. #737

C₃²×C₁₃

semidirect

80. C₃×3₋¹⁺²

C₃²×C₁₃

semidirect

82. C₃²×C₉

C₃²×C₁₃

semidirect

83. #733

C₃²×C₁₃

semidirect

84. #732

C₃²×C₁₃

semidirect

85. #731

C₃²×C₁₃

semidirect

86. AΣL(1,𝔽₂₇)

C₃²×C₁₃

semidirect

87. AΣL(1,𝔽₂₇)

C₃²×C₁₃

semidirect

116. C₁₃

#10624934

direct (N × H)

Subgroups — C₁₃×#10624934 #23745085

Conjugacy classes of subgroups

Complements

Subgroups — C13×#10624934 #23745085

Conjugacy classes of subgroups

Complements

Subgroups — C₁₃×#10624934 #23745085