Group #23744896 — subgroups

Subgroups — C₁₃×#10624745 #23744896

Order 9477.

500 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 104 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	C₃²×C₁₃×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×C₁₃×#731	C₃	1	9	yes	maximal, complemented
3	C₁₃×#6891	C₃	1	1	yes	maximal, characteristic, complemented
4	C₁₃×C₃²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
5	C₃×C₁₃×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃⁴×C₁₃	C₃²	1	1	yes	characteristic
7	C₃×C₁₃×3₋¹⁺²	—	3	2	no	—
8	C₃×C₁₃×3₋¹⁺²	—	3	9	no	—
9	C₃×C₁₃×3₋¹⁺²	C₃²	1	3	yes	—
10	C₃×C₁₃×3₋¹⁺²	C₃²	1	3	yes	—
11	C₃×C₁₃×H(3,𝔽₃)	—	3	2	no	—
12	C₃×C₁₃×H(3,𝔽₃)	—	3	1	no	—
13	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	3	yes	—
14	C₃²×C₉×C₁₃	—	3	1	no	—
15	C₃²×C₉×C₁₃	C₃²	1	3	yes	—
16	C₁₃×#731	—	3	27	no	—
17	C₁₃×C₃²⋊C₉	—	3	2	no	—
18	C₁₃×C₃²⋊C₉	—	3	1	no	—
19	C₁₃×C₃²⋊C₉	—	3	2	no	—
20	C₁₃×C₃²⋊C₉	—	3	1	no	—
21	#10624745	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₃³×C₁₃	—	3	3	no	—
23	C₃³×C₁₃	—	3	3	no	—
24	C₃³×C₁₃	—	3	3	no	—
25	C₃³×C₁₃	—	9	2	no	—
26	C₃³×C₁₃	H(3,𝔽₃)	1	2	yes	—
27	C₃³×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
28	C₃³×C₁₃	C₃³	1	1	yes	characteristic
29	C₁₃×3₋¹⁺²	—	3	18	no	—
30	C₁₃×3₋¹⁺²	—	3	6	no	—
31	C₁₃×3₋¹⁺²	—	3	9	no	—
32	C₁₃×3₋¹⁺²	H(3,𝔽₃)	1	9	yes	complemented
33	C₁₃×H(3,𝔽₃)	—	3	9	no	—
34	C₁₃×H(3,𝔽₃)	—	9	6	no	—
35	C₁₃×H(3,𝔽₃)	—	9	3	no	—
36	C₃×C₉×C₁₃	—	3	9	no	—
37	C₃×C₉×C₁₃	—	3	3	no	—
38	C₃×C₉×C₁₃	—	3	9	no	—
39	C₃×C₉×C₁₃	—	3	3	no	—
40	C₃×C₉×C₁₃	—	3	3	no	—
41	C₃²×3₋¹⁺²	C₃×C₁₃	1	1	yes	characteristic, complemented
42	C₃×#731	C₃×C₁₃	1	9	yes	complemented
43	#6891	C₃×C₁₃	1	1	yes	characteristic, complemented
44	C₃²⋊H(3,𝔽₃)	C₃×C₁₃	1	1	yes	characteristic
45	C₃×C₃²⋊C₉	C₃×C₁₃	1	1	yes	characteristic, complemented
46	C₃²×C₁₃	—	3	1	no	—
47	C₃²×C₁₃	—	3	1	no	—
48	C₃²×C₁₃	—	3	2	no	—
49	C₃²×C₁₃	—	3	2	no	—
50	C₃²×C₁₃	—	3	2	no	—
51	C₃²×C₁₃	—	3	2	no	—
52	C₃²×C₁₃	—	3	2	no	—
53	C₃²×C₁₃	—	3	2	no	—
54	C₃²×C₁₃	—	9	9	no	—
55	C₃²×C₁₃	—	9	3	no	—
56	C₃²×C₁₃	—	9	3	no	—
57	C₃²×C₁₃	—	9	6	no	—
58	C₃²×C₁₃	#731	1	3	yes	complemented
59	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	3	yes	—
60	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
61	C₉×C₁₃	—	3	9	no	—
62	C₉×C₁₃	—	9	3	no	—
63	C₉×C₁₃	—	9	3	no	—
64	C₃⁴	C₃²×C₁₃	1	1	yes	characteristic
65	C₃×3₋¹⁺²	—	3	2	no	—
66	C₃×3₋¹⁺²	—	3	9	no	—
67	C₃×3₋¹⁺²	C₃²×C₁₃	1	3	yes	—
68	C₃×3₋¹⁺²	C₃²×C₁₃	1	3	yes	—
69	C₃×H(3,𝔽₃)	—	3	2	no	—
70	C₃×H(3,𝔽₃)	—	3	1	no	—
71	C₃×H(3,𝔽₃)	C₃²×C₁₃	1	3	yes	—
72	C₃²×C₉	—	3	1	no	—
73	C₃²×C₉	C₃²×C₁₃	1	3	yes	—
74	#731	—	3	27	no	—
75	C₃²⋊C₉	—	3	2	no	—
76	C₃²⋊C₉	—	3	1	no	—
77	C₃²⋊C₉	—	3	2	no	—
78	C₃²⋊C₉	—	3	1	no	—
79	C₃×C₁₃	—	3	3	no	—
80	C₃×C₁₃	—	3	3	no	—
81	C₃×C₁₃	—	9	2	no	—
82	C₃×C₁₃	—	27	3	no	—
83	C₃×C₁₃	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
84	C₃×C₁₃	C₃×#731	1	1	yes	characteristic
85	C₃×C₁₃	#6910	1	2	yes	—
86	C₃³	—	3	3	no	—
87	C₃³	—	3	3	no	—
88	C₃³	—	3	3	no	—
89	C₃³	—	9	2	no	—
90	C₃³	C₁₃×H(3,𝔽₃)	1	2	yes	—
91	C₃³	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
92	C₃³	C₃³×C₁₃	1	1	yes	characteristic
93	3₋¹⁺²	—	3	18	no	—
94	3₋¹⁺²	—	3	6	no	—
95	3₋¹⁺²	—	3	9	no	—
96	3₋¹⁺²	C₁₃×H(3,𝔽₃)	1	9	yes	complemented
97	H(3,𝔽₃)	—	3	9	no	—
98	H(3,𝔽₃)	—	9	6	no	—
99	H(3,𝔽₃)	—	9	3	no	—
100	C₃×C₉	—	3	9	no	—
101	C₃×C₉	—	3	3	no	—
102	C₃×C₉	—	3	3	no	—
103	C₃×C₉	—	3	3	no	—
104	C₃×C₉	—	3	9	no	—
105	C₁₃	#10624745	1	1	yes	characteristic, direct product factor, Sylow
106	C₃²	—	3	1	no	—
107	C₃²	—	3	1	no	—
108	C₃²	—	3	2	no	—
109	C₃²	—	3	2	no	—
110	C₃²	—	3	2	no	—
111	C₃²	—	3	2	no	—
112	C₃²	—	3	2	no	—
113	C₃²	—	3	2	no	—
114	C₃²	—	9	9	no	—
115	C₃²	—	9	3	no	—
116	C₃²	—	9	3	no	—
117	C₃²	—	9	6	no	—
118	C₃²	C₁₃×#731	1	3	yes	complemented
119	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	3	yes	—
120	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
121	C₉	—	3	9	no	—
122	C₉	—	9	3	no	—
123	C₉	—	9	3	no	—
124	C₃	—	3	3	no	—
125	C₃	—	3	3	no	—
126	C₃	—	9	2	no	—
127	C₃	—	27	3	no	—
128	C₃	C₁₃×C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
129	C₃	C₃×C₁₃×#731	1	1	yes	characteristic
130	C₃	C₁₃×#6910	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₁₃×3₋¹⁺²

C₃

yes

maximal, characteristic, complemented

C₃×C₁₃×#731

C₃

yes

maximal, complemented

C₁₃×#6891

C₃

yes

maximal, characteristic, complemented

C₁₃×C₃²⋊H(3,𝔽₃)

C₃

yes

maximal, characteristic

C₃×C₁₃×C₃²⋊C₉

C₃

yes

maximal, characteristic, complemented

C₃⁴×C₁₃

C₃²

yes

characteristic

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)

—

C₃×C₁₃×H(3,𝔽₃)

—

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

—

C₃²×C₉×C₁₃

—

C₃²×C₉×C₁₃

C₃²

yes

—

C₁₃×#731

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

—

#10624745

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

H(3,𝔽₃)

yes

—

C₃³×C₁₃

H(3,𝔽₃)

yes

characteristic

C₃³×C₁₃

C₃³

yes

characteristic

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

H(3,𝔽₃)

yes

complemented

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃²×3₋¹⁺²

C₃×C₁₃

yes

characteristic, complemented

C₃×#731

C₃×C₁₃

yes

complemented

#6891

C₃×C₁₃

yes

characteristic, complemented

C₃²⋊H(3,𝔽₃)

C₃×C₁₃

yes

characteristic

C₃×C₃²⋊C₉

C₃×C₁₃

yes

characteristic, complemented

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

#731

yes

complemented

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

—

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₃⁴

C₃²×C₁₃

yes

characteristic

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)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃²×C₁₃

yes

—

C₃²×C₉

—

C₃²×C₉

C₃²×C₁₃

yes

—

#731

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

C₃²⋊H(3,𝔽₃)

yes

C₃×C₁₃

yes

C₃×C₁₃

yes

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₁₃×H(3,𝔽₃)

yes

—

C₃³

C₁₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃³×C₁₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

C₁₃×H(3,𝔽₃)

yes

—

—

—

100

C₃×C₉

—

101

C₃×C₉

—

102

C₃×C₉

—

103

C₃×C₉

—

104

C₃×C₉

—

105

C₁₃

#10624745

yes

characteristic, direct product factor, Sylow

106

C₃²

—

107

C₃²

—

108

C₃²

—

109

C₃²

—

110

C₃²

—

111

C₃²

—

112

C₃²

—

113

C₃²

—

114

C₃²

—

115

C₃²

—

116

C₃²

—

117

C₃²

—

118

C₃²

C₁₃×#731

yes

complemented

119

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

—

120

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

121

C₉

—

122

C₉

—

123

C₉

—

124

C₃

—

125

C₃

—

126

C₃

—

127

C₃

—

128

C₃

C₁₃×C₃²⋊H(3,𝔽₃)

yes

characteristic

129

C₃

C₃×C₁₃×#731

yes

characteristic

130

C₃

C₁₃×#6910

yes

—

	subgroup	G/H
center, socle	C₃²×C₁₃	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₁₃
prime radical	C₁₃×C₃²⋊H(3,𝔽₃)	C₃
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₁₃ ◁ C₃⁴×C₁₃ ◁ G

subgroup

G/H

center, socle

C₃²×C₁₃

C₃×H(3,𝔽₃)

commutator, Frattini, cosocle

C₃³

C₃³×C₁₃

prime radical

C₁₃×C₃²⋊H(3,𝔽₃)

C₃

LCS

G ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃²×C₁₃ ◁ C₃⁴×C₁₃ ◁ G

N	H ≅ G/N	type
1. C₃²×C₁₃×3₋¹⁺²	C₃	semidirect
2. C₃×C₁₃×#731	C₃	semidirect
3. C₁₃×#6891	C₃	semidirect
5. C₃×C₁₃×C₃²⋊C₉	C₃	semidirect
21. #10624745	C₁₃	direct (N × H)
32. C₁₃×3₋¹⁺²	H(3,𝔽₃)	semidirect
41. C₃²×3₋¹⁺²	C₃×C₁₃	semidirect
42. C₃×#731	C₃×C₁₃	semidirect
43. #6891	C₃×C₁₃	semidirect
45. C₃×C₃²⋊C₉	C₃×C₁₃	semidirect
58. C₃²×C₁₃	#731	semidirect
96. 3₋¹⁺²	C₁₃×H(3,𝔽₃)	semidirect
105. C₁₃	#10624745	direct (N × H)
118. C₃²	C₁₃×#731	semidirect

H ≅ G/N

type

1. C₃²×C₁₃×3₋¹⁺²

C₃

semidirect

2. C₃×C₁₃×#731

C₃

semidirect

3. C₁₃×#6891

C₃

semidirect

5. C₃×C₁₃×C₃²⋊C₉

C₃

semidirect

21. #10624745

C₁₃

direct (N × H)

32. C₁₃×3₋¹⁺²

H(3,𝔽₃)

semidirect

41. C₃²×3₋¹⁺²

C₃×C₁₃

semidirect

42. C₃×#731

C₃×C₁₃

semidirect

43. #6891

C₃×C₁₃

semidirect

45. C₃×C₃²⋊C₉

C₃×C₁₃

semidirect

58. C₃²×C₁₃

#731

semidirect

96. 3₋¹⁺²

C₁₃×H(3,𝔽₃)

semidirect

105. C₁₃

#10624745

direct (N × H)

118. C₃²

C₁₃×#731

semidirect

Subgroups — C₁₃×#10624745 #23744896

Conjugacy classes of subgroups

Complements

Subgroups — C13×#10624745 #23744896

Conjugacy classes of subgroups

Complements

Subgroups — C₁₃×#10624745 #23744896