Group #23744929 — subgroups

Subgroups — C₁₃×#10624778 #23744929

Order 9477.

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

normal subgroups only

This group has 476 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×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
2	C₁₃×#6891	C₃	1	2	yes	maximal, complemented
3	C₃×C₁₃×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₁₃×H(3,𝔽₃)⋊C₉	C₃	1	9	yes	maximal, complemented
5	C₃⁴×C₁₃	—	3	1	no	—
6	C₃⁴×C₁₃	C₃²	1	1	yes	characteristic
7	C₃×C₁₃×3₋¹⁺²	—	3	2	no	—
8	C₃×C₁₃×H(3,𝔽₃)	—	3	2	no	—
9	C₃×C₁₃×H(3,𝔽₃)	—	3	9	no	—
10	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	3	yes	—
11	C₃²×C₉×C₁₃	C₃²	1	3	yes	complemented
12	C₁₃×C₃²⋊C₉	—	3	2	no	—
13	C₁₃×C₃²⋊C₉	—	3	1	no	—
14	C₁₃×C₃²⋊C₉	—	3	2	no	—
15	C₁₃×C₃²⋊C₉	—	3	2	no	—
16	C₁₃×C₃²⋊C₉	C₃²	1	6	yes	complemented
17	#10624778	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₃³×C₁₃	—	3	9	no	—
19	C₃³×C₁₃	—	3	3	no	—
20	C₃³×C₁₃	—	3	3	no	—
21	C₃³×C₁₃	—	3	3	no	—
22	C₃³×C₁₃	—	9	1	no	—
23	C₃³×C₁₃	—	9	1	no	—
24	C₃³×C₁₃	—	9	3	no	—
25	C₃³×C₁₃	—	9	3	no	—
26	C₃³×C₁₃	—	9	3	no	—
27	C₃³×C₁₃	H(3,𝔽₃)	1	2	yes	—
28	C₃³×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
29	C₃³×C₁₃	C₃³	1	1	yes	characteristic
30	C₁₃×3₋¹⁺²	—	9	6	no	—
31	C₁₃×H(3,𝔽₃)	—	3	18	no	—
32	C₁₃×H(3,𝔽₃)	—	3	6	no	—
33	C₁₃×H(3,𝔽₃)	3₋¹⁺²	1	9	yes	complemented
34	C₃×C₉×C₁₃	—	3	3	no	—
35	C₃×C₉×C₁₃	—	3	6	no	—
36	C₃×C₉×C₁₃	—	9	3	no	—
37	C₃²×H(3,𝔽₃)	C₃×C₁₃	1	1	yes	characteristic
38	#6891	C₃×C₁₃	1	2	yes	complemented
39	C₃×C₃²⋊C₉	C₃×C₁₃	1	1	yes	characteristic, complemented
40	H(3,𝔽₃)⋊C₉	C₃×C₁₃	1	9	yes	complemented
41	C₃²×C₁₃	—	3	1	no	—
42	C₃²×C₁₃	—	3	1	no	—
43	C₃²×C₁₃	—	3	1	no	—
44	C₃²×C₁₃	—	3	1	no	—
45	C₃²×C₁₃	—	3	1	no	—
46	C₃²×C₁₃	—	3	1	no	—
47	C₃²×C₁₃	—	3	2	no	—
48	C₃²×C₁₃	—	3	9	no	—
49	C₃²×C₁₃	—	3	2	no	—
50	C₃²×C₁₃	—	3	2	no	—
51	C₃²×C₁₃	—	3	2	no	—
52	C₃²×C₁₃	—	9	9	no	—
53	C₃²×C₁₃	—	9	27	no	—
54	C₃²×C₁₃	—	9	3	no	—
55	C₃²×C₁₃	—	9	3	no	—
56	C₃²×C₁₃	—	9	3	no	—
57	C₃²×C₁₃	#731	1	3	yes	—
58	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
59	C₃²×C₁₃	C₃×3₋¹⁺²	1	3	yes	—
60	C₉×C₁₃	—	9	3	no	—
61	C₉×C₁₃	—	9	6	no	—
62	C₃⁴	—	3	1	no	—
63	C₃⁴	C₃²×C₁₃	1	1	yes	characteristic
64	C₃×3₋¹⁺²	—	3	2	no	—
65	C₃×H(3,𝔽₃)	—	3	2	no	—
66	C₃×H(3,𝔽₃)	—	3	9	no	—
67	C₃×H(3,𝔽₃)	C₃²×C₁₃	1	3	yes	—
68	C₃²×C₉	C₃²×C₁₃	1	3	yes	complemented
69	C₃²⋊C₉	—	3	2	no	—
70	C₃²⋊C₉	—	3	1	no	—
71	C₃²⋊C₉	—	3	2	no	—
72	C₃²⋊C₉	—	3	2	no	—
73	C₃²⋊C₉	C₃²×C₁₃	1	6	yes	complemented
74	C₃×C₁₃	—	3	3	no	—
75	C₃×C₁₃	—	3	3	no	—
76	C₃×C₁₃	—	9	2	no	—
77	C₃×C₁₃	—	9	9	no	—
78	C₃×C₁₃	#6891	1	1	yes	characteristic
79	C₃×C₁₃	C₃×#731	1	1	yes	characteristic
80	C₃×C₁₃	#6909	1	1	yes	characteristic
81	C₃×C₁₃	#6911	1	1	yes	characteristic
82	C₃³	—	3	9	no	—
83	C₃³	—	3	3	no	—
84	C₃³	—	3	3	no	—
85	C₃³	—	3	3	no	—
86	C₃³	—	9	1	no	—
87	C₃³	—	9	1	no	—
88	C₃³	—	9	3	no	—
89	C₃³	—	9	3	no	—
90	C₃³	—	9	3	no	—
91	C₃³	C₁₃×H(3,𝔽₃)	1	2	yes	—
92	C₃³	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
93	C₃³	C₃³×C₁₃	1	1	yes	characteristic
94	3₋¹⁺²	—	9	6	no	—
95	H(3,𝔽₃)	—	3	18	no	—
96	H(3,𝔽₃)	—	3	6	no	—
97	H(3,𝔽₃)	C₁₃×3₋¹⁺²	1	9	yes	complemented
98	C₃×C₉	—	3	3	no	—
99	C₃×C₉	—	3	6	no	—
100	C₃×C₉	—	9	3	no	—
101	C₁₃	#10624778	1	1	yes	characteristic, direct product factor, Sylow
102	C₃²	—	3	1	no	—
103	C₃²	—	3	1	no	—
104	C₃²	—	3	1	no	—
105	C₃²	—	3	1	no	—
106	C₃²	—	3	1	no	—
107	C₃²	—	3	1	no	—
108	C₃²	—	3	2	no	—
109	C₃²	—	3	9	no	—
110	C₃²	—	3	2	no	—
111	C₃²	—	3	2	no	—
112	C₃²	—	3	2	no	—
113	C₃²	—	9	9	no	—
114	C₃²	—	9	27	no	—
115	C₃²	—	9	3	no	—
116	C₃²	—	9	3	no	—
117	C₃²	—	9	3	no	—
118	C₃²	C₁₃×#731	1	3	yes	—
119	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
120	C₃²	C₃×C₁₃×3₋¹⁺²	1	3	yes	—
121	C₉	—	9	3	no	—
122	C₉	—	9	6	no	—
123	C₃	—	3	3	no	—
124	C₃	—	3	3	no	—
125	C₃	—	9	2	no	—
126	C₃	—	9	9	no	—
127	C₃	C₁₃×#6891	1	1	yes	characteristic
128	C₃	C₃×C₁₃×#731	1	1	yes	characteristic
129	C₃	C₁₃×#6909	1	1	yes	characteristic
130	C₃	C₁₃×#6911	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₁₃×H(3,𝔽₃)

C₃

yes

maximal, characteristic

C₁₃×#6891

C₃

yes

maximal, complemented

C₃×C₁₃×C₃²⋊C₉

C₃

yes

maximal, characteristic, complemented

C₁₃×H(3,𝔽₃)⋊C₉

C₃

yes

maximal, complemented

C₃⁴×C₁₃

—

C₃⁴×C₁₃

C₃²

yes

characteristic

C₃×C₁₃×3₋¹⁺²

—

C₃×C₁₃×H(3,𝔽₃)

—

C₃×C₁₃×H(3,𝔽₃)

—

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

—

C₃²×C₉×C₁₃

C₃²

yes

complemented

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

C₃²

yes

complemented

#10624778

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

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×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

3₋¹⁺²

yes

complemented

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃²×H(3,𝔽₃)

C₃×C₁₃

yes

characteristic

#6891

C₃×C₁₃

yes

complemented

C₃×C₃²⋊C₉

C₃×C₁₃

yes

characteristic, complemented

H(3,𝔽₃)⋊C₉

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

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

#731

yes

—

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₁₃

C₃×3₋¹⁺²

yes

—

C₉×C₁₃

—

C₉×C₁₃

—

C₃⁴

—

C₃⁴

C₃²×C₁₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃²×C₁₃

yes

—

C₃²×C₉

C₃²×C₁₃

yes

complemented

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃²×C₁₃

yes

complemented

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

yes

C₃×C₁₃

yes

C₃×C₁₃

yes

C₃×C₁₃

yes

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

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

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

H(3,𝔽₃)

C₁₃×3₋¹⁺²

yes

complemented

C₃×C₉

—

C₃×C₉

—

100

C₃×C₉

—

101

C₁₃

#10624778

yes

characteristic, direct product factor, Sylow

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

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

—

119

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

120

C₃²

C₃×C₁₃×3₋¹⁺²

yes

—

121

C₉

—

122

C₉

—

123

C₃

—

124

C₃

—

125

C₃

—

126

C₃

—

127

C₃

C₁₃×#6891

yes

characteristic

128

C₃

C₃×C₁₃×#731

yes

characteristic

129

C₃

C₁₃×#6909

yes

characteristic

130

C₃

C₁₃×#6911

yes

characteristic

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

C₃²×C₁₃×H(3,𝔽₃)

C₃

LCS

G ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃²×C₁₃ ◁ C₃⁴×C₁₃ ◁ G

N	H ≅ G/N	type
2. C₁₃×#6891	C₃	semidirect
3. C₃×C₁₃×C₃²⋊C₉	C₃	semidirect
4. C₁₃×H(3,𝔽₃)⋊C₉	C₃	semidirect
11. C₃²×C₉×C₁₃	C₃²	semidirect
16. C₁₃×C₃²⋊C₉	C₃²	semidirect
17. #10624778	C₁₃	direct (N × H)
33. C₁₃×H(3,𝔽₃)	3₋¹⁺²	semidirect
38. #6891	C₃×C₁₃	semidirect
39. C₃×C₃²⋊C₉	C₃×C₁₃	semidirect
40. H(3,𝔽₃)⋊C₉	C₃×C₁₃	semidirect
68. C₃²×C₉	C₃²×C₁₃	semidirect
73. C₃²⋊C₉	C₃²×C₁₃	semidirect
97. H(3,𝔽₃)	C₁₃×3₋¹⁺²	semidirect
101. C₁₃	#10624778	direct (N × H)

H ≅ G/N

type

2. C₁₃×#6891

C₃

semidirect

3. C₃×C₁₃×C₃²⋊C₉

C₃

semidirect

4. C₁₃×H(3,𝔽₃)⋊C₉

C₃

semidirect

11. C₃²×C₉×C₁₃

C₃²

semidirect

16. C₁₃×C₃²⋊C₉

C₃²

semidirect

17. #10624778

C₁₃

direct (N × H)

33. C₁₃×H(3,𝔽₃)

3₋¹⁺²

semidirect

38. #6891

C₃×C₁₃

semidirect

39. C₃×C₃²⋊C₉

C₃×C₁₃

semidirect

40. H(3,𝔽₃)⋊C₉

C₃×C₁₃

semidirect

68. C₃²×C₉

C₃²×C₁₃

semidirect

73. C₃²⋊C₉

C₃²×C₁₃

semidirect

97. H(3,𝔽₃)

C₁₃×3₋¹⁺²

semidirect

101. C₁₃

#10624778

direct (N × H)

Subgroups — C₁₃×#10624778 #23744929

Conjugacy classes of subgroups

Complements

Subgroups — C13×#10624778 #23744929

Conjugacy classes of subgroups

Complements

Subgroups — C₁₃×#10624778 #23744929