Group #23744891 — subgroups

Subgroups — C₁₃×#10624740 #23744891

Order 9477.

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

normal subgroups only

This group has 650 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, complemented
2	C₃×C₁₃×#732	C₃	1	9	yes	maximal, complemented
3	C₁₃×C₃²⋊H(3,𝔽₃)	C₃	1	2	yes	maximal, complemented
4	C₃×C₁₃×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃⁴×C₁₃	—	3	1	no	—
6	C₃⁴×C₁₃	C₃²	1	1	yes	characteristic
7	C₃×C₁₃×H(3,𝔽₃)	—	3	2	no	—
8	C₃×C₁₃×H(3,𝔽₃)	—	3	9	no	—
9	C₃×C₁₃×H(3,𝔽₃)	—	3	4	no	—
10	C₃×C₁₃×H(3,𝔽₃)	—	3	2	no	—
11	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	3	yes	—
12	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	6	yes	complemented
13	C₃²×C₉×C₁₃	C₃²	1	3	yes	complemented
14	C₁₃×#732	—	3	27	no	—
15	C₁₃×C₃²⋊C₉	—	3	2	no	—
16	C₁₃×C₃²⋊C₉	—	3	1	no	—
17	#10624740	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	—	3	6	no	—
23	C₃³×C₁₃	—	9	2	no	—
24	C₃³×C₁₃	—	9	6	no	—
25	C₃³×C₁₃	—	9	3	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×H(3,𝔽₃)	—	3	18	no	—
30	C₁₃×H(3,𝔽₃)	—	3	6	no	—
31	C₁₃×H(3,𝔽₃)	—	3	18	no	—
32	C₁₃×H(3,𝔽₃)	—	9	12	no	—
33	C₁₃×H(3,𝔽₃)	—	9	6	no	—
34	C₁₃×H(3,𝔽₃)	H(3,𝔽₃)	1	9	yes	complemented
35	C₃×C₉×C₁₃	—	3	9	no	—
36	C₃×C₉×C₁₃	—	3	3	no	—
37	C₃²×H(3,𝔽₃)	C₃×C₁₃	1	1	yes	characteristic, complemented
38	C₃×#732	C₃×C₁₃	1	9	yes	complemented
39	C₃²⋊H(3,𝔽₃)	C₃×C₁₃	1	2	yes	complemented
40	C₃×C₃²⋊C₉	C₃×C₁₃	1	1	yes	characteristic, complemented
41	C₃²×C₁₃	—	3	1	no	—
42	C₃²×C₁₃	—	3	1	no	—
43	C₃²×C₁₃	—	3	2	no	—
44	C₃²×C₁₃	—	3	2	no	—
45	C₃²×C₁₃	—	3	2	no	—
46	C₃²×C₁₃	—	3	9	no	—
47	C₃²×C₁₃	—	3	2	no	—
48	C₃²×C₁₃	—	3	4	no	—
49	C₃²×C₁₃	—	9	9	no	—
50	C₃²×C₁₃	—	9	27	no	—
51	C₃²×C₁₃	—	9	3	no	—
52	C₃²×C₁₃	—	9	6	no	—
53	C₃²×C₁₃	—	9	6	no	—
54	C₃²×C₁₃	—	9	6	no	—
55	C₃²×C₁₃	—	9	12	no	—
56	C₃²×C₁₃	#732	1	3	yes	complemented
57	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	3	yes	—
58	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
59	C₉×C₁₃	—	9	3	no	—
60	C₃⁴	—	3	1	no	—
61	C₃⁴	C₃²×C₁₃	1	1	yes	characteristic
62	C₃×H(3,𝔽₃)	—	3	2	no	—
63	C₃×H(3,𝔽₃)	—	3	9	no	—
64	C₃×H(3,𝔽₃)	—	3	4	no	—
65	C₃×H(3,𝔽₃)	—	3	2	no	—
66	C₃×H(3,𝔽₃)	C₃²×C₁₃	1	3	yes	—
67	C₃×H(3,𝔽₃)	C₃²×C₁₃	1	6	yes	complemented
68	C₃²×C₉	C₃²×C₁₃	1	3	yes	complemented
69	#732	—	3	27	no	—
70	C₃²⋊C₉	—	3	2	no	—
71	C₃²⋊C₉	—	3	1	no	—
72	C₃×C₁₃	—	3	3	no	—
73	C₃×C₁₃	—	3	3	no	—
74	C₃×C₁₃	—	9	2	no	—
75	C₃×C₁₃	—	9	9	no	—
76	C₃×C₁₃	—	27	6	no	—
77	C₃×C₁₃	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
78	C₃×C₁₃	C₃×#732	1	1	yes	characteristic
79	C₃×C₁₃	#6911	1	2	yes	—
80	C₃³	—	3	9	no	—
81	C₃³	—	3	3	no	—
82	C₃³	—	3	3	no	—
83	C₃³	—	3	3	no	—
84	C₃³	—	3	6	no	—
85	C₃³	—	9	2	no	—
86	C₃³	—	9	6	no	—
87	C₃³	—	9	3	no	—
88	C₃³	C₁₃×H(3,𝔽₃)	1	2	yes	—
89	C₃³	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
90	C₃³	C₃³×C₁₃	1	1	yes	characteristic
91	H(3,𝔽₃)	—	3	18	no	—
92	H(3,𝔽₃)	—	3	6	no	—
93	H(3,𝔽₃)	—	3	18	no	—
94	H(3,𝔽₃)	—	9	12	no	—
95	H(3,𝔽₃)	—	9	6	no	—
96	H(3,𝔽₃)	C₁₃×H(3,𝔽₃)	1	9	yes	complemented
97	C₃×C₉	—	3	3	no	—
98	C₃×C₉	—	3	9	no	—
99	C₁₃	#10624740	1	1	yes	characteristic, direct product factor, Sylow
100	C₃²	—	3	1	no	—
101	C₃²	—	3	1	no	—
102	C₃²	—	3	2	no	—
103	C₃²	—	3	2	no	—
104	C₃²	—	3	2	no	—
105	C₃²	—	3	9	no	—
106	C₃²	—	3	2	no	—
107	C₃²	—	3	4	no	—
108	C₃²	—	9	9	no	—
109	C₃²	—	9	27	no	—
110	C₃²	—	9	3	no	—
111	C₃²	—	9	6	no	—
112	C₃²	—	9	6	no	—
113	C₃²	—	9	6	no	—
114	C₃²	—	9	12	no	—
115	C₃²	C₁₃×#732	1	3	yes	complemented
116	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	3	yes	—
117	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
118	C₉	—	9	3	no	—
119	C₃	—	3	3	no	—
120	C₃	—	3	3	no	—
121	C₃	—	9	2	no	—
122	C₃	—	9	9	no	—
123	C₃	—	27	6	no	—
124	C₃	C₁₃×C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
125	C₃	C₃×C₁₃×#732	1	1	yes	characteristic
126	C₃	C₁₃×#6911	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₁₃×H(3,𝔽₃)

C₃

yes

maximal, characteristic, complemented

C₃×C₁₃×#732

C₃

yes

maximal, complemented

C₁₃×C₃²⋊H(3,𝔽₃)

C₃

yes

maximal, complemented

C₃×C₁₃×C₃²⋊C₉

C₃

yes

maximal, characteristic, complemented

C₃⁴×C₁₃

—

C₃⁴×C₁₃

C₃²

yes

characteristic

C₃×C₁₃×H(3,𝔽₃)

—

C₃×C₁₃×H(3,𝔽₃)

—

C₃×C₁₃×H(3,𝔽₃)

—

C₃×C₁₃×H(3,𝔽₃)

—

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

—

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

complemented

C₃²×C₉×C₁₃

C₃²

yes

complemented

C₁₃×#732

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

—

#10624740

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

H(3,𝔽₃)

yes

—

C₃³×C₁₃

H(3,𝔽₃)

yes

characteristic

C₃³×C₁₃

C₃³

yes

characteristic

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

H(3,𝔽₃)

yes

complemented

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃²×H(3,𝔽₃)

C₃×C₁₃

yes

characteristic, complemented

C₃×#732

C₃×C₁₃

yes

complemented

C₃²⋊H(3,𝔽₃)

C₃×C₁₃

yes

complemented

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

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

#732

yes

complemented

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

—

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₁₃

—

C₃⁴

—

C₃⁴

C₃²×C₁₃

yes

characteristic

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃²×C₁₃

yes

—

C₃×H(3,𝔽₃)

C₃²×C₁₃

yes

complemented

C₃²×C₉

C₃²×C₁₃

yes

complemented

#732

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃×C₁₃

—

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

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₁₃×H(3,𝔽₃)

yes

—

C₃³

C₁₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃³×C₁₃

yes

—

—

—

—

—

C₁₃×H(3,𝔽₃)

yes

complemented

C₃×C₉

—

C₃×C₉

—

C₁₃

#10624740

yes

characteristic, direct product factor, Sylow

100

C₃²

—

101

C₃²

—

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

C₁₃×#732

yes

complemented

116

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

—

117

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

118

C₉

—

119

C₃

—

120

C₃

—

121

C₃

—

122

C₃

—

123

C₃

—

124

C₃

C₁₃×C₃²⋊H(3,𝔽₃)

yes

characteristic

125

C₃

C₃×C₁₃×#732

yes

characteristic

126

C₃

C₁₃×#6911

yes

—

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

LCS

G ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃²×C₁₃ ◁ C₃⁴×C₁₃ ◁ G

N	H ≅ G/N	type
1. C₃²×C₁₃×H(3,𝔽₃)	C₃	semidirect
2. C₃×C₁₃×#732	C₃	semidirect
3. C₁₃×C₃²⋊H(3,𝔽₃)	C₃	semidirect
4. C₃×C₁₃×C₃²⋊C₉	C₃	semidirect
12. C₃×C₁₃×H(3,𝔽₃)	C₃²	semidirect
13. C₃²×C₉×C₁₃	C₃²	semidirect
17. #10624740	C₁₃	direct (N × H)
34. C₁₃×H(3,𝔽₃)	H(3,𝔽₃)	semidirect
37. C₃²×H(3,𝔽₃)	C₃×C₁₃	semidirect
38. C₃×#732	C₃×C₁₃	semidirect
39. C₃²⋊H(3,𝔽₃)	C₃×C₁₃	semidirect
40. C₃×C₃²⋊C₉	C₃×C₁₃	semidirect
56. C₃²×C₁₃	#732	semidirect
67. C₃×H(3,𝔽₃)	C₃²×C₁₃	semidirect
68. C₃²×C₉	C₃²×C₁₃	semidirect
96. H(3,𝔽₃)	C₁₃×H(3,𝔽₃)	semidirect
99. C₁₃	#10624740	direct (N × H)
115. C₃²	C₁₃×#732	semidirect

H ≅ G/N

type

1. C₃²×C₁₃×H(3,𝔽₃)

C₃

semidirect

2. C₃×C₁₃×#732

C₃

semidirect

3. C₁₃×C₃²⋊H(3,𝔽₃)

C₃

semidirect

4. C₃×C₁₃×C₃²⋊C₉

C₃

semidirect

12. C₃×C₁₃×H(3,𝔽₃)

C₃²

semidirect

13. C₃²×C₉×C₁₃

C₃²

semidirect

17. #10624740

C₁₃

direct (N × H)

34. C₁₃×H(3,𝔽₃)

H(3,𝔽₃)

semidirect

37. C₃²×H(3,𝔽₃)

C₃×C₁₃

semidirect

38. C₃×#732

C₃×C₁₃

semidirect

39. C₃²⋊H(3,𝔽₃)

C₃×C₁₃

semidirect

40. C₃×C₃²⋊C₉

C₃×C₁₃

semidirect

56. C₃²×C₁₃

#732

semidirect

67. C₃×H(3,𝔽₃)

C₃²×C₁₃

semidirect

68. C₃²×C₉

C₃²×C₁₃

semidirect

96. H(3,𝔽₃)

C₁₃×H(3,𝔽₃)

semidirect

99. C₁₃

#10624740

direct (N × H)

115. C₃²

C₁₃×#732

semidirect

Subgroups — C₁₃×#10624740 #23744891

Conjugacy classes of subgroups

Complements

Subgroups — C13×#10624740 #23744891

Conjugacy classes of subgroups

Complements

Subgroups — C₁₃×#10624740 #23744891