Group #23744711 — subgroups

Subgroups — C₁₃×#10624560 #23744711

Order 9477.

506 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 206 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
2	C₃×C₁₃×C₉⋊C₉	C₃	1	2	yes	maximal, complemented
3	C₃×C₁₃×C₃²⋊C₉	C₃	1	3	yes	maximal
4	C₃×C₉²×C₁₃	C₃	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#6855	C₃	1	6	yes	maximal, complemented
6	C₃⁴×C₁₃	C₃²	1	1	yes	characteristic
7	C₃×C₁₃×3₋¹⁺²	—	3	3	no	—
8	C₃×C₁₃×3₋¹⁺²	—	3	3	no	—
9	C₃×C₁₃×3₋¹⁺²	—	3	3	no	—
10	C₃×C₁₃×3₋¹⁺²	C₉	1	9	yes	complemented
11	C₃²×C₉×C₁₃	C₃²	1	6	yes	—
12	C₃²×C₉×C₁₃	C₃²	1	2	yes	—
13	C₃²×C₉×C₁₃	C₃²	1	3	yes	—
14	C₃²×C₉×C₁₃	C₃²	1	1	yes	characteristic
15	C₁₃×C₉⋊C₉	—	3	6	no	—
16	C₁₃×C₃²⋊C₉	—	3	9	no	—
17	C₉²×C₁₃	—	3	3	no	—
18	#10624560	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₃³×C₁₃	—	3	1	no	—
20	C₃³×C₁₃	—	3	3	no	—
21	C₃³×C₁₃	—	3	1	no	—
22	C₃³×C₁₃	—	3	1	no	—
23	C₃³×C₁₃	—	3	6	no	—
24	C₃³×C₁₃	C₃×C₉	1	3	yes	—
25	C₃³×C₁₃	C₃³	1	1	yes	characteristic
26	C₁₃×3₋¹⁺²	—	3	27	no	—
27	C₃×C₉×C₁₃	—	3	3	no	—
28	C₃×C₉×C₁₃	—	3	2	no	—
29	C₃×C₉×C₁₃	—	3	2	no	—
30	C₃×C₉×C₁₃	—	3	2	no	—
31	C₃×C₉×C₁₃	—	3	6	no	—
32	C₃×C₉×C₁₃	—	3	6	no	—
33	C₃×C₉×C₁₃	—	3	6	no	—
34	C₃×C₉×C₁₃	—	3	12	no	—
35	C₃×C₉×C₁₃	C₃×C₉	1	6	yes	—
36	C₃×C₉×C₁₃	C₃×C₉	1	3	yes	—
37	C₃×C₉×C₁₃	H(3,𝔽₃)	1	3	yes	—
38	C₃×C₉×C₁₃	3₋¹⁺²	1	3	yes	—
39	C₃×C₉×C₁₃	3₋¹⁺²	1	3	yes	—
40	C₃×C₉×C₁₃	3₋¹⁺²	1	9	yes	complemented
41	C₃²×3₋¹⁺²	C₃×C₁₃	1	1	yes	characteristic
42	C₃×C₉⋊C₉	C₃×C₁₃	1	2	yes	complemented
43	C₃×C₃²⋊C₉	C₃×C₁₃	1	3	yes	—
44	C₃×C₉²	C₃×C₁₃	1	1	yes	characteristic, complemented
45	#6855	C₃×C₁₃	1	6	yes	complemented
46	C₃²×C₁₃	—	3	3	no	—
47	C₃²×C₁₃	—	3	9	no	—
48	C₃²×C₁₃	—	9	3	no	—
49	C₃²×C₁₃	—	9	3	no	—
50	C₃²×C₁₃	—	9	2	no	—
51	C₃²×C₁₃	—	9	1	no	—
52	C₃²×C₁₃	C₃²×C₉	1	1	yes	characteristic
53	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
54	C₃²×C₁₃	C₃×3₋¹⁺²	1	1	yes	characteristic
55	C₃²×C₁₃	C₃×3₋¹⁺²	1	1	yes	characteristic
56	C₃²×C₁₃	C₃×3₋¹⁺²	1	3	yes	—
57	C₃²×C₁₃	#737	1	6	yes	—
58	C₉×C₁₃	—	3	6	no	—
59	C₉×C₁₃	—	3	9	no	—
60	C₉×C₁₃	—	9	6	no	—
61	C₉×C₁₃	C₃²⋊C₉	1	9	yes	complemented
62	C₃⁴	C₃²×C₁₃	1	1	yes	characteristic
63	C₃×3₋¹⁺²	—	3	3	no	—
64	C₃×3₋¹⁺²	—	3	3	no	—
65	C₃×3₋¹⁺²	—	3	3	no	—
66	C₃×3₋¹⁺²	C₉×C₁₃	1	9	yes	complemented
67	C₃²×C₉	C₃²×C₁₃	1	6	yes	—
68	C₃²×C₉	C₃²×C₁₃	1	2	yes	—
69	C₃²×C₉	C₃²×C₁₃	1	3	yes	—
70	C₃²×C₉	C₃²×C₁₃	1	1	yes	characteristic
71	C₉⋊C₉	—	3	6	no	—
72	C₃²⋊C₉	—	3	9	no	—
73	C₉²	—	3	3	no	—
74	C₃×C₁₃	—	9	3	no	—
75	C₃×C₁₃	C₃×C₃²⋊C₉	1	1	yes	characteristic
76	C₃×C₁₃	C₉×3₋¹⁺²	1	3	yes	—
77	C₃×C₁₃	C₉⋊H(3,𝔽₃)	1	3	yes	—
78	C₃×C₁₃	#6896	1	3	yes	—
79	C₃×C₁₃	#6899	1	2	yes	—
80	C₃×C₁₃	#6900	1	1	yes	characteristic
81	C₃³	—	3	1	no	—
82	C₃³	—	3	3	no	—
83	C₃³	—	3	1	no	—
84	C₃³	—	3	1	no	—
85	C₃³	—	3	6	no	—
86	C₃³	C₃×C₉×C₁₃	1	3	yes	—
87	C₃³	C₃³×C₁₃	1	1	yes	characteristic
88	3₋¹⁺²	—	3	27	no	—
89	C₃×C₉	—	3	3	no	—
90	C₃×C₉	—	3	2	no	—
91	C₃×C₉	—	3	2	no	—
92	C₃×C₉	—	3	2	no	—
93	C₃×C₉	—	3	6	no	—
94	C₃×C₉	—	3	6	no	—
95	C₃×C₉	—	3	6	no	—
96	C₃×C₉	—	3	12	no	—
97	C₃×C₉	C₃×C₉×C₁₃	1	6	yes	—
98	C₃×C₉	C₃×C₉×C₁₃	1	3	yes	—
99	C₃×C₉	C₁₃×H(3,𝔽₃)	1	3	yes	—
100	C₃×C₉	C₁₃×3₋¹⁺²	1	3	yes	—
101	C₃×C₉	C₁₃×3₋¹⁺²	1	3	yes	—
102	C₃×C₉	C₁₃×3₋¹⁺²	1	9	yes	complemented
103	C₁₃	#10624560	1	1	yes	characteristic, direct product factor, Sylow
104	C₃²	—	3	3	no	—
105	C₃²	—	3	9	no	—
106	C₃²	—	9	3	no	—
107	C₃²	—	9	3	no	—
108	C₃²	—	9	2	no	—
109	C₃²	—	9	1	no	—
110	C₃²	C₃²×C₉×C₁₃	1	1	yes	characteristic
111	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
112	C₃²	C₃×C₁₃×3₋¹⁺²	1	1	yes	characteristic
113	C₃²	C₃×C₁₃×3₋¹⁺²	1	1	yes	characteristic
114	C₃²	C₃×C₁₃×3₋¹⁺²	1	3	yes	—
115	C₃²	C₁₃×#737	1	6	yes	—
116	C₉	—	3	6	no	—
117	C₉	—	3	9	no	—
118	C₉	—	9	6	no	—
119	C₉	C₁₃×C₃²⋊C₉	1	9	yes	complemented
120	C₃	—	9	3	no	—
121	C₃	C₃×C₁₃×C₃²⋊C₉	1	1	yes	characteristic
122	C₃	C₉×C₁₃×3₋¹⁺²	1	3	yes	—
123	C₃	C₁₃×C₉⋊H(3,𝔽₃)	1	3	yes	—
124	C₃	C₁₃×#6896	1	3	yes	—
125	C₃	C₁₃×#6899	1	2	yes	—
126	C₃	C₁₃×#6900	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₁₃×3₋¹⁺²

C₃

yes

maximal, characteristic

C₃×C₁₃×C₉⋊C₉

C₃

yes

maximal, complemented

C₃×C₁₃×C₃²⋊C₉

C₃

yes

maximal

C₃×C₉²×C₁₃

C₃

yes

maximal, characteristic, complemented

C₁₃×#6855

C₃

yes

maximal, complemented

C₃⁴×C₁₃

C₃²

yes

characteristic

C₃×C₁₃×3₋¹⁺²

—

C₃×C₁₃×3₋¹⁺²

—

C₃×C₁₃×3₋¹⁺²

—

C₃×C₁₃×3₋¹⁺²

C₉

yes

complemented

C₃²×C₉×C₁₃

C₃²

yes

—

C₃²×C₉×C₁₃

C₃²

yes

—

C₃²×C₉×C₁₃

C₃²

yes

—

C₃²×C₉×C₁₃

C₃²

yes

characteristic

C₁₃×C₉⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₉²×C₁₃

—

#10624560

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

C₃×C₉

yes

—

C₃³×C₁₃

C₃³

yes

characteristic

C₁₃×3₋¹⁺²

—

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

yes

—

C₃×C₉×C₁₃

C₃×C₉

yes

—

C₃×C₉×C₁₃

H(3,𝔽₃)

yes

—

C₃×C₉×C₁₃

3₋¹⁺²

yes

—

C₃×C₉×C₁₃

3₋¹⁺²

yes

—

C₃×C₉×C₁₃

3₋¹⁺²

yes

complemented

C₃²×3₋¹⁺²

C₃×C₁₃

yes

characteristic

C₃×C₉⋊C₉

C₃×C₁₃

yes

complemented

C₃×C₃²⋊C₉

C₃×C₁₃

yes

—

C₃×C₉²

C₃×C₁₃

yes

characteristic, complemented

#6855

C₃×C₁₃

yes

complemented

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

C₃²×C₉

yes

characteristic

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₁₃

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₁₃

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₁₃

C₃×3₋¹⁺²

yes

—

C₃²×C₁₃

#737

yes

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

C₃²⋊C₉

yes

complemented

C₃⁴

C₃²×C₁₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₉×C₁₃

yes

complemented

C₃²×C₉

C₃²×C₁₃

yes

—

C₃²×C₉

C₃²×C₁₃

yes

—

C₃²×C₉

C₃²×C₁₃

yes

—

C₃²×C₉

C₃²×C₁₃

yes

characteristic

C₉⋊C₉

—

C₃²⋊C₉

—

C₉²

—

C₃×C₁₃

—

C₃×C₁₃

C₃×C₃²⋊C₉

yes

characteristic

C₃×C₁₃

C₉×3₋¹⁺²

yes

—

C₃×C₁₃

C₉⋊H(3,𝔽₃)

yes

—

C₃×C₁₃

#6896

yes

—

C₃×C₁₃

#6899

yes

—

C₃×C₁₃

#6900

yes

characteristic

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×C₉×C₁₃

yes

—

C₃³

C₃³×C₁₃

yes

characteristic

3₋¹⁺²

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃×C₉×C₁₃

yes

—

C₃×C₉

C₃×C₉×C₁₃

yes

—

C₃×C₉

C₁₃×H(3,𝔽₃)

yes

—

100

C₃×C₉

C₁₃×3₋¹⁺²

yes

—

101

C₃×C₉

C₁₃×3₋¹⁺²

yes

—

102

C₃×C₉

C₁₃×3₋¹⁺²

yes

complemented

103

C₁₃

#10624560

yes

characteristic, direct product factor, Sylow

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

—

108

C₃²

—

109

C₃²

—

110

C₃²

C₃²×C₉×C₁₃

yes

characteristic

111

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

112

C₃²

C₃×C₁₃×3₋¹⁺²

yes

characteristic

113

C₃²

C₃×C₁₃×3₋¹⁺²

yes

characteristic

114

C₃²

C₃×C₁₃×3₋¹⁺²

yes

—

115

C₃²

C₁₃×#737

yes

—

116

C₉

—

117

C₉

—

118

C₉

—

119

C₉

C₁₃×C₃²⋊C₉

yes

complemented

120

C₃

—

121

C₃

C₃×C₁₃×C₃²⋊C₉

yes

characteristic

122

C₃

C₉×C₁₃×3₋¹⁺²

yes

—

123

C₃

C₁₃×C₉⋊H(3,𝔽₃)

yes

—

124

C₃

C₁₃×#6896

yes

—

125

C₃

C₁₃×#6899

yes

—

126

C₃

C₁₃×#6900

yes

characteristic

	subgroup	G/H
center, socle	C₃³×C₁₃	C₃³
commutator	C₃²	C₃²×C₉×C₁₃
Frattini, cosocle	C₃³	C₃³×C₁₃
prime radical	C₃⁴×C₁₃	C₃²
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃³×C₁₃ ◁ G

subgroup

G/H

center, socle

C₃³×C₁₃

C₃³

commutator

C₃²

C₃²×C₉×C₁₃

Frattini, cosocle

C₃³

C₃³×C₁₃

prime radical

C₃⁴×C₁₃

C₃²

LCS

G ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₃³×C₁₃ ◁ G

N	H ≅ G/N	type
2. C₃×C₁₃×C₉⋊C₉	C₃	semidirect
4. C₃×C₉²×C₁₃	C₃	semidirect
5. C₁₃×#6855	C₃	semidirect
10. C₃×C₁₃×3₋¹⁺²	C₉	semidirect
18. #10624560	C₁₃	direct (N × H)
40. C₃×C₉×C₁₃	3₋¹⁺²	semidirect
42. C₃×C₉⋊C₉	C₃×C₁₃	semidirect
44. C₃×C₉²	C₃×C₁₃	semidirect
45. #6855	C₃×C₁₃	semidirect
61. C₉×C₁₃	C₃²⋊C₉	semidirect
66. C₃×3₋¹⁺²	C₉×C₁₃	semidirect
102. C₃×C₉	C₁₃×3₋¹⁺²	semidirect
103. C₁₃	#10624560	direct (N × H)
119. C₉	C₁₃×C₃²⋊C₉	semidirect

H ≅ G/N

type

2. C₃×C₁₃×C₉⋊C₉

C₃

semidirect

4. C₃×C₉²×C₁₃

C₃

semidirect

5. C₁₃×#6855

C₃

semidirect

10. C₃×C₁₃×3₋¹⁺²

C₉

semidirect

18. #10624560

C₁₃

direct (N × H)

40. C₃×C₉×C₁₃

3₋¹⁺²

semidirect

42. C₃×C₉⋊C₉

C₃×C₁₃

semidirect

44. C₃×C₉²

C₃×C₁₃

semidirect

45. #6855

C₃×C₁₃

semidirect

61. C₉×C₁₃

C₃²⋊C₉

semidirect

66. C₃×3₋¹⁺²

C₉×C₁₃

semidirect

102. C₃×C₉

C₁₃×3₋¹⁺²

semidirect

103. C₁₃

#10624560

direct (N × H)

119. C₉

C₁₃×C₃²⋊C₉

semidirect

Subgroups — C₁₃×#10624560 #23744711

Conjugacy classes of subgroups

Complements

Subgroups — C13×#10624560 #23744711

Conjugacy classes of subgroups

Complements

Subgroups — C₁₃×#10624560 #23744711