Group #23744943 — subgroups

Subgroups — C₁₃×#10624792 #23744943

Order 9477.

268 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 92 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	C₃×C₁₃×#732	C₃	1	1	yes	maximal, characteristic
2	C₁₃×C₉⋊H(3,𝔽₃)	C₃	1	2	yes	maximal, complemented
3	C₉×C₁₃×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₁₃×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
5	C₁₃×H(3,𝔽₃)⋊C₉	C₃	1	4	yes	maximal, complemented
6	C₁₃×H(3,𝔽₃)⋊C₉	C₃	1	2	yes	maximal, complemented
7	C₁₃×#6868	C₃	1	2	yes	maximal, complemented
8	C₃×C₁₃×3₋¹⁺²	—	3	2	no	—
9	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	2	yes	—
10	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
11	C₃²×C₉×C₁₃	—	3	1	no	—
12	C₃²×C₉×C₁₃	C₃²	1	1	yes	characteristic
13	C₃²×C₉×C₁₃	C₃²	1	2	yes	—
14	C₃²×C₉×C₁₃	C₃²	1	1	yes	characteristic
15	C₁₃×#732	—	3	3	no	—
16	C₁₃×C₉⋊C₉	—	3	2	no	—
17	C₁₃×C₉⋊C₉	—	3	1	no	—
18	C₁₃×C₃²⋊C₉	—	3	2	no	—
19	C₁₃×C₃²⋊C₉	—	3	4	no	—
20	C₁₃×C₃²⋊C₉	C₃²	1	2	yes	—
21	C₁₃×C₃²⋊C₉	C₃²	1	4	yes	—
22	#10624792	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₃³×C₁₃	—	3	1	no	—
24	C₃³×C₁₃	—	3	2	no	—
25	C₃³×C₁₃	C₃³	1	1	yes	characteristic
26	C₁₃×3₋¹⁺²	—	9	6	no	—
27	C₁₃×H(3,𝔽₃)	—	3	2	no	—
28	C₁₃×H(3,𝔽₃)	—	3	2	no	—
29	C₁₃×H(3,𝔽₃)	—	3	4	no	—
30	C₁₃×H(3,𝔽₃)	3₋¹⁺²	1	3	yes	complemented
31	C₃×C₉×C₁₃	—	3	3	no	—
32	C₃×C₉×C₁₃	—	3	2	no	—
33	C₃×C₉×C₁₃	—	3	4	no	—
34	C₃×C₉×C₁₃	—	3	2	no	—
35	C₃×C₉×C₁₃	—	3	1	no	—
36	C₃×C₉×C₁₃	—	3	1	no	—
37	C₃×C₉×C₁₃	—	3	2	no	—
38	C₃×C₉×C₁₃	—	9	3	no	—
39	C₃×C₉×C₁₃	—	9	2	no	—
40	C₃×C₉×C₁₃	H(3,𝔽₃)	1	2	yes	—
41	C₃×C₉×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
42	C₃×#732	C₃×C₁₃	1	1	yes	characteristic
43	C₉⋊H(3,𝔽₃)	C₃×C₁₃	1	2	yes	complemented
44	C₉×H(3,𝔽₃)	C₃×C₁₃	1	1	yes	characteristic, complemented
45	C₃×C₉⋊C₉	C₃×C₁₃	1	1	yes	characteristic, complemented
46	H(3,𝔽₃)⋊C₉	C₃×C₁₃	1	4	yes	complemented
47	H(3,𝔽₃)⋊C₉	C₃×C₁₃	1	2	yes	complemented
48	#6868	C₃×C₁₃	1	2	yes	complemented
49	C₃²×C₁₃	—	3	2	no	—
50	C₃²×C₁₃	—	3	1	no	—
51	C₃²×C₁₃	—	3	3	no	—
52	C₃²×C₁₃	—	9	2	no	—
53	C₃²×C₁₃	—	9	1	no	—
54	C₃²×C₁₃	—	9	4	no	—
55	C₃²×C₁₃	—	9	2	no	—
56	C₃²×C₁₃	—	9	2	no	—
57	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
58	C₃²×C₁₃	C₃×3₋¹⁺²	1	1	yes	characteristic
59	C₃²×C₁₃	#737	1	2	yes	—
60	C₉×C₁₃	—	3	2	no	—
61	C₉×C₁₃	—	9	2	no	—
62	C₉×C₁₃	—	9	1	no	—
63	C₉×C₁₃	—	9	2	no	—
64	C₉×C₁₃	—	9	4	no	—
65	C₉×C₁₃	#732	1	3	yes	complemented
66	C₃×3₋¹⁺²	—	3	2	no	—
67	C₃×H(3,𝔽₃)	C₃²×C₁₃	1	2	yes	—
68	C₃×H(3,𝔽₃)	C₃²×C₁₃	1	1	yes	characteristic
69	C₃²×C₉	—	3	1	no	—
70	C₃²×C₉	C₃²×C₁₃	1	1	yes	characteristic
71	C₃²×C₉	C₃²×C₁₃	1	1	yes	characteristic
72	C₃²×C₉	C₃²×C₁₃	1	2	yes	—
73	#732	—	3	3	no	—
74	C₉⋊C₉	—	3	2	no	—
75	C₉⋊C₉	—	3	1	no	—
76	C₃²⋊C₉	—	3	2	no	—
77	C₃²⋊C₉	—	3	4	no	—
78	C₃²⋊C₉	C₃²×C₁₃	1	2	yes	—
79	C₃²⋊C₉	C₃²×C₁₃	1	4	yes	—
80	C₃×C₁₃	—	3	1	no	—
81	C₃×C₁₃	—	3	2	no	—
82	C₃×C₁₃	—	9	3	no	—
83	C₃×C₁₃	—	27	2	no	—
84	C₃×C₁₃	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
85	C₃×C₁₃	C₃×#732	1	1	yes	characteristic
86	C₃×C₁₃	#6913	1	2	yes	—
87	C₃³	—	3	1	no	—
88	C₃³	—	3	2	no	—
89	C₃³	C₃³×C₁₃	1	1	yes	characteristic
90	3₋¹⁺²	—	9	6	no	—
91	H(3,𝔽₃)	—	3	2	no	—
92	H(3,𝔽₃)	—	3	2	no	—
93	H(3,𝔽₃)	—	3	4	no	—
94	H(3,𝔽₃)	C₁₃×3₋¹⁺²	1	3	yes	complemented
95	C₃×C₉	—	3	3	no	—
96	C₃×C₉	—	3	2	no	—
97	C₃×C₉	—	3	4	no	—
98	C₃×C₉	—	3	1	no	—
99	C₃×C₉	—	3	2	no	—
100	C₃×C₉	—	3	1	no	—
101	C₃×C₉	—	3	2	no	—
102	C₃×C₉	—	9	3	no	—
103	C₃×C₉	—	9	2	no	—
104	C₃×C₉	C₁₃×H(3,𝔽₃)	1	2	yes	—
105	C₃×C₉	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
106	C₁₃	#10624792	1	1	yes	characteristic, direct product factor, Sylow
107	C₃²	—	3	1	no	—
108	C₃²	—	3	2	no	—
109	C₃²	—	3	3	no	—
110	C₃²	—	9	1	no	—
111	C₃²	—	9	2	no	—
112	C₃²	—	9	2	no	—
113	C₃²	—	9	2	no	—
114	C₃²	—	9	4	no	—
115	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
116	C₃²	C₃×C₁₃×3₋¹⁺²	1	1	yes	characteristic
117	C₃²	C₁₃×#737	1	2	yes	—
118	C₉	—	3	2	no	—
119	C₉	—	9	1	no	—
120	C₉	—	9	2	no	—
121	C₉	—	9	4	no	—
122	C₉	—	9	2	no	—
123	C₉	C₁₃×#732	1	3	yes	complemented
124	C₃	—	3	1	no	—
125	C₃	—	3	2	no	—
126	C₃	—	9	3	no	—
127	C₃	—	27	2	no	—
128	C₃	C₁₃×C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
129	C₃	C₃×C₁₃×#732	1	1	yes	characteristic
130	C₃	C₁₃×#6913	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₁₃×#732

C₃

yes

maximal, characteristic

C₁₃×C₉⋊H(3,𝔽₃)

C₃

yes

maximal, complemented

C₉×C₁₃×H(3,𝔽₃)

C₃

yes

maximal, characteristic, complemented

C₃×C₁₃×C₉⋊C₉

C₃

yes

maximal, characteristic, complemented

C₁₃×H(3,𝔽₃)⋊C₉

C₃

yes

maximal, complemented

C₁₃×H(3,𝔽₃)⋊C₉

C₃

yes

maximal, complemented

C₁₃×#6868

C₃

yes

maximal, complemented

C₃×C₁₃×3₋¹⁺²

—

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

—

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

characteristic

C₃²×C₉×C₁₃

—

C₃²×C₉×C₁₃

C₃²

yes

characteristic

C₃²×C₉×C₁₃

C₃²

yes

—

C₃²×C₉×C₁₃

C₃²

yes

characteristic

C₁₃×#732

—

C₁₃×C₉⋊C₉

—

C₁₃×C₉⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

C₃²

yes

—

C₁₃×C₃²⋊C₉

C₃²

yes

—

#10624792

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

C₃³

yes

characteristic

C₁₃×3₋¹⁺²

—

C₁₃×H(3,𝔽₃)

—

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

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

H(3,𝔽₃)

yes

—

C₃×C₉×C₁₃

H(3,𝔽₃)

yes

characteristic

C₃×#732

C₃×C₁₃

yes

characteristic

C₉⋊H(3,𝔽₃)

C₃×C₁₃

yes

complemented

C₉×H(3,𝔽₃)

C₃×C₁₃

yes

characteristic, complemented

C₃×C₉⋊C₉

C₃×C₁₃

yes

characteristic, complemented

H(3,𝔽₃)⋊C₉

C₃×C₁₃

yes

complemented

H(3,𝔽₃)⋊C₉

C₃×C₁₃

yes

complemented

#6868

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

characteristic

C₃²×C₁₃

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₁₃

#737

yes

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

#732

yes

complemented

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

C₃²×C₁₃

yes

—

C₃×H(3,𝔽₃)

C₃²×C₁₃

yes

characteristic

C₃²×C₉

—

C₃²×C₉

C₃²×C₁₃

yes

characteristic

C₃²×C₉

C₃²×C₁₃

yes

characteristic

C₃²×C₉

C₃²×C₁₃

yes

—

#732

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃²×C₁₃

yes

—

C₃²⋊C₉

C₃²×C₁₃

yes

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

C₉⋊H(3,𝔽₃)

yes

C₃×C₁₃

yes

C₃×C₁₃

yes

—

C₃³

—

C₃³

—

C₃³

C₃³×C₁₃

yes

characteristic

3₋¹⁺²

—

—

—

—

C₁₃×3₋¹⁺²

yes

complemented

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

100

C₃×C₉

—

101

C₃×C₉

—

102

C₃×C₉

—

103

C₃×C₉

—

104

C₃×C₉

C₁₃×H(3,𝔽₃)

yes

—

105

C₃×C₉

C₁₃×H(3,𝔽₃)

yes

characteristic

106

C₁₃

#10624792

yes

characteristic, direct product factor, Sylow

107

C₃²

—

108

C₃²

—

109

C₃²

—

110

C₃²

—

111

C₃²

—

112

C₃²

—

113

C₃²

—

114

C₃²

—

115

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

116

C₃²

C₃×C₁₃×3₋¹⁺²

yes

characteristic

117

C₃²

C₁₃×#737

yes

—

118

C₉

—

119

C₉

—

120

C₉

—

121

C₉

—

122

C₉

—

123

C₉

C₁₃×#732

yes

complemented

124

C₃

—

125

C₃

—

126

C₃

—

127

C₃

—

128

C₃

C₁₃×C₉⋊H(3,𝔽₃)

yes

characteristic

129

C₃

C₃×C₁₃×#732

yes

characteristic

130

C₃

C₁₃×#6913

yes

—

	subgroup	G/H
center, socle	C₃²×C₁₃	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₁₃
prime radical	C₃×C₁₃×#732	C₃
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₁₃ ◁ C₃²×C₉×C₁₃ ◁ G

subgroup

G/H

center, socle

C₃²×C₁₃

C₃×H(3,𝔽₃)

commutator, Frattini, cosocle

C₃³

C₃³×C₁₃

prime radical

C₃×C₁₃×#732

C₃

LCS

G ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃²×C₁₃ ◁ C₃²×C₉×C₁₃ ◁ G

N	H ≅ G/N	type
2. C₁₃×C₉⋊H(3,𝔽₃)	C₃	semidirect
3. C₉×C₁₃×H(3,𝔽₃)	C₃	semidirect
4. C₃×C₁₃×C₉⋊C₉	C₃	semidirect
5. C₁₃×H(3,𝔽₃)⋊C₉	C₃	semidirect
6. C₁₃×H(3,𝔽₃)⋊C₉	C₃	semidirect
7. C₁₃×#6868	C₃	semidirect
22. #10624792	C₁₃	direct (N × H)
30. C₁₃×H(3,𝔽₃)	3₋¹⁺²	semidirect
43. C₉⋊H(3,𝔽₃)	C₃×C₁₃	semidirect
44. C₉×H(3,𝔽₃)	C₃×C₁₃	semidirect
45. C₃×C₉⋊C₉	C₃×C₁₃	semidirect
46. H(3,𝔽₃)⋊C₉	C₃×C₁₃	semidirect
47. H(3,𝔽₃)⋊C₉	C₃×C₁₃	semidirect
48. #6868	C₃×C₁₃	semidirect
65. C₉×C₁₃	#732	semidirect
94. H(3,𝔽₃)	C₁₃×3₋¹⁺²	semidirect
106. C₁₃	#10624792	direct (N × H)
123. C₉	C₁₃×#732	semidirect

H ≅ G/N

type

2. C₁₃×C₉⋊H(3,𝔽₃)

C₃

semidirect

3. C₉×C₁₃×H(3,𝔽₃)

C₃

semidirect

4. C₃×C₁₃×C₉⋊C₉

C₃

semidirect

5. C₁₃×H(3,𝔽₃)⋊C₉

C₃

semidirect

6. C₁₃×H(3,𝔽₃)⋊C₉

C₃

semidirect

7. C₁₃×#6868

C₃

semidirect

22. #10624792

C₁₃

direct (N × H)

30. C₁₃×H(3,𝔽₃)

3₋¹⁺²

semidirect

43. C₉⋊H(3,𝔽₃)

C₃×C₁₃

semidirect

44. C₉×H(3,𝔽₃)

C₃×C₁₃

semidirect

45. C₃×C₉⋊C₉

C₃×C₁₃

semidirect

46. H(3,𝔽₃)⋊C₉

C₃×C₁₃

semidirect

47. H(3,𝔽₃)⋊C₉

C₃×C₁₃

semidirect

48. #6868

C₃×C₁₃

semidirect

65. C₉×C₁₃

#732

semidirect

94. H(3,𝔽₃)

C₁₃×3₋¹⁺²

semidirect

106. C₁₃

#10624792

direct (N × H)

123. C₉

C₁₃×#732

semidirect

Subgroups — C₁₃×#10624792 #23744943

Conjugacy classes of subgroups

Complements

Subgroups — C13×#10624792 #23744943

Conjugacy classes of subgroups

Complements

Subgroups — C₁₃×#10624792 #23744943