Group #23744815 — subgroups

Subgroups — C₁₃×#10624664 #23744815

Order 9477.

198 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 74 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	C₁₃×#6894	C₃	1	2	yes	maximal, complemented
2	C₁₃×#6893	C₃	1	1	yes	maximal, characteristic, complemented
3	C₁₃×C₉⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#6860	C₃	1	2	yes	maximal, complemented
5	C₁₃×#6859	C₃	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#6858	C₃	1	2	yes	maximal, complemented
7	C₁₃×#6857	C₃	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#6857	C₃	1	2	yes	maximal, complemented
9	C₁₃×#6856	C₃	1	1	yes	maximal, characteristic
10	C₃×C₁₃×3₋¹⁺²	—	3	1	no	—
11	C₃×C₁₃×3₋¹⁺²	C₃²	1	2	yes	—
12	C₃×C₁₃×3₋¹⁺²	C₃²	1	2	yes	—
13	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
14	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
15	C₃²×C₉×C₁₃	C₃²	1	1	yes	characteristic
16	C₁₃×C₉⋊C₉	—	3	4	no	—
17	C₁₃×C₉⋊C₉	—	3	2	no	—
18	C₁₃×C₃²⋊C₉	—	3	2	no	—
19	C₁₃×C₃²⋊C₉	—	3	2	no	—
20	C₁₃×C₃²⋊C₉	—	3	1	no	—
21	C₁₃×C₃²⋊C₉	C₃²	1	2	yes	—
22	C₁₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
23	C₁₃×C₃²⋊C₉	C₃²	1	2	yes	—
24	C₁₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
25	#10624664	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₃³×C₁₃	—	3	1	no	—
27	C₃³×C₁₃	—	3	1	no	—
28	C₃³×C₁₃	C₃³	1	1	yes	characteristic
29	C₁₃×3₋¹⁺²	—	9	3	no	—
30	C₁₃×3₋¹⁺²	—	9	2	no	—
31	C₁₃×3₋¹⁺²	—	9	2	no	—
32	C₁₃×H(3,𝔽₃)	—	9	1	no	—
33	C₁₃×H(3,𝔽₃)	—	9	1	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	—	3	2	no	—
39	C₃×C₉×C₁₃	—	3	2	no	—
40	C₃×C₉×C₁₃	—	9	1	no	—
41	C₃×C₉×C₁₃	H(3,𝔽₃)	1	2	yes	—
42	C₃×C₉×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
43	#6894	C₃×C₁₃	1	2	yes	complemented
44	#6893	C₃×C₁₃	1	1	yes	characteristic, complemented
45	C₉⋊H(3,𝔽₃)	C₃×C₁₃	1	1	yes	characteristic, complemented
46	#6860	C₃×C₁₃	1	2	yes	complemented
47	#6859	C₃×C₁₃	1	1	yes	characteristic, complemented
48	#6858	C₃×C₁₃	1	2	yes	complemented
49	#6857	C₃×C₁₃	1	1	yes	characteristic, complemented
50	#6857	C₃×C₁₃	1	2	yes	complemented
51	#6856	C₃×C₁₃	1	1	yes	characteristic
52	C₃²×C₁₃	—	3	1	no	—
53	C₃²×C₁₃	—	3	2	no	—
54	C₃²×C₁₃	—	3	1	no	—
55	C₃²×C₁₃	—	9	1	no	—
56	C₃²×C₁₃	—	9	1	no	—
57	C₃²×C₁₃	—	9	2	no	—
58	C₃²×C₁₃	—	9	2	no	—
59	C₃²×C₁₃	—	9	1	no	—
60	C₃²×C₁₃	—	9	1	no	—
61	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
62	C₉×C₁₃	—	3	2	no	—
63	C₉×C₁₃	—	3	1	no	—
64	C₉×C₁₃	—	9	2	no	—
65	C₉×C₁₃	—	9	2	no	—
66	C₉×C₁₃	—	9	2	no	—
67	C₉×C₁₃	—	9	2	no	—
68	C₉×C₁₃	—	9	1	no	—
69	C₉×C₁₃	—	9	1	no	—
70	C₃×3₋¹⁺²	—	3	1	no	—
71	C₃×3₋¹⁺²	C₃²×C₁₃	1	2	yes	—
72	C₃×3₋¹⁺²	C₃²×C₁₃	1	2	yes	—
73	C₃×H(3,𝔽₃)	C₃²×C₁₃	1	1	yes	characteristic
74	C₃×H(3,𝔽₃)	C₃²×C₁₃	1	1	yes	characteristic
75	C₃²×C₉	C₃²×C₁₃	1	1	yes	characteristic
76	C₉⋊C₉	—	3	4	no	—
77	C₉⋊C₉	—	3	2	no	—
78	C₃²⋊C₉	—	3	2	no	—
79	C₃²⋊C₉	—	3	2	no	—
80	C₃²⋊C₉	—	3	1	no	—
81	C₃²⋊C₉	C₃²×C₁₃	1	2	yes	—
82	C₃²⋊C₉	C₃²×C₁₃	1	1	yes	characteristic
83	C₃²⋊C₉	C₃²×C₁₃	1	2	yes	—
84	C₃²⋊C₉	C₃²×C₁₃	1	1	yes	characteristic
85	C₃×C₁₃	—	9	1	no	—
86	C₃×C₁₃	—	27	1	no	—
87	C₃×C₁₃	—	27	1	no	—
88	C₃×C₁₃	#6909	1	1	yes	characteristic
89	C₃×C₁₃	#6913	1	1	yes	characteristic
90	C₃×C₁₃	#6913	1	2	yes	—
91	C₃³	—	3	1	no	—
92	C₃³	—	3	1	no	—
93	C₃³	C₃³×C₁₃	1	1	yes	characteristic
94	3₋¹⁺²	—	9	3	no	—
95	3₋¹⁺²	—	9	2	no	—
96	3₋¹⁺²	—	9	2	no	—
97	H(3,𝔽₃)	—	9	1	no	—
98	H(3,𝔽₃)	—	9	1	no	—
99	C₃×C₉	—	3	2	no	—
100	C₃×C₉	—	3	1	no	—
101	C₃×C₉	—	3	1	no	—
102	C₃×C₉	—	3	2	no	—
103	C₃×C₉	—	3	2	no	—
104	C₃×C₉	—	3	2	no	—
105	C₃×C₉	—	9	1	no	—
106	C₃×C₉	C₁₃×H(3,𝔽₃)	1	2	yes	—
107	C₃×C₉	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
108	C₁₃	#10624664	1	1	yes	characteristic, direct product factor, Sylow
109	C₃²	—	3	1	no	—
110	C₃²	—	3	1	no	—
111	C₃²	—	3	2	no	—
112	C₃²	—	9	1	no	—
113	C₃²	—	9	1	no	—
114	C₃²	—	9	1	no	—
115	C₃²	—	9	2	no	—
116	C₃²	—	9	1	no	—
117	C₃²	—	9	2	no	—
118	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
119	C₉	—	3	1	no	—
120	C₉	—	3	2	no	—
121	C₉	—	9	1	no	—
122	C₉	—	9	1	no	—
123	C₉	—	9	2	no	—
124	C₉	—	9	2	no	—
125	C₉	—	9	2	no	—
126	C₉	—	9	2	no	—
127	C₃	—	9	1	no	—
128	C₃	—	27	1	no	—
129	C₃	—	27	1	no	—
130	C₃	C₁₃×#6909	1	1	yes	characteristic
131	C₃	C₁₃×#6913	1	2	yes	—
132	C₃	C₁₃×#6913	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#6894

C₃

yes

maximal, complemented

C₁₃×#6893

C₃

yes

maximal, characteristic, complemented

C₁₃×C₉⋊H(3,𝔽₃)

C₃

yes

maximal, characteristic, complemented

C₁₃×#6860

C₃

yes

maximal, complemented

C₁₃×#6859

C₃

yes

maximal, characteristic, complemented

C₁₃×#6858

C₃

yes

maximal, complemented

C₁₃×#6857

C₃

yes

maximal, characteristic, complemented

C₁₃×#6857

C₃

yes

maximal, complemented

C₁₃×#6856

C₃

yes

maximal, characteristic

C₃×C₁₃×3₋¹⁺²

—

C₃×C₁₃×3₋¹⁺²

C₃²

yes

—

C₃×C₁₃×3₋¹⁺²

C₃²

yes

—

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

characteristic

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

characteristic

C₃²×C₉×C₁₃

C₃²

yes

characteristic

C₁₃×C₉⋊C₉

—

C₁₃×C₉⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

C₃²

yes

—

C₁₃×C₃²⋊C₉

C₃²

yes

characteristic

C₁₃×C₃²⋊C₉

C₃²

yes

—

C₁₃×C₃²⋊C₉

C₃²

yes

characteristic

#10624664

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

C₃³

yes

characteristic

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

—

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

#6894

C₃×C₁₃

yes

complemented

#6893

C₃×C₁₃

yes

characteristic, complemented

C₉⋊H(3,𝔽₃)

C₃×C₁₃

yes

characteristic, complemented

#6860

C₃×C₁₃

yes

complemented

#6859

C₃×C₁₃

yes

characteristic, complemented

#6858

C₃×C₁₃

yes

complemented

#6857

C₃×C₁₃

yes

characteristic, complemented

#6857

C₃×C₁₃

yes

complemented

#6856

C₃×C₁₃

yes

characteristic

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃²×C₁₃

yes

—

C₃×3₋¹⁺²

C₃²×C₁₃

yes

—

C₃×H(3,𝔽₃)

C₃²×C₁₃

yes

characteristic

C₃×H(3,𝔽₃)

C₃²×C₁₃

yes

characteristic

C₃²×C₉

C₃²×C₁₃

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃²×C₁₃

yes

—

C₃²⋊C₉

C₃²×C₁₃

yes

characteristic

C₃²⋊C₉

C₃²×C₁₃

yes

—

C₃²⋊C₉

C₃²×C₁₃

yes

characteristic

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

yes

C₃×C₁₃

yes

C₃×C₁₃

yes

—

C₃³

—

C₃³

—

C₃³

C₃³×C₁₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

C₃×C₉

—

100

C₃×C₉

—

101

C₃×C₉

—

102

C₃×C₉

—

103

C₃×C₉

—

104

C₃×C₉

—

105

C₃×C₉

—

106

C₃×C₉

C₁₃×H(3,𝔽₃)

yes

—

107

C₃×C₉

C₁₃×H(3,𝔽₃)

yes

characteristic

108

C₁₃

#10624664

yes

characteristic, direct product factor, Sylow

109

C₃²

—

110

C₃²

—

111

C₃²

—

112

C₃²

—

113

C₃²

—

114

C₃²

—

115

C₃²

—

116

C₃²

—

117

C₃²

—

118

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

119

C₉

—

120

C₉

—

121

C₉

—

122

C₉

—

123

C₉

—

124

C₉

—

125

C₉

—

126

C₉

—

127

C₃

—

128

C₃

—

129

C₃

—

130

C₃

C₁₃×#6909

yes

characteristic

131

C₃

C₁₃×#6913

yes

—

132

C₃

C₁₃×#6913

yes

characteristic

	subgroup	G/H
center, socle	C₃²×C₁₃	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₁₃
prime radical	C₁₃×#6856	C₃
LCS	G ▷ C₃³ ▷ C₃² ▷ 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

C₁₃×#6856

C₃

LCS

G ▷ C₃³ ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₃²×C₁₃ ◁ C₃²×C₉×C₁₃ ◁ G

N	H ≅ G/N	type
1. C₁₃×#6894	C₃	semidirect
2. C₁₃×#6893	C₃	semidirect
3. C₁₃×C₉⋊H(3,𝔽₃)	C₃	semidirect
4. C₁₃×#6860	C₃	semidirect
5. C₁₃×#6859	C₃	semidirect
6. C₁₃×#6858	C₃	semidirect
7. C₁₃×#6857	C₃	semidirect
8. C₁₃×#6857	C₃	semidirect
25. #10624664	C₁₃	direct (N × H)
43. #6894	C₃×C₁₃	semidirect
44. #6893	C₃×C₁₃	semidirect
45. C₉⋊H(3,𝔽₃)	C₃×C₁₃	semidirect
46. #6860	C₃×C₁₃	semidirect
47. #6859	C₃×C₁₃	semidirect
48. #6858	C₃×C₁₃	semidirect
49. #6857	C₃×C₁₃	semidirect
50. #6857	C₃×C₁₃	semidirect
108. C₁₃	#10624664	direct (N × H)

H ≅ G/N

type

1. C₁₃×#6894

C₃

semidirect

2. C₁₃×#6893

C₃

semidirect

3. C₁₃×C₉⋊H(3,𝔽₃)

C₃

semidirect

4. C₁₃×#6860

C₃

semidirect

5. C₁₃×#6859

C₃

semidirect

6. C₁₃×#6858

C₃

semidirect

7. C₁₃×#6857

C₃

semidirect

8. C₁₃×#6857

C₃

semidirect

25. #10624664

C₁₃

direct (N × H)

43. #6894

C₃×C₁₃

semidirect

44. #6893

C₃×C₁₃

semidirect

45. C₉⋊H(3,𝔽₃)

C₃×C₁₃

semidirect

46. #6860

C₃×C₁₃

semidirect

47. #6859

C₃×C₁₃

semidirect

48. #6858

C₃×C₁₃

semidirect

49. #6857

C₃×C₁₃

semidirect

50. #6857

C₃×C₁₃

semidirect

108. C₁₃

#10624664

direct (N × H)

Subgroups — C₁₃×#10624664 #23744815

Conjugacy classes of subgroups

Complements

Subgroups — C13×#10624664 #23744815

Conjugacy classes of subgroups

Complements

Subgroups — C₁₃×#10624664 #23744815