Group #23744716 — subgroups

Subgroups — C₁₃×#10624565 #23744716

Order 9477.

470 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 152 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	3	yes	maximal, complemented
3	C₃×C₁₃×C₃²⋊C₉	C₃	1	3	yes	maximal
4	C₁₃×#6855	C₃	1	2	yes	maximal, complemented
5	C₁₃×#6855	C₃	1	3	yes	maximal, complemented
6	C₁₃×#6855	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃⁴×C₁₃	C₃²	1	1	yes	characteristic
8	C₃×C₁₃×3₋¹⁺²	—	3	3	no	—
9	C₃×C₁₃×3₋¹⁺²	—	3	3	no	—
10	C₃×C₁₃×3₋¹⁺²	—	3	3	no	—
11	C₃×C₁₃×3₋¹⁺²	—	3	3	no	—
12	C₃²×C₉×C₁₃	C₃²	1	6	yes	—
13	C₃²×C₉×C₁₃	C₃²	1	1	yes	characteristic
14	C₃²×C₉×C₁₃	C₃²	1	1	yes	characteristic
15	C₃²×C₉×C₁₃	C₃²	1	1	yes	characteristic
16	C₃²×C₉×C₁₃	C₃²	1	3	yes	—
17	C₁₃×C₉⋊C₉	—	3	9	no	—
18	C₁₃×C₃²⋊C₉	—	3	9	no	—
19	#10624565	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₃³×C₁₃	—	3	1	no	—
21	C₃³×C₁₃	—	3	3	no	—
22	C₃³×C₁₃	—	3	1	no	—
23	C₃³×C₁₃	—	3	1	no	—
24	C₃³×C₁₃	—	3	6	no	—
25	C₃³×C₁₃	3₋¹⁺²	1	3	yes	—
26	C₃³×C₁₃	C₃³	1	1	yes	characteristic
27	C₁₃×3₋¹⁺²	—	3	27	no	—
28	C₃×C₉×C₁₃	—	3	3	no	—
29	C₃×C₉×C₁₃	—	3	1	no	—
30	C₃×C₉×C₁₃	—	3	1	no	—
31	C₃×C₉×C₁₃	—	3	1	no	—
32	C₃×C₉×C₁₃	—	3	1	no	—
33	C₃×C₉×C₁₃	—	3	1	no	—
34	C₃×C₉×C₁₃	—	3	1	no	—
35	C₃×C₉×C₁₃	—	3	1	no	—
36	C₃×C₉×C₁₃	—	3	1	no	—
37	C₃×C₉×C₁₃	—	3	1	no	—
38	C₃×C₉×C₁₃	—	3	6	no	—
39	C₃×C₉×C₁₃	—	3	6	no	—
40	C₃×C₉×C₁₃	—	3	6	no	—
41	C₃×C₉×C₁₃	—	3	12	no	—
42	C₃×C₉×C₁₃	H(3,𝔽₃)	1	3	yes	—
43	C₃×C₉×C₁₃	3₋¹⁺²	1	3	yes	—
44	C₃×C₉×C₁₃	3₋¹⁺²	1	3	yes	—
45	C₃×C₉×C₁₃	3₋¹⁺²	1	9	yes	complemented
46	C₃²×3₋¹⁺²	C₃×C₁₃	1	1	yes	characteristic
47	C₃×C₉⋊C₉	C₃×C₁₃	1	3	yes	complemented
48	C₃×C₃²⋊C₉	C₃×C₁₃	1	3	yes	—
49	#6855	C₃×C₁₃	1	2	yes	complemented
50	#6855	C₃×C₁₃	1	3	yes	complemented
51	#6855	C₃×C₁₃	1	1	yes	characteristic, complemented
52	C₃²×C₁₃	—	3	3	no	—
53	C₃²×C₁₃	—	3	9	no	—
54	C₃²×C₁₃	—	9	3	no	—
55	C₃²×C₁₃	—	9	3	no	—
56	C₃²×C₁₃	—	9	2	no	—
57	C₃²×C₁₃	—	9	1	no	—
58	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
59	C₃²×C₁₃	C₃×3₋¹⁺²	1	1	yes	characteristic
60	C₃²×C₁₃	C₃×3₋¹⁺²	1	1	yes	characteristic
61	C₃²×C₁₃	C₃×3₋¹⁺²	1	1	yes	characteristic
62	C₃²×C₁₃	C₃×3₋¹⁺²	1	3	yes	—
63	C₃²×C₁₃	#737	1	6	yes	—
64	C₉×C₁₃	—	3	3	no	—
65	C₉×C₁₃	—	3	3	no	—
66	C₉×C₁₃	—	3	3	no	—
67	C₉×C₁₃	—	3	9	no	—
68	C₉×C₁₃	—	9	6	no	—
69	C₃⁴	C₃²×C₁₃	1	1	yes	characteristic
70	C₃×3₋¹⁺²	—	3	3	no	—
71	C₃×3₋¹⁺²	—	3	3	no	—
72	C₃×3₋¹⁺²	—	3	3	no	—
73	C₃×3₋¹⁺²	—	3	3	no	—
74	C₃²×C₉	C₃²×C₁₃	1	6	yes	—
75	C₃²×C₉	C₃²×C₁₃	1	1	yes	characteristic
76	C₃²×C₉	C₃²×C₁₃	1	1	yes	characteristic
77	C₃²×C₉	C₃²×C₁₃	1	1	yes	characteristic
78	C₃²×C₉	C₃²×C₁₃	1	3	yes	—
79	C₉⋊C₉	—	3	9	no	—
80	C₃²⋊C₉	—	3	9	no	—
81	C₃×C₁₃	—	9	3	no	—
82	C₃×C₁₃	#6891	1	1	yes	characteristic
83	C₃×C₁₃	C₉⋊H(3,𝔽₃)	1	3	yes	—
84	C₃×C₁₃	#6894	1	3	yes	—
85	C₃×C₁₃	#6896	1	3	yes	—
86	C₃×C₁₃	#6899	1	2	yes	—
87	C₃×C₁₃	#6900	1	1	yes	characteristic
88	C₃³	—	3	1	no	—
89	C₃³	—	3	3	no	—
90	C₃³	—	3	1	no	—
91	C₃³	—	3	1	no	—
92	C₃³	—	3	6	no	—
93	C₃³	C₁₃×3₋¹⁺²	1	3	yes	—
94	C₃³	C₃³×C₁₃	1	1	yes	characteristic
95	3₋¹⁺²	—	3	27	no	—
96	C₃×C₉	—	3	3	no	—
97	C₃×C₉	—	3	1	no	—
98	C₃×C₉	—	3	1	no	—
99	C₃×C₉	—	3	1	no	—
100	C₃×C₉	—	3	1	no	—
101	C₃×C₉	—	3	1	no	—
102	C₃×C₉	—	3	1	no	—
103	C₃×C₉	—	3	1	no	—
104	C₃×C₉	—	3	1	no	—
105	C₃×C₉	—	3	1	no	—
106	C₃×C₉	—	3	6	no	—
107	C₃×C₉	—	3	6	no	—
108	C₃×C₉	—	3	6	no	—
109	C₃×C₉	—	3	12	no	—
110	C₃×C₉	C₁₃×H(3,𝔽₃)	1	3	yes	—
111	C₃×C₉	C₁₃×3₋¹⁺²	1	3	yes	—
112	C₃×C₉	C₁₃×3₋¹⁺²	1	3	yes	—
113	C₃×C₉	C₁₃×3₋¹⁺²	1	9	yes	complemented
114	C₁₃	#10624565	1	1	yes	characteristic, direct product factor, Sylow
115	C₃²	—	3	3	no	—
116	C₃²	—	3	9	no	—
117	C₃²	—	9	3	no	—
118	C₃²	—	9	3	no	—
119	C₃²	—	9	2	no	—
120	C₃²	—	9	1	no	—
121	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
122	C₃²	C₃×C₁₃×3₋¹⁺²	1	1	yes	characteristic
123	C₃²	C₃×C₁₃×3₋¹⁺²	1	1	yes	characteristic
124	C₃²	C₃×C₁₃×3₋¹⁺²	1	1	yes	characteristic
125	C₃²	C₃×C₁₃×3₋¹⁺²	1	3	yes	—
126	C₃²	C₁₃×#737	1	6	yes	—
127	C₉	—	3	3	no	—
128	C₉	—	3	3	no	—
129	C₉	—	3	3	no	—
130	C₉	—	3	9	no	—
131	C₉	—	9	6	no	—
132	C₃	—	9	3	no	—
133	C₃	C₁₃×#6891	1	1	yes	characteristic
134	C₃	C₁₃×C₉⋊H(3,𝔽₃)	1	3	yes	—
135	C₃	C₁₃×#6894	1	3	yes	—
136	C₃	C₁₃×#6896	1	3	yes	—
137	C₃	C₁₃×#6899	1	2	yes	—
138	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

C₁₃×#6855

C₃

yes

maximal, complemented

C₁₃×#6855

C₃

yes

maximal, complemented

C₁₃×#6855

C₃

yes

maximal, characteristic, 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

—

C₃²×C₉×C₁₃

C₃²

yes

—

C₃²×C₉×C₁₃

C₃²

yes

characteristic

C₃²×C₉×C₁₃

C₃²

yes

characteristic

C₃²×C₉×C₁₃

C₃²

yes

characteristic

C₃²×C₉×C₁₃

C₃²

yes

—

C₁₃×C₉⋊C₉

—

C₁₃×C₃²⋊C₉

—

#10624565

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

3₋¹⁺²

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×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

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

—

#6855

C₃×C₁₃

yes

complemented

#6855

C₃×C₁₃

yes

complemented

#6855

C₃×C₁₃

yes

characteristic, complemented

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

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

—

C₉×C₁₃

—

C₃⁴

C₃²×C₁₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃²×C₉

C₃²×C₁₃

yes

—

C₃²×C₉

C₃²×C₁₃

yes

characteristic

C₃²×C₉

C₃²×C₁₃

yes

characteristic

C₃²×C₉

C₃²×C₁₃

yes

characteristic

C₃²×C₉

C₃²×C₁₃

yes

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃×C₁₃

—

C₃×C₁₃

#6891

yes

characteristic

C₃×C₁₃

C₉⋊H(3,𝔽₃)

yes

—

C₃×C₁₃

#6894

yes

—

C₃×C₁₃

#6896

yes

—

C₃×C₁₃

#6899

yes

—

C₃×C₁₃

#6900

yes

characteristic

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₁₃×3₋¹⁺²

yes

—

C₃³

C₃³×C₁₃

yes

characteristic

3₋¹⁺²

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

100

C₃×C₉

—

101

C₃×C₉

—

102

C₃×C₉

—

103

C₃×C₉

—

104

C₃×C₉

—

105

C₃×C₉

—

106

C₃×C₉

—

107

C₃×C₉

—

108

C₃×C₉

—

109

C₃×C₉

—

110

C₃×C₉

C₁₃×H(3,𝔽₃)

yes

—

111

C₃×C₉

C₁₃×3₋¹⁺²

yes

—

112

C₃×C₉

C₁₃×3₋¹⁺²

yes

—

113

C₃×C₉

C₁₃×3₋¹⁺²

yes

complemented

114

C₁₃

#10624565

yes

characteristic, direct product factor, Sylow

115

C₃²

—

116

C₃²

—

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

122

C₃²

C₃×C₁₃×3₋¹⁺²

yes

characteristic

123

C₃²

C₃×C₁₃×3₋¹⁺²

yes

characteristic

124

C₃²

C₃×C₁₃×3₋¹⁺²

yes

characteristic

125

C₃²

C₃×C₁₃×3₋¹⁺²

yes

—

126

C₃²

C₁₃×#737

yes

—

127

C₉

—

128

C₉

—

129

C₉

—

130

C₉

—

131

C₉

—

132

C₃

—

133

C₃

C₁₃×#6891

yes

characteristic

134

C₃

C₁₃×C₉⋊H(3,𝔽₃)

yes

—

135

C₃

C₁₃×#6894

yes

—

136

C₃

C₁₃×#6896

yes

—

137

C₃

C₁₃×#6899

yes

—

138

C₃

C₁₃×#6900

yes

characteristic

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

subgroup

G/H

center, socle

C₃³×C₁₃

C₃³

commutator, Frattini, cosocle

C₃³

C₃³×C₁₃

prime radical

C₃⁴×C₁₃

C₃²

LCS

G ▷ C₃³ ▷ C₁

UCS

C₁ ◁ C₃³×C₁₃ ◁ G

N	H ≅ G/N	type
2. C₃×C₁₃×C₉⋊C₉	C₃	semidirect
4. C₁₃×#6855	C₃	semidirect
5. C₁₃×#6855	C₃	semidirect
6. C₁₃×#6855	C₃	semidirect
19. #10624565	C₁₃	direct (N × H)
45. C₃×C₉×C₁₃	3₋¹⁺²	semidirect
47. C₃×C₉⋊C₉	C₃×C₁₃	semidirect
49. #6855	C₃×C₁₃	semidirect
50. #6855	C₃×C₁₃	semidirect
51. #6855	C₃×C₁₃	semidirect
113. C₃×C₉	C₁₃×3₋¹⁺²	semidirect
114. C₁₃	#10624565	direct (N × H)

H ≅ G/N

type

2. C₃×C₁₃×C₉⋊C₉

C₃

semidirect

4. C₁₃×#6855

C₃

semidirect

5. C₁₃×#6855

C₃

semidirect

6. C₁₃×#6855

C₃

semidirect

19. #10624565

C₁₃

direct (N × H)

45. C₃×C₉×C₁₃

3₋¹⁺²

semidirect

47. C₃×C₉⋊C₉

C₃×C₁₃

semidirect

49. #6855

C₃×C₁₃

semidirect

50. #6855

C₃×C₁₃

semidirect

51. #6855

C₃×C₁₃

semidirect

113. C₃×C₉

C₁₃×3₋¹⁺²

semidirect

114. C₁₃

#10624565

direct (N × H)

Subgroups — C₁₃×#10624565 #23744716

Conjugacy classes of subgroups

Complements

Subgroups — C13×#10624565 #23744716

Conjugacy classes of subgroups

Complements

Subgroups — C₁₃×#10624565 #23744716