Group #23744932 — subgroups

Subgroups — C₁₃×#10624781 #23744932

Order 9477.

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

normal subgroups only

This group has 464 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×C₁₃×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
2	C₃²×C₁₃×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
3	C₁₃×#6891	C₃	1	1	yes	maximal, characteristic
4	C₁₃×#6891	C₃	1	1	yes	maximal, characteristic
5	C₁₃×3₋¹⁺²⋊C₉	C₃	1	9	yes	maximal, complemented
6	C₃⁴×C₁₃	C₃²	1	1	yes	characteristic
7	C₃×C₁₃×3₋¹⁺²	—	3	1	no	—
8	C₃×C₁₃×3₋¹⁺²	—	3	1	no	—
9	C₃×C₁₃×3₋¹⁺²	—	3	1	no	—
10	C₃×C₁₃×3₋¹⁺²	—	3	1	no	—
11	C₃×C₁₃×3₋¹⁺²	—	3	1	no	—
12	C₃×C₁₃×3₋¹⁺²	—	3	9	no	—
13	C₃×C₁₃×3₋¹⁺²	—	3	9	no	—
14	C₃×C₁₃×3₋¹⁺²	—	3	1	no	—
15	C₃×C₁₃×3₋¹⁺²	—	3	1	no	—
16	C₃×C₁₃×3₋¹⁺²	C₃²	1	3	yes	—
17	C₃²×C₉×C₁₃	—	3	1	no	—
18	C₃²×C₉×C₁₃	C₃²	1	3	yes	—
19	C₁₃×C₃²⋊C₉	—	3	1	no	—
20	C₁₃×C₃²⋊C₉	—	3	1	no	—
21	C₁₃×C₃²⋊C₉	—	3	1	no	—
22	C₁₃×C₃²⋊C₉	—	3	1	no	—
23	C₁₃×C₃²⋊C₉	C₃²	1	3	yes	—
24	C₁₃×C₃²⋊C₉	C₃²	1	3	yes	—
25	#10624781	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₃³×C₁₃	—	3	3	no	—
27	C₃³×C₁₃	—	3	3	no	—
28	C₃³×C₁₃	—	9	1	no	—
29	C₃³×C₁₃	—	9	1	no	—
30	C₃³×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
31	C₃³×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
32	C₃³×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
33	C₃³×C₁₃	C₃³	1	1	yes	characteristic
34	C₁₃×3₋¹⁺²	—	3	9	no	—
35	C₁₃×3₋¹⁺²	—	3	9	no	—
36	C₁₃×3₋¹⁺²	—	3	9	no	—
37	C₁₃×3₋¹⁺²	—	3	9	no	—
38	C₁₃×3₋¹⁺²	—	3	9	no	—
39	C₁₃×3₋¹⁺²	—	3	6	no	—
40	C₁₃×3₋¹⁺²	—	9	3	no	—
41	C₁₃×3₋¹⁺²	—	9	3	no	—
42	C₁₃×3₋¹⁺²	3₋¹⁺²	1	9	yes	complemented
43	C₃×C₉×C₁₃	—	3	9	no	—
44	C₃×C₉×C₁₃	—	3	3	no	—
45	C₃×C₉×C₁₃	—	3	3	no	—
46	C₃×C₉×C₁₃	—	3	9	no	—
47	C₃×C₉×C₁₃	—	3	3	no	—
48	C₃×C₉×C₁₃	—	3	3	no	—
49	C₃²×3₋¹⁺²	C₃×C₁₃	1	1	yes	characteristic
50	C₃²×3₋¹⁺²	C₃×C₁₃	1	1	yes	characteristic
51	#6891	C₃×C₁₃	1	1	yes	characteristic
52	#6891	C₃×C₁₃	1	1	yes	characteristic
53	3₋¹⁺²⋊C₉	C₃×C₁₃	1	9	yes	complemented
54	C₃²×C₁₃	—	3	1	no	—
55	C₃²×C₁₃	—	3	1	no	—
56	C₃²×C₁₃	—	3	1	no	—
57	C₃²×C₁₃	—	3	1	no	—
58	C₃²×C₁₃	—	3	1	no	—
59	C₃²×C₁₃	—	3	1	no	—
60	C₃²×C₁₃	—	3	1	no	—
61	C₃²×C₁₃	—	3	1	no	—
62	C₃²×C₁₃	—	3	1	no	—
63	C₃²×C₁₃	—	3	1	no	—
64	C₃²×C₁₃	—	3	1	no	—
65	C₃²×C₁₃	—	3	1	no	—
66	C₃²×C₁₃	—	3	1	no	—
67	C₃²×C₁₃	—	3	1	no	—
68	C₃²×C₁₃	—	9	9	no	—
69	C₃²×C₁₃	AΣL(1,𝔽₂₇)	1	3	yes	—
70	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
71	C₃²×C₁₃	C₃×3₋¹⁺²	1	3	yes	—
72	C₉×C₁₃	—	3	9	no	—
73	C₉×C₁₃	—	3	9	no	—
74	C₉×C₁₃	—	9	3	no	—
75	C₉×C₁₃	—	9	3	no	—
76	C₃⁴	C₃²×C₁₃	1	1	yes	characteristic
77	C₃×3₋¹⁺²	—	3	1	no	—
78	C₃×3₋¹⁺²	—	3	1	no	—
79	C₃×3₋¹⁺²	—	3	1	no	—
80	C₃×3₋¹⁺²	—	3	1	no	—
81	C₃×3₋¹⁺²	—	3	1	no	—
82	C₃×3₋¹⁺²	—	3	9	no	—
83	C₃×3₋¹⁺²	—	3	9	no	—
84	C₃×3₋¹⁺²	—	3	1	no	—
85	C₃×3₋¹⁺²	—	3	1	no	—
86	C₃×3₋¹⁺²	C₃²×C₁₃	1	3	yes	—
87	C₃²×C₉	—	3	1	no	—
88	C₃²×C₉	C₃²×C₁₃	1	3	yes	—
89	C₃²⋊C₉	—	3	1	no	—
90	C₃²⋊C₉	—	3	1	no	—
91	C₃²⋊C₉	—	3	1	no	—
92	C₃²⋊C₉	—	3	1	no	—
93	C₃²⋊C₉	C₃²×C₁₃	1	3	yes	—
94	C₃²⋊C₉	C₃²×C₁₃	1	3	yes	—
95	C₃×C₁₃	—	3	3	no	—
96	C₃×C₁₃	—	3	3	no	—
97	C₃×C₁₃	—	9	1	no	—
98	C₃×C₁₃	—	9	1	no	—
99	C₃×C₁₃	#6891	1	1	yes	characteristic
100	C₃×C₁₃	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
101	C₃×C₁₃	#6910	1	1	yes	characteristic
102	C₃×C₁₃	#6912	1	1	yes	characteristic
103	C₃³	—	3	3	no	—
104	C₃³	—	3	3	no	—
105	C₃³	—	9	1	no	—
106	C₃³	—	9	1	no	—
107	C₃³	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
108	C₃³	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
109	C₃³	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
110	C₃³	C₃³×C₁₃	1	1	yes	characteristic
111	3₋¹⁺²	—	3	9	no	—
112	3₋¹⁺²	—	3	9	no	—
113	3₋¹⁺²	—	3	9	no	—
114	3₋¹⁺²	—	3	9	no	—
115	3₋¹⁺²	—	3	9	no	—
116	3₋¹⁺²	—	3	6	no	—
117	3₋¹⁺²	—	9	3	no	—
118	3₋¹⁺²	—	9	3	no	—
119	3₋¹⁺²	C₁₃×3₋¹⁺²	1	9	yes	complemented
120	C₃×C₉	—	3	9	no	—
121	C₃×C₉	—	3	3	no	—
122	C₃×C₉	—	3	3	no	—
123	C₃×C₉	—	3	9	no	—
124	C₃×C₉	—	3	3	no	—
125	C₃×C₉	—	3	3	no	—
126	C₁₃	#10624781	1	1	yes	characteristic, direct product factor, Sylow
127	C₃²	—	3	1	no	—
128	C₃²	—	3	1	no	—
129	C₃²	—	3	1	no	—
130	C₃²	—	3	1	no	—
131	C₃²	—	3	1	no	—
132	C₃²	—	3	1	no	—
133	C₃²	—	3	1	no	—
134	C₃²	—	3	1	no	—
135	C₃²	—	3	1	no	—
136	C₃²	—	3	1	no	—
137	C₃²	—	3	1	no	—
138	C₃²	—	3	1	no	—
139	C₃²	—	3	1	no	—
140	C₃²	—	3	1	no	—
141	C₃²	—	9	9	no	—
142	C₃²	C₁₃×AΣL(1,𝔽₂₇)	1	3	yes	—
143	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
144	C₃²	C₃×C₁₃×3₋¹⁺²	1	3	yes	—
145	C₉	—	3	9	no	—
146	C₉	—	3	9	no	—
147	C₉	—	9	3	no	—
148	C₉	—	9	3	no	—
149	C₃	—	3	3	no	—
150	C₃	—	3	3	no	—
151	C₃	—	9	1	no	—
152	C₃	—	9	1	no	—
153	C₃	C₁₃×#6891	1	1	yes	characteristic
154	C₃	C₃×C₁₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
155	C₃	C₁₃×#6910	1	1	yes	characteristic
156	C₃	C₁₃×#6912	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₁₃×3₋¹⁺²

C₃

yes

maximal, characteristic

C₃²×C₁₃×3₋¹⁺²

C₃

yes

maximal, characteristic

C₁₃×#6891

C₃

yes

maximal, characteristic

C₁₃×#6891

C₃

yes

maximal, characteristic

C₁₃×3₋¹⁺²⋊C₉

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

—

C₃×C₁₃×3₋¹⁺²

—

C₃×C₁₃×3₋¹⁺²

—

C₃×C₁₃×3₋¹⁺²

—

C₃×C₁₃×3₋¹⁺²

—

C₃×C₁₃×3₋¹⁺²

—

C₃×C₁₃×3₋¹⁺²

C₃²

yes

—

C₃²×C₉×C₁₃

—

C₃²×C₉×C₁₃

C₃²

yes

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

C₃²

yes

—

C₁₃×C₃²⋊C₉

C₃²

yes

—

#10624781

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

yes

C₃³×C₁₃

yes

C₃³×C₁₃

yes

C₃³×C₁₃

C₃³

yes

characteristic

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

3₋¹⁺²

yes

complemented

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃²×3₋¹⁺²

C₃×C₁₃

yes

characteristic

C₃²×3₋¹⁺²

C₃×C₁₃

yes

characteristic

#6891

C₃×C₁₃

yes

characteristic

#6891

C₃×C₁₃

yes

characteristic

3₋¹⁺²⋊C₉

C₃×C₁₃

yes

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

AΣL(1,𝔽₂₇)

yes

—

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₁₃

C₃×3₋¹⁺²

yes

—

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

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃²×C₁₃

yes

—

C₃²×C₉

—

C₃²×C₉

C₃²×C₁₃

yes

—

C₃²⋊C₉

—

C₃²⋊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

#6891

yes

characteristic

100

C₃×C₁₃

C₃×AΣL(1,𝔽₂₇)

yes

characteristic

101

C₃×C₁₃

#6910

yes

characteristic

102

C₃×C₁₃

#6912

yes

characteristic

103

C₃³

—

104

C₃³

—

105

C₃³

—

106

C₃³

—

107

C₃³

C₁₃×H(3,𝔽₃)

yes

characteristic

108

C₃³

C₁₃×H(3,𝔽₃)

yes

characteristic

109

C₃³

C₁₃×H(3,𝔽₃)

yes

characteristic

110

C₃³

C₃³×C₁₃

yes

characteristic

111

3₋¹⁺²

—

112

3₋¹⁺²

—

113

3₋¹⁺²

—

114

3₋¹⁺²

—

115

3₋¹⁺²

—

116

3₋¹⁺²

—

117

3₋¹⁺²

—

118

3₋¹⁺²

—

119

3₋¹⁺²

C₁₃×3₋¹⁺²

yes

complemented

120

C₃×C₉

—

121

C₃×C₉

—

122

C₃×C₉

—

123

C₃×C₉

—

124

C₃×C₉

—

125

C₃×C₉

—

126

C₁₃

#10624781

yes

characteristic, direct product factor, Sylow

127

C₃²

—

128

C₃²

—

129

C₃²

—

130

C₃²

—

131

C₃²

—

132

C₃²

—

133

C₃²

—

134

C₃²

—

135

C₃²

—

136

C₃²

—

137

C₃²

—

138

C₃²

—

139

C₃²

—

140

C₃²

—

141

C₃²

—

142

C₃²

C₁₃×AΣL(1,𝔽₂₇)

yes

—

143

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

144

C₃²

C₃×C₁₃×3₋¹⁺²

yes

—

145

C₉

—

146

C₉

—

147

C₉

—

148

C₉

—

149

C₃

—

150

C₃

—

151

C₃

—

152

C₃

—

153

C₃

C₁₃×#6891

yes

characteristic

154

C₃

C₃×C₁₃×AΣL(1,𝔽₂₇)

yes

characteristic

155

C₃

C₁₃×#6910

yes

characteristic

156

C₃

C₁₃×#6912

yes

characteristic

	subgroup	G/H
center, socle	C₃²×C₁₃	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₁₃
prime radical	C₃⁴×C₁₃	C₃²
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

prime radical

C₃⁴×C₁₃

C₃²

LCS

G ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃²×C₁₃ ◁ C₃⁴×C₁₃ ◁ G

N	H ≅ G/N	type
5. C₁₃×3₋¹⁺²⋊C₉	C₃	semidirect
25. #10624781	C₁₃	direct (N × H)
42. C₁₃×3₋¹⁺²	3₋¹⁺²	semidirect
53. 3₋¹⁺²⋊C₉	C₃×C₁₃	semidirect
119. 3₋¹⁺²	C₁₃×3₋¹⁺²	semidirect
126. C₁₃	#10624781	direct (N × H)

H ≅ G/N

type

5. C₁₃×3₋¹⁺²⋊C₉

C₃

semidirect

25. #10624781

C₁₃

direct (N × H)

42. C₁₃×3₋¹⁺²

3₋¹⁺²

semidirect

53. 3₋¹⁺²⋊C₉

C₃×C₁₃

semidirect

119. 3₋¹⁺²

C₁₃×3₋¹⁺²

semidirect

126. C₁₃

#10624781

direct (N × H)

Subgroups — C₁₃×#10624781 #23744932

Conjugacy classes of subgroups

Complements

Subgroups — C13×#10624781 #23744932

Conjugacy classes of subgroups

Complements

Subgroups — C₁₃×#10624781 #23744932