Group #15699186 — subgroups

Subgroups — C₁₃×#12847 #15699186

Order 3328.

434 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 166 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	C₂×C₁₃×#405	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2112	C₂	1	2	yes	maximal
3	C₁₃×#1377	C₂	1	4	yes	maximal, complemented
4	C₂²×C₁₃×M₁₆	C₄	1	1	yes	characteristic
5	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
6	C₂×C₁₃×#117	C₄	1	1	yes	characteristic
7	C₂×C₁₃×M₃₂	C₂²	1	4	yes	—
8	C₁₃×#405	—	2	4	no	—
9	C₂×C₁₃×#97	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₃	C₄	1	2	yes	—
11	C₁₃×C₂²⋊C₁₆	—	2	4	no	—
12	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
13	C₂×C₁₃×M₁₆	—	2	2	no	—
14	C₂×C₁₃×M₁₆	—	2	4	no	—
15	C₂²×C₈×C₁₃	—	2	4	no	—
16	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₁₃	D₄	1	4	yes	—
19	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
20	C₁₃×#117	—	2	2	no	—
21	C₁₃×#117	—	2	2	no	—
22	C₂×C₁₃×C₄⋊C₄	C₈	1	2	yes	—
23	C₂×C₁₃×C₂²⋊C₄	C₈	1	2	yes	—
24	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
25	C₁₃×M₃₂	—	2	8	no	—
26	C₂×C₁₃×C₁₆	—	2	4	no	—
27	C₁₃×#97	—	2	4	no	—
28	C₄×C₈×C₁₃	—	2	4	no	—
29	#12847	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
31	C₂²×C₄×C₁₃	—	2	2	no	—
32	C₂²×C₄×C₁₃	—	2	2	no	—
33	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
34	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
36	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
37	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
38	C₁₃×M₁₆	—	2	8	no	—
39	C₂×C₈×C₁₃	—	2	2	no	—
40	C₂×C₈×C₁₃	—	2	2	no	—
41	C₂×C₈×C₁₃	—	2	2	no	—
42	C₂×C₈×C₁₃	—	2	8	no	—
43	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
44	C₂×C₈×C₁₃	M₁₆	1	4	yes	—
45	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
46	C₂×C₈×C₁₃	#41	1	4	yes	—
47	C₁₃×C₄⋊C₄	—	2	4	no	—
48	C₁₃×C₂²⋊C₄	—	2	4	no	—
49	C₄²×C₁₃	—	2	2	no	—
50	C₄²×C₁₃	—	2	2	no	—
51	C₁₃×C₁₆	—	4	4	no	—
52	C₂×#405	C₂×C₁₃	1	1	yes	characteristic
53	#2112	C₂×C₁₃	1	2	yes	—
54	#1377	C₂×C₁₃	1	4	yes	complemented
55	C₂³×C₁₃	—	2	2	no	—
56	C₂³×C₁₃	—	4	2	no	—
57	C₂³×C₁₃	C₂²⋊C₈	1	2	yes	—
58	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
59	C₂×C₄×C₁₃	—	2	2	no	—
60	C₂×C₄×C₁₃	—	2	2	no	—
61	C₂×C₄×C₁₃	—	2	2	no	—
62	C₂×C₄×C₁₃	—	2	4	no	—
63	C₂×C₄×C₁₃	—	2	4	no	—
64	C₂×C₄×C₁₃	—	4	4	no	—
65	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
66	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₃	#117	1	2	yes	—
68	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
69	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
70	C₈×C₁₃	—	2	2	no	—
71	C₈×C₁₃	—	2	4	no	—
72	C₈×C₁₃	#123	1	4	yes	—
73	C₂²×M₁₆	C₄×C₁₃	1	1	yes	characteristic
74	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
75	C₂×#117	C₄×C₁₃	1	1	yes	characteristic
76	C₂×M₃₂	C₂²×C₁₃	1	4	yes	—
77	#405	—	2	4	no	—
78	C₂×#97	C₂²×C₁₃	1	2	yes	—
79	C₂×C₄×C₈	C₄×C₁₃	1	2	yes	—
80	C₂²⋊C₁₆	—	2	4	no	—
81	C₂²×C₁₃	—	2	2	no	—
82	C₂²×C₁₃	—	2	2	no	—
83	C₂²×C₁₃	—	2	4	no	—
84	C₂²×C₁₃	—	4	4	no	—
85	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
86	C₂²×C₁₃	#431	1	2	yes	—
87	C₄×C₁₃	—	2	2	no	—
88	C₄×C₁₃	—	4	4	no	—
89	C₄×C₁₃	#381	1	1	yes	characteristic
90	C₄×C₁₃	#407	1	1	yes	characteristic
91	C₄×C₁₃	#432	1	2	yes	—
92	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
93	C₂×M₁₆	—	2	2	no	—
94	C₂×M₁₆	—	2	4	no	—
95	C₂²×C₈	—	2	4	no	—
96	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
97	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
98	C₂²×C₈	D₄×C₁₃	1	4	yes	—
99	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
100	#117	—	2	2	no	—
101	#117	—	2	2	no	—
102	C₂×C₄⋊C₄	C₈×C₁₃	1	2	yes	—
103	C₂×C₂²⋊C₄	C₈×C₁₃	1	2	yes	—
104	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
105	M₃₂	—	2	8	no	—
106	C₂×C₁₆	—	2	4	no	—
107	#97	—	2	4	no	—
108	C₄×C₈	—	2	4	no	—
109	C₂×C₁₃	—	2	2	no	—
110	C₂×C₁₃	—	4	2	no	—
111	C₂×C₁₃	#1817	1	1	yes	characteristic
112	C₂×C₁₃	#2115	1	2	yes	—
113	C₂⁴	C₂×C₈×C₁₃	1	1	yes	characteristic
114	C₂²×C₄	—	2	2	no	—
115	C₂²×C₄	—	2	2	no	—
116	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
117	C₂²×C₄	C₂×C₈×C₁₃	1	1	yes	characteristic
118	C₂²×C₄	C₂×C₈×C₁₃	1	4	yes	—
119	C₂²×C₄	C₁₃×M₁₆	1	2	yes	—
120	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
121	M₁₆	—	2	8	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	8	no	—
126	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
127	C₂×C₈	C₁₃×M₁₆	1	4	yes	—
128	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
129	C₂×C₈	C₁₃×#41	1	4	yes	—
130	C₄⋊C₄	—	2	4	no	—
131	C₂²⋊C₄	—	2	4	no	—
132	C₄²	—	2	2	no	—
133	C₄²	—	2	2	no	—
134	C₁₆	—	4	4	no	—
135	C₁₃	#12847	1	1	yes	characteristic, direct product factor, Sylow
136	C₂³	—	2	2	no	—
137	C₂³	—	4	2	no	—
138	C₂³	C₁₃×C₂²⋊C₈	1	2	yes	—
139	C₂³	C₂²×C₈×C₁₃	1	1	yes	characteristic
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	4	4	no	—
146	C₂×C₄	C₁₃×C₂²⋊C₈	1	2	yes	—
147	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
148	C₂×C₄	C₁₃×#117	1	2	yes	—
149	C₂×C₄	C₂×C₁₃×M₁₆	1	2	yes	—
150	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
151	C₈	—	2	2	no	—
152	C₈	—	2	4	no	—
153	C₈	C₁₃×#123	1	4	yes	—
154	C₂²	—	2	2	no	—
155	C₂²	—	2	2	no	—
156	C₂²	—	2	4	no	—
157	C₂²	—	4	4	no	—
158	C₂²	C₂×C₁₃×C₂²⋊C₈	1	1	yes	characteristic
159	C₂²	C₁₃×#431	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	4	4	no	—
162	C₄	C₁₃×#381	1	1	yes	characteristic
163	C₄	C₁₃×#407	1	1	yes	characteristic
164	C₄	C₁₃×#432	1	2	yes	—
165	C₂	—	2	2	no	—
166	C₂	—	4	2	no	—
167	C₂	C₁₃×#1817	1	1	yes	characteristic
168	C₂	C₁₃×#2115	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×#405

C₂

yes

maximal, characteristic

C₁₃×#2112

C₂

yes

maximal

C₁₃×#1377

C₂

yes

maximal, complemented

C₂²×C₁₃×M₁₆

C₄

yes

characteristic

C₂³×C₈×C₁₃

C₂²

yes

characteristic

C₂×C₁₃×#117

C₄

yes

characteristic

C₂×C₁₃×M₃₂

C₂²

yes

—

C₁₃×#405

—

C₂×C₁₃×#97

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₄

yes

—

C₁₃×C₂²⋊C₁₆

—

C₂³×C₄×C₁₃

C₂×C₄

yes

characteristic

C₂×C₁₃×M₁₆

—

C₂×C₁₃×M₁₆

—

C₂²×C₈×C₁₃

—

C₂²×C₈×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₃

C₂×C₄

yes

—

C₂²×C₈×C₁₃

D₄

yes

—

C₂²×C₈×C₁₃

C₂³

yes

characteristic

C₁₃×#117

—

C₁₃×#117

—

C₂×C₁₃×C₄⋊C₄

C₈

yes

—

C₂×C₁₃×C₂²⋊C₄

C₈

yes

—

C₂×C₄²×C₁₃

C₂×C₄

yes

—

C₁₃×M₃₂

—

C₂×C₁₃×C₁₆

—

C₁₃×#97

—

C₄×C₈×C₁₃

—

#12847

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₂⁴×C₁₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₃

C₂×C₈

yes

—

C₂²×C₄×C₁₃

M₁₆

yes

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂²⋊C₄

yes

—

C₂×C₈×C₁₃

M₁₆

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₂×C₈×C₁₃

#41

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₁₃×C₁₆

—

C₂×#405

C₂×C₁₃

yes

characteristic

#2112

C₂×C₁₃

yes

—

#1377

C₂×C₁₃

yes

complemented

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂²⋊C₈

yes

—

C₂³×C₁₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₈

yes

—

C₂×C₄×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

#117

yes

—

C₂×C₄×C₁₃

C₂×M₁₆

yes

—

C₂×C₄×C₁₃

C₂×M₁₆

yes

characteristic

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

#123

yes

—

C₂²×M₁₆

C₄×C₁₃

yes

characteristic

C₂³×C₈

C₂²×C₁₃

yes

characteristic

C₂×#117

C₄×C₁₃

yes

characteristic

C₂×M₃₂

C₂²×C₁₃

yes

—

#405

—

C₂×#97

C₂²×C₁₃

yes

—

C₂×C₄×C₈

C₄×C₁₃

yes

—

C₂²⋊C₁₆

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₁₃

#431

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

—

C₂³×C₄

C₂×C₄×C₁₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄×C₁₃

yes

characteristic

C₂²×C₈

C₂×C₄×C₁₃

yes

—

C₂²×C₈

D₄×C₁₃

yes

—

C₂²×C₈

C₂³×C₁₃

yes

characteristic

100

#117

—

101

#117

—

102

C₂×C₄⋊C₄

C₈×C₁₃

yes

—

103

C₂×C₂²⋊C₄

C₈×C₁₃

yes

—

104

C₂×C₄²

C₂×C₄×C₁₃

yes

—

105

M₃₂

—

106

C₂×C₁₆

—

107

#97

—

108

C₄×C₈

—

109

C₂×C₁₃

—

110

C₂×C₁₃

—

111

C₂×C₁₃

#1817

yes

characteristic

112

C₂×C₁₃

#2115

yes

—

113

C₂⁴

C₂×C₈×C₁₃

yes

characteristic

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

C₁₃×C₂²⋊C₄

yes

—

117

C₂²×C₄

C₂×C₈×C₁₃

yes

characteristic

118

C₂²×C₄

C₂×C₈×C₁₃

yes

—

119

C₂²×C₄

C₁₃×M₁₆

yes

—

120

C₂²×C₄

C₂²×C₄×C₁₃

yes

characteristic

121

M₁₆

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

C₁₃×C₂²⋊C₄

yes

—

127

C₂×C₈

C₁₃×M₁₆

yes

—

128

C₂×C₈

C₂×D₄×C₁₃

yes

—

129

C₂×C₈

C₁₃×#41

yes

—

130

C₄⋊C₄

—

131

C₂²⋊C₄

—

132

C₄²

—

133

C₄²

—

134

C₁₆

—

135

C₁₃

#12847

yes

characteristic, direct product factor, Sylow

136

C₂³

—

137

C₂³

—

138

C₂³

C₁₃×C₂²⋊C₈

yes

—

139

C₂³

C₂²×C₈×C₁₃

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

C₁₃×C₂²⋊C₈

yes

—

147

C₂×C₄

C₂×C₁₃×C₂²⋊C₄

yes

characteristic

148

C₂×C₄

C₁₃×#117

yes

—

149

C₂×C₄

C₂×C₁₃×M₁₆

yes

—

150

C₂×C₄

C₂×C₁₃×M₁₆

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

C₁₃×#123

yes

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

C₂×C₁₃×C₂²⋊C₈

yes

characteristic

159

C₂²

C₁₃×#431

yes

—

160

C₄

—

161

C₄

—

162

C₄

C₁₃×#381

yes

characteristic

163

C₄

C₁₃×#407

yes

characteristic

164

C₄

C₁₃×#432

yes

—

165

C₂

—

166

C₂

—

167

C₂

C₁₃×#1817

yes

characteristic

168

C₂

C₁₃×#2115

yes

—

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₈×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂²×C₁₃	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₁₃	C₂×C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂³×C₈×C₁₃ ◁ G

subgroup

G/H

center

C₂×C₄×C₁₃

C₂×C₂²⋊C₄

commutator

C₂³

C₂²×C₈×C₁₃

Frattini, cosocle

C₂²×C₈

C₂³×C₁₃

socle

C₂²×C₁₃

C₂×C₂²⋊C₈

prime radical

C₂⁴×C₁₃

C₂×C₈

LCS

G ▷ C₂³ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₁₃ ◁ C₂³×C₈×C₁₃ ◁ G

N	H ≅ G/N	type
3. C₁₃×#1377	C₂	semidirect
29. #12847	C₁₃	direct (N × H)
54. #1377	C₂×C₁₃	semidirect
135. C₁₃	#12847	direct (N × H)

H ≅ G/N

type

3. C₁₃×#1377

C₂

semidirect

29. #12847

C₁₃

direct (N × H)

54. #1377

C₂×C₁₃

semidirect

135. C₁₃

#12847

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₁₃×#405	2
C₁₃×#2112	2
C₁₃×#1377	2
#12847	13

subgroup

index

C₂×C₁₃×#405

C₁₃×#2112

C₁₃×#1377

#12847

Subgroups — C₁₃×#12847 #15699186

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C13×#12847 #15699186

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#12847 #15699186