Group #15699179 — subgroups

Subgroups — C₁₃×#12840 #15699179

Order 3328.

354 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 154 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×#504	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#368	C₂	1	2	yes	maximal, complemented
3	C₁₃×#1813	C₂	1	2	yes	maximal
4	C₁₃×#1385	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
6	C₁₃×#504	—	2	4	no	—
7	C₂×C₁₃×M₃₂	C₄	1	2	yes	—
8	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃×C₁₆	C₄	1	2	yes	—
10	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#108	C₂²	1	4	yes	—
12	C₁₃×#413	—	2	2	no	—
13	C₁₃×#368	—	2	4	no	—
14	C₁₃×#338	—	2	2	no	—
15	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
16	C₁₃×#131	—	2	2	no	—
17	C₁₃×#131	—	2	2	no	—
18	C₂×C₁₃×M₁₆	—	2	4	no	—
19	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
20	C₂×C₁₃×M₁₆	Q₈	1	2	yes	—
21	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
23	C₁₃×M₃₂	—	2	2	no	—
24	C₁₃×M₃₂	—	2	4	no	—
25	C₂×C₁₃×C₁₆	—	2	4	no	—
26	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₃×C₁₆	D₄	1	4	yes	—
30	C₁₃×#108	—	2	4	no	—
31	C₁₃×#108	—	2	4	no	—
32	C₁₃×#101	—	4	2	no	—
33	C₁₃×#100	—	4	2	no	—
34	#12840	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₁₃×#41	—	2	2	no	—
36	C₁₃×#41	—	2	2	no	—
37	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
38	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
39	C₂×D₄×C₁₃	Q₁₆	1	2	yes	—
40	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
41	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
42	C₁₃×M₁₆	—	2	2	no	—
43	C₁₃×M₁₆	—	2	4	no	—
44	C₁₃×M₁₆	—	4	4	no	—
45	C₁₃×M₁₆	—	4	4	no	—
46	C₂×C₈×C₁₃	—	2	2	no	—
47	C₂×C₈×C₁₃	—	2	2	no	—
48	C₂×C₈×C₁₃	—	4	4	no	—
49	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
50	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
52	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₁₃	#41	1	2	yes	—
54	C₁₃×C₁₆	—	2	2	no	—
55	C₁₃×C₁₆	C₂²⋊C₄	1	2	yes	—
56	C₁₃×C₁₆	C₂²⋊C₄	1	2	yes	—
57	C₂×#504	C₂×C₁₃	1	1	yes	characteristic
58	C₂×#368	C₂×C₁₃	1	2	yes	complemented
59	#1813	C₂×C₁₃	1	2	yes	—
60	#1385	C₂×C₁₃	1	2	yes	complemented
61	C₂³×C₁₃	#107	1	2	yes	—
62	C₂³×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
63	Q₈×C₁₃	—	2	2	no	—
64	D₄×C₁₃	—	2	4	no	—
65	D₄×C₁₃	—	2	2	no	—
66	C₂×C₄×C₁₃	—	2	2	no	—
67	C₂×C₄×C₁₃	—	2	2	no	—
68	C₂×C₄×C₁₃	#107	1	2	yes	—
69	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₃	#121	1	2	yes	—
71	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
72	C₂×C₄×C₁₃	#135	1	2	yes	—
73	C₈×C₁₃	—	2	2	no	—
74	C₈×C₁₃	—	8	4	no	—
75	C₈×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₈×C₁₃	#117	1	1	yes	characteristic
77	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
78	C₂×#131	C₂²×C₁₃	1	1	yes	characteristic
79	#504	—	2	4	no	—
80	C₂×M₃₂	C₄×C₁₃	1	2	yes	—
81	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
82	C₂²×C₁₆	C₄×C₁₃	1	2	yes	—
83	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
84	C₂×#108	C₂²×C₁₃	1	4	yes	—
85	#413	—	2	2	no	—
86	#368	—	2	4	no	—
87	#338	—	2	2	no	—
88	C₂²×C₁₃	—	2	2	no	—
89	C₂²×C₁₃	—	4	1	no	—
90	C₂²×C₁₃	—	4	2	no	—
91	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
92	C₂²×C₁₃	#467	1	2	yes	—
93	C₄×C₁₃	—	2	2	no	—
94	C₄×C₁₃	#380	1	1	yes	characteristic
95	C₄×C₁₃	#427	1	1	yes	characteristic
96	C₄×C₁₃	#466	1	2	yes	—
97	C₂×#41	D₄×C₁₃	1	1	yes	characteristic
98	#131	—	2	2	no	—
99	#131	—	2	2	no	—
100	C₂×M₁₆	—	2	4	no	—
101	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
102	C₂×M₁₆	Q₈×C₁₃	1	2	yes	—
103	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
104	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
105	M₃₂	—	2	2	no	—
106	M₃₂	—	2	4	no	—
107	C₂×C₁₆	—	2	4	no	—
108	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	—
109	C₂×C₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
110	C₂×C₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
111	C₂×C₁₆	D₄×C₁₃	1	4	yes	—
112	#108	—	2	4	no	—
113	#108	—	2	4	no	—
114	#101	—	4	2	no	—
115	#100	—	4	2	no	—
116	C₂×C₁₃	—	2	1	no	—
117	C₂×C₁₃	—	2	2	no	—
118	C₂×C₁₃	—	4	2	no	—
119	C₂×C₁₃	#1808	1	1	yes	characteristic
120	#41	—	2	2	no	—
121	#41	—	2	2	no	—
122	C₂×Q₈	C₁₃×D₈	1	1	yes	characteristic
123	C₂×D₄	C₁₃×D₈	1	1	yes	characteristic
124	C₂×D₄	C₁₃×Q₁₆	1	2	yes	—
125	C₂²×C₄	C₁₃×C₄⋊C₄	1	2	yes	—
126	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
127	M₁₆	—	2	2	no	—
128	M₁₆	—	2	4	no	—
129	M₁₆	—	4	4	no	—
130	M₁₆	—	4	4	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	4	4	no	—
134	C₂×C₈	C₁₃×C₄⋊C₄	1	2	yes	—
135	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
136	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
137	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
138	C₂×C₈	C₁₃×#41	1	2	yes	—
139	C₁₆	—	2	2	no	—
140	C₁₆	C₁₃×C₂²⋊C₄	1	2	yes	—
141	C₁₆	C₁₃×C₂²⋊C₄	1	2	yes	—
142	C₁₃	#12840	1	1	yes	characteristic, direct product factor, Sylow
143	C₂³	C₁₃×#107	1	2	yes	—
144	C₂³	C₂×C₁₃×Q₁₆	1	1	yes	characteristic
145	Q₈	—	2	2	no	—
146	D₄	—	2	4	no	—
147	D₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	C₁₃×#107	1	2	yes	—
151	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₄	C₁₃×#121	1	2	yes	—
153	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
154	C₂×C₄	C₁₃×#135	1	2	yes	—
155	C₈	—	2	2	no	—
156	C₈	—	8	4	no	—
157	C₈	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
158	C₈	C₁₃×#117	1	1	yes	characteristic
159	C₈	C₁₃×C₂²⋊Q₈	1	2	yes	—
160	C₂²	—	2	2	no	—
161	C₂²	—	4	1	no	—
162	C₂²	—	4	2	no	—
163	C₂²	C₂×C₁₃×#107	1	1	yes	characteristic
164	C₂²	C₁₃×#467	1	2	yes	—
165	C₄	—	2	2	no	—
166	C₄	C₁₃×#380	1	1	yes	characteristic
167	C₄	C₁₃×#427	1	1	yes	characteristic
168	C₄	C₁₃×#466	1	2	yes	—
169	C₂	—	2	1	no	—
170	C₂	—	2	2	no	—
171	C₂	—	4	2	no	—
172	C₂	C₁₃×#1808	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×#504

C₂

yes

maximal, characteristic

C₂×C₁₃×#368

C₂

yes

maximal, complemented

C₁₃×#1813

C₂

yes

maximal

C₁₃×#1385

C₂

yes

maximal, complemented

C₂×C₁₃×#131

C₂²

yes

characteristic

C₁₃×#504

—

C₂×C₁₃×M₃₂

C₄

yes

—

C₂×C₁₃×M₃₂

C₂²

yes

characteristic

C₂²×C₁₃×C₁₆

C₄

yes

—

C₂²×C₁₃×C₁₆

C₂²

yes

characteristic

C₂×C₁₃×#108

C₂²

yes

—

C₁₃×#413

—

C₁₃×#368

—

C₁₃×#338

—

C₂×C₁₃×#41

D₄

yes

characteristic

C₁₃×#131

—

C₁₃×#131

—

C₂×C₁₃×M₁₆

—

C₂×C₁₃×M₁₆

D₄

yes

characteristic

C₂×C₁₃×M₁₆

Q₈

yes

—

C₂²×C₈×C₁₃

C₂×C₄

yes

—

C₂²×C₈×C₁₃

C₂³

yes

characteristic

C₁₃×M₃₂

—

C₁₃×M₃₂

—

C₂×C₁₃×C₁₆

—

C₂×C₁₃×C₁₆

C₂×C₄

yes

—

C₂×C₁₃×C₁₆

C₂×C₄

yes

characteristic

C₂×C₁₃×C₁₆

C₂×C₄

yes

characteristic

C₂×C₁₃×C₁₆

D₄

yes

—

C₁₃×#108

—

C₁₃×#108

—

C₁₃×#101

—

C₁₃×#100

—

#12840

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₁₃×#41

—

C₁₃×#41

—

C₂×Q₈×C₁₃

D₈

yes

characteristic

C₂×D₄×C₁₃

D₈

yes

characteristic

C₂×D₄×C₁₃

Q₁₆

yes

—

C₂²×C₄×C₁₃

C₄⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₂×C₈×C₁₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₃

#41

yes

—

C₁₃×C₁₆

—

C₁₃×C₁₆

C₂²⋊C₄

yes

—

C₁₃×C₁₆

C₂²⋊C₄

yes

—

C₂×#504

C₂×C₁₃

yes

characteristic

C₂×#368

C₂×C₁₃

yes

complemented

#1813

C₂×C₁₃

yes

—

#1385

C₂×C₁₃

yes

complemented

C₂³×C₁₃

#107

yes

—

C₂³×C₁₃

C₂×Q₁₆

yes

characteristic

Q₈×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#107

yes

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

#121

yes

—

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

#135

yes

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₁₃

#117

yes

characteristic

C₈×C₁₃

C₂²⋊Q₈

yes

—

C₂×#131

C₂²×C₁₃

yes

characteristic

#504

—

C₂×M₃₂

C₄×C₁₃

yes

—

C₂×M₃₂

C₂²×C₁₃

yes

characteristic

C₂²×C₁₆

C₄×C₁₃

yes

—

C₂²×C₁₆

C₂²×C₁₃

yes

characteristic

C₂×#108

C₂²×C₁₃

yes

—

#413

—

#368

—

#338

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×#107

yes

characteristic

C₂²×C₁₃

#467

yes

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

—

C₂×#41

D₄×C₁₃

yes

characteristic

#131

—

#131

—

100

C₂×M₁₆

—

101

C₂×M₁₆

D₄×C₁₃

yes

characteristic

102

C₂×M₁₆

Q₈×C₁₃

yes

—

103

C₂²×C₈

C₂×C₄×C₁₃

yes

—

104

C₂²×C₈

C₂³×C₁₃

yes

characteristic

105

M₃₂

—

106

M₃₂

—

107

C₂×C₁₆

—

108

C₂×C₁₆

C₂×C₄×C₁₃

yes

—

109

C₂×C₁₆

C₂×C₄×C₁₃

yes

characteristic

110

C₂×C₁₆

C₂×C₄×C₁₃

yes

characteristic

111

C₂×C₁₆

D₄×C₁₃

yes

—

112

#108

—

113

#108

—

114

#101

—

115

#100

—

116

C₂×C₁₃

—

117

C₂×C₁₃

—

118

C₂×C₁₃

—

119

C₂×C₁₃

#1808

yes

characteristic

120

#41

—

121

#41

—

122

C₂×Q₈

C₁₃×D₈

yes

characteristic

123

C₂×D₄

C₁₃×D₈

yes

characteristic

124

C₂×D₄

C₁₃×Q₁₆

yes

—

125

C₂²×C₄

C₁₃×C₄⋊C₄

yes

—

126

C₂²×C₄

C₂×D₄×C₁₃

yes

characteristic

127

M₁₆

—

128

M₁₆

—

129

M₁₆

—

130

M₁₆

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

C₁₃×C₄⋊C₄

yes

—

135

C₂×C₈

C₂²×C₄×C₁₃

yes

characteristic

136

C₂×C₈

C₂×D₄×C₁₃

yes

—

137

C₂×C₈

C₂×Q₈×C₁₃

yes

characteristic

138

C₂×C₈

C₁₃×#41

yes

—

139

C₁₆

—

140

C₁₆

C₁₃×C₂²⋊C₄

yes

—

141

C₁₆

C₁₃×C₂²⋊C₄

yes

—

142

C₁₃

#12840

yes

characteristic, direct product factor, Sylow

143

C₂³

C₁₃×#107

yes

—

144

C₂³

C₂×C₁₃×Q₁₆

yes

characteristic

145

Q₈

—

146

D₄

—

147

D₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

C₁₃×#107

yes

—

151

C₂×C₄

C₂×C₁₃×C₄⋊C₄

yes

characteristic

152

C₂×C₄

C₁₃×#121

yes

—

153

C₂×C₄

C₂×C₁₃×D₈

yes

characteristic

154

C₂×C₄

C₁₃×#135

yes

—

155

C₈

—

156

C₈

—

157

C₈

C₂×C₁₃×C₂²⋊C₄

yes

characteristic

158

C₈

C₁₃×#117

yes

characteristic

159

C₈

C₁₃×C₂²⋊Q₈

yes

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

C₂×C₁₃×#107

yes

characteristic

164

C₂²

C₁₃×#467

yes

—

165

C₄

—

166

C₄

C₁₃×#380

yes

characteristic

167

C₄

C₁₃×#427

yes

characteristic

168

C₄

C₁₃×#466

yes

—

169

C₂

—

170

C₂

—

171

C₂

—

172

C₂

C₁₃×#1808

yes

characteristic

	subgroup	G/H
center	C₄×C₁₃	#427
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂×C₁₃	#1808
prime radical	C₂×C₁₃×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃ ◁ C₂×C₁₃×M₁₆ ◁ C₂×C₁₃×#131 ◁ G

subgroup

G/H

center

C₄×C₁₃

#427

commutator

C₂×C₈

C₂²×C₄×C₁₃

Frattini, cosocle

C₂²×C₈

C₂³×C₁₃

socle

C₂×C₁₃

#1808

prime radical

C₂×C₁₃×#41

D₄

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄×C₁₃ ◁ C₂×C₁₃×M₁₆ ◁ C₂×C₁₃×#131 ◁ G

N	H ≅ G/N	type
2. C₂×C₁₃×#368	C₂	semidirect
4. C₁₃×#1385	C₂	semidirect
34. #12840	C₁₃	direct (N × H)
58. C₂×#368	C₂×C₁₃	semidirect
60. #1385	C₂×C₁₃	semidirect
142. C₁₃	#12840	direct (N × H)

H ≅ G/N

type

2. C₂×C₁₃×#368

C₂

semidirect

4. C₁₃×#1385

C₂

semidirect

34. #12840

C₁₃

direct (N × H)

58. C₂×#368

C₂×C₁₃

semidirect

60. #1385

C₂×C₁₃

semidirect

142. C₁₃

#12840

direct (N × H)

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

subgroup	index
C₂×C₁₃×#504	2
C₂×C₁₃×#368	2
C₁₃×#1813	2
C₁₃×#1385	2
#12840	13

subgroup

index

C₂×C₁₃×#504

C₂×C₁₃×#368

C₁₃×#1813

C₁₃×#1385

#12840

Subgroups — C₁₃×#12840 #15699179

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C13×#12840 #15699179

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#12840 #15699179