Group #15779258 — subgroups

Subgroups — #15779258

Order 3328.

366 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 119 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	#13123447	C₂	1	1	yes	maximal, characteristic
2	#13114882	C₂	1	1	yes	maximal, characteristic
3	#13112951	C₂	1	2	yes	maximal, complemented
4	#13112932	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2113	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
7	#11719524	—	2	1	no	—
8	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
10	#11718901	—	2	1	no	—
11	#11718786	—	2	1	no	—
12	#11718786	C₄	1	4	yes	complemented
13	C₂×#88059	—	2	2	no	—
14	C₂×#88059	C₂²	1	2	yes	—
15	#11718767	C₂²	1	2	yes	—
16	C₁₃×C₂²⋊C₁₆	—	2	2	no	—
17	#11718149	—	2	1	no	—
18	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
19	C₁₃×#131	—	2	2	no	—
20	C₁₃×#131	—	2	2	no	—
21	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
22	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₃	D₄	1	2	yes	—
24	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
25	#88202	—	4	4	no	—
26	#88180	—	2	2	no	—
27	C₂×#6099	—	2	4	no	—
28	C₂×#6099	—	2	2	no	—
29	C₁₃×M₃₂	—	2	2	no	—
30	C₂×C₁₃×C₁₆	—	2	1	no	—
31	C₂×C₁₃×C₁₆	—	2	1	no	—
32	C₂×C₁₃×C₁₆	—	2	2	no	—
33	#88074	—	4	2	no	—
34	#88059	—	2	4	no	—
35	#88059	—	2	2	no	—
36	#88059	C₂×C₄	1	4	yes	—
37	#88058	—	2	2	no	—
38	#88057	—	2	2	no	—
39	#12877	—	13	1	no	maximal, Sylow
40	C₁₃×#41	—	2	2	no	—
41	C₁₃×#41	—	2	2	no	—
42	C₂×Q₈×C₁₃	Q₁₆	1	1	yes	characteristic
43	C₂×D₄×C₁₃	SD₁₆	1	2	yes	—
44	C₂×D₄×C₁₃	Q₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
47	C₁₃×M₁₆	—	2	2	no	—
48	C₁₃×M₁₆	—	2	4	no	—
49	C₂×C₈×C₁₃	—	2	2	no	—
50	C₂×C₈×C₁₃	—	2	2	no	—
51	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
52	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₁₃	#41	1	2	yes	—
55	C₂×C₈×C₁₃	#41	1	2	yes	—
56	C₂²⋊Dic₁₃	—	4	2	no	—
57	C₄⋊Dic₁₃	—	4	2	no	—
58	C₄×Dic₁₃	—	4	2	no	—
59	#6099	—	2	4	no	—
60	#6099	—	4	4	no	—
61	#6099	—	4	2	no	—
62	C₂×#1055	—	4	4	no	—
63	C₂×#1055	—	4	2	no	—
64	C₁₃×C₁₆	—	4	1	no	—
65	C₁₃×C₁₆	—	4	1	no	—
66	#2916	—	13	1	no	—
67	#2113	D₁₃	1	1	yes	characteristic
68	#1812	—	13	1	no	—
69	#1389	—	13	2	no	—
70	#1382	—	13	2	no	—
71	C₂³×C₁₃	#103	1	2	yes	—
72	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
73	Q₈×C₁₃	—	2	2	no	—
74	D₄×C₁₃	—	2	4	no	—
75	D₄×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	#103	1	2	yes	—
79	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃	#123	1	2	yes	—
81	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
82	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
83	C₂×Dic₁₃	—	8	2	no	—
84	C₈×C₁₃	—	2	2	no	—
85	C₈×C₁₃	#117	1	2	yes	—
86	C₈×C₁₃	#123	1	1	yes	characteristic
87	C₈×C₁₃	#123	1	1	yes	characteristic
88	#1055	—	4	4	no	—
89	#568	—	26	1	no	—
90	C₂×#131	C₂×D₁₃	1	1	yes	characteristic
91	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
92	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
93	#430	—	13	4	no	—
94	#430	—	26	1	no	—
95	C₂×#108	—	13	2	no	—
96	C₂×#108	—	26	2	no	—
97	#427	—	13	2	no	—
98	#410	—	26	1	no	—
99	C₂²⋊C₁₆	—	2	2	no	—
100	H(3,ℤ/4)	—	26	1	no	—
101	C₂²×C₁₃	—	2	2	no	—
102	C₂²×C₁₃	—	4	1	no	—
103	C₂²×C₁₃	—	4	2	no	—
104	C₂²×C₁₃	C₂×#103	1	1	yes	characteristic
105	C₂²×C₁₃	#481	1	2	yes	—
106	C₄×C₁₃	—	2	2	no	—
107	C₄×C₁₃	#381	1	1	yes	characteristic
108	C₄×C₁₃	#418	1	1	yes	characteristic
109	C₄×C₁₃	#484	1	2	yes	—
110	Dic₁₃	—	16	2	no	—
111	C₂×#41	D₅₂	1	1	yes	characteristic
112	#131	—	2	2	no	—
113	#131	—	2	2	no	—
114	#131	—	52	4	no	—
115	C₂×M₁₆	—	26	4	no	—
116	C₂×M₁₆	—	26	2	no	—
117	C₂×M₁₆	C₄×D₁₃	1	2	yes	—
118	C₂×M₁₆	D₅₂	1	1	yes	characteristic
119	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
120	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
121	#117	—	26	2	no	—
122	M₃₂	—	2	2	no	—
123	C₂×C₁₆	—	2	2	no	—
124	C₂×C₁₆	—	2	1	no	—
125	C₂×C₁₆	—	2	1	no	—
126	#108	—	13	4	no	—
127	#108	—	26	4	no	—
128	#108	—	26	2	no	—
129	#107	—	26	2	no	—
130	#106	—	26	2	no	—
131	#99	—	52	2	no	—
132	C₂×C₁₃	—	2	2	no	—
133	C₂×C₁₃	—	2	1	no	—
134	C₂×C₁₃	—	4	2	no	—
135	C₂×C₁₃	#1824	1	1	yes	characteristic
136	#41	—	2	2	no	—
137	#41	—	2	2	no	—
138	C₂×Q₈	Dic₅₂	1	1	yes	characteristic
139	C₂×D₄	#6095	1	2	yes	—
140	C₂×D₄	Dic₅₂	1	1	yes	characteristic
141	C₂²×C₄	#6103	1	2	yes	—
142	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
143	M₁₆	—	2	2	no	—
144	M₁₆	—	2	4	no	—
145	M₁₆	—	26	4	no	—
146	M₁₆	—	52	4	no	—
147	M₁₆	—	52	2	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	52	4	no	—
151	C₂×C₈	—	52	2	no	—
152	C₂×C₈	#6103	1	2	yes	—
153	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
154	C₂×C₈	#6129	1	2	yes	—
155	C₂×C₈	#6129	1	2	yes	—
156	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
157	C₄⋊C₄	—	52	2	no	—
158	C₂²⋊C₄	—	52	2	no	—
159	C₄²	—	52	2	no	—
160	C₁₆	—	4	1	no	—
161	C₁₆	—	4	1	no	—
162	C₁₃	#12877	1	1	yes	characteristic, complemented, Sylow
163	C₂³	#88056	1	2	yes	—
164	C₂³	C₂×#6095	1	1	yes	characteristic
165	Q₈	—	2	2	no	—
166	D₄	—	2	4	no	—
167	D₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	104	2	no	—
171	C₂×C₄	#88056	1	2	yes	—
172	C₂×C₄	#88140	1	2	yes	—
173	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
174	C₂×C₄	#88162	1	2	yes	—
175	C₂×C₄	C₂×#6103	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	52	4	no	—
178	C₈	#88131	1	2	yes	—
179	C₈	#88189	1	1	yes	characteristic
180	C₈	#88189	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	—	4	1	no	—
183	C₂²	—	4	2	no	—
184	C₂²	#11718396	1	2	yes	—
185	C₂²	C₂×#88056	1	1	yes	characteristic
186	C₄	—	2	2	no	—
187	C₄	—	208	2	no	—
188	C₄	#11718383	1	2	yes	—
189	C₄	#11718604	1	1	yes	characteristic
190	C₄	#11718792	1	1	yes	characteristic
191	C₂	—	2	2	no	—
192	C₂	—	2	1	no	—
193	C₂	—	4	2	no	—
194	C₂	#13114927	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13123447

C₂

yes

maximal, characteristic

#13114882

C₂

yes

maximal, characteristic

#13112951

C₂

yes

maximal, complemented

#13112932

C₂

yes

maximal, complemented

C₁₃×#2113

C₂

yes

maximal, characteristic

C₂×C₁₃×#131

C₂²

yes

characteristic

#11719524

—

C₂×C₁₃×M₃₂

C₂²

yes

characteristic

C₂²×C₁₃×C₁₆

C₂²

yes

—

—

C₄

yes

complemented

C₂×#88059

—

C₂×#88059

C₂²

yes

—

#11718767

C₂²

yes

—

C₁₃×C₂²⋊C₁₆

—

#11718149

—

C₂×C₁₃×#41

D₄

yes

characteristic

C₁₃×#131

—

C₁₃×#131

—

C₂×C₁₃×M₁₆

C₂×C₄

yes

—

C₂×C₁₃×M₁₆

D₄

yes

characteristic

C₂²×C₈×C₁₃

D₄

yes

—

C₂²×C₈×C₁₃

C₂³

yes

—

—

—

—

C₁₃×M₃₂

—

C₂×C₁₃×C₁₆

—

C₂×C₁₃×C₁₆

—

C₂×C₁₃×C₁₆

—

—

—

—

C₂×C₄

yes

—

—

—

—

C₁₃×#41

—

C₁₃×#41

—

C₂×Q₈×C₁₃

Q₁₆

yes

characteristic

C₂×D₄×C₁₃

SD₁₆

yes

—

C₂×D₄×C₁₃

Q₁₆

yes

characteristic

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂²⋊C₄

yes

—

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

#41

yes

—

C₂×C₈×C₁₃

#41

yes

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

—

—

—

—

—

—

C₁₃×C₁₆

—

C₁₃×C₁₆

—

#2916

—

#2113

D₁₃

yes

—

—

—

C₂³×C₁₃

#103

yes

—

C₂³×C₁₃

C₂×SD₁₆

yes

characteristic

Q₈×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#103

yes

—

C₂×C₄×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

#123

yes

—

C₂×C₄×C₁₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₁₃

—

C₈×C₁₃

—

C₈×C₁₃

#117

yes

—

C₈×C₁₃

yes

C₈×C₁₃

yes

—

—

C₂×D₁₃

yes

characteristic

C₂×M₃₂

C₂×D₁₃

yes

characteristic

C₂²×C₁₆

C₂×D₁₃

yes

—

—

—

—

—

—

C₂²⋊C₁₆

—

100

H(3,ℤ/4)

—

101

C₂²×C₁₃

—

102

C₂²×C₁₃

—

103

C₂²×C₁₃

—

104

C₂²×C₁₃

C₂×#103

yes

characteristic

105

C₂²×C₁₃

#481

yes

—

106

C₄×C₁₃

—

107

C₄×C₁₃

#381

yes

characteristic

108

C₄×C₁₃

#418

yes

characteristic

109

C₄×C₁₃

#484

yes

—

110

Dic₁₃

—

111

C₂×#41

D₅₂

yes

characteristic

112

#131

—

113

#131

—

114

#131

—

115

C₂×M₁₆

—

116

C₂×M₁₆

—

117

C₂×M₁₆

C₄×D₁₃

yes

—

118

C₂×M₁₆

D₅₂

yes

characteristic

119

C₂²×C₈

C₂²⋊D₁₃

yes

—

120

C₂²×C₈

C₂²×D₁₃

yes

characteristic

121

#117

—

122

M₃₂

—

123

C₂×C₁₆

—

124

C₂×C₁₆

—

125

C₂×C₁₆

—

126

#108

—

127

#108

—

128

#108

—

129

#107

—

130

#106

—

131

#99

—

132

C₂×C₁₃

—

133

C₂×C₁₃

—

134

C₂×C₁₃

—

135

C₂×C₁₃

#1824

yes

characteristic

136

#41

—

137

#41

—

138

C₂×Q₈

Dic₅₂

yes

characteristic

139

C₂×D₄

#6095

yes

—

140

C₂×D₄

Dic₅₂

yes

characteristic

141

C₂²×C₄

#6103

yes

—

142

C₂²×C₄

C₂×D₅₂

yes

characteristic

143

M₁₆

—

144

M₁₆

—

145

M₁₆

—

146

M₁₆

—

147

M₁₆

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

#6103

yes

—

153

C₂×C₈

C₂×C₄×D₁₃

yes

characteristic

154

C₂×C₈

#6129

yes

—

155

C₂×C₈

#6129

yes

—

156

C₂×C₈

C₂×C₂²⋊D₁₃

yes

characteristic

157

C₄⋊C₄

—

158

C₂²⋊C₄

—

159

C₄²

—

160

C₁₆

—

161

C₁₆

—

162

C₁₃

#12877

yes

characteristic, complemented, Sylow

163

C₂³

#88056

yes

—

164

C₂³

C₂×#6095

yes

characteristic

165

Q₈

—

166

D₄

—

167

D₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

104

—

171

C₂×C₄

#88056

yes

—

172

C₂×C₄

#88140

yes

—

173

C₂×C₄

C₂×Dic₅₂

yes

characteristic

174

C₂×C₄

#88162

yes

—

175

C₂×C₄

C₂×#6103

yes

characteristic

176

C₈

—

177

C₈

—

178

C₈

#88131

yes

—

179

C₈

#88189

yes

characteristic

180

C₈

#88189

yes

characteristic

181

C₂²

—

182

C₂²

—

183

C₂²

—

184

C₂²

#11718396

yes

—

185

C₂²

C₂×#88056

yes

characteristic

186

C₄

—

187

C₄

—

208

—

188

C₄

#11718383

yes

—

189

C₄

#11718604

yes

characteristic

190

C₄

#11718792

yes

characteristic

191

C₂

—

192

C₂

—

193

C₂

—

194

C₂

#13114927

yes

characteristic

	subgroup	G/H
center	C₄	#11718792
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#2113	C₂
socle	C₂×C₁₃	#1824
prime radical	C₂×C₁₃×#41	D₄
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#2113	D₁₃
hypocenter	C₁₃	#12877
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #2113

subgroup

G/H

center

C₄

#11718792

commutator

C₂×C₈×C₁₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₁₃

Fitting

C₁₃×#2113

C₂

socle

C₂×C₁₃

#1824

prime radical

C₂×C₁₃×#41

D₄

cosocle

C₂²×C₈×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #2113

N	H ≅ G/N	type
3. #13112951	C₂	semidirect
4. #13112932	C₂	semidirect
12. #11718786	C₄	semidirect
162. C₁₃	#12877	semidirect

H ≅ G/N

type

3. #13112951

C₂

semidirect

4. #13112932

C₂

semidirect

12. #11718786

C₄

semidirect

162. C₁₃

#12877

semidirect

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

subgroup	index
#13123447	2
#13114882	2
#13112951	2
#13112932	2
C₁₃×#2113	2
#12877	13

subgroup

index

C₁₃×#2113

Subgroups — #15779258

Conjugacy classes of subgroups

Complements

Maximal subgroups