Group #16015523 — subgroups

Subgroups — #16015523

Order 3328.

822 conjugacy classes of subgroups; only the 267 normal subgroups are shown.

normal subgroups only

This group has 822 conjugacy classes of subgroups — too many to draw. Only its 267 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×#11718726	C₂	1	4	yes	maximal
2	#13123214	C₂	1	1	yes	maximal, characteristic
3	#13115002	C₂	1	4	yes	maximal
4	#13114998	C₂	1	4	yes	maximal
5	C₄×#88043	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#2859	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₃×#125	C₂²	1	2	yes	—
8	C₂×#88123	C₂²	1	2	yes	—
9	C₁₃×#389	—	2	4	no	—
10	C₁₃×#384	C₂²	1	2	yes	—
11	C₁₃×#382	C₂²	1	2	yes	—
12	C₄³×C₁₃	C₂²	1	1	yes	characteristic
13	#11718726	—	2	16	no	—
14	C₂×#88048	C₂²	1	8	yes	—
15	C₂×#88047	C₂²	1	8	yes	—
16	#11718597	C₂²	1	2	yes	—
17	#11718597	C₂²	1	2	yes	—
18	#11718590	—	2	4	no	—
19	C₂×#88043	C₂²	1	2	yes	—
20	C₂×#88043	C₂²	1	2	yes	—
21	C₂×C₄×#1055	C₂²	1	2	yes	—
22	C₂²×Dic₂₆	—	2	2	no	—
23	C₁₃×#125	—	2	8	no	—
24	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
25	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
26	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
27	C₂×C₄²×C₁₃	D₄	1	4	yes	—
28	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
31	C₂×C₄⋊Dic₁₃	—	2	2	no	—
32	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
33	C₂²×#1055	C₂³	1	4	yes	—
34	#88123	—	2	8	no	—
35	C₁₃×#95	—	2	4	no	—
36	#88071	—	2	4	no	—
37	#88071	—	2	4	no	—
38	#88048	—	2	16	no	—
39	#88047	—	2	16	no	—
40	#88043	D₄	1	8	yes	—
41	#88043	Q₈	1	8	yes	complemented
42	C₄×#1055	Q₈	1	8	yes	complemented
43	#33317	—	13	1	no	maximal, Sylow
44	C₂²×C₄×C₁₃	—	2	4	no	—
45	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
46	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
49	C₂²×Dic₁₃	—	2	4	no	—
50	C₂×Dic₂₆	—	4	8	no	—
51	C₂×Dic₂₆	—	4	4	no	—
52	C₁₃×C₄⋊C₄	—	2	4	no	—
53	C₁₃×C₄⋊C₄	—	2	4	no	—
54	C₁₃×C₄⋊C₄	—	4	8	no	—
55	C₁₃×C₄⋊C₄	—	4	8	no	—
56	C₄²×C₁₃	—	4	4	no	—
57	C₄²×C₁₃	C₂×D₄	1	4	yes	—
58	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
59	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
60	C₄⋊Dic₁₃	—	2	8	no	—
61	C₄⋊Dic₁₃	—	4	4	no	—
62	C₂×#1055	C₂×D₄	1	8	yes	—
63	C₂×#1055	C₂×Q₈	1	8	yes	—
64	C₂×#1055	C₂×Q₈	1	8	yes	—
65	C₂×#501	—	13	4	no	—
66	#2862	—	13	1	no	—
67	#2859	D₁₃	1	1	yes	characteristic
68	#1847	—	13	4	no	—
69	#1846	—	13	4	no	—
70	C₄×#97	—	13	1	no	—
71	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
72	C₂×C₄×C₁₃	—	2	4	no	—
73	C₂×C₄×C₁₃	—	2	4	no	—
74	C₂×C₄×C₁₃	—	2	4	no	—
75	C₂×C₄×C₁₃	—	4	4	no	—
76	C₂×C₄×C₁₃	—	4	8	no	—
77	C₂×C₄×C₁₃	#127	1	4	yes	—
78	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
79	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
80	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
81	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
82	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
83	C₂×Dic₁₃	—	4	4	no	—
84	C₂×Dic₁₃	—	4	8	no	—
85	Dic₂₆	—	8	8	no	—
86	#1055	C₄⋊Q₈	1	16	yes	—
87	C₂×C₄⋊Q₈	—	13	2	no	—
88	C₂×#125	C₂×D₁₃	1	2	yes	—
89	#501	—	26	16	no	—
90	C₂×#107	—	13	8	no	—
91	C₂×#106	—	13	8	no	—
92	C₂×#97	—	13	2	no	—
93	C₂×#97	—	13	2	no	—
94	C₂×C₄×C₈	—	13	2	no	—
95	#391	—	26	4	no	—
96	#389	—	2	4	no	—
97	#384	C₂×D₁₃	1	2	yes	—
98	#382	—	13	2	no	—
99	#382	—	13	2	no	—
100	#382	C₂×D₁₃	1	2	yes	—
101	C₄³	C₂×D₁₃	1	1	yes	characteristic
102	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
103	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
104	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
105	C₂²×C₁₃	#575	1	4	yes	—
106	C₄×C₁₃	—	2	4	no	—
107	C₄×C₁₃	—	2	4	no	—
108	C₄×C₁₃	—	2	4	no	—
109	C₄×C₁₃	—	8	4	no	—
110	C₄×C₁₃	C₂×C₄⋊Q₈	1	4	yes	—
111	Dic₁₃	—	8	4	no	—
112	C₂²×Q₈	—	26	2	no	—
113	C₂²×C₈	—	13	4	no	—
114	C₄⋊Q₈	—	26	8	no	—
115	#125	—	2	8	no	—
116	C₂×C₄⋊C₄	—	2	4	no	—
117	C₂×C₄⋊C₄	—	2	4	no	—
118	C₂×C₄⋊C₄	—	13	4	no	—
119	C₂×C₄⋊C₄	—	26	2	no	—
120	C₂×C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
121	C₂×C₄²	C₂²⋊D₁₃	1	4	yes	—
122	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
123	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
124	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
125	#107	—	26	16	no	—
126	#106	—	26	16	no	—
127	#97	—	13	8	no	—
128	#97	—	13	8	no	—
129	C₄×C₈	—	13	8	no	—
130	#95	—	2	4	no	—
131	#95	—	26	4	no	—
132	#95	—	26	4	no	—
133	C₂×C₁₃	#2868	1	1	yes	characteristic
134	C₂×C₁₃	#3153	1	2	yes	—
135	C₂×C₁₃	#3162	1	2	yes	—
136	C₂×C₁₃	#3166	1	2	yes	—
137	C₂×Q₈	—	52	8	no	—
138	C₂×Q₈	—	52	4	no	—
139	C₂²×C₄	—	2	4	no	—
140	C₂²×C₄	—	26	4	no	—
141	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
142	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
143	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
144	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
145	C₂×C₈	—	13	8	no	—
146	C₂×C₈	—	13	8	no	—
147	C₂×C₈	—	13	8	no	—
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	4	8	no	—
151	C₄⋊C₄	—	4	8	no	—
152	C₄⋊C₄	—	26	8	no	—
153	C₄⋊C₄	—	52	4	no	—
154	C₄²	—	4	4	no	—
155	C₄²	D₄×D₁₃	1	4	yes	—
156	C₄²	Q₈×D₁₃	1	4	yes	—
157	C₄²	Q₈×D₁₃	1	4	yes	—
158	C₁₃	#33317	1	1	yes	characteristic, complemented, Sylow
159	C₂³	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
160	Q₈	—	104	8	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	4	4	no	—
165	C₂×C₄	—	4	8	no	—
166	C₂×C₄	—	52	4	no	—
167	C₂×C₄	—	52	8	no	—
168	C₂×C₄	#88194	1	4	yes	—
169	C₂×C₄	#88198	1	4	yes	—
170	C₂×C₄	#88198	1	4	yes	—
171	C₂×C₄	C₂×D₄×D₁₃	1	2	yes	—
172	C₂×C₄	C₂×Q₈×D₁₃	1	2	yes	—
173	C₂×C₄	C₂×Q₈×D₁₃	1	2	yes	—
174	C₈	—	13	16	no	—
175	C₂²	C₂×#88194	1	1	yes	characteristic
176	C₂²	C₂×#88198	1	1	yes	characteristic
177	C₂²	C₂×#88198	1	1	yes	characteristic
178	C₂²	#11719527	1	4	yes	—
179	C₄	—	2	4	no	—
180	C₄	—	2	4	no	—
181	C₄	—	2	4	no	—
182	C₄	—	8	4	no	—
183	C₄	—	104	4	no	—
184	C₄	D₁₃×C₄⋊Q₈	1	4	yes	—
185	C₂	#13123232	1	1	yes	characteristic
186	C₂	#13125304	1	2	yes	—
187	C₂	#13125358	1	2	yes	—
188	C₂	#13125385	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11718726

C₂

yes

maximal

#13123214

C₂

yes

maximal, characteristic

#13115002

C₂

yes

maximal

#13114998

C₂

yes

maximal

C₄×#88043

C₂

yes

maximal, characteristic

C₁₃×#2859

C₂

yes

maximal, characteristic

C₂×C₁₃×#125

C₂²

yes

—

C₂×#88123

C₂²

yes

—

C₁₃×#389

—

C₁₃×#384

C₂²

yes

—

C₁₃×#382

C₂²

yes

—

C₄³×C₁₃

C₂²

yes

characteristic

#11718726

—

C₂×#88048

C₂²

yes

—

C₂×#88047

C₂²

yes

—

#11718597

C₂²

yes

—

#11718597

C₂²

yes

—

#11718590

—

C₂×#88043

C₂²

yes

—

C₂×#88043

C₂²

yes

—

C₂×C₄×#1055

C₂²

yes

—

C₂²×Dic₂₆

—

C₁₃×#125

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

C₂³

yes

—

C₂×C₄²×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

C₂³

yes

—

C₂²×#1055

C₂³

yes

—

#88123

—

C₁₃×#95

—

—

—

—

—

D₄

yes

—

#88043

Q₈

yes

complemented

C₄×#1055

Q₈

yes

complemented

#33317

—

maximal, Sylow

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×C₄×C₁₃

C₂⁴

yes

characteristic

C₂²×Dic₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

C₂×Q₈

yes

—

C₄²×C₁₃

C₂×Q₈

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₂×#1055

C₂×D₄

yes

—

C₂×#1055

C₂×Q₈

yes

—

C₂×#1055

C₂×Q₈

yes

—

C₂×#501

—

#2862

—

#2859

D₁₃

yes

characteristic

#1847

—

#1846

—

C₄×#97

—

C₂³×C₁₃

C₂²×D₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#127

yes

—

C₂×C₄×C₁₃

C₄⋊Q₈

yes

—

C₂×C₄×C₁₃

C₄⋊Q₈

yes

—

C₂×C₄×C₁₃

C₂²×D₄

yes

—

C₂×C₄×C₁₃

C₂²×Q₈

yes

—

C₂×C₄×C₁₃

C₂²×Q₈

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

#1055

C₄⋊Q₈

yes

—

C₂×C₄⋊Q₈

—

C₂×#125

C₂×D₁₃

yes

—

#501

—

C₂×#107

—

C₂×#106

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

#391

—

#389

—

#384

C₂×D₁₃

yes

—

#382

—

#382

—

100

#382

C₂×D₁₃

yes

—

101

C₄³

C₂×D₁₃

yes

characteristic

102

C₂²×C₁₃

C₂×#127

yes

characteristic

103

C₂²×C₁₃

C₂×C₄⋊Q₈

yes

characteristic

104

C₂²×C₁₃

C₂×C₄⋊Q₈

yes

characteristic

105

C₂²×C₁₃

#575

yes

—

106

C₄×C₁₃

—

107

C₄×C₁₃

—

108

C₄×C₁₃

—

109

C₄×C₁₃

—

110

C₄×C₁₃

C₂×C₄⋊Q₈

yes

—

111

Dic₁₃

—

112

C₂²×Q₈

—

113

C₂²×C₈

—

114

C₄⋊Q₈

—

115

#125

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

C₂²×D₁₃

yes

—

121

C₂×C₄²

C₂²⋊D₁₃

yes

—

122

C₂×C₄²

C₂²×D₁₃

yes

characteristic

123

C₂×C₄²

C₂²×D₁₃

yes

characteristic

124

C₂×C₄²

C₂²×D₁₃

yes

characteristic

125

#107

—

126

#106

—

127

#97

—

128

#97

—

129

C₄×C₈

—

130

#95

—

131

#95

—

132

#95

—

133

C₂×C₁₃

#2868

yes

characteristic

134

C₂×C₁₃

#3153

yes

—

135

C₂×C₁₃

#3162

yes

—

136

C₂×C₁₃

#3166

yes

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

—

142

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

—

143

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

—

144

C₂²×C₄

C₂³×D₁₃

yes

characteristic

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄²

—

155

C₄²

D₄×D₁₃

yes

—

156

C₄²

Q₈×D₁₃

yes

—

157

C₄²

Q₈×D₁₃

yes

—

158

C₁₃

#33317

yes

characteristic, complemented, Sylow

159

C₂³

C₂²×C₂²⋊D₁₃

yes

characteristic

160

Q₈

—

104

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

#88194

yes

—

169

C₂×C₄

#88198

yes

—

170

C₂×C₄

#88198

yes

—

171

C₂×C₄

C₂×D₄×D₁₃

yes

—

172

C₂×C₄

C₂×Q₈×D₁₃

yes

—

173

C₂×C₄

C₂×Q₈×D₁₃

yes

—

174

C₈

—

175

C₂²

C₂×#88194

yes

characteristic

176

C₂²

C₂×#88198

yes

characteristic

177

C₂²

C₂×#88198

yes

characteristic

178

C₂²

#11719527

yes

—

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

104

—

184

C₄

D₁₃×C₄⋊Q₈

yes

—

185

C₂

#13123232

yes

characteristic

186

C₂

#13125304

yes

—

187

C₂

#13125358

yes

—

188

C₂

#13125385

yes

—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₁₃
commutator, cosocle	C₂²×C₄×C₁₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₁₃×#2859	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
hypercenter	#2859	D₁₃
hypocenter	C₁₃	#33317
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

subgroup

G/H

center

C₂³

C₂²×C₂²⋊D₁₃

commutator, cosocle

C₂²×C₄×C₁₃

C₂⁴

Frattini

C₂²×C₄

C₂³×D₁₃

Fitting

C₁₃×#2859

C₂

socle, prime radical

C₂³×C₁₃

C₂²×D₄

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

N	H ≅ G/N	type
41. #88043	Q₈	semidirect
42. C₄×#1055	Q₈	semidirect
158. C₁₃	#33317	semidirect

H ≅ G/N

type

41. #88043

Q₈

semidirect

42. C₄×#1055

Q₈

semidirect

158. C₁₃

#33317

semidirect

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

subgroup	index
C₂×#11718726	2
#13123214	2
#13115002	2
#13114998	2
C₄×#88043	2
C₁₃×#2859	2
#33317	13

subgroup

index

C₁₃×#2859

Subgroups — #16015523

Conjugacy classes of subgroups

Complements

Maximal subgroups