Group #15761716 — subgroups

Subgroups — #15761716

Order 3328.

546 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 179 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	C₄×#88123	C₂	1	1	yes	maximal, characteristic
2	#13114582	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718076	C₂	1	4	yes	maximal
4	C₁₃×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×#88123	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₂₆	C₂²	1	2	yes	—
7	C₂×C₁₃×C₄⋊C₈	—	2	2	no	—
8	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
9	C₄³×C₁₃	C₂²	1	1	yes	characteristic
10	#11718596	C₄	1	1	yes	characteristic
11	C₄×C₄⋊Dic₁₃	—	2	2	no	—
12	#11718590	C₄	1	2	yes	—
13	#11718589	—	2	2	no	—
14	C₂×C₄×#1055	C₂²	1	2	yes	—
15	#11718116	—	2	2	no	—
16	#11718076	—	2	8	no	—
17	C₂²×Dic₂₆	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₁₃	—	2	2	no	—
19	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	D₄	1	2	yes	—
22	C₂×C₄²×C₁₃	D₄	1	2	yes	—
23	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₃	—	2	2	no	—
25	C₂×C₄⋊Dic₁₃	—	2	2	no	—
26	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
27	C₂×#6101	—	2	2	no	—
28	C₂×C₄×Dic₁₃	—	2	2	no	—
29	C₂²×#1055	—	2	2	no	—
30	#88123	—	2	4	no	—
31	C₄×Dic₂₆	—	2	4	no	—
32	C₄×Dic₂₆	—	2	4	no	—
33	C₁₃×C₄⋊C₈	—	2	4	no	—
34	C₁₃×C₄⋊C₈	—	4	4	no	—
35	#88071	—	2	4	no	—
36	C₄×#1055	—	2	2	no	—
37	C₄×#1055	D₄	1	4	yes	—
38	#9643	—	13	1	no	maximal, Sylow
39	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₁₃	—	2	2	no	—
45	C₂²×Dic₁₃	—	2	2	no	—
46	C₂×Dic₂₆	—	2	4	no	—
47	C₂×Dic₂₆	—	2	4	no	—
48	C₂×Dic₂₆	—	2	2	no	—
49	C₂×Dic₂₆	M₁₆	1	4	yes	—
50	C₂×C₈×C₁₃	—	4	2	no	—
51	C₂×C₈×C₁₃	—	4	4	no	—
52	C₄²×C₁₃	—	2	2	no	—
53	C₄²×C₁₃	—	2	2	no	—
54	C₄²×C₁₃	—	4	2	no	—
55	C₄²×C₁₃	SD₁₆	1	4	yes	—
56	C₄²×C₁₃	Q₁₆	1	4	yes	—
57	C₄²×C₁₃	C₂×D₄	1	2	yes	—
58	C₄²×C₁₃	#41	1	2	yes	—
59	C₄⋊Dic₁₃	—	2	8	no	—
60	C₄⋊Dic₁₃	—	2	4	no	—
61	C₄⋊Dic₁₃	—	4	4	no	—
62	#6101	—	4	4	no	—
63	C₄×Dic₁₃	—	4	2	no	—
64	C₄×Dic₁₃	—	4	2	no	—
65	C₂×#1055	—	2	4	no	—
66	C₂×#1055	—	4	4	no	—
67	C₄×C₄⋊Q₈	—	13	1	no	—
68	#1842	D₁₃	1	1	yes	characteristic
69	#1744	—	13	1	no	—
70	C₂×Q₈⋊C₈	—	13	4	no	—
71	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₃	—	2	2	no	—
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	—	2	2	no	—
75	C₂×C₄×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	#103	1	4	yes	—
78	C₂×C₄×C₁₃	#103	1	4	yes	—
79	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
80	C₂×C₄×C₁₃	#121	1	2	yes	—
81	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
82	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
84	C₂×C₄×C₁₃	#131	1	2	yes	—
85	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
86	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
87	C₂×Dic₁₃	—	2	4	no	—
88	C₂×Dic₁₃	—	4	2	no	—
89	C₂×Dic₁₃	—	4	2	no	—
90	C₂×Dic₁₃	—	4	2	no	—
91	C₂×Dic₁₃	—	4	4	no	—
92	Dic₂₆	—	2	8	no	—
93	Dic₂₆	—	4	4	no	—
94	C₈×C₁₃	—	8	2	no	—
95	#1055	—	4	4	no	—
96	C₂×C₄⋊Q₈	—	13	1	no	—
97	C₂×C₄×Q₈	—	13	2	no	—
98	C₂×C₄⋊C₈	—	2	2	no	—
99	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
100	C₂×C₄×C₈	—	13	2	no	—
101	#391	—	13	2	no	—
102	#389	—	26	2	no	—
103	#384	—	13	1	no	—
104	C₄×C₄⋊C₄	—	26	2	no	—
105	C₄³	C₂×D₁₃	1	1	yes	characteristic
106	#336	—	26	2	no	—
107	Q₈⋊C₈	—	26	8	no	—
108	C₂²×C₁₃	#390	1	1	yes	characteristic
109	C₂²×C₁₃	#407	1	1	yes	characteristic
110	C₂²×C₁₃	#408	1	1	yes	characteristic
111	C₂²×C₁₃	C₂×#103	1	2	yes	—
112	C₂²×C₁₃	#421	1	2	yes	—
113	C₄×C₁₃	—	2	2	no	—
114	C₄×C₁₃	—	2	4	no	—
115	C₄×C₁₃	—	2	4	no	—
116	C₄×C₁₃	#435	1	4	yes	—
117	C₄×C₁₃	#487	1	2	yes	—
118	C₄×C₁₃	#492	1	1	yes	characteristic
119	C₄×C₁₃	#494	1	1	yes	characteristic
120	Dic₁₃	—	4	4	no	—
121	Dic₁₃	—	8	2	no	—
122	C₂²×Q₈	—	13	2	no	—
123	C₂²×C₈	—	2	2	no	—
124	C₂²×C₈	—	26	2	no	—
125	C₄⋊Q₈	—	26	4	no	—
126	C₄×Q₈	—	26	4	no	—
127	C₄×Q₈	—	26	4	no	—
128	C₂×C₄⋊C₄	—	13	2	no	—
129	C₂×C₄⋊C₄	—	26	2	no	—
130	C₂×C₄⋊C₄	—	26	2	no	—
131	C₂×C₄⋊C₄	—	26	2	no	—
132	C₂×C₄²	—	26	2	no	—
133	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
134	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
135	C₂×C₄²	D₅₂	1	2	yes	—
136	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
137	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
138	C₄⋊C₈	—	2	4	no	—
139	C₄⋊C₈	—	4	4	no	—
140	C₄×C₈	—	13	4	no	—
141	C₄×C₈	—	26	2	no	—
142	#95	—	26	4	no	—
143	C₂×C₁₃	#1481	1	2	yes	—
144	C₂×C₁₃	#1487	1	2	yes	—
145	C₂×C₁₃	#1955	1	1	yes	characteristic
146	C₂×C₁₃	#1956	1	1	yes	characteristic
147	C₂×C₁₃	#1963	1	1	yes	characteristic
148	C₂×Q₈	—	13	4	no	—
149	C₂×Q₈	—	26	2	no	—
150	C₂×Q₈	—	26	4	no	—
151	C₂×Q₈	—	26	4	no	—
152	C₂²×C₄	—	26	2	no	—
153	C₂²×C₄	—	26	2	no	—
154	C₂²×C₄	#6103	1	2	yes	—
155	C₂²×C₄	#6103	1	2	yes	—
156	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
157	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
159	C₂×C₈	—	4	4	no	—
160	C₂×C₈	—	4	2	no	—
161	C₂×C₈	—	26	4	no	—
162	C₂×C₈	—	52	4	no	—
163	C₄⋊C₄	—	26	8	no	—
164	C₄⋊C₄	—	26	4	no	—
165	C₄⋊C₄	—	52	4	no	—
166	C₄⋊C₄	—	52	4	no	—
167	C₄²	—	2	2	no	—
168	C₄²	—	2	2	no	—
169	C₄²	—	4	2	no	—
170	C₄²	—	52	2	no	—
171	C₄²	—	52	2	no	—
172	C₄²	#6105	1	4	yes	—
173	C₄²	#6107	1	4	yes	—
174	C₄²	D₄×D₁₃	1	2	yes	—
175	C₄²	#6131	1	2	yes	—
176	C₁₃	#9643	1	1	yes	characteristic, complemented, Sylow
177	C₂³	C₂×#6103	1	1	yes	characteristic
178	Q₈	—	26	8	no	—
179	Q₈	—	52	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	26	4	no	—
186	C₂×C₄	—	52	2	no	—
187	C₂×C₄	—	52	2	no	—
188	C₂×C₄	—	52	4	no	—
189	C₂×C₄	—	52	2	no	—
190	C₂×C₄	#88050	1	4	yes	—
191	C₂×C₄	#88050	1	4	yes	—
192	C₂×C₄	#88147	1	2	yes	—
193	C₂×C₄	#88149	1	2	yes	—
194	C₂×C₄	#88155	1	2	yes	—
195	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
196	C₂×C₄	C₂×#6105	1	2	yes	—
197	C₂×C₄	#88194	1	1	yes	characteristic
198	C₂×C₄	C₂×#6107	1	2	yes	—
199	C₂×C₄	#88201	1	1	yes	characteristic
200	C₈	—	8	2	no	—
201	C₈	—	52	4	no	—
202	C₂²	C₂×#88050	1	2	yes	—
203	C₂²	#11718649	1	1	yes	characteristic
204	C₂²	#11718771	1	1	yes	characteristic
205	C₂²	#11718787	1	1	yes	characteristic
206	C₂²	#11718800	1	2	yes	—
207	C₄	—	2	2	no	—
208	C₄	—	2	4	no	—
209	C₄	—	2	4	no	—
210	C₄	—	52	4	no	—
211	C₄	—	104	2	no	—
212	C₄	#11718497	1	4	yes	—
213	C₄	#11718737	1	1	yes	characteristic
214	C₄	#11718753	1	1	yes	characteristic
215	C₄	#11718754	1	2	yes	—
216	C₂	#13113238	1	2	yes	—
217	C₂	#13113264	1	2	yes	—
218	C₂	#13115633	1	1	yes	characteristic
219	C₂	#13115637	1	1	yes	characteristic
220	C₂	#13115667	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#88123

C₂

yes

maximal, characteristic

#13114582

C₂

yes

maximal, characteristic

C₂×#11718076

C₂

yes

maximal

C₁₃×#1842

C₂

yes

maximal, characteristic

C₂×#88123

C₄

yes

characteristic

C₂×C₄×Dic₂₆

C₂²

yes

—

C₂×C₁₃×C₄⋊C₈

—

C₂×C₁₃×C₄⋊C₈

C₂²

yes

—

C₄³×C₁₃

C₂²

yes

characteristic

#11718596

C₄

yes

characteristic

C₄×C₄⋊Dic₁₃

—

#11718590

C₄

yes

—

#11718589

—

C₂×C₄×#1055

C₂²

yes

—

#11718116

—

#11718076

—

C₂²×Dic₂₆

C₂×C₄

yes

—

C₂²×C₈×C₁₃

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

C₂×C₄

yes

—

C₂×#6101

—

C₂×C₄×Dic₁₃

—

C₂²×#1055

—

#88123

—

C₄×Dic₂₆

—

C₄×Dic₂₆

—

C₁₃×C₄⋊C₈

—

C₁₃×C₄⋊C₈

—

#88071

—

C₄×#1055

—

C₄×#1055

D₄

yes

—

#9643

—

maximal, Sylow

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

M₁₆

yes

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

SD₁₆

yes

—

C₄²×C₁₃

Q₁₆

yes

—

C₄²×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

#41

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₂×#1055

—

C₂×#1055

—

C₄×C₄⋊Q₈

—

#1842

D₁₃

yes

characteristic

#1744

—

C₂×Q₈⋊C₈

—

C₂³×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#103

yes

—

C₂×C₄×C₁₃

#103

yes

—

C₂×C₄×C₁₃

C₄×D₄

yes

—

C₂×C₄×C₁₃

#121

yes

—

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

C₂×M₁₆

yes

—

C₂×C₄×C₁₃

#131

yes

—

C₂×C₄×C₁₃

C₂×SD₁₆

yes

—

C₂×C₄×C₁₃

C₂×Q₁₆

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

#1055

—

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×D₁₃

yes

—

100

C₂×C₄×C₈

—

101

#391

—

102

#389

—

103

#384

—

104

C₄×C₄⋊C₄

—

105

C₄³

C₂×D₁₃

yes

characteristic

106

#336

—

107

Q₈⋊C₈

—

108

C₂²×C₁₃

#390

yes

characteristic

109

C₂²×C₁₃

#407

yes

characteristic

110

C₂²×C₁₃

#408

yes

characteristic

111

C₂²×C₁₃

C₂×#103

yes

—

112

C₂²×C₁₃

#421

yes

—

113

C₄×C₁₃

—

114

C₄×C₁₃

—

115

C₄×C₁₃

—

116

C₄×C₁₃

#435

yes

—

117

C₄×C₁₃

#487

yes

—

118

C₄×C₁₃

#492

yes

characteristic

119

C₄×C₁₃

#494

yes

characteristic

120

Dic₁₃

—

121

Dic₁₃

—

122

C₂²×Q₈

—

123

C₂²×C₈

—

124

C₂²×C₈

—

125

C₄⋊Q₈

—

126

C₄×Q₈

—

127

C₄×Q₈

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

131

C₂×C₄⋊C₄

—

132

C₂×C₄²

—

133

C₂×C₄²

C₄×D₁₃

yes

characteristic

134

C₂×C₄²

C₄×D₁₃

yes

characteristic

135

C₂×C₄²

D₅₂

yes

—

136

C₂×C₄²

C₂²⋊D₁₃

yes

—

137

C₂×C₄²

C₂²×D₁₃

yes

characteristic

138

C₄⋊C₈

—

139

C₄⋊C₈

—

140

C₄×C₈

—

141

C₄×C₈

—

142

#95

—

143

C₂×C₁₃

#1481

yes

—

144

C₂×C₁₃

#1487

yes

—

145

C₂×C₁₃

#1955

yes

characteristic

146

C₂×C₁₃

#1956

yes

characteristic

147

C₂×C₁₃

#1963

yes

characteristic

148

C₂×Q₈

—

149

C₂×Q₈

—

150

C₂×Q₈

—

151

C₂×Q₈

—

152

C₂²×C₄

—

153

C₂²×C₄

—

154

C₂²×C₄

#6103

yes

—

155

C₂²×C₄

#6103

yes

—

156

C₂²×C₄

C₂×C₄×D₁₃

yes

characteristic

157

C₂²×C₄

C₂×D₅₂

yes

characteristic

158

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄²

—

168

C₄²

—

169

C₄²

—

170

C₄²

—

171

C₄²

—

172

C₄²

#6105

yes

—

173

C₄²

#6107

yes

—

174

C₄²

D₄×D₁₃

yes

—

175

C₄²

#6131

yes

—

176

C₁₃

#9643

yes

characteristic, complemented, Sylow

177

C₂³

C₂×#6103

yes

characteristic

178

Q₈

—

179

Q₈

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

#88050

yes

—

191

C₂×C₄

#88050

yes

—

192

C₂×C₄

#88147

yes

—

193

C₂×C₄

#88149

yes

—

194

C₂×C₄

#88155

yes

—

195

C₂×C₄

M₁₆×D₁₃

yes

—

196

C₂×C₄

C₂×#6105

yes

—

197

C₂×C₄

#88194

yes

characteristic

198

C₂×C₄

C₂×#6107

yes

—

199

C₂×C₄

#88201

yes

characteristic

200

C₈

—

201

C₈

—

202

C₂²

C₂×#88050

yes

—

203

C₂²

#11718649

yes

characteristic

204

C₂²

#11718771

yes

characteristic

205

C₂²

#11718787

yes

characteristic

206

C₂²

#11718800

yes

—

207

C₄

—

208

C₄

—

209

C₄

—

210

C₄

—

211

C₄

—

104

—

212

C₄

#11718497

yes

—

213

C₄

#11718737

yes

characteristic

214

C₄

#11718753

yes

characteristic

215

C₄

#11718754

yes

—

216

C₂

#13113238

yes

—

217

C₂

#13113264

yes

—

218

C₂

#13115633

yes

characteristic

219

C₂

#13115637

yes

characteristic

220

C₂

#13115667

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₃
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1842	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1842	D₁₃
hypocenter	C₁₃	#9643
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

subgroup

G/H

center

C₂²×C₄

C₂×C₂²⋊D₁₃

commutator

C₂²×C₄×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1842

C₂

socle, prime radical

C₂³×C₁₃

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

N	H ≅ G/N	type
176. C₁₃	#9643	semidirect

H ≅ G/N

type

176. C₁₃

#9643

semidirect

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

subgroup	index
C₄×#88123	2
#13114582	2
C₂×#11718076	2
C₁₃×#1842	2
#9643	13

subgroup

index

C₁₃×#1842

Subgroups — #15761716

Conjugacy classes of subgroups

Complements

Maximal subgroups