Group #11998414 — subgroups

Subgroups — #11998414

Order 1280.

466 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 151 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	C₅×#2862	C₂	1	1	yes	maximal, characteristic
2	C₂×#64766	C₂	1	2	yes	maximal
3	C₂×#64767	C₂	1	2	yes	maximal
4	#10600185	C₂	1	1	yes	maximal, characteristic
5	#10600184	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	C₅×#391	—	2	1	no	—
9	C₅×#391	—	2	1	no	—
10	C₅×#391	C₄	1	4	yes	—
11	C₅×#382	—	2	2	no	—
12	C₄³×C₅	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊#168	C₂²	1	2	yes	—
14	C₂×C₄⋊#168	C₂²	1	2	yes	—
15	C₂×#3935	—	2	1	no	—
16	#64767	—	2	4	no	—
17	#64766	—	2	4	no	—
18	#64744	—	2	1	no	—
19	#64744	—	2	2	no	—
20	#9709	—	5	1	no	maximal, Sylow
21	C₂²×C₅×Q₈	—	2	1	no	—
22	C₂²×C₅×Q₈	C₂×C₄	1	2	yes	—
23	C₅×C₄⋊Q₈	—	2	2	no	—
24	C₅×C₄⋊Q₈	—	2	2	no	—
25	C₅×C₄⋊Q₈	—	2	2	no	—
26	C₅×C₄⋊Q₈	—	2	2	no	—
27	C₂×C₅×C₄⋊C₄	—	2	1	no	—
28	C₂×C₅×C₄⋊C₄	—	2	1	no	—
29	C₂×C₅×C₄⋊C₄	—	2	1	no	—
30	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₅	D₄	1	2	yes	—
33	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
36	C₂²×#168	—	2	2	no	—
37	C₂²×#168	—	2	2	no	—
38	C₅×#95	—	2	4	no	—
39	C₅×#95	—	2	4	no	—
40	C₄⋊#168	—	2	2	no	—
41	C₄⋊#168	—	2	2	no	—
42	C₄⋊#168	—	2	2	no	—
43	C₄⋊#168	D₄	1	4	yes	—
44	#3935	—	4	2	no	—
45	#2862	D₅	1	1	yes	characteristic
46	#1831	—	5	1	no	—
47	#1767	—	5	1	no	—
48	C₂×#333	—	5	2	no	—
49	C₂×#332	—	5	2	no	—
50	C₂×C₅×Q₈	—	2	4	no	—
51	C₂×C₅×Q₈	—	4	4	no	—
52	C₂×C₅×Q₈	—	4	2	no	—
53	C₂×C₅×Q₈	—	4	2	no	—
54	C₂×C₅×Q₈	—	4	1	no	—
55	C₂×C₅×Q₈	—	4	1	no	—
56	C₂²×C₄×C₅	—	2	2	no	—
57	C₂²×C₄×C₅	—	2	2	no	—
58	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₅×C₄⋊C₄	—	2	4	no	—
64	C₅×C₄⋊C₄	—	4	2	no	—
65	C₅×C₄⋊C₄	—	4	1	no	—
66	C₅×C₄⋊C₄	—	4	1	no	—
67	C₅×C₄⋊C₄	—	4	1	no	—
68	C₅×C₄⋊C₄	—	4	1	no	—
69	C₄²×C₅	—	2	2	no	—
70	C₄²×C₅	—	2	1	no	—
71	C₄²×C₅	—	2	1	no	—
72	C₄²×C₅	—	2	4	no	—
73	C₄²×C₅	—	4	2	no	—
74	C₄²×C₅	C₂×D₄	1	2	yes	—
75	C₄²×C₅	#41	1	2	yes	—
76	C₂×#168	—	2	4	no	—
77	C₂×#168	—	4	2	no	—
78	C₂×#168	—	4	2	no	—
79	C₂×#168	—	4	2	no	—
80	C₂×#168	—	4	4	no	—
81	C₂×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
82	C₂×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
83	C₂×C₄⋊C₈	—	5	2	no	—
84	C₂×C₄⋊C₈	—	5	2	no	—
85	C₂×#97	—	10	1	no	—
86	#391	—	2	1	no	—
87	#391	—	2	1	no	—
88	#391	Dic₅	1	4	yes	—
89	#382	—	2	2	no	—
90	C₄³	C₂×D₅	1	1	yes	characteristic
91	#336	—	10	2	no	—
92	#336	—	10	1	no	—
93	#333	—	10	4	no	—
94	#332	—	10	4	no	—
95	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₅×Q₈	—	8	4	no	—
97	C₅×Q₈	—	8	2	no	—
98	C₅×Q₈	—	8	2	no	—
99	C₂×C₄×C₅	—	2	2	no	—
100	C₂×C₄×C₅	—	2	2	no	—
101	C₂×C₄×C₅	—	2	2	no	—
102	C₂×C₄×C₅	—	2	2	no	—
103	C₂×C₄×C₅	—	2	2	no	—
104	C₂×C₄×C₅	—	2	2	no	—
105	C₂×C₄×C₅	—	2	2	no	—
106	C₂×C₄×C₅	—	4	2	no	—
107	C₂×C₄×C₅	—	4	2	no	—
108	C₂×C₄×C₅	—	4	4	no	—
109	C₂×C₄×C₅	—	4	2	no	—
110	C₂×C₄×C₅	—	4	2	no	—
111	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₅	#121	1	2	yes	—
113	C₂×C₄×C₅	#124	1	1	yes	characteristic
114	C₂×C₄×C₅	#127	1	1	yes	characteristic
115	C₂×C₄×C₅	#135	1	2	yes	—
116	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₂×C₄×C₅	#137	1	1	yes	characteristic
118	C₂×C₄×C₅	#137	1	4	yes	—
119	#168	—	4	4	no	—
120	#168	—	8	2	no	—
121	C₂²×Q₈	—	2	1	no	—
122	C₂²×Q₈	C₂×Dic₅	1	2	yes	—
123	C₂²×C₈	—	10	2	no	—
124	C₂²×C₈	—	10	2	no	—
125	C₄⋊Q₈	—	2	2	no	—
126	C₄⋊Q₈	—	2	2	no	—
127	C₄⋊Q₈	—	2	2	no	—
128	C₄⋊Q₈	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	1	no	—
130	C₂×C₄⋊C₄	—	2	1	no	—
131	C₂×C₄⋊C₄	—	2	1	no	—
132	C₂×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
133	C₂×C₄²	C₂×Dic₅	1	2	yes	—
134	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
135	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
136	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
137	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
138	C₄⋊C₈	—	5	4	no	—
139	C₄⋊C₈	—	10	2	no	—
140	C₄⋊C₈	—	10	2	no	—
141	C₄⋊C₈	—	10	2	no	—
142	#97	—	20	2	no	—
143	#95	—	2	4	no	—
144	#95	—	2	4	no	—
145	C₂²×C₅	#390	1	1	yes	characteristic
146	C₂²×C₅	#413	1	2	yes	—
147	C₂²×C₅	#416	1	1	yes	characteristic
148	C₂²×C₅	#417	1	1	yes	characteristic
149	C₂²×C₅	#419	1	2	yes	—
150	C₄×C₅	—	2	2	no	—
151	C₄×C₅	—	4	2	no	—
152	C₄×C₅	—	4	2	no	—
153	C₄×C₅	—	8	2	no	—
154	C₄×C₅	—	8	2	no	—
155	C₄×C₅	#463	1	2	yes	—
156	C₄×C₅	#464	1	2	yes	—
157	C₄×C₅	#491	1	2	yes	—
158	C₄×C₅	#497	1	2	yes	—
159	C₂×Q₈	—	2	4	no	—
160	C₂×Q₈	—	4	4	no	—
161	C₂×Q₈	—	4	2	no	—
162	C₂×Q₈	—	4	1	no	—
163	C₂×Q₈	—	4	2	no	—
164	C₂×Q₈	—	4	1	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
168	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
169	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
170	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
171	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
172	C₂×C₈	—	10	4	no	—
173	C₂×C₈	—	20	2	no	—
174	C₂×C₈	—	20	2	no	—
175	C₂×C₈	—	20	2	no	—
176	C₂×C₈	—	20	4	no	—
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	4	1	no	—
179	C₄⋊C₄	—	4	1	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	1	no	—
182	C₄⋊C₄	—	4	1	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	2	1	no	—
185	C₄²	—	2	4	no	—
186	C₄²	—	2	1	no	—
187	C₄²	—	4	2	no	—
188	C₄²	D₄×D₅	1	2	yes	—
189	C₄²	#711	1	2	yes	—
190	C₂×C₅	#1554	1	2	yes	—
191	C₂×C₅	#1559	1	2	yes	—
192	C₂×C₅	#1960	1	1	yes	characteristic
193	C₂×C₅	#1967	1	1	yes	characteristic
194	C₂×C₅	#1983	1	1	yes	characteristic
195	C₂³	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
196	Q₈	—	8	4	no	—
197	Q₈	—	8	2	no	—
198	Q₈	—	8	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	D₄×Dic₅	1	2	yes	—
212	C₂×C₄	#4082	1	1	yes	characteristic
213	C₂×C₄	#4084	1	2	yes	—
214	C₂×C₄	#4086	1	1	yes	characteristic
215	C₂×C₄	#4088	1	4	yes	—
216	C₂×C₄	#4095	1	1	yes	characteristic
217	C₂×C₄	#4096	1	2	yes	—
218	C₂×C₄	#4097	1	1	yes	characteristic
219	C₈	—	20	4	no	—
220	C₈	—	40	2	no	—
221	C₅	#9709	1	1	yes	characteristic, complemented, Sylow
222	C₂²	#65510	1	1	yes	characteristic
223	C₂²	#65514	1	2	yes	—
224	C₂²	#65521	1	1	yes	characteristic
225	C₂²	#65522	1	1	yes	characteristic
226	C₂²	#65526	1	2	yes	—
227	C₄	—	2	2	no	—
228	C₄	—	4	2	no	—
229	C₄	—	4	2	no	—
230	C₄	—	8	2	no	—
231	C₄	—	8	2	no	—
232	C₄	#65343	1	2	yes	—
233	C₄	#65346	1	2	yes	—
234	C₄	#65371	1	2	yes	—
235	C₄	#65375	1	2	yes	—
236	C₂	#10602514	1	1	yes	characteristic
237	C₂	#10602520	1	1	yes	characteristic
238	C₂	#10602579	1	1	yes	characteristic
239	C₂	#10602625	1	2	yes	—
240	C₂	#10602626	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#2862

C₂

yes

maximal, characteristic

C₂×#64766

C₂

yes

maximal

C₂×#64767

C₂

yes

maximal

#10600185

C₂

yes

maximal, characteristic

#10600184

C₂

yes

maximal, characteristic

C₂×C₅×C₄⋊Q₈

C₂²

yes

characteristic

C₂×C₅×C₄⋊Q₈

C₂²

yes

—

—

C₄

yes

—

C₅×#382

—

C₄³×C₅

C₂²

yes

characteristic

C₂×C₄⋊#168

C₂²

yes

—

C₂×C₄⋊#168

C₂²

yes

—

—

—

—

—

—

—

C₂²×C₅×Q₈

—

C₂²×C₅×Q₈

C₂×C₄

yes

—

C₅×C₄⋊Q₈

—

C₅×C₄⋊Q₈

—

C₅×C₄⋊Q₈

—

C₅×C₄⋊Q₈

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²×C₅

C₂×C₄

yes

—

C₂×C₄²×C₅

D₄

yes

—

C₂×C₄²×C₅

D₄

yes

characteristic

C₂×C₄²×C₅

D₄

yes

characteristic

C₂×C₄²×C₅

C₂³

yes

characteristic

C₂²×#168

—

C₂²×#168

—

C₅×#95

—

C₅×#95

—

—

—

—

D₄

yes

—

#3935

—

#2862

D₅

yes

—

—

—

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂²⋊C₄

yes

—

C₂²×C₄×C₅

C₂²⋊C₄

yes

—

C₂²×C₄×C₅

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

C₂×D₄

yes

—

C₄²×C₅

#41

yes

—

—

—

—

—

—

C₂×C₄⋊Q₈

C₂×D₅

yes

characteristic

C₂×C₄⋊Q₈

C₂×D₅

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#97

—

#391

—

#391

—

#391

Dic₅

yes

—

#382

—

C₄³

C₂×D₅

yes

—

—

—

—

C₂³×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₅×Q₈

—

C₅×Q₈

—

C₅×Q₈

—

C₂×C₄×C₅

—

100

C₂×C₄×C₅

—

101

C₂×C₄×C₅

—

102

C₂×C₄×C₅

—

103

C₂×C₄×C₅

—

104

C₂×C₄×C₅

—

105

C₂×C₄×C₅

—

106

C₂×C₄×C₅

—

107

C₂×C₄×C₅

—

108

C₂×C₄×C₅

—

109

C₂×C₄×C₅

—

110

C₂×C₄×C₅

—

111

C₂×C₄×C₅

C₄×D₄

yes

—

112

C₂×C₄×C₅

#121

yes

—

113

C₂×C₄×C₅

#124

yes

characteristic

114

C₂×C₄×C₅

#127

yes

characteristic

115

C₂×C₄×C₅

#135

yes

—

116

C₂×C₄×C₅

AGL(1,ℤ/8)

yes

characteristic

117

C₂×C₄×C₅

#137

yes

characteristic

118

C₂×C₄×C₅

#137

yes

—

119

#168

—

120

#168

—

121

C₂²×Q₈

—

122

C₂²×Q₈

C₂×Dic₅

yes

—

123

C₂²×C₈

—

124

C₂²×C₈

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

131

C₂×C₄⋊C₄

—

132

C₂×C₄⋊C₄

C₂×Dic₅

yes

—

133

C₂×C₄²

C₂×Dic₅

yes

—

134

C₂×C₄²

C₂²⋊D₅

yes

—

135

C₂×C₄²

C₂²⋊D₅

yes

characteristic

136

C₂×C₄²

C₂²⋊D₅

yes

characteristic

137

C₂×C₄²

C₂²×D₅

yes

characteristic

138

C₄⋊C₈

—

139

C₄⋊C₈

—

140

C₄⋊C₈

—

141

C₄⋊C₈

—

142

#97

—

143

#95

—

144

#95

—

145

C₂²×C₅

#390

yes

characteristic

146

C₂²×C₅

#413

yes

—

147

C₂²×C₅

#416

yes

characteristic

148

C₂²×C₅

#417

yes

characteristic

149

C₂²×C₅

#419

yes

—

150

C₄×C₅

—

151

C₄×C₅

—

152

C₄×C₅

—

153

C₄×C₅

—

154

C₄×C₅

—

155

C₄×C₅

#463

yes

—

156

C₄×C₅

#464

yes

—

157

C₄×C₅

#491

yes

—

158

C₄×C₅

#497

yes

—

159

C₂×Q₈

—

160

C₂×Q₈

—

161

C₂×Q₈

—

162

C₂×Q₈

—

163

C₂×Q₈

—

164

C₂×Q₈

—

165

C₂²×C₄

—

166

C₂²×C₄

—

167

C₂²×C₄

C₂²⋊Dic₅

yes

—

168

C₂²×C₄

C₂²⋊Dic₅

yes

—

169

C₂²×C₄

C₂²×Dic₅

yes

characteristic

170

C₂²×C₄

C₂×C₂²⋊D₅

yes

characteristic

171

C₂²×C₄

C₂×C₂²⋊D₅

yes

characteristic

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

—

176

C₂×C₈

—

177

C₄⋊C₄

—

178

C₄⋊C₄

—

179

C₄⋊C₄

—

180

C₄⋊C₄

—

181

C₄⋊C₄

—

182

C₄⋊C₄

—

183

C₄²

—

184

C₄²

—

185

C₄²

—

186

C₄²

—

187

C₄²

—

188

C₄²

D₄×D₅

yes

—

189

C₄²

#711

yes

—

190

C₂×C₅

#1554

yes

—

191

C₂×C₅

#1559

yes

—

192

C₂×C₅

#1960

yes

characteristic

193

C₂×C₅

#1967

yes

characteristic

194

C₂×C₅

#1983

yes

characteristic

195

C₂³

C₂×C₂²⋊Dic₅

yes

characteristic

196

Q₈

—

197

Q₈

—

198

Q₈

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

—

204

C₂×C₄

—

205

C₂×C₄

—

206

C₂×C₄

—

207

C₂×C₄

—

208

C₂×C₄

—

209

C₂×C₄

—

210

C₂×C₄

—

211

C₂×C₄

D₄×Dic₅

yes

—

212

C₂×C₄

#4082

yes

characteristic

213

C₂×C₄

#4084

yes

—

214

C₂×C₄

#4086

yes

characteristic

215

C₂×C₄

#4088

yes

—

216

C₂×C₄

#4095

yes

characteristic

217

C₂×C₄

#4096

yes

—

218

C₂×C₄

#4097

yes

characteristic

219

C₈

—

220

C₈

—

221

C₅

#9709

yes

characteristic, complemented, Sylow

222

C₂²

#65510

yes

characteristic

223

C₂²

#65514

yes

—

224

C₂²

#65521

yes

characteristic

225

C₂²

#65522

yes

characteristic

226

C₂²

#65526

yes

—

227

C₄

—

228

C₄

—

229

C₄

—

230

C₄

—

231

C₄

—

232

C₄

#65343

yes

—

233

C₄

#65346

yes

—

234

C₄

#65371

yes

—

235

C₄

#65375

yes

—

236

C₂

#10602514

yes

characteristic

237

C₂

#10602520

yes

characteristic

238

C₂

#10602579

yes

characteristic

239

C₂

#10602625

yes

—

240

C₂

#10602626

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅
commutator	C₂²×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#2862	C₂
socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#2862	D₅
hypocenter	C₅	#9709
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2862

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₅

commutator

C₂²×C₄×C₅

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅

Fitting

C₅×#2862

C₂

socle, prime radical

C₂³×C₅

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₅

C₂³

D₅

C₅

G ▷ C₂²×C₄×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #2862

N	H ≅ G/N	type
221. C₅	#9709	semidirect

H ≅ G/N

type

221. C₅

#9709

semidirect

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

subgroup	index
C₅×#2862	2
C₂×#64766	2
C₂×#64767	2
#10600185	2
#10600184	2
#9709	5

subgroup

index

Subgroups — #11998414

Conjugacy classes of subgroups

Complements

Maximal subgroups