Group #11930325 — subgroups

Subgroups — C₅×#7605 #11930325

Order 1280.

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

normal subgroups only

This group has 440 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	C₄×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₅×#1839	C₂	1	2	yes	maximal
3	C₅×#1272	C₂	1	4	yes	maximal
4	C₂×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic
5	C₂×C₄×C₅×Q₈	C₄	1	2	yes	—
6	C₅×#391	C₂²	1	2	yes	—
7	C₅×#389	—	2	2	no	—
8	C₅×#384	C₄	1	1	yes	characteristic
9	C₄×C₅×C₄⋊C₄	—	2	2	no	—
10	C₄³×C₅	C₂²	1	1	yes	characteristic
11	C₅×#336	—	2	2	no	—
12	C₅×#336	—	2	2	no	—
13	C₅×#336	C₂²	1	4	yes	—
14	#7605	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₂²×C₅×Q₈	C₂×C₄	1	2	yes	—
16	C₂²×C₅×C₈	—	2	4	no	—
17	C₅×C₄⋊Q₈	—	2	2	no	—
18	C₅×C₄⋊Q₈	C₈	1	4	yes	complemented
19	C₄×C₅×Q₈	—	4	4	no	—
20	C₂×C₅×C₄⋊C₄	—	2	2	no	—
21	C₂×C₅×C₄⋊C₄	—	2	2	no	—
22	C₂×C₅×C₄⋊C₄	—	2	2	no	—
23	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₅	—	2	2	no	—
25	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅	D₄	1	2	yes	—
28	C₂×C₄²×C₅	D₄	1	2	yes	—
29	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
30	C₅×#95	—	2	4	no	—
31	C₄×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
32	#1839	C₂×C₅	1	2	yes	—
33	#1272	C₂×C₅	1	4	yes	—
34	C₂×C₅×Q₈	—	2	2	no	—
35	C₂×C₅×Q₈	—	4	2	no	—
36	C₂×C₅×Q₈	—	4	2	no	—
37	C₂×C₅×Q₈	C₂×C₈	1	4	yes	—
38	C₂²×C₄×C₅	—	2	2	no	—
39	C₂²×C₄×C₅	—	2	2	no	—
40	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
45	C₂×C₅×C₈	—	4	4	no	—
46	C₂×C₅×C₈	—	4	4	no	—
47	C₂×C₅×C₈	—	4	4	no	—
48	C₅×C₄⋊C₄	—	2	8	no	—
49	C₅×C₄⋊C₄	—	4	2	no	—
50	C₅×C₄⋊C₄	—	4	4	no	—
51	C₅×C₄⋊C₄	—	4	4	no	—
52	C₄²×C₅	—	2	2	no	—
53	C₄²×C₅	—	2	4	no	—
54	C₄²×C₅	—	4	2	no	—
55	C₄²×C₅	—	4	1	no	—
56	C₄²×C₅	—	4	4	no	—
57	C₄²×C₅	C₂×C₈	1	2	yes	—
58	C₄²×C₅	M₁₆	1	2	yes	—
59	C₂×C₄⋊Q₈	C₄×C₅	1	1	yes	characteristic
60	C₂×C₄×Q₈	C₄×C₅	1	2	yes	—
61	#391	C₂²×C₅	1	2	yes	—
62	#389	—	2	2	no	—
63	#384	C₄×C₅	1	1	yes	characteristic
64	C₄×C₄⋊C₄	—	2	2	no	—
65	C₄³	C₂²×C₅	1	1	yes	characteristic
66	#336	—	2	2	no	—
67	#336	—	2	2	no	—
68	#336	C₂²×C₅	1	4	yes	—
69	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₅×Q₈	—	4	4	no	—
71	C₅×Q₈	—	4	4	no	—
72	C₂×C₄×C₅	—	2	2	no	—
73	C₂×C₄×C₅	—	2	1	no	—
74	C₂×C₄×C₅	—	2	1	no	—
75	C₂×C₄×C₅	—	2	2	no	—
76	C₂×C₄×C₅	—	2	2	no	—
77	C₂×C₄×C₅	—	2	2	no	—
78	C₂×C₄×C₅	—	2	2	no	—
79	C₂×C₄×C₅	—	2	4	no	—
80	C₂×C₄×C₅	—	2	1	no	—
81	C₂×C₄×C₅	—	2	2	no	—
82	C₂×C₄×C₅	—	4	2	no	—
83	C₂×C₄×C₅	—	4	2	no	—
84	C₂×C₄×C₅	—	4	2	no	—
85	C₂×C₄×C₅	—	4	4	no	—
86	C₂×C₄×C₅	C₂²⋊C₈	1	4	yes	—
87	C₂×C₄×C₅	#100	1	2	yes	—
88	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₅	#137	1	4	yes	—
91	C₅×C₈	—	8	4	no	—
92	C₂²×Q₈	C₂×C₄×C₅	1	2	yes	—
93	C₂²×C₈	—	2	4	no	—
94	C₄⋊Q₈	—	2	2	no	—
95	C₄⋊Q₈	C₅×C₈	1	4	yes	complemented
96	C₄×Q₈	—	4	4	no	—
97	C₂×C₄⋊C₄	—	2	2	no	—
98	C₂×C₄⋊C₄	—	2	2	no	—
99	C₂×C₄⋊C₄	—	2	2	no	—
100	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
101	C₂×C₄²	—	2	2	no	—
102	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
103	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
104	C₂×C₄²	C₅×D₄	1	2	yes	—
105	C₂×C₄²	C₅×D₄	1	2	yes	—
106	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
107	#95	—	2	4	no	—
108	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
109	C₂²×C₅	#410	1	1	yes	characteristic
110	C₂²×C₅	C₂×#100	1	1	yes	characteristic
111	C₂²×C₅	#419	1	2	yes	—
112	C₂²×C₅	#421	1	2	yes	—
113	C₄×C₅	—	2	2	no	—
114	C₄×C₅	—	2	1	no	—
115	C₄×C₅	—	2	1	no	—
116	C₄×C₅	—	2	4	no	—
117	C₄×C₅	—	2	4	no	—
118	C₄×C₅	—	4	1	no	—
119	C₄×C₅	—	4	4	no	—
120	C₄×C₅	—	8	2	no	—
121	C₂×Q₈	—	2	2	no	—
122	C₂×Q₈	—	4	2	no	—
123	C₂×Q₈	—	4	2	no	—
124	C₂×Q₈	C₂×C₅×C₈	1	4	yes	—
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	—	2	2	no	—
127	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
128	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
129	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
130	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
131	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
132	C₂×C₈	—	4	4	no	—
133	C₂×C₈	—	4	4	no	—
134	C₂×C₈	—	4	4	no	—
135	C₄⋊C₄	—	2	8	no	—
136	C₄⋊C₄	—	4	2	no	—
137	C₄⋊C₄	—	4	4	no	—
138	C₄⋊C₄	—	4	4	no	—
139	C₄²	—	2	4	no	—
140	C₄²	—	2	2	no	—
141	C₄²	—	4	2	no	—
142	C₄²	—	4	1	no	—
143	C₄²	—	4	4	no	—
144	C₄²	C₂×C₅×C₈	1	2	yes	—
145	C₄²	C₅×M₁₆	1	2	yes	—
146	C₂×C₅	#1460	1	1	yes	characteristic
147	C₂×C₅	#1479	1	2	yes	—
148	C₂×C₅	#1515	1	2	yes	—
149	C₂×C₅	#1536	1	1	yes	characteristic
150	C₂×C₅	#1558	1	1	yes	characteristic
151	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
152	Q₈	—	4	4	no	—
153	Q₈	—	4	4	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	C₅×C₂²⋊C₈	1	4	yes	—
169	C₂×C₄	C₅×#100	1	2	yes	—
170	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
171	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
172	C₂×C₄	C₅×#137	1	4	yes	—
173	C₈	—	8	4	no	—
174	C₅	#7605	1	1	yes	characteristic, direct product factor, Sylow
175	C₂²	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
176	C₂²	C₅×#410	1	1	yes	characteristic
177	C₂²	C₂×C₅×#100	1	1	yes	characteristic
178	C₂²	C₅×#419	1	2	yes	—
179	C₂²	C₅×#421	1	2	yes	—
180	C₄	—	2	4	no	—
181	C₄	—	2	2	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	2	4	no	—
185	C₄	—	4	1	no	—
186	C₄	—	4	4	no	—
187	C₄	—	8	2	no	—
188	C₂	C₅×#1460	1	1	yes	characteristic
189	C₂	C₅×#1479	1	2	yes	—
190	C₂	C₅×#1515	1	2	yes	—
191	C₂	C₅×#1536	1	1	yes	characteristic
192	C₂	C₅×#1558	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₅×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₅×#1839

C₂

yes

maximal

C₅×#1272

C₂

yes

maximal

C₂×C₅×C₄⋊Q₈

C₄

yes

characteristic

C₂×C₄×C₅×Q₈

C₄

yes

—

C₅×#391

C₂²

yes

—

C₅×#389

—

C₅×#384

C₄

yes

characteristic

C₄×C₅×C₄⋊C₄

—

C₄³×C₅

C₂²

yes

—

—

C₂²

yes

—

#7605

C₅

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅×Q₈

C₂×C₄

yes

—

C₂²×C₅×C₈

—

C₅×C₄⋊Q₈

—

C₅×C₄⋊Q₈

C₈

yes

complemented

C₄×C₅×Q₈

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²×C₅

—

C₂×C₄²×C₅

C₂×C₄

yes

characteristic

C₂×C₄²×C₅

C₂×C₄

yes

characteristic

C₂×C₄²×C₅

D₄

yes

—

C₂×C₄²×C₅

D₄

yes

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₅×#95

—

C₄×C₄⋊Q₈

C₂×C₅

yes

characteristic

#1839

C₂×C₅

yes

—

#1272

C₂×C₅

yes

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

C₂×C₈

yes

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅

C₂²⋊C₄

yes

—

C₂²×C₄×C₅

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅

C₂×D₄

yes

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

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×C₈

yes

—

C₄²×C₅

M₁₆

yes

—

C₂×C₄⋊Q₈

C₄×C₅

yes

characteristic

C₂×C₄×Q₈

C₄×C₅

yes

—

#391

C₂²×C₅

yes

—

#389

—

#384

C₄×C₅

yes

characteristic

C₄×C₄⋊C₄

—

C₄³

C₂²×C₅

yes

—

—

C₂²×C₅

yes

—

C₂³×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₅×Q₈

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊C₈

yes

—

C₂×C₄×C₅

#100

yes

—

C₂×C₄×C₅

C₂²×C₈

yes

characteristic

C₂×C₄×C₅

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅

#137

yes

—

C₅×C₈

—

C₂²×Q₈

C₂×C₄×C₅

yes

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₅×C₈

yes

complemented

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

100

C₂×C₄⋊C₄

C₂×C₄×C₅

yes

—

101

C₂×C₄²

—

102

C₂×C₄²

C₂×C₄×C₅

yes

characteristic

103

C₂×C₄²

C₂×C₄×C₅

yes

characteristic

104

C₂×C₄²

C₅×D₄

yes

—

105

C₂×C₄²

C₅×D₄

yes

—

106

C₂×C₄²

C₂³×C₅

yes

characteristic

107

#95

—

108

C₂²×C₅

C₂×C₂²⋊C₈

yes

characteristic

109

C₂²×C₅

#410

yes

characteristic

110

C₂²×C₅

C₂×#100

yes

characteristic

111

C₂²×C₅

#419

yes

—

112

C₂²×C₅

#421

yes

—

113

C₄×C₅

—

114

C₄×C₅

—

115

C₄×C₅

—

116

C₄×C₅

—

117

C₄×C₅

—

118

C₄×C₅

—

119

C₄×C₅

—

120

C₄×C₅

—

121

C₂×Q₈

—

122

C₂×Q₈

—

123

C₂×Q₈

—

124

C₂×Q₈

C₂×C₅×C₈

yes

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

C₅×C₂²⋊C₄

yes

characteristic

128

C₂²×C₄

C₅×C₂²⋊C₄

yes

—

129

C₂²×C₄

C₅×C₂²⋊C₄

yes

characteristic

130

C₂²×C₄

C₂²×C₄×C₅

yes

characteristic

131

C₂²×C₄

C₂×C₅×D₄

yes

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄²

—

140

C₄²

—

141

C₄²

—

142

C₄²

—

143

C₄²

—

144

C₄²

C₂×C₅×C₈

yes

—

145

C₄²

C₅×M₁₆

yes

—

146

C₂×C₅

#1460

yes

characteristic

147

C₂×C₅

#1479

yes

—

148

C₂×C₅

#1515

yes

—

149

C₂×C₅

#1536

yes

characteristic

150

C₂×C₅

#1558

yes

characteristic

151

C₂³

C₂×C₅×C₂²⋊C₄

yes

characteristic

152

Q₈

—

153

Q₈

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

C₅×C₂²⋊C₈

yes

—

169

C₂×C₄

C₅×#100

yes

—

170

C₂×C₄

C₂²×C₅×C₈

yes

characteristic

171

C₂×C₄

C₂×C₅×M₁₆

yes

characteristic

172

C₂×C₄

C₅×#137

yes

—

173

C₈

—

174

C₅

#7605

yes

characteristic, direct product factor, Sylow

175

C₂²

C₂×C₅×C₂²⋊C₈

yes

characteristic

176

C₂²

C₅×#410

yes

characteristic

177

C₂²

C₂×C₅×#100

yes

characteristic

178

C₂²

C₅×#419

yes

—

179

C₂²

C₅×#421

yes

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₂

C₅×#1460

yes

characteristic

189

C₂

C₅×#1479

yes

—

190

C₂

C₅×#1515

yes

—

191

C₂

C₅×#1536

yes

characteristic

192

C₂

C₅×#1558

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₂×C₄²	C₂³×C₅
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₄³×C₅ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₅

C₂×C₂²⋊C₄

commutator

C₂×C₄

C₂²×C₅×C₈

Frattini, cosocle

C₂×C₄²

C₂³×C₅

LCS

G ▷ C₂×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₅ ◁ C₄³×C₅ ◁ G

N	H ≅ G/N	type
14. #7605	C₅	direct (N × H)
18. C₅×C₄⋊Q₈	C₈	semidirect
95. C₄⋊Q₈	C₅×C₈	semidirect
174. C₅	#7605	direct (N × H)

H ≅ G/N

type

14. #7605

C₅

direct (N × H)

18. C₅×C₄⋊Q₈

C₈

semidirect

95. C₄⋊Q₈

C₅×C₈

semidirect

174. C₅

#7605

direct (N × H)

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

subgroup	index
C₄×C₅×C₄⋊Q₈	2
C₅×#1839	2
C₅×#1272	2
#7605	5

subgroup

index

C₄×C₅×C₄⋊Q₈

C₅×#1839

C₅×#1272

#7605

Subgroups — C₅×#7605 #11930325

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C5×#7605 #11930325

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#7605 #11930325