Group #14285341 — subgroups

Subgroups — #14285341

Order 1920.

456 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 159 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×C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#11751027	C₂	1	2	yes	maximal
3	#11746367	C₂	1	4	yes	maximal, complemented
4	#10602570	—	3	1	no	maximal, Hall
5	C₂²×C₃×C₅×Q₈	C₄	1	2	yes	—
6	C₃×C₅×C₄⋊Q₈	—	2	2	no	—
7	C₃×C₅×C₄⋊Q₈	C₄	1	2	yes	—
8	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	#91114	—	2	2	no	—
12	C₄⋊(C₃⋊#168)	C₂²	1	4	yes	—
13	#91087	—	2	2	no	—
14	#71777	—	5	1	no	maximal, Hall
15	C₂×C₅×C₄⋊Q₈	D₃	1	1	yes	characteristic
16	#65215	—	3	2	no	—
17	#64766	—	3	4	no	—
18	C₂×C₃×C₅×Q₈	—	2	2	no	—
19	C₂×C₃×C₅×Q₈	—	4	4	no	—
20	C₂×C₃×C₅×Q₈	C₂×C₄	1	4	yes	—
21	C₂²×C₃×C₄×C₅	—	2	2	no	—
22	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
24	C₃×C₅×C₄⋊C₄	—	2	4	no	—
25	C₃×C₅×C₄⋊C₄	—	2	4	no	—
26	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
27	C₃×C₄²×C₅	D₄	1	2	yes	—
28	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
29	C₂×C₃⋊#168	—	2	4	no	—
30	C₂×C₃×C₄⋊Q₈	D₅	1	1	yes	characteristic
31	#4926	—	5	2	no	—
32	#4546	—	5	4	no	—
33	C₂²×C₅×Q₈	Dic₃	1	2	yes	—
34	C₅×C₄⋊Q₈	—	2	2	no	—
35	C₅×C₄⋊Q₈	Dic₃	1	2	yes	—
36	C₅×C₄⋊Q₈	C₂×D₃	1	2	yes	—
37	C₂×C₅×C₄⋊C₄	—	2	2	no	—
38	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
39	C₂²⋊#168	—	6	2	no	—
40	C₄⋊#168	—	3	4	no	—
41	#3935	—	6	2	no	—
42	#1558	—	15	1	no	Sylow
43	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
44	C₃×C₅×Q₈	—	4	4	no	—
45	C₃×C₅×Q₈	—	4	4	no	—
46	C₂×C₃×C₄×C₅	—	2	2	no	—
47	C₂×C₃×C₄×C₅	—	2	4	no	—
48	C₂×C₃×C₄×C₅	—	2	4	no	—
49	C₂×C₃×C₄×C₅	—	4	4	no	—
50	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
51	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
54	C₃⋊#168	—	4	4	no	—
55	C₂²×C₃×Q₈	Dic₅	1	2	yes	—
56	C₃×C₄⋊Q₈	—	2	2	no	—
57	C₃×C₄⋊Q₈	Dic₅	1	2	yes	—
58	C₃×C₄⋊Q₈	C₂×D₅	1	2	yes	—
59	C₂×C₃×C₄⋊C₄	—	2	2	no	—
60	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
61	#830	—	10	2	no	—
62	C₄⋊(C₃⋊C₈)	—	5	4	no	—
63	#803	—	10	2	no	—
64	C₂×C₅×Q₈	—	2	2	no	—
65	C₂×C₅×Q₈	—	4	4	no	—
66	C₂×C₅×Q₈	C₂×Dic₃	1	4	yes	—
67	C₂²×C₄×C₅	—	2	2	no	—
68	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
69	C₂²×C₄×C₅	#67	1	2	yes	—
70	C₅×C₄⋊C₄	—	2	4	no	—
71	C₅×C₄⋊C₄	—	2	4	no	—
72	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
73	C₄²×C₅	#67	1	2	yes	—
74	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
75	C₂×#168	—	6	4	no	—
76	C₂×C₄⋊Q₈	D₁₅	1	1	yes	characteristic
77	#432	—	15	2	no	—
78	#333	—	15	4	no	—
79	C₂²×C₃×C₅	—	2	2	no	—
80	C₂²×C₃×C₅	#100	1	2	yes	—
81	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₃×C₄×C₅	—	2	1	no	—
83	C₃×C₄×C₅	—	2	1	no	—
84	C₃×C₄×C₅	—	2	2	no	—
85	C₃×C₄×C₅	—	4	4	no	—
86	C₃×C₄×C₅	#137	1	4	yes	—
87	C₂×C₃×Q₈	—	2	2	no	—
88	C₂×C₃×Q₈	—	4	4	no	—
89	C₂×C₃×Q₈	C₂×Dic₅	1	4	yes	—
90	C₂²×C₃×C₄	—	2	2	no	—
91	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
92	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
93	C₃×C₄⋊C₄	—	2	4	no	—
94	C₃×C₄⋊C₄	—	2	4	no	—
95	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
96	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
97	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
98	C₂×C₃⋊C₈	—	10	4	no	—
99	C₂³×C₅	#217	1	1	yes	characteristic
100	C₅×Q₈	—	4	4	no	—
101	C₅×Q₈	—	4	4	no	—
102	C₂×C₄×C₅	—	2	2	no	—
103	C₂×C₄×C₅	—	2	4	no	—
104	C₂×C₄×C₅	—	2	4	no	—
105	C₂×C₄×C₅	—	4	4	no	—
106	C₂×C₄×C₅	#217	1	2	yes	—
107	C₂×C₄×C₅	#217	1	1	yes	characteristic
108	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
109	C₂×C₄×C₅	C₂×#67	1	2	yes	—
110	#168	—	12	4	no	—
111	C₂²×Q₈	Dic₁₅	1	2	yes	—
112	C₄⋊Q₈	—	2	2	no	—
113	C₄⋊Q₈	Dic₁₅	1	2	yes	—
114	C₄⋊Q₈	C₂×D₁₅	1	2	yes	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
117	C₄⋊C₈	—	15	4	no	—
118	C₂²⋊C₈	—	30	2	no	—
119	#97	—	30	2	no	—
120	C₂×C₃×C₅	—	2	2	no	—
121	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
122	C₂×C₃×C₅	#419	1	2	yes	—
123	C₂³×C₃	C₂²⋊Dic₅	1	1	yes	characteristic
124	C₃×Q₈	—	4	4	no	—
125	C₃×Q₈	—	4	4	no	—
126	C₂×C₃×C₄	—	2	2	no	—
127	C₂×C₃×C₄	—	2	4	no	—
128	C₂×C₃×C₄	—	2	4	no	—
129	C₂×C₃×C₄	—	4	4	no	—
130	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
131	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
132	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
134	C₃⋊C₈	—	20	4	no	—
135	C₂²×C₅	—	2	2	no	—
136	C₂²×C₅	#833	1	2	yes	—
137	C₂²×C₅	C₂×#217	1	1	yes	characteristic
138	C₄×C₅	—	2	1	no	—
139	C₄×C₅	—	2	1	no	—
140	C₄×C₅	—	2	2	no	—
141	C₄×C₅	—	4	4	no	—
142	C₄×C₅	#942	1	4	yes	—
143	C₂×Q₈	—	2	2	no	—
144	C₂×Q₈	—	4	4	no	—
145	C₂×Q₈	C₂×Dic₁₅	1	4	yes	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	C₂×Dic₁₅	1	1	yes	characteristic
148	C₂²×C₄	#1220	1	2	yes	—
149	C₂×C₈	—	30	4	no	—
150	C₄⋊C₄	—	2	4	no	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄²	C₂×Dic₁₅	1	1	yes	characteristic
153	C₄²	#1220	1	2	yes	—
154	C₄²	C₂²×D₁₅	1	1	yes	characteristic
155	C₃×C₅	#1558	1	1	yes	characteristic, complemented, Hall
156	C₂²×C₃	—	2	2	no	—
157	C₂²×C₃	#3965	1	2	yes	—
158	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
159	C₃×C₄	—	2	1	no	—
160	C₃×C₄	—	2	1	no	—
161	C₃×C₄	—	2	2	no	—
162	C₃×C₄	—	4	4	no	—
163	C₃×C₄	#4088	1	4	yes	—
164	C₂×C₅	—	2	2	no	—
165	C₂×C₅	C₂×#833	1	1	yes	characteristic
166	C₂×C₅	#5225	1	2	yes	—
167	C₂³	#6719	1	1	yes	characteristic
168	Q₈	—	4	4	no	—
169	Q₈	—	4	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	#6719	1	2	yes	—
175	C₂×C₄	#6719	1	1	yes	characteristic
176	C₂×C₄	C₂²×Dic₁₅	1	1	yes	characteristic
177	C₂×C₄	C₂×#1220	1	2	yes	—
178	C₈	—	60	4	no	—
179	C₂×C₃	—	2	2	no	—
180	C₂×C₃	C₂×#3965	1	1	yes	characteristic
181	C₂×C₃	#65514	1	2	yes	—
182	C₅	#71777	1	1	yes	characteristic, complemented, Sylow
183	C₂²	—	2	2	no	—
184	C₂²	#91117	1	2	yes	—
185	C₂²	C₂×#6719	1	1	yes	characteristic
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	—	2	2	no	—
189	C₄	—	4	4	no	—
190	C₄	#91831	1	4	yes	—
191	C₃	#10602570	1	1	yes	characteristic, complemented, Sylow
192	C₂	—	2	2	no	—
193	C₂	C₂×#91117	1	1	yes	characteristic
194	C₂	#11751326	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×C₅×C₄⋊Q₈

C₂

yes

maximal, characteristic

#11751027

C₂

yes

maximal

#11746367

C₂

yes

maximal, complemented

#10602570

—

maximal, Hall

C₂²×C₃×C₅×Q₈

C₄

yes

—

C₃×C₅×C₄⋊Q₈

—

C₃×C₅×C₄⋊Q₈

C₄

yes

—

C₃×C₅×C₄⋊Q₈

C₂²

yes

—

C₂×C₃×C₅×C₄⋊C₄

—

C₂×C₃×C₄²×C₅

C₂²

yes

characteristic

#91114

—

C₄⋊(C₃⋊#168)

C₂²

yes

—

#91087

—

#71777

—

maximal, Hall

C₂×C₅×C₄⋊Q₈

D₃

yes

characteristic

#65215

—

#64766

—

C₂×C₃×C₅×Q₈

—

C₂×C₃×C₅×Q₈

—

C₂×C₃×C₅×Q₈

C₂×C₄

yes

—

C₂²×C₃×C₄×C₅

—

C₂²×C₃×C₄×C₅

C₂×C₄

yes

characteristic

C₂²×C₃×C₄×C₅

D₄

yes

—

C₃×C₅×C₄⋊C₄

—

C₃×C₅×C₄⋊C₄

—

C₃×C₄²×C₅

C₂×C₄

yes

characteristic

C₃×C₄²×C₅

D₄

yes

—

C₃×C₄²×C₅

C₂³

yes

characteristic

C₂×C₃⋊#168

—

C₂×C₃×C₄⋊Q₈

D₅

yes

characteristic

#4926

—

#4546

—

C₂²×C₅×Q₈

Dic₃

yes

—

C₅×C₄⋊Q₈

—

C₅×C₄⋊Q₈

Dic₃

yes

—

C₅×C₄⋊Q₈

C₂×D₃

yes

—

C₂×C₅×C₄⋊C₄

—

C₂×C₄²×C₅

C₂×D₃

yes

characteristic

C₂²⋊#168

—

—

—

—

C₂³×C₃×C₅

C₂²⋊C₄

yes

characteristic

C₃×C₅×Q₈

—

C₃×C₅×Q₈

—

C₂×C₃×C₄×C₅

—

C₂×C₃×C₄×C₅

—

C₂×C₃×C₄×C₅

—

C₂×C₃×C₄×C₅

—

C₂×C₃×C₄×C₅

C₂²⋊C₄

yes

—

C₂×C₃×C₄×C₅

C₂²⋊C₄

yes

characteristic

C₂×C₃×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×C₃×C₄×C₅

C₂×D₄

yes

—

C₃⋊#168

—

C₂²×C₃×Q₈

Dic₅

yes

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

Dic₅

yes

—

C₃×C₄⋊Q₈

C₂×D₅

yes

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄²

C₂×D₅

yes

characteristic

#830

—

C₄⋊(C₃⋊C₈)

—

#803

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

C₂×Dic₃

yes

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂×Dic₃

yes

characteristic

C₂²×C₄×C₅

#67

yes

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂×Dic₃

yes

characteristic

C₄²×C₅

#67

yes

—

C₄²×C₅

C₂²×D₃

yes

characteristic

C₂×#168

—

C₂×C₄⋊Q₈

D₁₅

yes

characteristic

#432

—

#333

—

C₂²×C₃×C₅

—

C₂²×C₃×C₅

#100

yes

—

C₂²×C₃×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₃×C₄×C₅

—

C₃×C₄×C₅

—

C₃×C₄×C₅

—

C₃×C₄×C₅

—

C₃×C₄×C₅

#137

yes

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

C₂×Dic₅

yes

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×Dic₅

yes

characteristic

C₂²×C₃×C₄

C₂²⋊D₅

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂×Dic₅

yes

characteristic

C₃×C₄²

C₂²⋊D₅

yes

—

C₃×C₄²

C₂²×D₅

yes

characteristic

C₂×C₃⋊C₈

—

C₂³×C₅

#217

yes

characteristic

100

C₅×Q₈

—

101

C₅×Q₈

—

102

C₂×C₄×C₅

—

103

C₂×C₄×C₅

—

104

C₂×C₄×C₅

—

105

C₂×C₄×C₅

—

106

C₂×C₄×C₅

#217

yes

—

107

C₂×C₄×C₅

#217

yes

characteristic

108

C₂×C₄×C₅

C₂²×Dic₃

yes

characteristic

109

C₂×C₄×C₅

C₂×#67

yes

—

110

#168

—

111

C₂²×Q₈

Dic₁₅

yes

—

112

C₄⋊Q₈

—

113

C₄⋊Q₈

Dic₁₅

yes

—

114

C₄⋊Q₈

C₂×D₁₅

yes

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄²

C₂×D₁₅

yes

characteristic

117

C₄⋊C₈

—

118

C₂²⋊C₈

—

119

#97

—

120

C₂×C₃×C₅

—

121

C₂×C₃×C₅

C₂×#100

yes

characteristic

122

C₂×C₃×C₅

#419

yes

—

123

C₂³×C₃

C₂²⋊Dic₅

yes

characteristic

124

C₃×Q₈

—

125

C₃×Q₈

—

126

C₂×C₃×C₄

—

127

C₂×C₃×C₄

—

128

C₂×C₃×C₄

—

129

C₂×C₃×C₄

—

130

C₂×C₃×C₄

C₂²⋊Dic₅

yes

—

131

C₂×C₃×C₄

C₂²⋊Dic₅

yes

characteristic

132

C₂×C₃×C₄

C₂²×Dic₅

yes

characteristic

133

C₂×C₃×C₄

C₂×C₂²⋊D₅

yes

—

134

C₃⋊C₈

—

135

C₂²×C₅

—

136

C₂²×C₅

#833

yes

—

137

C₂²×C₅

C₂×#217

yes

characteristic

138

C₄×C₅

—

139

C₄×C₅

—

140

C₄×C₅

—

141

C₄×C₅

—

142

C₄×C₅

#942

yes

—

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂×Q₈

C₂×Dic₁₅

yes

—

146

C₂²×C₄

—

147

C₂²×C₄

C₂×Dic₁₅

yes

characteristic

148

C₂²×C₄

#1220

yes

—

149

C₂×C₈

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄²

C₂×Dic₁₅

yes

characteristic

153

C₄²

#1220

yes

—

154

C₄²

C₂²×D₁₅

yes

characteristic

155

C₃×C₅

#1558

yes

characteristic, complemented, Hall

156

C₂²×C₃

—

157

C₂²×C₃

#3965

yes

—

158

C₂²×C₃

C₂×C₂²⋊Dic₅

yes

characteristic

159

C₃×C₄

—

160

C₃×C₄

—

161

C₃×C₄

—

162

C₃×C₄

—

163

C₃×C₄

#4088

yes

—

164

C₂×C₅

—

165

C₂×C₅

C₂×#833

yes

characteristic

166

C₂×C₅

#5225

yes

—

167

C₂³

#6719

yes

characteristic

168

Q₈

—

169

Q₈

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

#6719

yes

—

175

C₂×C₄

#6719

yes

characteristic

176

C₂×C₄

C₂²×Dic₁₅

yes

characteristic

177

C₂×C₄

C₂×#1220

yes

—

178

C₈

—

179

C₂×C₃

—

180

C₂×C₃

C₂×#3965

yes

characteristic

181

C₂×C₃

#65514

yes

—

182

C₅

#71777

yes

characteristic, complemented, Sylow

183

C₂²

—

184

C₂²

#91117

yes

—

185

C₂²

C₂×#6719

yes

characteristic

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

#91831

yes

—

191

C₃

#10602570

yes

characteristic, complemented, Sylow

192

C₂

—

193

C₂

C₂×#91117

yes

characteristic

194

C₂

#11751326

yes

—

	subgroup	G/H
center	C₂²	C₂×#6719
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂²×D₁₅
Fitting	C₂×C₃×C₅×C₄⋊Q₈	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂³×C₃×C₅	C₂²⋊C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₅
hypocenter	C₃×C₅	#1558
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×#6719

commutator

C₂×C₃×C₄×C₅

C₂²×C₄

Frattini

C₄²

C₂²×D₁₅

Fitting

C₂×C₃×C₅×C₄⋊Q₈

C₂

socle

C₂²×C₃×C₅

C₂×C₂²⋊C₄

prime radical

C₂³×C₃×C₅

C₂²⋊C₄

cosocle

C₃×C₄²×C₅

C₂³

hypercenter

C₂×C₄⋊Q₈

D₁₅

hypocenter

C₃×C₅

#1558

LCS

G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
3. #11746367	C₂	semidirect
155. C₃×C₅	#1558	semidirect
182. C₅	#71777	semidirect
191. C₃	#10602570	semidirect

H ≅ G/N

type

3. #11746367

C₂

semidirect

155. C₃×C₅

#1558

semidirect

182. C₅

#71777

semidirect

191. C₃

#10602570

semidirect

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

subgroup	index
C₂×C₃×C₅×C₄⋊Q₈	2
#11751027	2
#11746367	2
#10602570	3
#71777	5

subgroup

index

C₂×C₃×C₅×C₄⋊Q₈

Subgroups — #14285341

Conjugacy classes of subgroups

Complements

Maximal subgroups