Group #14284742 — subgroups

Subgroups — C₅×#71829 #14284742

Order 1920.

404 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 170 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	C₃×C₅×#533	C₂	1	1	yes	maximal, characteristic
2	C₅×#4925	C₂	1	2	yes	maximal
3	C₅×#4547	C₂	1	4	yes	maximal, complemented
4	C₅×#1544	—	3	1	no	maximal, Hall
5	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
6	C₃×C₅×#125	C₄	1	2	yes	—
7	C₃×C₅×C₂²⋊Q₈	—	2	2	no	—
8	C₃×C₄×C₅×Q₈	—	2	2	no	—
9	C₃×C₅×#117	C₄	1	2	yes	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₅×#830	—	2	2	no	—
12	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	4	yes	—
13	C₄×C₅×C₃⋊C₈	—	2	2	no	—
14	#71829	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₅×#533	D₃	1	1	yes	characteristic
16	C₅×#431	—	3	2	no	—
17	C₅×#332	—	3	4	no	—
18	C₂×C₃×C₅×Q₈	—	2	2	no	—
19	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
21	C₃×C₅×C₄⋊C₄	—	2	4	no	—
22	C₃×C₅×C₄⋊C₄	—	2	2	no	—
23	C₃×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
24	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
25	C₃×C₄²×C₅	—	2	2	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×C₅×C₃⋊C₈	—	2	4	no	—
30	C₃×#533	C₂×C₅	1	1	yes	characteristic
31	#4925	C₂×C₅	1	2	yes	—
32	#4547	C₂×C₅	1	4	yes	complemented
33	C₅×C₄⋊Q₈	C₂×D₃	1	2	yes	—
34	C₅×#125	Dic₃	1	2	yes	—
35	C₅×C₂²⋊Q₈	—	2	2	no	—
36	C₄×C₅×Q₈	—	2	2	no	—
37	C₅×#117	Dic₃	1	2	yes	—
38	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
39	C₅×C₄⋊C₈	—	3	4	no	—
40	C₅×C₂²⋊C₈	—	6	2	no	—
41	C₄×C₅×C₈	—	6	2	no	—
42	#1544	—	3	1	no	Sylow
43	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
44	C₃×C₅×Q₈	—	4	4	no	—
45	C₂×C₃×C₄×C₅	—	2	2	no	—
46	C₂×C₃×C₄×C₅	—	2	4	no	—
47	C₂×C₃×C₄×C₅	—	2	4	no	—
48	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
49	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
52	C₅×C₃⋊C₈	—	4	4	no	—
53	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
54	C₃×#125	C₄×C₅	1	2	yes	—
55	C₃×C₂²⋊Q₈	—	2	2	no	—
56	C₃×C₄×Q₈	—	2	2	no	—
57	C₃×#117	C₄×C₅	1	2	yes	—
58	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
59	#830	—	2	2	no	—
60	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	4	yes	—
61	C₄×C₃⋊C₈	—	2	2	no	—
62	C₂×C₅×Q₈	—	2	2	no	—
63	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
64	C₂²×C₄×C₅	#67	1	2	yes	—
65	C₂×C₅×C₈	—	6	4	no	—
66	C₅×C₄⋊C₄	—	2	4	no	—
67	C₅×C₄⋊C₄	—	2	2	no	—
68	C₅×C₄⋊C₄	C₂×Dic₃	1	4	yes	—
69	C₅×C₂²⋊C₄	—	2	2	no	—
70	C₄²×C₅	—	2	2	no	—
71	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
72	C₄²×C₅	#67	1	2	yes	—
73	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
74	#533	C₅×D₃	1	1	yes	characteristic
75	#431	—	3	2	no	—
76	#332	—	3	4	no	—
77	C₂²×C₃×C₅	—	2	1	no	—
78	C₂²×C₃×C₅	—	2	2	no	—
79	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₃×C₄×C₅	—	2	1	no	—
81	C₃×C₄×C₅	—	2	2	no	—
82	C₃×C₄×C₅	—	4	4	no	—
83	C₃×C₄×C₅	#101	1	2	yes	—
84	C₃×C₄×C₅	#135	1	4	yes	—
85	C₂×C₃×Q₈	—	2	2	no	—
86	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
87	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
88	C₃×C₄⋊C₄	—	2	4	no	—
89	C₃×C₄⋊C₄	—	2	2	no	—
90	C₃×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
91	C₃×C₂²⋊C₄	—	2	2	no	—
92	C₃×C₄²	—	2	2	no	—
93	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
94	C₃×C₄²	C₅×D₄	1	2	yes	—
95	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
96	C₂×C₃⋊C₈	—	2	4	no	—
97	C₂³×C₅	#217	1	1	yes	characteristic
98	C₅×Q₈	—	4	4	no	—
99	C₂×C₄×C₅	—	2	2	no	—
100	C₂×C₄×C₅	—	2	4	no	—
101	C₂×C₄×C₅	—	2	4	no	—
102	C₂×C₄×C₅	#217	1	2	yes	—
103	C₂×C₄×C₅	#217	1	1	yes	characteristic
104	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
105	C₂×C₄×C₅	C₂×#67	1	2	yes	—
106	C₅×C₈	—	12	4	no	—
107	C₄⋊Q₈	C₂×C₅×D₃	1	2	yes	—
108	#125	C₅×Dic₃	1	2	yes	—
109	C₂²⋊Q₈	—	2	2	no	—
110	C₄×Q₈	—	2	2	no	—
111	#117	C₅×Dic₃	1	2	yes	—
112	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
113	C₄⋊C₈	—	3	4	no	—
114	C₂²⋊C₈	—	6	2	no	—
115	C₄×C₈	—	6	2	no	—
116	C₂×C₃×C₅	—	4	1	no	—
117	C₂×C₃×C₅	C₂×#101	1	1	yes	characteristic
118	C₂×C₃×C₅	#416	1	2	yes	—
119	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
120	C₃×Q₈	—	4	4	no	—
121	C₂×C₃×C₄	—	2	4	no	—
122	C₂×C₃×C₄	—	2	2	no	—
123	C₂×C₃×C₄	—	2	4	no	—
124	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
125	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
127	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
128	C₃⋊C₈	—	4	4	no	—
129	C₂²×C₅	—	2	1	no	—
130	C₂²×C₅	—	2	2	no	—
131	C₂²×C₅	C₂×#217	1	1	yes	characteristic
132	C₄×C₅	—	2	1	no	—
133	C₄×C₅	—	2	2	no	—
134	C₄×C₅	—	4	4	no	—
135	C₄×C₅	#836	1	2	yes	—
136	C₄×C₅	#950	1	4	yes	—
137	C₂×Q₈	—	2	2	no	—
138	C₂²×C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
139	C₂²×C₄	C₅×#67	1	2	yes	—
140	C₂×C₈	—	6	4	no	—
141	C₄⋊C₄	—	2	4	no	—
142	C₄⋊C₄	—	2	2	no	—
143	C₄⋊C₄	C₂×C₅×Dic₃	1	4	yes	—
144	C₂²⋊C₄	—	2	2	no	—
145	C₄²	—	2	2	no	—
146	C₄²	C₂×C₅×Dic₃	1	1	yes	characteristic
147	C₄²	C₅×#67	1	2	yes	—
148	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
149	C₃×C₅	#1544	1	1	yes	characteristic, complemented, Hall
150	C₂²×C₃	—	2	1	no	—
151	C₂²×C₃	—	2	2	no	—
152	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
153	C₃×C₄	—	2	1	no	—
154	C₃×C₄	—	2	2	no	—
155	C₃×C₄	—	4	4	no	—
156	C₃×C₄	C₅×#101	1	2	yes	—
157	C₃×C₄	C₅×#135	1	4	yes	—
158	C₂×C₅	—	4	1	no	—
159	C₂×C₅	C₂×#836	1	1	yes	characteristic
160	C₂×C₅	#5233	1	2	yes	—
161	C₂³	C₅×#217	1	1	yes	characteristic
162	Q₈	—	4	4	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	C₅×#217	1	2	yes	—
167	C₂×C₄	C₅×#217	1	1	yes	characteristic
168	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
169	C₂×C₄	C₂×C₅×#67	1	2	yes	—
170	C₈	—	12	4	no	—
171	C₂×C₃	—	4	1	no	—
172	C₂×C₃	C₂×C₅×#101	1	1	yes	characteristic
173	C₂×C₃	C₅×#416	1	2	yes	—
174	C₅	#71829	1	1	yes	characteristic, direct product factor, Sylow
175	C₂²	—	2	1	no	—
176	C₂²	—	2	2	no	—
177	C₂²	C₂×C₅×#217	1	1	yes	characteristic
178	C₄	—	2	1	no	—
179	C₄	—	2	2	no	—
180	C₄	—	4	4	no	—
181	C₄	C₅×#836	1	2	yes	—
182	C₄	C₅×#950	1	4	yes	—
183	C₃	C₅×#1544	1	1	yes	characteristic, complemented, Sylow
184	C₂	—	4	1	no	—
185	C₂	C₂×C₅×#836	1	1	yes	characteristic
186	C₂	C₅×#5233	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₅×#533

C₂

yes

maximal, characteristic

C₅×#4925

C₂

yes

maximal

C₅×#4547

C₂

yes

maximal, complemented

C₅×#1544

—

maximal, Hall

C₃×C₅×C₄⋊Q₈

C₂²

yes

—

C₃×C₅×#125

C₄

yes

—

C₃×C₅×C₂²⋊Q₈

—

C₃×C₄×C₅×Q₈

—

C₃×C₅×#117

C₄

yes

—

C₂×C₃×C₄²×C₅

C₂²

yes

characteristic

C₅×#830

—

C₅×C₄⋊(C₃⋊C₈)

C₂²

yes

—

C₄×C₅×C₃⋊C₈

—

#71829

C₅

yes

maximal, characteristic, direct product factor, Hall

C₅×#533

D₃

yes

characteristic

C₅×#431

—

C₅×#332

—

C₂×C₃×C₅×Q₈

—

C₂²×C₃×C₄×C₅

C₂×C₄

yes

characteristic

C₂²×C₃×C₄×C₅

D₄

yes

—

C₃×C₅×C₄⋊C₄

—

C₃×C₅×C₄⋊C₄

—

C₃×C₅×C₄⋊C₄

C₂×C₄

yes

—

C₃×C₅×C₂²⋊C₄

—

C₃×C₄²×C₅

—

C₃×C₄²×C₅

C₂×C₄

yes

characteristic

C₃×C₄²×C₅

D₄

yes

—

C₃×C₄²×C₅

C₂³

yes

characteristic

C₂×C₅×C₃⋊C₈

—

C₃×#533

C₂×C₅

yes

characteristic

#4925

C₂×C₅

yes

—

#4547

C₂×C₅

yes

complemented

C₅×C₄⋊Q₈

C₂×D₃

yes

—

C₅×#125

Dic₃

yes

—

C₅×C₂²⋊Q₈

—

C₄×C₅×Q₈

—

C₅×#117

Dic₃

yes

—

C₂×C₄²×C₅

C₂×D₃

yes

characteristic

C₅×C₄⋊C₈

—

C₅×C₂²⋊C₈

—

C₄×C₅×C₈

—

#1544

—

Sylow

C₂³×C₃×C₅

C₂²⋊C₄

yes

characteristic

C₃×C₅×Q₈

—

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

—

C₅×C₃⋊C₈

—

C₃×C₄⋊Q₈

C₂²×C₅

yes

—

C₃×#125

C₄×C₅

yes

—

C₃×C₂²⋊Q₈

—

C₃×C₄×Q₈

—

C₃×#117

C₄×C₅

yes

—

C₂×C₃×C₄²

C₂²×C₅

yes

characteristic

#830

—

C₄⋊(C₃⋊C₈)

C₂²×C₅

yes

—

C₄×C₃⋊C₈

—

C₂×C₅×Q₈

—

C₂²×C₄×C₅

C₂×Dic₃

yes

characteristic

C₂²×C₄×C₅

#67

yes

—

C₂×C₅×C₈

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

C₂×Dic₃

yes

—

C₅×C₂²⋊C₄

—

C₄²×C₅

—

C₄²×C₅

C₂×Dic₃

yes

characteristic

C₄²×C₅

#67

yes

—

C₄²×C₅

C₂²×D₃

yes

characteristic

#533

C₅×D₃

yes

characteristic

#431

—

#332

—

C₂²×C₃×C₅

—

C₂²×C₃×C₅

—

C₂²×C₃×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₃×C₄×C₅

—

C₃×C₄×C₅

—

C₃×C₄×C₅

—

C₃×C₄×C₅

#101

yes

—

C₃×C₄×C₅

#135

yes

—

C₂×C₃×Q₈

—

C₂²×C₃×C₄

C₂×C₄×C₅

yes

characteristic

C₂²×C₃×C₄

C₅×D₄

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

C₂×C₄×C₅

yes

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₂×C₄×C₅

yes

characteristic

C₃×C₄²

C₅×D₄

yes

—

C₃×C₄²

C₂³×C₅

yes

characteristic

C₂×C₃⋊C₈

—

C₂³×C₅

#217

yes

characteristic

C₅×Q₈

—

C₂×C₄×C₅

—

100

C₂×C₄×C₅

—

101

C₂×C₄×C₅

—

102

C₂×C₄×C₅

#217

yes

—

103

C₂×C₄×C₅

#217

yes

characteristic

104

C₂×C₄×C₅

C₂²×Dic₃

yes

characteristic

105

C₂×C₄×C₅

C₂×#67

yes

—

106

C₅×C₈

—

107

C₄⋊Q₈

C₂×C₅×D₃

yes

—

108

#125

C₅×Dic₃

yes

—

109

C₂²⋊Q₈

—

110

C₄×Q₈

—

111

#117

C₅×Dic₃

yes

—

112

C₂×C₄²

C₂×C₅×D₃

yes

characteristic

113

C₄⋊C₈

—

114

C₂²⋊C₈

—

115

C₄×C₈

—

116

C₂×C₃×C₅

—

117

C₂×C₃×C₅

C₂×#101

yes

characteristic

118

C₂×C₃×C₅

#416

yes

—

119

C₂³×C₃

C₅×C₂²⋊C₄

yes

characteristic

120

C₃×Q₈

—

121

C₂×C₃×C₄

—

122

C₂×C₃×C₄

—

123

C₂×C₃×C₄

—

124

C₂×C₃×C₄

C₅×C₂²⋊C₄

yes

—

125

C₂×C₃×C₄

C₅×C₂²⋊C₄

yes

characteristic

126

C₂×C₃×C₄

C₂²×C₄×C₅

yes

characteristic

127

C₂×C₃×C₄

C₂×C₅×D₄

yes

—

128

C₃⋊C₈

—

129

C₂²×C₅

—

130

C₂²×C₅

—

131

C₂²×C₅

C₂×#217

yes

characteristic

132

C₄×C₅

—

133

C₄×C₅

—

134

C₄×C₅

—

135

C₄×C₅

#836

yes

—

136

C₄×C₅

#950

yes

—

137

C₂×Q₈

—

138

C₂²×C₄

C₂×C₅×Dic₃

yes

characteristic

139

C₂²×C₄

C₅×#67

yes

—

140

C₂×C₈

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

C₂×C₅×Dic₃

yes

—

144

C₂²⋊C₄

—

145

C₄²

—

146

C₄²

C₂×C₅×Dic₃

yes

characteristic

147

C₄²

C₅×#67

yes

—

148

C₄²

C₂²×C₅×D₃

yes

characteristic

149

C₃×C₅

#1544

yes

characteristic, complemented, Hall

150

C₂²×C₃

—

151

C₂²×C₃

—

152

C₂²×C₃

C₂×C₅×C₂²⋊C₄

yes

characteristic

153

C₃×C₄

—

154

C₃×C₄

—

155

C₃×C₄

—

156

C₃×C₄

C₅×#101

yes

—

157

C₃×C₄

C₅×#135

yes

—

158

C₂×C₅

—

159

C₂×C₅

C₂×#836

yes

characteristic

160

C₂×C₅

#5233

yes

—

161

C₂³

C₅×#217

yes

characteristic

162

Q₈

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

C₅×#217

yes

—

167

C₂×C₄

C₅×#217

yes

characteristic

168

C₂×C₄

C₂²×C₅×Dic₃

yes

characteristic

169

C₂×C₄

C₂×C₅×#67

yes

—

170

C₈

—

171

C₂×C₃

—

172

C₂×C₃

C₂×C₅×#101

yes

characteristic

173

C₂×C₃

C₅×#416

yes

—

174

C₅

#71829

yes

characteristic, direct product factor, Sylow

175

C₂²

—

176

C₂²

—

177

C₂²

C₂×C₅×#217

yes

characteristic

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

C₅×#836

yes

—

182

C₄

C₅×#950

yes

—

183

C₃

C₅×#1544

yes

characteristic, complemented, Sylow

184

C₂

—

185

C₂

C₂×C₅×#836

yes

characteristic

186

C₂

C₅×#5233

yes

—

	subgroup	G/H
center	C₂×C₄×C₅	#217
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#533	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂³×C₃×C₅	C₂²⋊C₄
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×#533	D₃
hypocenter	C₃	C₅×#1544
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₅×#533

subgroup

G/H

center

C₂×C₄×C₅

#217

commutator

C₂×C₃×C₄

C₂²×C₄×C₅

Frattini

C₄²

C₂²×C₅×D₃

Fitting

C₃×C₅×#533

C₂

socle

C₂²×C₃×C₅

C₂×C₂²⋊C₄

prime radical

C₂³×C₃×C₅

C₂²⋊C₄

cosocle

C₃×C₄²

C₂³×C₅

D₃

C₃

G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₅×#533

N	H ≅ G/N	type
3. C₅×#4547	C₂	semidirect
14. #71829	C₅	direct (N × H)
32. #4547	C₂×C₅	semidirect
149. C₃×C₅	#1544	semidirect
174. C₅	#71829	direct (N × H)
183. C₃	C₅×#1544	semidirect

H ≅ G/N

type

3. C₅×#4547

C₂

semidirect

14. #71829

C₅

direct (N × H)

32. #4547

C₂×C₅

semidirect

149. C₃×C₅

#1544

semidirect

174. C₅

#71829

direct (N × H)

183. C₃

C₅×#1544

semidirect

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

subgroup	index
C₃×C₅×#533	2
C₅×#4925	2
C₅×#4547	2
C₅×#1544	3
#71829	5

subgroup

index

C₃×C₅×#533

Subgroups — C₅×#71829 #14284742

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C5×#71829 #14284742

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71829 #14284742