Group #23763592 — subgroups

Subgroups — #23763592

Order 10112.

360 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 121 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	C₂²×C₄⋊Dic₇₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#14579092	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊(C₇₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#19098325	C₂	1	2	yes	maximal, complemented
5	#380×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₇₉	—	2	1	no	—
7	C₂²×C₇₉⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₇₉	—	2	1	no	—
9	C₂×C₄⋊Dic₇₉	—	2	4	no	—
10	C₂×C₄⋊Dic₇₉	C₂²	1	2	yes	—
11	#14579092	—	2	4	no	—
12	C₂²⋊(C₇₉⋊C₈)	C₄	1	4	yes	complemented
13	C₂³×C₄×C₇₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
17	#95×C₇₉	—	2	1	no	—
18	C₂²×Dic₇₉	—	2	4	no	—
19	C₂²×Dic₇₉	—	2	2	no	—
20	C₂²×Dic₇₉	—	4	2	no	—
21	C₂²×Dic₇₉	—	4	2	no	—
22	C₂×C₇₉⋊C₈	—	2	2	no	—
23	C₂×C₇₉⋊C₈	—	2	2	no	—
24	C₂×C₇₉⋊C₈	C₂×C₄	1	4	yes	complemented
25	C₄⋊Dic₇₉	—	2	4	no	—
26	C₄⋊Dic₇₉	—	2	2	no	—
27	C₄⋊Dic₇₉	D₄	1	4	yes	—
28	C₂⁴×C₇₉	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₇₉	—	2	2	no	—
30	C₂²×C₄×C₇₉	—	2	2	no	—
31	C₂²×C₄×C₇₉	—	2	1	no	—
32	C₂²×C₄×C₇₉	—	2	1	no	—
33	C₂²×C₄×C₇₉	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₇₉	Q₈	1	2	yes	—
36	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₇₉	—	2	2	no	—
38	C₄⋊C₄×C₇₉	—	2	2	no	—
39	C₂²⋊C₄×C₇₉	—	4	2	no	—
40	C₂³×C₇₉	—	2	1	no	—
41	C₂³×C₇₉	—	2	1	no	—
42	C₂³×C₇₉	—	2	1	no	—
43	C₂³×C₇₉	—	2	1	no	—
44	C₂³×C₇₉	C₄⋊C₄	1	2	yes	—
45	C₂³×C₇₉	SD₁₆	1	2	yes	—
46	C₂³×C₇₉	SD₁₆	1	2	yes	—
47	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₇₉	—	2	1	no	—
49	C₂×C₄×C₇₉	—	2	1	no	—
50	C₂×C₄×C₇₉	—	2	4	no	—
51	C₂×C₄×C₇₉	—	2	4	no	—
52	C₂×C₄×C₇₉	—	4	1	no	—
53	C₂×C₄×C₇₉	—	4	1	no	—
54	C₂×C₄×C₇₉	—	4	1	no	—
55	C₂×C₄×C₇₉	—	4	1	no	—
56	C₂×C₄×C₇₉	—	4	1	no	—
57	C₂×C₄×C₇₉	—	4	1	no	—
58	C₂×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
59	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₄×C₇₉	#41	1	2	yes	—
63	C₂×Dic₇₉	—	2	4	no	—
64	C₂×Dic₇₉	—	4	8	no	—
65	C₂×Dic₇₉	—	4	2	no	—
66	C₂×Dic₇₉	—	4	2	no	—
67	C₇₉⋊C₈	—	2	4	no	—
68	C₂²×C₇₉	—	2	2	no	—
69	C₂²×C₇₉	—	2	2	no	—
70	C₂²×C₇₉	—	2	2	no	—
71	C₂²×C₇₉	—	2	2	no	—
72	C₂²×C₇₉	—	2	2	no	—
73	C₂²×C₇₉	—	2	2	no	—
74	C₂²×C₇₉	#106	1	4	yes	complemented
75	C₂²×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₇₉	#120	1	1	yes	characteristic
77	C₂²×C₇₉	#123	1	1	yes	characteristic
78	C₂²×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
79	C₂²×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
80	C₂²×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₂²×C₇₉	#137	1	1	yes	characteristic
82	C₄×C₇₉	—	2	2	no	—
83	C₄×C₇₉	—	8	1	no	—
84	C₄×C₇₉	—	8	1	no	—
85	C₄×C₇₉	C₄×D₄	1	2	yes	—
86	C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
87	Dic₇₉	—	4	4	no	—
88	C₂×C₇₉	—	2	4	no	—
89	C₂×C₇₉	#385	1	1	yes	characteristic
90	C₂×C₇₉	C₂×#106	1	1	yes	characteristic
91	C₂×C₇₉	#428	1	1	yes	characteristic
92	C₂×C₇₉	#448	1	1	yes	characteristic
93	C₂×C₇₉	#450	1	1	yes	characteristic
94	C₂×C₇₉	#481	1	1	yes	characteristic
95	C₂×C₇₉	#483	1	1	yes	characteristic
96	#1853	—	79	1	no	maximal, Sylow
97	C₇₉	#1853	1	1	yes	characteristic, complemented, Sylow
98	C₂²×C₄⋊C₄	—	79	1	no	—
99	C₂×#106	—	79	2	no	—
100	C₂×C₂²⋊C₈	—	79	1	no	—
101	#380	D₇₉	1	1	yes	characteristic
102	#340	—	79	2	no	—
103	C₂³×C₄	—	158	1	no	—
104	C₂³×C₄	C₂×D₇₉	1	1	yes	characteristic
105	C₂²×C₈	—	79	2	no	—
106	C₂×C₄⋊C₄	—	79	2	no	—
107	C₂×C₄⋊C₄	—	158	4	no	—
108	C₂×C₄⋊C₄	—	158	1	no	—
109	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
110	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
111	C₂×C₂²⋊C₄	—	2	1	no	—
112	#106	—	158	4	no	—
113	C₂²⋊C₈	—	79	4	no	—
114	#95	—	2	1	no	—
115	C₂⁴	C₂²⋊D₇₉	1	1	yes	characteristic
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	—	2	2	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	—	316	2	no	—
121	C₂²×C₄	—	316	2	no	—
122	C₂²×C₄	—	158	4	no	—
123	C₂²×C₄	—	158	2	no	—
124	C₂²×C₄	Dic₁₅₈	1	2	yes	—
125	C₂²×C₄	C₄×D₇₉	1	2	yes	—
126	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
128	C₂×C₈	—	79	4	no	—
129	C₂×C₈	—	158	2	no	—
130	C₂×C₈	—	158	2	no	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	2	2	no	—
133	C₄⋊C₄	—	79	4	no	—
134	C₄⋊C₄	—	158	4	no	—
135	C₄⋊C₄	—	158	2	no	—
136	C₂²⋊C₄	—	4	2	no	—
137	C₂³	—	2	1	no	—
138	C₂³	—	2	1	no	—
139	C₂³	—	2	1	no	—
140	C₂³	—	2	1	no	—
141	C₂³	Dic₇₉⋊C₄	1	2	yes	—
142	C₂³	#11929611	1	2	yes	—
143	C₂³	#11929612	1	2	yes	—
144	C₂³	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	316	8	no	—
156	C₂×C₄	—	316	2	no	—
157	C₂×C₄	—	316	2	no	—
158	C₂×C₄	—	158	4	no	—
159	C₂×C₄	Dic₇₉⋊C₄	1	2	yes	—
160	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
161	C₂×C₄	D₄×D₇₉	1	2	yes	—
162	C₂×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
163	C₂×C₄	#11929623	1	2	yes	—
164	C₈	—	158	4	no	—
165	C₂²	—	2	2	no	—
166	C₂²	—	2	2	no	—
167	C₂²	—	2	2	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	2	no	—
170	C₂²	—	2	2	no	—
171	C₂²	#14579092	1	4	yes	complemented
172	C₂²	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
173	C₂²	#14579129	1	1	yes	characteristic
174	C₂²	#14579142	1	1	yes	characteristic
175	C₂²	C₂×#11929611	1	1	yes	characteristic
176	C₂²	C₂×#11929612	1	1	yes	characteristic
177	C₂²	#14579186	1	1	yes	characteristic
178	C₂²	#14579191	1	1	yes	characteristic
179	C₄	—	2	2	no	—
180	C₄	—	8	1	no	—
181	C₄	—	8	1	no	—
182	C₄	—	316	4	no	—
183	C₄	(C₂²⋊D₇₉)⋊C₄	1	2	yes	—
184	C₄	#14579138	1	2	yes	—
185	C₂	—	2	4	no	—
186	C₂	#19098435	1	1	yes	characteristic
187	C₂	C₂×#14579092	1	1	yes	characteristic
188	C₂	#19098606	1	1	yes	characteristic
189	C₂	#19098701	1	1	yes	characteristic
190	C₂	#19098713	1	1	yes	characteristic
191	C₂	#19098879	1	1	yes	characteristic
192	C₂	#19098891	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊Dic₇₉

C₂

yes

maximal, characteristic

C₂×#14579092

C₂

yes

maximal, complemented

C₂×C₂²⋊(C₇₉⋊C₈)

C₂

yes

maximal, characteristic

#19098325

C₂

yes

maximal, complemented

#380×C₇₉

C₂

yes

maximal, characteristic

C₂³×Dic₇₉

—

C₂²×C₇₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

C₂²

yes

—

#14579092

—

C₂²⋊(C₇₉⋊C₈)

C₄

yes

complemented

C₂³×C₄×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₉

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₇₉

—

#95×C₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₂×C₄

yes

complemented

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

D₄

yes

—

C₂⁴×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×C₄

yes

—

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

Q₈

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

C₄⋊C₄

yes

—

C₂³×C₇₉

SD₁₆

yes

—

C₂³×C₇₉

SD₁₆

yes

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₄⋊C₄

yes

—

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₉

#41

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#106

yes

complemented

C₂²×C₇₉

C₂×C₄⋊C₄

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

C₂×SD₁₆

yes

characteristic

C₂²×C₇₉

C₂×SD₁₆

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₄×D₄

yes

—

C₄×C₇₉

C₂²⋊Q₈

yes

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

C₂²×C₄⋊C₄

—

C₂×#106

—

100

C₂×C₂²⋊C₈

—

101

#380

D₇₉

yes

characteristic

102

#340

—

103

C₂³×C₄

—

158

—

104

C₂³×C₄

C₂×D₇₉

yes

characteristic

105

C₂²×C₈

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

158

—

108

C₂×C₄⋊C₄

—

158

—

109

C₂×C₄⋊C₄

C₂×D₇₉

yes

characteristic

110

C₂×C₄⋊C₄

C₂×D₇₉

yes

characteristic

111

C₂×C₂²⋊C₄

—

112

#106

—

158

—

113

C₂²⋊C₈

—

114

#95

—

115

C₂⁴

C₂²⋊D₇₉

yes

characteristic

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

316

—

121

C₂²×C₄

—

316

—

122

C₂²×C₄

—

158

—

123

C₂²×C₄

—

158

—

124

C₂²×C₄

Dic₁₅₈

yes

—

125

C₂²×C₄

C₄×D₇₉

yes

—

126

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

127

C₂²×C₄

C₂²×D₇₉

yes

characteristic

128

C₂×C₈

—

129

C₂×C₈

—

158

—

130

C₂×C₈

—

158

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

158

—

135

C₄⋊C₄

—

158

—

136

C₂²⋊C₄

—

137

C₂³

—

138

C₂³

—

139

C₂³

—

140

C₂³

—

141

C₂³

Dic₇₉⋊C₄

yes

—

142

C₂³

#11929611

yes

—

143

C₂³

#11929612

yes

—

144

C₂³

C₂×C₂²⋊D₇₉

yes

characteristic

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

316

—

156

C₂×C₄

—

316

—

157

C₂×C₄

—

316

—

158

C₂×C₄

—

158

—

159

C₂×C₄

Dic₇₉⋊C₄

yes

—

160

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

161

C₂×C₄

D₄×D₇₉

yes

—

162

C₂×C₄

C₂×Dic₁₅₈

yes

characteristic

163

C₂×C₄

#11929623

yes

—

164

C₈

—

158

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

#14579092

yes

complemented

172

C₂²

C₂×Dic₇₉⋊C₄

yes

characteristic

173

C₂²

#14579129

yes

characteristic

174

C₂²

#14579142

yes

characteristic

175

C₂²

C₂×#11929611

yes

characteristic

176

C₂²

C₂×#11929612

yes

characteristic

177

C₂²

#14579186

yes

characteristic

178

C₂²

#14579191

yes

characteristic

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

316

—

183

C₄

(C₂²⋊D₇₉)⋊C₄

yes

—

184

C₄

#14579138

yes

—

185

C₂

—

186

C₂

#19098435

yes

characteristic

187

C₂

C₂×#14579092

yes

characteristic

188

C₂

#19098606

yes

characteristic

189

C₂

#19098701

yes

characteristic

190

C₂

#19098713

yes

characteristic

191

C₂

#19098879

yes

characteristic

192

C₂

#19098891

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	#380×C₇₉	C₂
socle	C₂³×C₇₉	C₂×D₄
prime radical	C₂⁴×C₇₉	D₄
cosocle	C₂²×C₄×C₇₉	C₂³
hypercenter	#380	D₇₉
hypocenter	C₇₉	#1853
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #380

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₉

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

#380×C₇₉

C₂

socle

C₂³×C₇₉

C₂×D₄

prime radical

C₂⁴×C₇₉

D₄

cosocle

C₂²×C₄×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #380

N	H ≅ G/N	type
2. C₂×#14579092	C₂	semidirect
4. #19098325	C₂	semidirect
12. C₂²⋊(C₇₉⋊C₈)	C₄	semidirect
24. C₂×C₇₉⋊C₈	C₂×C₄	semidirect
74. C₂²×C₇₉	#106	semidirect
97. C₇₉	#1853	semidirect
171. C₂²	#14579092	semidirect

H ≅ G/N

type

2. C₂×#14579092

C₂

semidirect

4. #19098325

C₂

semidirect

12. C₂²⋊(C₇₉⋊C₈)

C₄

semidirect

24. C₂×C₇₉⋊C₈

C₂×C₄

semidirect

74. C₂²×C₇₉

#106

semidirect

97. C₇₉

#1853

semidirect

171. C₂²

#14579092

semidirect

Subgroups — #23763592

Conjugacy classes of subgroups

Complements