Group #34181725 — subgroups

Subgroups — #34181725

Order 20224.

404 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 115 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	#23761714	C₂	1	2	yes	maximal
2	#23761233	C₂	1	2	yes	maximal, complemented
3	#23761229	C₂	1	2	yes	maximal, complemented
4	C₇₉×#2298	C₂	1	1	yes	maximal, characteristic
5	#19098616	C₂²	1	4	yes	—
6	#19098289	—	2	2	no	—
7	#19098287	—	2	2	no	—
8	C₂×#117×C₇₉	C₂²	1	1	yes	characteristic
9	#388×C₇₉	—	2	2	no	—
10	#387×C₇₉	—	2	2	no	—
11	C₄×C₄⋊C₄×C₇₉	C₄	1	2	yes	—
12	C₄×C₂²⋊C₄×C₇₉	C₄	1	2	yes	—
13	#376×C₇₉	C₂²	1	1	yes	characteristic
14	C₄³×C₇₉	C₂²	1	1	yes	characteristic
15	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
16	C₂²⋊(C₇₉⋊C₈)	—	2	4	no	—
17	C₄×C₇₉⋊C₈	—	2	4	no	—
18	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
19	#117×C₇₉	—	2	2	no	—
20	#117×C₇₉	—	2	2	no	—
21	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
22	C₂×C₄⋊C₄×C₇₉	D₄	1	2	yes	—
23	C₂×C₂²⋊C₄×C₇₉	—	2	2	no	—
24	C₂×C₂²⋊C₄×C₇₉	D₄	1	2	yes	—
25	C₂×C₄²×C₇₉	—	2	2	no	—
26	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
30	#95×C₇₉	—	2	2	no	—
31	#95×C₇₉	—	2	2	no	—
32	#95×C₇₉	—	2	2	no	—
33	C₂×C₇₉⋊C₈	—	4	4	no	—
34	C₂⁴×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₇₉	—	2	2	no	—
36	C₂²×C₄×C₇₉	—	2	1	no	—
37	C₂²×C₄×C₇₉	—	2	1	no	—
38	C₂²×C₄×C₇₉	—	2	4	no	—
39	C₂²×C₄×C₇₉	—	2	2	no	—
40	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
44	C₄⋊C₄×C₇₉	—	2	4	no	—
45	C₄⋊C₄×C₇₉	—	4	4	no	—
46	C₂²⋊C₄×C₇₉	—	2	4	no	—
47	C₂²⋊C₄×C₇₉	—	4	4	no	—
48	C₄²×C₇₉	—	2	1	no	—
49	C₄²×C₇₉	—	2	1	no	—
50	C₄²×C₇₉	—	2	4	no	—
51	C₄²×C₇₉	—	2	4	no	—
52	C₄²×C₇₉	—	4	4	no	—
53	C₄²×C₇₉	#41	1	2	yes	—
54	C₄²×C₇₉	#41	1	2	yes	—
55	C₂³×C₇₉	—	2	1	no	—
56	C₂³×C₇₉	—	2	2	no	—
57	C₂³×C₇₉	—	4	2	no	—
58	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₇₉	—	2	4	no	—
60	C₂×C₄×C₇₉	—	2	2	no	—
61	C₂×C₄×C₇₉	—	2	1	no	—
62	C₂×C₄×C₇₉	—	2	2	no	—
63	C₂×C₄×C₇₉	—	4	2	no	—
64	C₂×C₄×C₇₉	—	4	2	no	—
65	C₂×C₄×C₇₉	—	4	2	no	—
66	C₂×C₄×C₇₉	—	4	2	no	—
67	C₂×C₄×C₇₉	—	4	2	no	—
68	C₂×C₄×C₇₉	—	4	2	no	—
69	C₂×C₄×C₇₉	—	4	2	no	—
70	C₂×C₄×C₇₉	—	4	2	no	—
71	C₂×C₄×C₇₉	—	4	2	no	—
72	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
73	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
74	C₂×C₄×C₇₉	#123	1	4	yes	—
75	C₂×C₄×C₇₉	#131	1	4	yes	—
76	C₇₉⋊C₈	—	8	4	no	—
77	C₂²×C₇₉	—	2	2	no	—
78	C₂²×C₇₉	—	4	1	no	—
79	C₂²×C₇₉	—	4	2	no	—
80	C₂²×C₇₉	—	8	2	no	—
81	C₂²×C₇₉	#381	1	1	yes	characteristic
82	C₂²×C₇₉	#408	1	2	yes	—
83	C₄×C₇₉	—	2	4	no	—
84	C₄×C₇₉	—	4	4	no	—
85	C₄×C₇₉	—	4	1	no	—
86	C₄×C₇₉	—	4	2	no	—
87	C₄×C₇₉	—	8	2	no	—
88	C₄×C₇₉	#433	1	2	yes	—
89	C₄×C₇₉	#433	1	2	yes	—
90	#8733	—	79	1	no	maximal, Sylow
91	C₂×C₇₉	—	2	2	no	—
92	C₂×C₇₉	—	8	1	no	—
93	C₂×C₇₉	#1818	1	1	yes	characteristic
94	C₂×C₇₉	#1827	1	2	yes	—
95	#2298	D₇₉	1	1	yes	characteristic
96	#1458	—	79	2	no	—
97	#1276	—	79	2	no	—
98	#1274	—	79	2	no	—
99	C₇₉	#8733	1	1	yes	characteristic, complemented, Sylow
100	C₂×#117	C₂×D₇₉	1	1	yes	characteristic
101	#431	—	79	4	no	—
102	#388	—	2	2	no	—
103	#387	—	2	2	no	—
104	C₄×C₄⋊C₄	Dic₇₉	1	2	yes	—
105	C₄×C₂²⋊C₄	Dic₇₉	1	2	yes	—
106	#376	C₂×D₇₉	1	1	yes	characteristic
107	C₄³	C₂×D₇₉	1	1	yes	characteristic
108	#324	—	158	2	no	—
109	#323	—	158	2	no	—
110	C₂³×C₄	C₂×Dic₇₉	1	1	yes	characteristic
111	#117	—	2	2	no	—
112	#117	—	2	2	no	—
113	C₂×C₄⋊C₄	—	2	2	no	—
114	C₂×C₄⋊C₄	C₂²⋊D₇₉	1	2	yes	—
115	C₂×C₂²⋊C₄	—	2	2	no	—
116	C₂×C₂²⋊C₄	C₂²⋊D₇₉	1	2	yes	—
117	C₂×C₄²	—	2	2	no	—
118	C₂×C₄²	C₂×Dic₇₉	1	2	yes	—
119	C₂×C₄²	C₂×Dic₇₉	1	2	yes	—
120	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
121	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
122	C₄⋊C₈	—	158	4	no	—
123	C₂²⋊C₈	—	158	4	no	—
124	C₄×C₈	—	158	4	no	—
125	#95	—	2	2	no	—
126	#95	—	2	2	no	—
127	#95	—	2	2	no	—
128	C₂⁴	C₂²⋊Dic₇₉	1	1	yes	characteristic
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	—	2	1	no	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	—	2	4	no	—
133	C₂²×C₄	—	2	2	no	—
134	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
135	C₂²×C₄	C₂²⋊Dic₇₉	1	2	yes	—
136	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
137	C₂²×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
138	C₂×C₈	—	316	4	no	—
139	C₄⋊C₄	—	2	4	no	—
140	C₄⋊C₄	—	4	4	no	—
141	C₂²⋊C₄	—	2	4	no	—
142	C₂²⋊C₄	—	4	4	no	—
143	C₄²	—	2	1	no	—
144	C₄²	—	2	1	no	—
145	C₄²	—	2	4	no	—
146	C₄²	—	2	4	no	—
147	C₄²	—	4	4	no	—
148	C₄²	#11929624	1	2	yes	—
149	C₄²	#11929624	1	2	yes	—
150	C₂³	—	2	1	no	—
151	C₂³	—	2	2	no	—
152	C₂³	—	4	2	no	—
153	C₂³	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	#14579120	1	1	yes	characteristic
168	C₂×C₄	#14579120	1	1	yes	characteristic
169	C₂×C₄	#14579144	1	4	yes	—
170	C₂×C₄	#14579168	1	4	yes	—
171	C₈	—	632	4	no	—
172	C₂²	—	2	2	no	—
173	C₂²	—	4	1	no	—
174	C₂²	—	4	2	no	—
175	C₂²	—	8	2	no	—
176	C₂²	#19098426	1	1	yes	characteristic
177	C₂²	#19098530	1	2	yes	—
178	C₄	—	2	4	no	—
179	C₄	—	4	4	no	—
180	C₄	—	4	1	no	—
181	C₄	—	4	2	no	—
182	C₄	—	8	2	no	—
183	C₄	#19098624	1	2	yes	—
184	C₄	#19098624	1	2	yes	—
185	C₂	—	2	2	no	—
186	C₂	—	8	1	no	—
187	C₂	#23763471	1	1	yes	characteristic
188	C₂	#23763504	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23761714

C₂

yes

maximal

#23761233

C₂

yes

maximal, complemented

#23761229

C₂

yes

maximal, complemented

C₇₉×#2298

C₂

yes

maximal, characteristic

#19098616

C₂²

yes

—

#19098289

—

#19098287

—

C₂×#117×C₇₉

C₂²

yes

characteristic

#388×C₇₉

—

#387×C₇₉

—

C₄×C₄⋊C₄×C₇₉

C₄

yes

—

C₄×C₂²⋊C₄×C₇₉

C₄

yes

—

#376×C₇₉

C₂²

yes

characteristic

C₄³×C₇₉

C₂²

yes

characteristic

C₄⋊(C₇₉⋊C₈)

—

C₂²⋊(C₇₉⋊C₈)

—

C₄×C₇₉⋊C₈

—

C₂³×C₄×C₇₉

C₂×C₄

yes

characteristic

#117×C₇₉

—

#117×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

D₄

yes

—

C₂×C₂²⋊C₄×C₇₉

—

C₂×C₂²⋊C₄×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

—

C₂×C₄²×C₇₉

C₂×C₄

yes

—

C₂×C₄²×C₇₉

C₂×C₄

yes

—

C₂×C₄²×C₇₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₉

C₂³

yes

characteristic

#95×C₇₉

—

#95×C₇₉

—

#95×C₇₉

—

C₂×C₇₉⋊C₈

—

C₂⁴×C₇₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×D₄

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

#41

yes

—

C₄²×C₇₉

#41

yes

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

C₂×C₂²⋊C₄

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

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

—

C₂×C₄×C₇₉

#131

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#381

yes

characteristic

C₂²×C₇₉

#408

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#433

yes

—

C₄×C₇₉

#433

yes

—

#8733

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#1818

yes

characteristic

C₂×C₇₉

#1827

yes

—

#2298

D₇₉

yes

—

—

—

C₇₉

#8733

yes

characteristic, complemented, Sylow

100

C₂×#117

C₂×D₇₉

yes

characteristic

101

#431

—

102

#388

—

103

#387

—

104

C₄×C₄⋊C₄

Dic₇₉

yes

—

105

C₄×C₂²⋊C₄

Dic₇₉

yes

—

106

#376

C₂×D₇₉

yes

characteristic

107

C₄³

C₂×D₇₉

yes

characteristic

108

#324

—

158

—

109

#323

—

158

—

110

C₂³×C₄

C₂×Dic₇₉

yes

characteristic

111

#117

—

112

#117

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

C₂²⋊D₇₉

yes

—

115

C₂×C₂²⋊C₄

—

116

C₂×C₂²⋊C₄

C₂²⋊D₇₉

yes

—

117

C₂×C₄²

—

118

C₂×C₄²

C₂×Dic₇₉

yes

—

119

C₂×C₄²

C₂×Dic₇₉

yes

—

120

C₂×C₄²

C₂×Dic₇₉

yes

characteristic

121

C₂×C₄²

C₂²×D₇₉

yes

characteristic

122

C₄⋊C₈

—

158

—

123

C₂²⋊C₈

—

158

—

124

C₄×C₈

—

158

—

125

#95

—

126

#95

—

127

#95

—

128

C₂⁴

C₂²⋊Dic₇₉

yes

characteristic

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂²×C₄

C₂²⋊Dic₇₉

yes

characteristic

135

C₂²×C₄

C₂²⋊Dic₇₉

yes

—

136

C₂²×C₄

C₂²×Dic₇₉

yes

characteristic

137

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

—

138

C₂×C₈

—

316

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₂²⋊C₄

—

142

C₂²⋊C₄

—

143

C₄²

—

144

C₄²

—

145

C₄²

—

146

C₄²

—

147

C₄²

—

148

C₄²

#11929624

yes

—

149

C₄²

#11929624

yes

—

150

C₂³

—

151

C₂³

—

152

C₂³

—

153

C₂³

C₂×C₂²⋊Dic₇₉

yes

characteristic

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

#14579120

yes

characteristic

168

C₂×C₄

#14579120

yes

characteristic

169

C₂×C₄

#14579144

yes

—

170

C₂×C₄

#14579168

yes

—

171

C₈

—

632

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

—

176

C₂²

#19098426

yes

characteristic

177

C₂²

#19098530

yes

—

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

#19098624

yes

—

184

C₄

#19098624

yes

—

185

C₂

—

186

C₂

—

187

C₂

#23763471

yes

characteristic

188

C₂

#23763504

yes

—

	subgroup	G/H
center	C₂×C₄	#14579120
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#2298	C₂
socle	C₂²×C₇₉	#381
prime radical	C₂⁴×C₇₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#2298	D₇₉
hypocenter	C₇₉	#8733
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ #376 ◁ #2298

subgroup

G/H

center

C₂×C₄

#14579120

commutator

C₂²×C₄×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#2298

C₂

socle

C₂²×C₇₉

#381

prime radical

C₂⁴×C₇₉

C₂²⋊C₄

cosocle

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ #376 ◁ #2298

N	H ≅ G/N	type
2. #23761233	C₂	semidirect
3. #23761229	C₂	semidirect
99. C₇₉	#8733	semidirect

H ≅ G/N

type

2. #23761233

C₂

semidirect

3. #23761229

C₂

semidirect

99. C₇₉

#8733

semidirect

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

subgroup	index
#23761714	2
#23761233	2
#23761229	2
C₇₉×#2298	2
#8733	79

subgroup

index

C₇₉×#2298

Subgroups — #34181725

Conjugacy classes of subgroups

Complements

Maximal subgroups