Group #34192722 — subgroups

Subgroups — #34192722

Order 20224.

406 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 129 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	#23768642	C₂	1	1	yes	maximal, characteristic
2	#23764164	C₂	1	2	yes	maximal
3	#23763559	C₂	1	1	yes	maximal, characteristic
4	#23761239	C₂	1	2	yes	maximal
5	C₇₉×#1296	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579094	—	2	2	no	—
7	C₂×#14579093	—	2	2	no	—
8	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
9	#19098502	—	2	2	no	—
10	#19098462	—	2	2	no	—
11	#19098433	C₂²	1	1	yes	characteristic
12	#19098428	—	2	2	no	—
13	C₄×C₄⋊Dic₇₉	C₂²	1	2	yes	—
14	#19098326	—	2	2	no	—
15	C₂×C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
16	#384×C₇₉	C₂²	1	1	yes	characteristic
17	C₂²×C₇₉⋊C₈	—	2	1	no	—
18	C₂×C₄⋊Dic₇₉	—	2	2	no	—
19	C₂×C₄⋊Dic₇₉	—	2	1	no	—
20	C₂×C₄⋊Dic₇₉	—	2	2	no	—
21	C₂×C₄⋊Dic₇₉	D₄	1	2	yes	—
22	C₂×C₄⋊Dic₇₉	D₄	1	2	yes	—
23	C₂×C₄×Dic₇₉	—	2	2	no	—
24	#14579094	—	4	4	no	—
25	#14579093	—	4	4	no	—
26	C₄×C₇₉⋊C₈	—	2	1	no	—
27	C₄×C₇₉⋊C₈	Q₈	1	2	yes	—
28	C₇₉⋊#95	—	2	2	no	—
29	C₇₉⋊#95	—	2	4	no	—
30	C₇₉⋊#95	—	2	2	no	—
31	C₂²×C₈×C₇₉	—	2	2	no	—
32	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
33	C₂×C₄⋊C₄×C₇₉	D₄	1	2	yes	—
34	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₇₉	—	2	2	no	—
36	C₄⋊C₈×C₇₉	Q₈	1	4	yes	—
37	C₂²×Dic₇₉	—	2	2	no	—
38	C₂²×Dic₇₉	—	2	2	no	—
39	C₂²×Dic₇₉	—	4	1	no	—
40	C₂×C₇₉⋊C₈	—	2	2	no	—
41	C₂×C₇₉⋊C₈	—	2	2	no	—
42	C₂×C₇₉⋊C₈	—	2	2	no	—
43	C₄⋊Dic₇₉	—	2	2	no	—
44	C₄⋊Dic₇₉	—	2	2	no	—
45	C₄⋊Dic₇₉	—	2	2	no	—
46	C₄⋊Dic₇₉	—	2	2	no	—
47	C₄⋊Dic₇₉	—	4	2	no	—
48	C₄⋊Dic₇₉	—	4	2	no	—
49	C₄⋊Dic₇₉	—	4	2	no	—
50	C₄⋊Dic₇₉	—	8	2	no	—
51	C₄×Dic₇₉	—	4	2	no	—
52	C₄×Dic₇₉	—	4	2	no	—
53	C₂²×C₄×C₇₉	—	2	2	no	—
54	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₇₉	—	2	4	no	—
57	C₂×C₈×C₇₉	—	4	2	no	—
58	C₂×C₈×C₇₉	—	4	2	no	—
59	C₄⋊C₄×C₇₉	—	4	2	no	—
60	C₄⋊C₄×C₇₉	—	4	2	no	—
61	C₄⋊C₄×C₇₉	—	4	2	no	—
62	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
63	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
64	C₄²×C₇₉	#41	1	1	yes	characteristic
65	C₂³×C₇₉	#120	1	1	yes	characteristic
66	C₂×C₄×C₇₉	—	2	1	no	—
67	C₂×C₄×C₇₉	—	2	1	no	—
68	C₂×C₄×C₇₉	—	4	4	no	—
69	C₂×C₄×C₇₉	—	4	2	no	—
70	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
71	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
73	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₇₉	#135	1	2	yes	—
75	C₂×C₄×C₇₉	#135	1	2	yes	—
76	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
77	C₂×C₄×C₇₉	#137	1	2	yes	—
78	C₂×Dic₇₉	—	4	2	no	—
79	C₂×Dic₇₉	—	4	2	no	—
80	C₂×Dic₇₉	—	4	2	no	—
81	C₂×Dic₇₉	—	4	2	no	—
82	C₂×Dic₇₉	—	4	2	no	—
83	C₂×Dic₇₉	—	4	2	no	—
84	C₂×Dic₇₉	—	8	2	no	—
85	C₂×Dic₇₉	—	8	1	no	—
86	C₈×C₇₉	—	4	4	no	—
87	C₇₉⋊C₈	—	2	4	no	—
88	C₂²×C₇₉	#395	1	1	yes	characteristic
89	C₂²×C₇₉	#449	1	2	yes	—
90	C₂²×C₇₉	#452	1	2	yes	—
91	C₂²×C₇₉	#453	1	1	yes	characteristic
92	C₂²×C₇₉	#456	1	1	yes	characteristic
93	C₄×C₇₉	—	2	1	no	—
94	C₄×C₇₉	—	2	1	no	—
95	C₄×C₇₉	—	8	2	no	—
96	C₄×C₇₉	#478	1	2	yes	—
97	C₄×C₇₉	#479	1	2	yes	—
98	C₄×C₇₉	#499	1	2	yes	—
99	C₄×C₇₉	#501	1	2	yes	—
100	Dic₇₉	—	8	2	no	—
101	Dic₇₉	—	8	2	no	—
102	Dic₇₉	—	16	1	no	—
103	#10576	—	79	1	no	maximal, Sylow
104	C₂×C₇₉	#1620	1	1	yes	characteristic
105	C₂×C₇₉	#1634	1	1	yes	characteristic
106	C₂×C₇₉	#1650	1	2	yes	—
107	C₂×C₇₉	#2059	1	2	yes	—
108	C₂×C₇₉	#2060	1	1	yes	characteristic
109	#2571	—	79	1	no	—
110	#1948	—	79	2	no	—
111	#1845	—	79	1	no	—
112	#1296	D₇₉	1	1	yes	characteristic
113	#1279	—	79	2	no	—
114	C₇₉	#10576	1	1	yes	characteristic, complemented, Sylow
115	C₂×#107	—	158	2	no	—
116	C₂×#106	—	158	2	no	—
117	C₂×C₄⋊C₈	C₂×D₇₉	1	2	yes	—
118	C₂×C₄×C₈	—	79	1	no	—
119	#400	—	158	2	no	—
120	#389	—	158	2	no	—
121	#384	—	79	1	no	—
122	#384	C₂×D₇₉	1	1	yes	characteristic
123	#382	—	158	2	no	—
124	C₄×C₄⋊C₄	—	79	2	no	—
125	#340	—	158	2	no	—
126	C₂²×C₈	—	2	2	no	—
127	C₂²×C₈	—	158	1	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	79	2	no	—
130	C₂×C₄⋊C₄	—	79	2	no	—
131	C₂×C₄⋊C₄	—	158	2	no	—
132	C₂×C₄⋊C₄	—	158	1	no	—
133	C₂×C₄⋊C₄	—	158	2	no	—
134	C₂×C₄⋊C₄	D₃₁₆	1	2	yes	—
135	C₂×C₄²	—	158	2	no	—
136	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
137	#107	—	316	4	no	—
138	#106	—	316	4	no	—
139	C₄⋊C₈	—	2	2	no	—
140	C₄⋊C₈	Dic₁₅₈	1	4	yes	—
141	C₄×C₈	—	79	2	no	—
142	C₄×C₈	—	158	1	no	—
143	#95	—	158	2	no	—
144	#95	—	158	4	no	—
145	#95	—	158	2	no	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	316	1	no	—
148	C₂²×C₄	—	158	2	no	—
149	C₂²×C₄	—	158	2	no	—
150	C₂²×C₄	D₄×D₇₉	1	2	yes	—
151	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	—	158	2	no	—
156	C₂×C₈	—	158	2	no	—
157	C₂×C₈	—	158	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	316	2	no	—
162	C₄⋊C₄	—	316	2	no	—
163	C₄⋊C₄	—	316	2	no	—
164	C₄⋊C₄	—	632	2	no	—
165	C₄⋊C₄	—	158	2	no	—
166	C₄⋊C₄	—	158	2	no	—
167	C₄⋊C₄	—	158	2	no	—
168	C₄⋊C₄	—	158	2	no	—
169	C₄²	—	316	2	no	—
170	C₄²	—	316	2	no	—
171	C₄²	Q₈×D₇₉	1	1	yes	characteristic
172	C₄²	C₂×Dic₁₅₈	1	2	yes	—
173	C₄²	#11929623	1	1	yes	characteristic
174	C₂³	#14579131	1	1	yes	characteristic
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	4	4	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	316	2	no	—
180	C₂×C₄	—	316	2	no	—
181	C₂×C₄	—	316	2	no	—
182	C₂×C₄	—	316	2	no	—
183	C₂×C₄	—	316	2	no	—
184	C₂×C₄	—	316	2	no	—
185	C₂×C₄	—	632	2	no	—
186	C₂×C₄	—	632	1	no	—
187	C₂×C₄	#14579138	1	2	yes	—
188	C₂×C₄	#14579141	1	1	yes	characteristic
189	C₂×C₄	#14579160	1	2	yes	—
190	C₂×C₄	#14579161	1	1	yes	characteristic
191	C₂×C₄	#14579182	1	2	yes	—
192	C₂×C₄	#14579182	1	2	yes	—
193	C₂×C₄	#14579185	1	2	yes	—
194	C₂×C₄	#14579193	1	2	yes	—
195	C₈	—	4	4	no	—
196	C₈	—	158	4	no	—
197	C₂²	#19098484	1	1	yes	characteristic
198	C₂²	#19098709	1	2	yes	—
199	C₂²	#19098728	1	2	yes	—
200	C₂²	#19098733	1	1	yes	characteristic
201	C₂²	#19098748	1	1	yes	characteristic
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	8	2	no	—
205	C₄	—	632	2	no	—
206	C₄	—	632	2	no	—
207	C₄	—	1264	1	no	—
208	C₄	#19098865	1	2	yes	—
209	C₄	#19098868	1	2	yes	—
210	C₄	#19098963	1	2	yes	—
211	C₄	#19098971	1	2	yes	—
212	C₂	#23762460	1	1	yes	characteristic
213	C₂	#23762546	1	1	yes	characteristic
214	C₂	#23762657	1	2	yes	—
215	C₂	#23764737	1	2	yes	—
216	C₂	#23764746	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23768642

C₂

yes

maximal, characteristic

#23764164

C₂

yes

maximal

#23763559

C₂

yes

maximal, characteristic

#23761239

C₂

yes

maximal

C₇₉×#1296

C₂

yes

maximal, characteristic

C₂×#14579094

—

C₂×#14579093

—

C₂×C₄×C₇₉⋊C₈

C₂²

yes

—

—

C₂²

yes

characteristic

#19098428

—

C₄×C₄⋊Dic₇₉

C₂²

yes

—

#19098326

—

C₂×C₄⋊C₈×C₇₉

C₂²

yes

—

#384×C₇₉

C₂²

yes

characteristic

C₂²×C₇₉⋊C₈

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

D₄

yes

—

C₂×C₄⋊Dic₇₉

D₄

yes

—

C₂×C₄×Dic₇₉

—

#14579094

—

#14579093

—

C₄×C₇₉⋊C₈

—

C₄×C₇₉⋊C₈

Q₈

yes

—

C₇₉⋊#95

—

C₇₉⋊#95

—

C₇₉⋊#95

—

C₂²×C₈×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

Q₈

yes

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂×Q₈

yes

characteristic

C₄²×C₇₉

C₂×Q₈

yes

—

C₄²×C₇₉

yes

C₂³×C₇₉

yes

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

C₄⋊Q₈

yes

—

C₂×C₄×C₇₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

#135

yes

—

C₂×C₄×C₇₉

#135

yes

—

C₂×C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇₉

#137

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

#395

yes

characteristic

C₂²×C₇₉

#449

yes

—

C₂²×C₇₉

#452

yes

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#478

yes

—

C₄×C₇₉

#479

yes

—

C₄×C₇₉

#499

yes

—

C₄×C₇₉

#501

yes

—

100

Dic₇₉

—

101

Dic₇₉

—

102

Dic₇₉

—

103

#10576

—

maximal, Sylow

104

C₂×C₇₉

#1620

yes

characteristic

105

C₂×C₇₉

#1634

yes

characteristic

106

C₂×C₇₉

#1650

yes

—

107

C₂×C₇₉

#2059

yes

—

108

C₂×C₇₉

#2060

yes

characteristic

109

#2571

—

110

#1948

—

111

#1845

—

112

#1296

D₇₉

yes

characteristic

113

#1279

—

114

C₇₉

#10576

yes

characteristic, complemented, Sylow

115

C₂×#107

—

158

—

116

C₂×#106

—

158

—

117

C₂×C₄⋊C₈

C₂×D₇₉

yes

—

118

C₂×C₄×C₈

—

119

#400

—

158

—

120

#389

—

158

—

121

#384

—

122

#384

C₂×D₇₉

yes

characteristic

123

#382

—

158

—

124

C₄×C₄⋊C₄

—

125

#340

—

158

—

126

C₂²×C₈

—

127

C₂²×C₈

—

158

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

131

C₂×C₄⋊C₄

—

158

—

132

C₂×C₄⋊C₄

—

158

—

133

C₂×C₄⋊C₄

—

158

—

134

C₂×C₄⋊C₄

D₃₁₆

yes

—

135

C₂×C₄²

—

158

—

136

C₂×C₄²

C₂²×D₇₉

yes

characteristic

137

#107

—

316

—

138

#106

—

316

—

139

C₄⋊C₈

—

140

C₄⋊C₈

Dic₁₅₈

yes

—

141

C₄×C₈

—

142

C₄×C₈

—

158

—

143

#95

—

158

—

144

#95

—

158

—

145

#95

—

158

—

146

C₂²×C₄

—

147

C₂²×C₄

—

316

—

148

C₂²×C₄

—

158

—

149

C₂²×C₄

—

158

—

150

C₂²×C₄

D₄×D₇₉

yes

—

151

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

158

—

156

C₂×C₈

—

158

—

157

C₂×C₈

—

158

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

316

—

162

C₄⋊C₄

—

316

—

163

C₄⋊C₄

—

316

—

164

C₄⋊C₄

—

632

—

165

C₄⋊C₄

—

158

—

166

C₄⋊C₄

—

158

—

167

C₄⋊C₄

—

158

—

168

C₄⋊C₄

—

158

—

169

C₄²

—

316

—

170

C₄²

—

316

—

171

C₄²

Q₈×D₇₉

yes

characteristic

172

C₄²

C₂×Dic₁₅₈

yes

—

173

C₄²

#11929623

yes

characteristic

174

C₂³

#14579131

yes

characteristic

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

316

—

180

C₂×C₄

—

316

—

181

C₂×C₄

—

316

—

182

C₂×C₄

—

316

—

183

C₂×C₄

—

316

—

184

C₂×C₄

—

316

—

185

C₂×C₄

—

632

—

186

C₂×C₄

—

632

—

187

C₂×C₄

#14579138

yes

—

188

C₂×C₄

#14579141

yes

characteristic

189

C₂×C₄

#14579160

yes

—

190

C₂×C₄

#14579161

yes

characteristic

191

C₂×C₄

#14579182

yes

—

192

C₂×C₄

#14579182

yes

—

193

C₂×C₄

#14579185

yes

—

194

C₂×C₄

#14579193

yes

—

195

C₈

—

196

C₈

—

158

—

197

C₂²

#19098484

yes

characteristic

198

C₂²

#19098709

yes

—

199

C₂²

#19098728

yes

—

200

C₂²

#19098733

yes

characteristic

201

C₂²

#19098748

yes

characteristic

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

—

632

—

206

C₄

—

632

—

207

C₄

—

1264

—

208

C₄

#19098865

yes

—

209

C₄

#19098868

yes

—

210

C₄

#19098963

yes

—

211

C₄

#19098971

yes

—

212

C₂

#23762460

yes

characteristic

213

C₂

#23762546

yes

characteristic

214

C₂

#23762657

yes

—

215

C₂

#23764737

yes

—

216

C₂

#23764746

yes

characteristic

	subgroup	G/H
center	C₂³	#14579131
commutator, cosocle	C₂×C₄²×C₇₉	C₂³
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1296	C₂
socle, prime radical	C₂³×C₇₉	#120
hypercenter	#1296	D₇₉
hypocenter	C₇₉	#10576
LCS	G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1296

subgroup

G/H

center

C₂³

#14579131

commutator, cosocle

C₂×C₄²×C₇₉

C₂³

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1296

C₂

C₂³×C₇₉

D₇₉

C₇₉

G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1296

N	H ≅ G/N	type
114. C₇₉	#10576	semidirect

H ≅ G/N

type

114. C₇₉

#10576

semidirect

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

subgroup	index
#23768642	2
#23764164	2
#23763559	2
#23761239	2
C₇₉×#1296	2
#10576	79

subgroup

index

C₇₉×#1296

Subgroups — #34192722

Conjugacy classes of subgroups

Complements

Maximal subgroups