Group #34122365 — subgroups

Subgroups — C₇₉×#10676 #34122365

Order 20224.

410 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 162 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	C₇₉×#1990	C₂	1	2	yes	maximal
2	C₇₉×#1956	C₂	1	1	yes	maximal, characteristic
3	C₇₉×#1845	C₂	1	1	yes	maximal, characteristic
4	C₇₉×#1834	C₂	1	2	yes	maximal
5	C₇₉×#1273	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
7	C₂×#107×C₇₉	—	2	2	no	—
8	C₂×#103×C₇₉	—	2	2	no	—
9	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
11	#384×C₇₉	C₂²	1	1	yes	characteristic
12	#384×C₇₉	C₂²	1	2	yes	—
13	#340×C₇₉	—	2	4	no	—
14	#340×C₇₉	—	2	4	no	—
15	C₂²×Q₈×C₇₉	—	2	1	no	—
16	C₂²×C₈×C₇₉	D₄	1	4	yes	—
17	C₂²×C₈×C₇₉	D₄	1	2	yes	—
18	C₄⋊Q₈×C₇₉	—	2	2	no	—
19	C₄⋊Q₈×C₇₉	—	2	1	no	—
20	C₄⋊Q₈×C₇₉	Q₈	1	2	yes	—
21	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
22	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
23	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
24	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
25	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
26	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
27	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
28	#107×C₇₉	—	4	4	no	—
29	#103×C₇₉	—	4	4	no	—
30	C₄×C₈×C₇₉	—	2	2	no	—
31	C₄×C₈×C₇₉	—	2	2	no	—
32	C₄×C₈×C₇₉	—	2	2	no	—
33	C₂×Q₈×C₇₉	—	2	2	no	—
34	C₂×Q₈×C₇₉	—	4	1	no	—
35	C₂×Q₈×C₇₉	—	4	2	no	—
36	C₂×Q₈×C₇₉	—	4	2	no	—
37	C₂²×C₄×C₇₉	—	4	1	no	—
38	C₂²×C₄×C₇₉	—	4	2	no	—
39	C₂²×C₄×C₇₉	—	4	1	no	—
40	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
42	C₂×C₈×C₇₉	—	2	2	no	—
43	C₂×C₈×C₇₉	—	2	4	no	—
44	C₂×C₈×C₇₉	—	2	4	no	—
45	C₂×C₈×C₇₉	—	2	2	no	—
46	C₂×C₈×C₇₉	—	2	4	no	—
47	C₂×C₈×C₇₉	D₈	1	4	yes	—
48	C₄⋊C₄×C₇₉	—	2	4	no	—
49	C₄⋊C₄×C₇₉	—	4	2	no	—
50	C₄⋊C₄×C₇₉	—	4	2	no	—
51	C₄⋊C₄×C₇₉	—	4	4	no	—
52	C₄⋊C₄×C₇₉	—	4	4	no	—
53	C₄⋊C₄×C₇₉	—	4	4	no	—
54	C₄⋊C₄×C₇₉	—	4	4	no	—
55	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₇₉	#41	1	1	yes	characteristic
57	C₄²×C₇₉	#41	1	2	yes	—
58	C₂³×C₇₉	#127	1	1	yes	characteristic
59	Q₈×C₇₉	—	4	4	no	—
60	Q₈×C₇₉	—	8	2	no	—
61	C₂×C₄×C₇₉	—	4	2	no	—
62	C₂×C₄×C₇₉	—	8	1	no	—
63	C₂×C₄×C₇₉	—	8	1	no	—
64	C₂×C₄×C₇₉	—	8	2	no	—
65	C₂×C₄×C₇₉	—	8	1	no	—
66	C₂×C₄×C₇₉	—	8	2	no	—
67	C₂×C₄×C₇₉	—	8	2	no	—
68	C₂×C₄×C₇₉	—	8	2	no	—
69	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
71	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
72	C₂×C₄×C₇₉	#123	1	2	yes	—
73	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
74	C₂×C₄×C₇₉	#135	1	4	yes	—
75	C₂×C₄×C₇₉	#137	1	2	yes	—
76	C₂×C₄×C₇₉	#137	1	4	yes	—
77	C₈×C₇₉	—	2	4	no	—
78	C₈×C₇₉	—	4	4	no	—
79	C₂²×C₇₉	#399	1	1	yes	characteristic
80	C₂²×C₇₉	#493	1	1	yes	characteristic
81	C₂²×C₇₉	#495	1	2	yes	—
82	C₂²×C₇₉	#497	1	1	yes	characteristic
83	C₂²×C₇₉	#497	1	2	yes	—
84	C₄×C₇₉	—	8	2	no	—
85	C₄×C₇₉	—	16	2	no	—
86	C₄×C₇₉	—	16	1	no	—
87	C₄×C₇₉	#474	1	2	yes	—
88	C₄×C₇₉	#479	1	4	yes	—
89	C₄×C₇₉	#480	1	2	yes	—
90	C₄×C₇₉	#485	1	4	yes	—
91	#10676	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂×C₇₉	#1709	1	1	yes	characteristic
93	C₂×C₇₉	#1718	1	2	yes	—
94	C₂×C₇₉	#1722	1	1	yes	characteristic
95	C₂×C₇₉	#2093	1	1	yes	characteristic
96	C₂×C₇₉	#2097	1	2	yes	—
97	#1990	C₂×C₇₉	1	2	yes	—
98	#1956	C₂×C₇₉	1	1	yes	characteristic
99	#1845	C₂×C₇₉	1	1	yes	characteristic
100	#1834	C₂×C₇₉	1	2	yes	—
101	#1273	C₂×C₇₉	1	1	yes	characteristic
102	C₇₉	#10676	1	1	yes	characteristic, direct product factor, Sylow
103	C₂×C₄⋊Q₈	C₂²×C₇₉	1	1	yes	characteristic
104	C₂×#107	—	2	2	no	—
105	C₂×#103	—	2	2	no	—
106	C₂×C₄×C₈	C₂²×C₇₉	1	2	yes	—
107	C₂×C₄×C₈	C₂²×C₇₉	1	1	yes	characteristic
108	#384	C₂²×C₇₉	1	1	yes	characteristic
109	#384	C₂²×C₇₉	1	2	yes	—
110	#340	—	2	4	no	—
111	#340	—	2	4	no	—
112	C₂²×Q₈	—	2	1	no	—
113	C₂²×C₈	D₄×C₇₉	1	4	yes	—
114	C₂²×C₈	D₄×C₇₉	1	2	yes	—
115	C₄⋊Q₈	—	2	2	no	—
116	C₄⋊Q₈	—	2	1	no	—
117	C₄⋊Q₈	Q₈×C₇₉	1	2	yes	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄⋊C₄	—	2	1	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄²	C₂³×C₇₉	1	1	yes	characteristic
125	#107	—	4	4	no	—
126	#103	—	4	4	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	—	2	2	no	—
130	C₂×Q₈	—	2	2	no	—
131	C₂×Q₈	—	4	1	no	—
132	C₂×Q₈	—	4	2	no	—
133	C₂×Q₈	—	4	2	no	—
134	C₂²×C₄	—	4	1	no	—
135	C₂²×C₄	—	4	2	no	—
136	C₂²×C₄	—	4	1	no	—
137	C₂²×C₄	C₂×D₄×C₇₉	1	1	yes	characteristic
138	C₂²×C₄	C₂×D₄×C₇₉	1	2	yes	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	D₈×C₇₉	1	4	yes	—
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	4	4	no	—
149	C₄⋊C₄	—	4	4	no	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄⋊C₄	—	4	4	no	—
152	C₄²	C₂×Q₈×C₇₉	1	1	yes	characteristic
153	C₄²	#41×C₇₉	1	1	yes	characteristic
154	C₄²	#41×C₇₉	1	2	yes	—
155	C₂³	#127×C₇₉	1	1	yes	characteristic
156	Q₈	—	4	4	no	—
157	Q₈	—	8	2	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	8	1	no	—
160	C₂×C₄	—	8	1	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	—	8	1	no	—
163	C₂×C₄	—	8	2	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	C₂²⋊Q₈×C₇₉	1	1	yes	characteristic
167	C₂×C₄	C₂²⋊Q₈×C₇₉	1	2	yes	—
168	C₂×C₄	#123×C₇₉	1	1	yes	characteristic
169	C₂×C₄	#123×C₇₉	1	2	yes	—
170	C₂×C₄	C₂×D₈×C₇₉	1	2	yes	—
171	C₂×C₄	#135×C₇₉	1	4	yes	—
172	C₂×C₄	#137×C₇₉	1	2	yes	—
173	C₂×C₄	#137×C₇₉	1	4	yes	—
174	C₈	—	2	4	no	—
175	C₈	—	4	4	no	—
176	C₂²	#399×C₇₉	1	1	yes	characteristic
177	C₂²	#493×C₇₉	1	1	yes	characteristic
178	C₂²	#495×C₇₉	1	2	yes	—
179	C₂²	#497×C₇₉	1	1	yes	characteristic
180	C₂²	#497×C₇₉	1	2	yes	—
181	C₄	—	8	2	no	—
182	C₄	—	16	2	no	—
183	C₄	—	16	1	no	—
184	C₄	#474×C₇₉	1	2	yes	—
185	C₄	#479×C₇₉	1	4	yes	—
186	C₄	#480×C₇₉	1	2	yes	—
187	C₄	#485×C₇₉	1	4	yes	—
188	C₂	C₇₉×#1709	1	1	yes	characteristic
189	C₂	C₇₉×#1718	1	2	yes	—
190	C₂	C₇₉×#1722	1	1	yes	characteristic
191	C₂	C₇₉×#2093	1	1	yes	characteristic
192	C₂	C₇₉×#2097	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₉×#1990

C₂

yes

maximal

C₇₉×#1956

C₂

yes

maximal, characteristic

C₇₉×#1845

C₂

yes

maximal, characteristic

C₇₉×#1834

C₂

yes

maximal

C₇₉×#1273

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₇₉

C₂²

yes

characteristic

C₂×#107×C₇₉

—

C₂×#103×C₇₉

—

C₂×C₄×C₈×C₇₉

C₂²

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#384×C₇₉

C₂²

yes

characteristic

#384×C₇₉

C₂²

yes

—

#340×C₇₉

—

#340×C₇₉

—

C₂²×Q₈×C₇₉

—

C₂²×C₈×C₇₉

D₄

yes

—

C₂²×C₈×C₇₉

D₄

yes

—

C₄⋊Q₈×C₇₉

—

C₄⋊Q₈×C₇₉

—

C₄⋊Q₈×C₇₉

Q₈

yes

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

#107×C₇₉

—

#103×C₇₉

—

C₄×C₈×C₇₉

—

C₄×C₈×C₇₉

—

C₄×C₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

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

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

D₈

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂×Q₈

yes

C₄²×C₇₉

yes

C₄²×C₇₉

yes

—

C₂³×C₇₉

#127

yes

characteristic

Q₈×C₇₉

—

Q₈×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

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₉

#123

yes

characteristic

C₂×C₄×C₇₉

#123

yes

—

C₂×C₄×C₇₉

C₂×D₈

yes

—

C₂×C₄×C₇₉

#135

yes

—

C₂×C₄×C₇₉

#137

yes

—

C₂×C₄×C₇₉

#137

yes

—

C₈×C₇₉

—

C₈×C₇₉

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

—

C₂²×C₇₉

#497

yes

characteristic

C₂²×C₇₉

#497

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#474

yes

—

C₄×C₇₉

#479

yes

—

C₄×C₇₉

#480

yes

—

C₄×C₇₉

#485

yes

—

#10676

C₇₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₉

#1709

yes

characteristic

C₂×C₇₉

#1718

yes

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

#1990

C₂×C₇₉

yes

—

#1956

C₂×C₇₉

yes

characteristic

#1845

C₂×C₇₉

yes

characteristic

100

#1834

C₂×C₇₉

yes

—

101

#1273

C₂×C₇₉

yes

characteristic

102

C₇₉

#10676

yes

characteristic, direct product factor, Sylow

103

C₂×C₄⋊Q₈

C₂²×C₇₉

yes

characteristic

104

C₂×#107

—

105

C₂×#103

—

106

C₂×C₄×C₈

C₂²×C₇₉

yes

—

107

C₂×C₄×C₈

C₂²×C₇₉

yes

characteristic

108

#384

C₂²×C₇₉

yes

characteristic

109

#384

C₂²×C₇₉

yes

—

110

#340

—

111

#340

—

112

C₂²×Q₈

—

113

C₂²×C₈

D₄×C₇₉

yes

—

114

C₂²×C₈

D₄×C₇₉

yes

—

115

C₄⋊Q₈

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

Q₈×C₇₉

yes

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄²

C₂³×C₇₉

yes

characteristic

125

#107

—

126

#103

—

127

C₄×C₈

—

128

C₄×C₈

—

129

C₄×C₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

C₂×D₄×C₇₉

yes

characteristic

138

C₂²×C₄

C₂×D₄×C₇₉

yes

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

D₈×C₇₉

yes

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄²

C₂×Q₈×C₇₉

yes

characteristic

153

C₄²

#41×C₇₉

yes

characteristic

154

C₄²

#41×C₇₉

yes

—

155

C₂³

#127×C₇₉

yes

characteristic

156

Q₈

—

157

Q₈

—

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

C₂²⋊Q₈×C₇₉

yes

characteristic

167

C₂×C₄

C₂²⋊Q₈×C₇₉

yes

—

168

C₂×C₄

#123×C₇₉

yes

characteristic

169

C₂×C₄

#123×C₇₉

yes

—

170

C₂×C₄

C₂×D₈×C₇₉

yes

—

171

C₂×C₄

#135×C₇₉

yes

—

172

C₂×C₄

#137×C₇₉

yes

—

173

C₂×C₄

#137×C₇₉

yes

—

174

C₈

—

175

C₈

—

176

C₂²

#399×C₇₉

yes

characteristic

177

C₂²

#493×C₇₉

yes

characteristic

178

C₂²

#495×C₇₉

yes

—

179

C₂²

#497×C₇₉

yes

characteristic

180

C₂²

#497×C₇₉

yes

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

#474×C₇₉

yes

—

185

C₄

#479×C₇₉

yes

—

186

C₄

#480×C₇₉

yes

—

187

C₄

#485×C₇₉

yes

—

188

C₂

C₇₉×#1709

yes

characteristic

189

C₂

C₇₉×#1718

yes

—

190

C₂

C₇₉×#1722

yes

characteristic

191

C₂

C₇₉×#2093

yes

characteristic

192

C₂

C₇₉×#2097

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₇₉	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇₉
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇₉ ◁ C₂×C₄²×C₇₉ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₇₉

#127

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₇₉

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₇₉ ◁ C₂×C₄²×C₇₉ ◁ G

N	H ≅ G/N	type
91. #10676	C₇₉	direct (N × H)
102. C₇₉	#10676	direct (N × H)

H ≅ G/N

type

91. #10676

C₇₉

direct (N × H)

102. C₇₉

#10676

direct (N × H)

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

subgroup	index
C₇₉×#1990	2
C₇₉×#1956	2
C₇₉×#1845	2
C₇₉×#1834	2
C₇₉×#1273	2
#10676	79

subgroup

index

C₇₉×#1990

C₇₉×#1956

C₇₉×#1845

C₇₉×#1834

C₇₉×#1273

#10676

Subgroups — C₇₉×#10676 #34122365

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C79×#10676 #34122365

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₉×#10676 #34122365