Group #34188229 — subgroups

Subgroups — #34188229

Order 20224.

390 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 127 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	#23763538	C₂	1	1	yes	maximal, characteristic
2	#23761855	C₂	1	2	yes	maximal
3	#23761855	C₂	1	2	yes	maximal
4	C₂×#19098285	C₂	1	1	yes	maximal, characteristic
5	C₇₉×#3160	C₂	1	1	yes	maximal, characteristic
6	#19098587	C₂²	1	2	yes	—
7	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
9	#19098328	—	2	2	no	—
10	Q₈⋊(C₇₉⋊C₈)	—	2	4	no	—
11	Q₈⋊(C₇₉⋊C₈)	—	2	4	no	—
12	#19098285	—	2	2	no	—
13	#533×C₇₉	C₂²	1	2	yes	—
14	#500×C₇₉	C₄	1	1	yes	characteristic
15	#500×C₇₉	C₄	1	1	yes	characteristic
16	#499×C₇₉	—	2	2	no	—
17	#498×C₇₉	C₄	1	2	yes	—
18	#427×C₇₉	—	2	2	no	—
19	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
20	C₂×#11929608	—	2	2	no	—
21	C₂²×C₇₉⋊C₈	—	2	1	no	—
22	C₂²×C₇₉⋊C₈	—	2	1	no	—
23	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
24	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
25	C₄×C₇₉⋊C₈	—	2	1	no	—
26	C₄×C₇₉⋊C₈	—	2	1	no	—
27	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
28	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
29	C₂²×C₈×C₇₉	D₄	1	2	yes	—
30	C₄⋊Q₈×C₇₉	C₂×C₄	1	2	yes	—
31	C₄⋊Q₈×C₇₉	C₂×C₄	1	2	yes	—
32	#125×C₇₉	—	2	2	no	—
33	C₂²⋊Q₈×C₇₉	—	2	2	no	—
34	C₂²⋊Q₈×C₇₉	—	2	2	no	—
35	C₄×Q₈×C₇₉	—	2	2	no	—
36	C₄×Q₈×C₇₉	—	2	2	no	—
37	#117×C₇₉	—	2	2	no	—
38	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
39	#107×C₇₉	—	2	2	no	—
40	#107×C₇₉	—	2	2	no	—
41	#106×C₇₉	—	2	4	no	—
42	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₇₉	D₄	1	2	yes	—
45	#11929608	—	4	4	no	—
46	C₂×C₇₉⋊C₈	—	2	4	no	—
47	C₂×C₇₉⋊C₈	—	2	2	no	—
48	C₂×C₇₉⋊C₈	—	2	2	no	—
49	C₂×C₇₉⋊C₈	—	4	2	no	—
50	C₂×C₇₉⋊C₈	—	4	2	no	—
51	C₂×Q₈×C₇₉	—	2	2	no	—
52	C₂×Q₈×C₇₉	—	2	2	no	—
53	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
55	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
56	C₂×C₈×C₇₉	—	2	4	no	—
57	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
58	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
59	C₄⋊C₄×C₇₉	—	2	2	no	—
60	C₄⋊C₄×C₇₉	—	2	2	no	—
61	C₄⋊C₄×C₇₉	—	2	2	no	—
62	C₄⋊C₄×C₇₉	—	2	2	no	—
63	C₄⋊C₄×C₇₉	—	4	2	no	—
64	C₄⋊C₄×C₇₉	—	4	2	no	—
65	C₂²⋊C₄×C₇₉	—	4	2	no	—
66	C₄²×C₇₉	—	4	2	no	—
67	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
71	C₂³×C₇₉	#124	1	1	yes	characteristic
72	Q₈×C₇₉	—	4	4	no	—
73	Q₈×C₇₉	—	4	4	no	—
74	C₂×C₄×C₇₉	—	2	2	no	—
75	C₂×C₄×C₇₉	—	2	2	no	—
76	C₂×C₄×C₇₉	—	4	2	no	—
77	C₂×C₄×C₇₉	—	4	2	no	—
78	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
81	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
82	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
83	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
84	C₂×C₄×C₇₉	#137	1	4	yes	—
85	C₈×C₇₉	—	2	2	no	—
86	C₈×C₇₉	—	2	2	no	—
87	C₇₉⋊C₈	—	2	4	no	—
88	C₇₉⋊C₈	—	4	2	no	—
89	C₇₉⋊C₈	—	8	2	no	—
90	C₂²×C₇₉	—	2	1	no	—
91	C₂²×C₇₉	—	2	1	no	—
92	C₂²×C₇₉	#390	1	1	yes	characteristic
93	C₂²×C₇₉	#491	1	2	yes	—
94	C₄×C₇₉	—	2	1	no	—
95	C₄×C₇₉	—	2	1	no	—
96	C₄×C₇₉	—	8	2	no	—
97	C₄×C₇₉	—	8	2	no	—
98	C₄×C₇₉	#419	1	2	yes	—
99	C₄×C₇₉	#443	1	2	yes	—
100	C₄×C₇₉	#444	1	2	yes	—
101	C₄×C₇₉	#497	1	2	yes	—
102	#9859	—	79	1	no	maximal, Sylow
103	C₂×C₇₉	—	2	2	no	—
104	C₂×C₇₉	#1960	1	1	yes	characteristic
105	C₂×C₇₉	#1974	1	1	yes	characteristic
106	C₂×C₇₉	#1975	1	1	yes	characteristic
107	#3160	D₇₉	1	1	yes	characteristic
108	#1837	—	79	1	no	—
109	#1493	—	79	2	no	—
110	#1493	—	79	2	no	—
111	C₂×#322	—	79	1	no	—
112	C₇₉	#9859	1	1	yes	characteristic, complemented, Sylow
113	#533	C₂×D₇₉	1	2	yes	—
114	#500	Dic₇₉	1	1	yes	characteristic
115	#500	Dic₇₉	1	1	yes	characteristic
116	#499	—	2	2	no	—
117	#498	Dic₇₉	1	2	yes	—
118	#427	—	2	2	no	—
119	#423	—	79	2	no	—
120	C₂×C₄×C₈	—	79	1	no	—
121	C₂×C₄×C₈	—	79	1	no	—
122	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
123	#341	—	158	2	no	—
124	Q₈⋊C₈	—	158	4	no	—
125	Q₈⋊C₈	—	158	4	no	—
126	#322	—	158	2	no	—
127	C₂×M₁₆	—	158	2	no	—
128	C₂²×C₈	—	158	1	no	—
129	C₂²×C₈	—	158	1	no	—
130	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
131	C₄⋊Q₈	C₂×Dic₇₉	1	2	yes	—
132	C₄⋊Q₈	C₂×Dic₇₉	1	2	yes	—
133	#125	—	2	2	no	—
134	C₂²⋊Q₈	—	2	2	no	—
135	C₂²⋊Q₈	—	2	2	no	—
136	C₄×Q₈	—	2	2	no	—
137	C₄×Q₈	—	2	2	no	—
138	#117	—	2	2	no	—
139	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
140	#107	—	2	2	no	—
141	#107	—	2	2	no	—
142	#106	—	2	4	no	—
143	C₄⋊C₈	—	158	2	no	—
144	C₄⋊C₈	—	158	2	no	—
145	C₄×C₈	—	79	2	no	—
146	C₄×C₈	—	79	2	no	—
147	C₄×C₈	—	158	1	no	—
148	C₄×C₈	—	158	1	no	—
149	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
150	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
151	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
152	C₂×Q₈	—	2	2	no	—
153	C₂×Q₈	—	2	2	no	—
154	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
155	C₂²×C₄	#11929623	1	1	yes	characteristic
156	C₂²×C₄	#11929623	1	1	yes	characteristic
157	M₁₆	—	316	4	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	316	2	no	—
160	C₂×C₈	—	316	2	no	—
161	C₂×C₈	—	158	2	no	—
162	C₂×C₈	—	158	4	no	—
163	C₂×C₈	—	158	2	no	—
164	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
165	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₂²⋊C₄	—	4	2	no	—
173	C₄²	—	4	2	no	—
174	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
175	C₄²	D₄×D₇₉	1	1	yes	characteristic
176	C₄²	D₄×D₇₉	1	1	yes	characteristic
177	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
178	C₂³	#14579148	1	1	yes	characteristic
179	Q₈	—	4	4	no	—
180	Q₈	—	4	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
186	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
187	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
188	C₂×C₄	#14579132	1	1	yes	characteristic
189	C₂×C₄	#14579132	1	1	yes	characteristic
190	C₂×C₄	#14579157	1	1	yes	characteristic
191	C₂×C₄	#14579192	1	4	yes	—
192	C₈	—	2	2	no	—
193	C₈	—	2	2	no	—
194	C₈	—	316	2	no	—
195	C₈	—	632	2	no	—
196	C₈	—	158	4	no	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	#19098466	1	1	yes	characteristic
200	C₂²	#19098931	1	2	yes	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	8	2	no	—
204	C₄	—	8	2	no	—
205	C₄	#19098572	1	2	yes	—
206	C₄	#19098680	1	2	yes	—
207	C₄	#19098685	1	2	yes	—
208	C₄	#19098955	1	2	yes	—
209	C₂	—	2	2	no	—
210	C₂	#23764221	1	1	yes	characteristic
211	C₂	#23764289	1	1	yes	characteristic
212	C₂	#23764294	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23763538

C₂

yes

maximal, characteristic

#23761855

C₂

yes

maximal

#23761855

C₂

yes

maximal

C₂×#19098285

C₂

yes

maximal, characteristic

C₇₉×#3160

C₂

yes

maximal, characteristic

#19098587

C₂²

yes

—

C₂×C₄×C₇₉⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₇₉⋊C₈

C₂²

yes

characteristic

#19098328

—

Q₈⋊(C₇₉⋊C₈)

—

Q₈⋊(C₇₉⋊C₈)

—

#19098285

—

#533×C₇₉

C₂²

yes

—

#500×C₇₉

C₄

yes

characteristic

#500×C₇₉

C₄

yes

characteristic

#499×C₇₉

—

#498×C₇₉

C₄

yes

—

#427×C₇₉

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×#11929608

—

C₂²×C₇₉⋊C₈

—

C₂²×C₇₉⋊C₈

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₄×C₇₉⋊C₈

—

C₄×C₇₉⋊C₈

—

C₄×C₇₉⋊C₈

D₄

yes

—

C₄×C₇₉⋊C₈

D₄

yes

—

C₂²×C₈×C₇₉

D₄

yes

—

C₄⋊Q₈×C₇₉

C₂×C₄

yes

—

C₄⋊Q₈×C₇₉

C₂×C₄

yes

—

#125×C₇₉

—

C₂²⋊Q₈×C₇₉

—

C₂²⋊Q₈×C₇₉

—

C₄×Q₈×C₇₉

—

C₄×Q₈×C₇₉

—

#117×C₇₉

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

#107×C₇₉

—

#107×C₇₉

—

#106×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

C₂²×C₄×C₇₉

yes

C₂²×C₄×C₇₉

yes

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₂³×C₇₉

#124

yes

characteristic

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄×D₄

yes

characteristic

C₂×C₄×C₇₉

C₄×D₄

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

—

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#390

yes

characteristic

C₂²×C₇₉

#491

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#419

yes

—

C₄×C₇₉

#443

yes

—

100

C₄×C₇₉

#444

yes

—

101

C₄×C₇₉

#497

yes

—

102

#9859

—

maximal, Sylow

103

C₂×C₇₉

—

104

C₂×C₇₉

#1960

yes

characteristic

105

C₂×C₇₉

#1974

yes

characteristic

106

C₂×C₇₉

#1975

yes

characteristic

107

#3160

D₇₉

yes

characteristic

108

#1837

—

109

#1493

—

110

#1493

—

111

C₂×#322

—

112

C₇₉

#9859

yes

characteristic, complemented, Sylow

113

#533

C₂×D₇₉

yes

—

114

#500

Dic₇₉

yes

characteristic

115

#500

Dic₇₉

yes

characteristic

116

#499

—

117

#498

Dic₇₉

yes

—

118

#427

—

119

#423

—

120

C₂×C₄×C₈

—

121

C₂×C₄×C₈

—

122

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

123

#341

—

158

—

124

Q₈⋊C₈

—

158

—

125

Q₈⋊C₈

—

158

—

126

#322

—

158

—

127

C₂×M₁₆

—

158

—

128

C₂²×C₈

—

158

—

129

C₂²×C₈

—

158

—

130

C₂²×C₈

C₂²⋊D₇₉

yes

—

131

C₄⋊Q₈

C₂×Dic₇₉

yes

—

132

C₄⋊Q₈

C₂×Dic₇₉

yes

—

133

#125

—

134

C₂²⋊Q₈

—

135

C₂²⋊Q₈

—

136

C₄×Q₈

—

137

C₄×Q₈

—

138

#117

—

139

C₂×C₄²

C₂²×D₇₉

yes

characteristic

140

#107

—

141

#107

—

142

#106

—

143

C₄⋊C₈

—

158

—

144

C₄⋊C₈

—

158

—

145

C₄×C₈

—

146

C₄×C₈

—

147

C₄×C₈

—

158

—

148

C₄×C₈

—

158

—

149

C₄×C₈

C₂×Dic₇₉

yes

characteristic

150

C₄×C₈

C₂×Dic₇₉

yes

characteristic

151

C₄×C₈

C₂²⋊D₇₉

yes

—

152

C₂×Q₈

—

153

C₂×Q₈

—

154

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

155

C₂²×C₄

#11929623

yes

characteristic

156

C₂²×C₄

#11929623

yes

characteristic

157

M₁₆

—

316

—

158

C₂×C₈

—

159

C₂×C₈

—

316

—

160

C₂×C₈

—

316

—

161

C₂×C₈

—

158

—

162

C₂×C₈

—

158

—

163

C₂×C₈

—

158

—

164

C₂×C₈

C₂²⋊Dic₇₉

yes

—

165

C₂×C₈

C₂²⋊Dic₇₉

yes

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₂²⋊C₄

—

173

C₄²

—

174

C₄²

C₂²×Dic₇₉

yes

characteristic

175

C₄²

D₄×D₇₉

yes

characteristic

176

C₄²

D₄×D₇₉

yes

characteristic

177

C₄²

C₂×C₂²⋊D₇₉

yes

characteristic

178

C₂³

#14579148

yes

characteristic

179

Q₈

—

180

Q₈

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

C₂×C₂²⋊Dic₇₉

yes

characteristic

186

C₂×C₄

D₄×Dic₇₉

yes

characteristic

187

C₂×C₄

D₄×Dic₇₉

yes

characteristic

188

C₂×C₄

#14579132

yes

characteristic

189

C₂×C₄

#14579132

yes

characteristic

190

C₂×C₄

#14579157

yes

characteristic

191

C₂×C₄

#14579192

yes

—

192

C₈

—

193

C₈

—

194

C₈

—

316

—

195

C₈

—

632

—

196

C₈

—

158

—

197

C₂²

—

198

C₂²

—

199

C₂²

#19098466

yes

characteristic

200

C₂²

#19098931

yes

—

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

#19098572

yes

—

206

C₄

#19098680

yes

—

207

C₄

#19098685

yes

—

208

C₄

#19098955

yes

—

209

C₂

—

210

C₂

#23764221

yes

characteristic

211

C₂

#23764289

yes

characteristic

212

C₂

#23764294

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14579157
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#3160	C₂
socle	C₂²×C₇₉	#390
prime radical	C₂³×C₇₉	#124
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#3160	D₇₉
hypocenter	C₇₉	#9859
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

subgroup

G/H

center

C₂×C₄

#14579157

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#3160

C₂

C₂²×C₇₉

C₂³×C₇₉

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

N	H ≅ G/N	type
112. C₇₉	#9859	semidirect

H ≅ G/N

type

112. C₇₉

#9859

semidirect

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

subgroup	index
#23763538	2
#23761855	2
C₂×#19098285	2
C₇₉×#3160	2
#9859	79

subgroup

index

C₇₉×#3160

Subgroups — #34188229

Conjugacy classes of subgroups

Complements

Maximal subgroups