Group #34188194 — subgroups

Subgroups — #34188194

Order 20224.

422 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 119 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	#23773986	C₂	1	1	yes	maximal, characteristic
2	#23764376	C₂	1	2	yes	maximal
3	#23763145	C₂	1	1	yes	maximal, characteristic
4	#23761854	C₂	1	2	yes	maximal
5	C₇₉×#1566	C₂	1	1	yes	maximal, characteristic
6	#19099120	C₂²	1	2	yes	—
7	#19098968	C₄	1	1	yes	characteristic, complemented
8	#19098943	C₄	1	1	yes	characteristic, complemented
9	#19098914	C₄	1	2	yes	complemented
10	#19098804	—	2	2	no	—
11	C₄×Dic₃₁₆	—	2	2	no	—
12	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
13	C₄²×Dic₇₉	C₂²	1	1	yes	characteristic
14	#19098324	—	2	4	no	—
15	#19098317	—	2	2	no	—
16	#19098293	—	2	4	no	—
17	#423×C₇₉	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
19	#334×C₇₉	—	2	2	no	—
20	C₂×Dic₃₁₆	—	2	2	no	—
21	C₂²×C₇₉⋊C₈	—	2	1	no	—
22	#14579161	C₂×C₄	1	2	yes	—
23	#14579161	C₂×C₄	1	2	yes	—
24	#14579153	—	2	2	no	—
25	#14579140	—	2	2	no	—
26	#14579140	—	2	2	no	—
27	C₄×Dic₁₅₈	—	2	2	no	—
28	C₄×Dic₁₅₈	—	2	2	no	—
29	#14579120	—	2	2	no	—
30	C₂×C₄×Dic₇₉	—	2	1	no	—
31	C₂×C₄×Dic₇₉	—	2	1	no	—
32	C₂×C₄×Dic₇₉	D₄	1	2	yes	—
33	#14579095	—	2	2	no	—
34	#14579086	—	2	4	no	—
35	C₄×C₇₉⋊C₈	—	2	1	no	—
36	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
37	C₂×M₁₆×C₇₉	—	2	2	no	—
38	C₂²×C₈×C₇₉	D₄	1	2	yes	—
39	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₇₉	—	2	2	no	—
41	C₄⋊C₈×C₇₉	—	2	2	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	C₂×Dic₁₅₈	—	2	2	no	—
46	C₂×Dic₁₅₈	—	2	2	no	—
47	C₂²×Dic₇₉	—	2	2	no	—
48	Dic₃₁₆	—	4	4	no	—
49	C₂×C₇₉⋊C₈	—	2	2	no	—
50	C₂×C₇₉⋊C₈	—	4	2	no	—
51	C₄⋊Dic₇₉	—	2	2	no	—
52	C₄⋊Dic₇₉	—	2	2	no	—
53	C₄⋊Dic₇₉	—	2	2	no	—
54	C₄⋊Dic₇₉	—	2	2	no	—
55	Dic₇₉⋊C₄	—	4	2	no	—
56	Dic₇₉⋊C₄	—	4	2	no	—
57	C₂²⋊Dic₇₉	—	4	2	no	—
58	C₄×Dic₇₉	—	2	4	no	—
59	C₄×Dic₇₉	—	2	4	no	—
60	C₄×Dic₇₉	—	4	2	no	—
61	C₄×Dic₇₉	—	4	2	no	—
62	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
65	M₁₆×C₇₉	—	4	4	no	—
66	C₂×C₈×C₇₉	—	4	2	no	—
67	C₂×C₈×C₇₉	—	4	2	no	—
68	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
69	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
71	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₇₉	#41	1	1	yes	characteristic
74	C₂³×C₇₉	#121	1	1	yes	characteristic
75	C₂×C₄×C₇₉	—	2	2	no	—
76	C₂×C₄×C₇₉	—	2	2	no	—
77	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
81	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
82	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
83	C₂×C₄×C₇₉	#137	1	4	yes	—
84	C₂×Dic₇₉	—	2	4	no	—
85	C₂×Dic₇₉	—	4	2	no	—
86	C₂×Dic₇₉	—	4	2	no	—
87	C₂×Dic₇₉	—	4	2	no	—
88	C₂×Dic₇₉	—	4	2	no	—
89	Dic₁₅₈	—	4	4	no	—
90	Dic₁₅₈	—	4	4	no	—
91	C₈×C₇₉	—	2	2	no	—
92	C₈×C₇₉	—	2	2	no	—
93	C₈×C₇₉	—	8	2	no	—
94	C₇₉⋊C₈	—	4	2	no	—
95	C₂²×C₇₉	—	2	1	no	—
96	C₂²×C₇₉	—	2	1	no	—
97	C₂²×C₇₉	#390	1	1	yes	characteristic
98	C₂²×C₇₉	#470	1	2	yes	—
99	C₄×C₇₉	—	2	1	no	—
100	C₄×C₇₉	—	2	1	no	—
101	C₄×C₇₉	#421	1	2	yes	—
102	C₄×C₇₉	#441	1	2	yes	—
103	C₄×C₇₉	#443	1	2	yes	—
104	C₄×C₇₉	#497	1	2	yes	—
105	Dic₇₉	—	4	4	no	—
106	Dic₇₉	—	8	2	no	—
107	Dic₇₉	—	8	2	no	—
108	#9853	—	79	1	no	maximal, Sylow
109	C₂×C₇₉	—	2	2	no	—
110	C₂×C₇₉	#1963	1	1	yes	characteristic
111	C₂×C₇₉	#1971	1	1	yes	characteristic
112	C₂×C₇₉	#1974	1	1	yes	characteristic
113	#3170	—	79	1	no	—
114	#1994	—	79	2	no	—
115	#1744	—	79	1	no	—
116	#1566	D₇₉	1	1	yes	characteristic
117	#1493	—	79	2	no	—
118	C₇₉	#9853	1	1	yes	characteristic, complemented, Sylow
119	#533	—	79	2	no	—
120	#500	—	79	1	no	—
121	#494	—	79	1	no	—
122	#487	—	79	2	no	—
123	#467	—	158	2	no	—
124	C₄×Q₁₆	—	158	2	no	—
125	#423	C₂×D₇₉	1	2	yes	—
126	C₂×C₄×C₈	—	79	1	no	—
127	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
128	C₄³	—	79	1	no	—
129	#339	—	158	4	no	—
130	#336	—	158	2	no	—
131	#334	—	2	2	no	—
132	Q₈⋊C₈	—	158	4	no	—
133	C₂×Q₁₆	—	158	2	no	—
134	C₂×M₁₆	—	2	2	no	—
135	C₂²×C₈	—	158	1	no	—
136	C₂²×C₈	D₃₁₆	1	2	yes	—
137	C₄⋊Q₈	—	79	2	no	—
138	C₄⋊Q₈	—	79	2	no	—
139	#125	—	158	2	no	—
140	C₂²⋊Q₈	—	158	2	no	—
141	C₂²⋊Q₈	—	158	2	no	—
142	C₄×Q₈	—	158	2	no	—
143	C₄×Q₈	—	158	2	no	—
144	#117	—	158	2	no	—
145	C₂×C₄²	—	79	2	no	—
146	C₂×C₄²	—	158	1	no	—
147	C₂×C₄²	—	158	1	no	—
148	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
149	#107	—	158	2	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	2	2	no	—
152	#103	—	158	4	no	—
153	C₄×C₈	—	79	2	no	—
154	C₄×C₈	—	158	1	no	—
155	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
156	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
157	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
158	C₂×Q₈	—	158	2	no	—
159	C₂×Q₈	—	158	2	no	—
160	C₂²×C₄	—	158	2	no	—
161	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
162	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
163	C₂²×C₄	#11929625	1	1	yes	characteristic
164	Q₁₆	—	316	4	no	—
165	M₁₆	—	4	4	no	—
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	—	4	2	no	—
168	C₂×C₈	—	316	2	no	—
169	C₂×C₈	—	158	2	no	—
170	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
171	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
172	C₄⋊C₄	—	316	2	no	—
173	C₄⋊C₄	—	316	2	no	—
174	C₄⋊C₄	—	158	2	no	—
175	C₄⋊C₄	—	158	2	no	—
176	C₄⋊C₄	—	158	2	no	—
177	C₄⋊C₄	—	158	2	no	—
178	C₂²⋊C₄	—	316	2	no	—
179	C₄²	—	316	2	no	—
180	C₄²	—	316	2	no	—
181	C₄²	—	158	4	no	—
182	C₄²	—	158	4	no	—
183	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
184	C₄²	D₄×D₇₉	1	1	yes	characteristic
185	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
186	C₄²	#11929625	1	1	yes	characteristic
187	C₂³	#14579136	1	1	yes	characteristic
188	Q₈	—	316	4	no	—
189	Q₈	—	316	4	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	316	2	no	—
193	C₂×C₄	—	316	2	no	—
194	C₂×C₄	—	316	2	no	—
195	C₂×C₄	—	316	2	no	—
196	C₂×C₄	—	158	4	no	—
197	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
198	C₂×C₄	#14579123	1	1	yes	characteristic
199	C₂×C₄	#14579123	1	1	yes	characteristic
200	C₂×C₄	#14579136	1	1	yes	characteristic
201	C₂×C₄	#14579150	1	1	yes	characteristic
202	C₂×C₄	#14579157	1	1	yes	characteristic
203	C₂×C₄	#14579190	1	4	yes	—
204	C₈	—	2	2	no	—
205	C₈	—	2	2	no	—
206	C₈	—	8	2	no	—
207	C₈	—	316	2	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	#19098464	1	1	yes	characteristic
211	C₂²	#19098822	1	2	yes	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	316	4	no	—
215	C₄	—	632	2	no	—
216	C₄	—	632	2	no	—
217	C₄	#19098579	1	2	yes	—
218	C₄	#19098669	1	2	yes	—
219	C₄	#19098679	1	2	yes	—
220	C₄	#19098954	1	2	yes	—
221	C₂	—	2	2	no	—
222	C₂	#23764231	1	1	yes	characteristic
223	C₂	#23764269	1	1	yes	characteristic
224	C₂	#23764286	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23773986

C₂

yes

maximal, characteristic

#23764376

C₂

yes

maximal

#23763145

C₂

yes

maximal, characteristic

#23761854

C₂

yes

maximal

C₇₉×#1566

C₂

yes

maximal, characteristic

#19099120

C₂²

yes

—

#19098968

C₄

yes

characteristic, complemented

#19098943

C₄

yes

characteristic, complemented

#19098914

C₄

yes

complemented

#19098804

—

C₄×Dic₃₁₆

—

C₂×C₄×C₇₉⋊C₈

C₂²

yes

characteristic

C₄²×Dic₇₉

C₂²

yes

—

—

—

#423×C₇₉

C₂²

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#334×C₇₉

—

C₂×Dic₃₁₆

—

C₂²×C₇₉⋊C₈

—

#14579161

C₂×C₄

yes

—

#14579161

C₂×C₄

yes

—

#14579153

—

#14579140

—

#14579140

—

C₄×Dic₁₅₈

—

C₄×Dic₁₅₈

—

#14579120

—

C₂×C₄×Dic₇₉

—

C₂×C₄×Dic₇₉

—

C₂×C₄×Dic₇₉

D₄

yes

—

#14579095

—

#14579086

—

C₄×C₇₉⋊C₈

—

C₄×C₇₉⋊C₈

D₄

yes

—

C₂×M₁₆×C₇₉

—

C₂²×C₈×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂²×Dic₇₉

—

Dic₃₁₆

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

#41

yes

characteristic

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

C₄²×C₇₉

yes

C₂³×C₇₉

yes

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

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#390

yes

characteristic

C₂²×C₇₉

#470

yes

—

C₄×C₇₉

—

100

C₄×C₇₉

—

101

C₄×C₇₉

#421

yes

—

102

C₄×C₇₉

#441

yes

—

103

C₄×C₇₉

#443

yes

—

104

C₄×C₇₉

#497

yes

—

105

Dic₇₉

—

106

Dic₇₉

—

107

Dic₇₉

—

108

#9853

—

maximal, Sylow

109

C₂×C₇₉

—

110

C₂×C₇₉

#1963

yes

characteristic

111

C₂×C₇₉

#1971

yes

characteristic

112

C₂×C₇₉

#1974

yes

characteristic

113

#3170

—

114

#1994

—

115

#1744

—

116

#1566

D₇₉

yes

characteristic

117

#1493

—

118

C₇₉

#9853

yes

characteristic, complemented, Sylow

119

#533

—

120

#500

—

121

#494

—

122

#487

—

123

#467

—

158

—

124

C₄×Q₁₆

—

158

—

125

#423

C₂×D₇₉

yes

—

126

C₂×C₄×C₈

—

127

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

128

C₄³

—

129

#339

—

158

—

130

#336

—

158

—

131

#334

—

132

Q₈⋊C₈

—

158

—

133

C₂×Q₁₆

—

158

—

134

C₂×M₁₆

—

135

C₂²×C₈

—

158

—

136

C₂²×C₈

D₃₁₆

yes

—

137

C₄⋊Q₈

—

138

C₄⋊Q₈

—

139

#125

—

158

—

140

C₂²⋊Q₈

—

158

—

141

C₂²⋊Q₈

—

158

—

142

C₄×Q₈

—

158

—

143

C₄×Q₈

—

158

—

144

#117

—

158

—

145

C₂×C₄²

—

146

C₂×C₄²

—

158

—

147

C₂×C₄²

—

158

—

148

C₂×C₄²

C₂²×D₇₉

yes

characteristic

149

#107

—

158

—

150

C₄⋊C₈

—

151

C₄⋊C₈

—

152

#103

—

158

—

153

C₄×C₈

—

154

C₄×C₈

—

158

—

155

C₄×C₈

C₄×D₇₉

yes

characteristic

156

C₄×C₈

C₄×D₇₉

yes

characteristic

157

C₄×C₈

C₂²⋊D₇₉

yes

—

158

C₂×Q₈

—

158

—

159

C₂×Q₈

—

158

—

160

C₂²×C₄

—

158

—

161

C₂²×C₄

D₄×D₇₉

yes

characteristic

162

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

163

C₂²×C₄

#11929625

yes

characteristic

164

Q₁₆

—

316

—

165

M₁₆

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

316

—

169

C₂×C₈

—

158

—

170

C₂×C₈

(C₂×D₇₉)⋊C₄

yes

—

171

C₂×C₈

(C₂×D₇₉)⋊C₄

yes

—

172

C₄⋊C₄

—

316

—

173

C₄⋊C₄

—

316

—

174

C₄⋊C₄

—

158

—

175

C₄⋊C₄

—

158

—

176

C₄⋊C₄

—

158

—

177

C₄⋊C₄

—

158

—

178

C₂²⋊C₄

—

316

—

179

C₄²

—

316

—

180

C₄²

—

316

—

181

C₄²

—

158

—

182

C₄²

—

158

—

183

C₄²

C₂×C₄×D₇₉

yes

characteristic

184

C₄²

D₄×D₇₉

yes

characteristic

185

C₄²

C₂×C₂²⋊D₇₉

yes

characteristic

186

C₄²

#11929625

yes

characteristic

187

C₂³

#14579136

yes

characteristic

188

Q₈

—

316

—

189

Q₈

—

316

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

316

—

193

C₂×C₄

—

316

—

194

C₂×C₄

—

316

—

195

C₂×C₄

—

316

—

196

C₂×C₄

—

158

—

197

C₂×C₄

C₂×(C₂×D₇₉)⋊C₄

yes

characteristic

198

C₂×C₄

#14579123

yes

characteristic

199

C₂×C₄

#14579123

yes

characteristic

200

C₂×C₄

#14579136

yes

characteristic

201

C₂×C₄

#14579150

yes

characteristic

202

C₂×C₄

#14579157

yes

characteristic

203

C₂×C₄

#14579190

yes

—

204

C₈

—

205

C₈

—

206

C₈

—

207

C₈

—

316

—

208

C₂²

—

209

C₂²

—

210

C₂²

#19098464

yes

characteristic

211

C₂²

#19098822

yes

—

212

C₄

—

213

C₄

—

214

C₄

—

316

—

215

C₄

—

632

—

216

C₄

—

632

—

217

C₄

#19098579

yes

—

218

C₄

#19098669

yes

—

219

C₄

#19098679

yes

—

220

C₄

#19098954

yes

—

221

C₂

—

222

C₂

#23764231

yes

characteristic

223

C₂

#23764269

yes

characteristic

224

C₂

#23764286

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14579157
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1566	C₂
socle	C₂²×C₇₉	#390
prime radical	C₂³×C₇₉	#121
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1566	D₇₉
hypocenter	C₇₉	#9853
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

subgroup

G/H

center

C₂×C₄

#14579157

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1566

C₂

C₂²×C₇₉

C₂³×C₇₉

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

N	H ≅ G/N	type
7. #19098968	C₄	semidirect
8. #19098943	C₄	semidirect
9. #19098914	C₄	semidirect
118. C₇₉	#9853	semidirect

H ≅ G/N

type

7. #19098968

C₄

semidirect

8. #19098943

C₄

semidirect

9. #19098914

C₄

semidirect

118. C₇₉

#9853

semidirect

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

subgroup	index
#23773986	2
#23764376	2
#23763145	2
#23761854	2
C₇₉×#1566	2
#9853	79

subgroup

index

C₇₉×#1566

Subgroups — #34188194

Conjugacy classes of subgroups

Complements

Maximal subgroups