Group #34182771 — subgroups

Subgroups — #34182771

Order 20224.

412 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 103 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	#23772467	C₂	1	1	yes	maximal, characteristic
2	#23762191	C₂	1	2	yes	maximal, complemented
3	#23762062	C₂	1	1	yes	maximal, characteristic
4	#23761233	C₂	1	2	yes	maximal, complemented
5	C₇₉×#1962	C₂	1	1	yes	maximal, characteristic
6	#19098641	—	2	2	no	—
7	D₄×C₇₉⋊C₈	—	2	2	no	—
8	#19098616	C₂²	1	2	yes	—
9	#19098616	C₂²	1	2	yes	—
10	#19098530	—	2	2	no	—
11	#19098315	—	2	2	no	—
12	#19098313	—	2	2	no	—
13	#19098312	C₄	1	2	yes	complemented
14	#19098307	C₄	1	1	yes	characteristic, complemented
15	#19098307	C₄	1	1	yes	characteristic, complemented
16	#532×C₇₉	C₂²	1	1	yes	characteristic
17	#423×C₇₉	C₂²	1	1	yes	characteristic
18	C₂×#104×C₇₉	—	2	2	no	—
19	C₄³×C₇₉	C₂²	1	1	yes	characteristic
20	C₂×#11929608	—	2	2	no	—
21	C₂²×C₇₉⋊C₈	—	2	2	no	—
22	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
23	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
24	C₄⋊(C₇₉⋊C₈)	D₄	1	2	yes	—
25	C₄⋊(C₇₉⋊C₈)	D₄	1	2	yes	—
26	C₂²⋊(C₇₉⋊C₈)	—	2	4	no	—
27	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
28	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
29	C₄×C₇₉⋊C₈	—	2	2	no	—
30	C₄×C₇₉⋊C₈	—	2	2	no	—
31	C₂×#41×C₇₉	D₄	1	2	yes	—
32	C₂×M₁₆×C₇₉	—	2	1	no	—
33	C₄⋊Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
34	#127×C₇₉	C₂×C₄	1	1	yes	characteristic
35	#124×C₇₉	Q₈	1	2	yes	—
36	#121×C₇₉	—	2	2	no	—
37	C₄×D₄×C₇₉	—	2	2	no	—
38	C₂×C₄²×C₇₉	—	2	1	no	—
39	C₂×C₄²×C₇₉	—	2	2	no	—
40	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₇₉	—	2	2	no	—
42	#104×C₇₉	—	4	4	no	—
43	#11929608	—	8	2	no	—
44	C₂×C₇₉⋊C₈	—	2	2	no	—
45	C₂×C₇₉⋊C₈	—	2	2	no	—
46	C₂×C₇₉⋊C₈	—	2	2	no	—
47	C₂×C₇₉⋊C₈	—	2	2	no	—
48	C₂×C₇₉⋊C₈	—	8	2	no	—
49	#41×C₇₉	—	4	2	no	—
50	#41×C₇₉	—	4	2	no	—
51	C₂×Q₈×C₇₉	C₄⋊C₄	1	2	yes	—
52	C₂×D₄×C₇₉	—	2	2	no	—
53	C₂×D₄×C₇₉	—	2	2	no	—
54	C₂×D₄×C₇₉	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₇₉	—	2	2	no	—
56	C₂²×C₄×C₇₉	—	4	1	no	—
57	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₇₉	#41	1	2	yes	—
59	M₁₆×C₇₉	—	4	2	no	—
60	C₂×C₈×C₇₉	—	4	1	no	—
61	C₄⋊C₄×C₇₉	—	2	2	no	—
62	C₂²⋊C₄×C₇₉	—	2	4	no	—
63	C₄²×C₇₉	—	2	2	no	—
64	C₄²×C₇₉	—	2	4	no	—
65	C₄²×C₇₉	—	4	2	no	—
66	C₄²×C₇₉	—	4	1	no	—
67	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₇₉	C₂×D₄	1	2	yes	—
69	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
70	C₂³×C₇₉	—	2	2	no	—
71	C₂³×C₇₉	#123	1	1	yes	characteristic
72	Q₈×C₇₉	—	4	2	no	—
73	D₄×C₇₉	—	4	2	no	—
74	D₄×C₇₉	—	4	2	no	—
75	D₄×C₇₉	—	4	2	no	—
76	D₄×C₇₉	—	8	2	no	—
77	C₂×C₄×C₇₉	—	2	2	no	—
78	C₂×C₄×C₇₉	—	2	2	no	—
79	C₂×C₄×C₇₉	—	2	1	no	—
80	C₂×C₄×C₇₉	—	2	1	no	—
81	C₂×C₄×C₇₉	—	2	2	no	—
82	C₂×C₄×C₇₉	—	4	2	no	—
83	C₂×C₄×C₇₉	—	4	4	no	—
84	C₂×C₄×C₇₉	—	4	2	no	—
85	C₂×C₄×C₇₉	—	4	2	no	—
86	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
88	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
89	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
90	C₈×C₇₉	—	8	1	no	—
91	C₇₉⋊C₈	—	4	2	no	—
92	C₇₉⋊C₈	—	4	2	no	—
93	C₇₉⋊C₈	—	4	2	no	—
94	C₇₉⋊C₈	—	4	2	no	—
95	C₂²×C₇₉	—	2	1	no	—
96	C₂²×C₇₉	—	2	2	no	—
97	C₂²×C₇₉	—	4	2	no	—
98	C₂²×C₇₉	—	4	4	no	—
99	C₂²×C₇₉	#385	1	1	yes	characteristic
100	C₄×C₇₉	—	2	1	no	—
101	C₄×C₇₉	—	2	2	no	—
102	C₄×C₇₉	—	4	4	no	—
103	C₄×C₇₉	—	4	2	no	—
104	C₄×C₇₉	#443	1	2	yes	—
105	C₄×C₇₉	#444	1	2	yes	—
106	C₄×C₇₉	#453	1	1	yes	characteristic
107	C₄×C₇₉	#455	1	1	yes	characteristic
108	#8934	—	79	1	no	maximal, Sylow
109	C₂×C₇₉	—	4	1	no	—
110	C₂×C₇₉	—	8	2	no	—
111	C₂×C₇₉	#1856	1	1	yes	characteristic
112	C₂×C₇₉	#1866	1	2	yes	—
113	#2979	—	79	1	no	—
114	#1962	D₇₉	1	1	yes	characteristic
115	#1574	—	79	2	no	—
116	#1543	—	79	1	no	—
117	#1276	—	79	2	no	—
118	C₇₉	#8934	1	1	yes	characteristic, complemented, Sylow
119	#532	C₂×D₇₉	1	1	yes	characteristic
120	#436	—	158	2	no	—
121	C₈×D₄	—	158	2	no	—
122	#431	—	79	2	no	—
123	#431	—	79	2	no	—
124	#423	C₂×D₇₉	1	1	yes	characteristic
125	C₂×#104	—	2	2	no	—
126	#408	—	158	2	no	—
127	C₄³	C₂×D₇₉	1	1	yes	characteristic
128	#335	—	158	2	no	—
129	#334	—	158	2	no	—
130	#333	—	79	2	no	—
131	#331	—	79	1	no	—
132	#331	—	79	1	no	—
133	C₂×#41	C₂²⋊D₇₉	1	2	yes	—
134	C₂×M₁₆	—	2	1	no	—
135	C₂×M₁₆	—	158	2	no	—
136	C₂²×C₈	—	158	2	no	—
137	C₄⋊Q₈	C₄×D₇₉	1	1	yes	characteristic
138	#127	C₄×D₇₉	1	1	yes	characteristic
139	#124	Dic₁₅₈	1	2	yes	—
140	#121	—	2	2	no	—
141	C₄×D₄	—	2	2	no	—
142	C₂×C₄²	—	2	1	no	—
143	C₂×C₄²	—	2	2	no	—
144	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
145	C₄⋊C₈	—	2	2	no	—
146	C₄⋊C₈	—	79	2	no	—
147	C₄⋊C₈	—	79	2	no	—
148	C₄⋊C₈	—	79	2	no	—
149	C₄⋊C₈	—	79	2	no	—
150	#104	—	4	4	no	—
151	C₂²⋊C₈	—	158	4	no	—
152	C₂²⋊C₈	—	158	2	no	—
153	C₂²⋊C₈	—	158	2	no	—
154	C₄×C₈	—	158	2	no	—
155	C₄×C₈	—	158	2	no	—
156	#41	—	4	2	no	—
157	#41	—	4	2	no	—
158	C₂×Q₈	Dic₇₉⋊C₄	1	2	yes	—
159	C₂×D₄	—	2	2	no	—
160	C₂×D₄	—	2	2	no	—
161	C₂×D₄	Dic₇₉⋊C₄	1	2	yes	—
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	—	4	1	no	—
164	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
165	C₂²×C₄	#11929623	1	2	yes	—
166	M₁₆	—	4	2	no	—
167	M₁₆	—	632	2	no	—
168	C₂×C₈	—	4	1	no	—
169	C₂×C₈	—	632	2	no	—
170	C₂×C₈	—	158	2	no	—
171	C₂×C₈	—	158	2	no	—
172	C₂×C₈	—	158	2	no	—
173	C₂×C₈	—	158	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₂²⋊C₄	—	2	4	no	—
176	C₄²	—	2	2	no	—
177	C₄²	—	2	4	no	—
178	C₄²	—	4	2	no	—
179	C₄²	—	4	1	no	—
180	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
181	C₄²	D₄×D₇₉	1	2	yes	—
182	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
183	C₂³	—	2	2	no	—
184	C₂³	#14579142	1	1	yes	characteristic
185	Q₈	—	4	2	no	—
186	D₄	—	4	2	no	—
187	D₄	—	4	2	no	—
188	D₄	—	4	2	no	—
189	D₄	—	8	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
200	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	2	yes	—
201	C₂×C₄	#14579129	1	1	yes	characteristic
202	C₂×C₄	#14579138	1	2	yes	—
203	C₈	—	8	1	no	—
204	C₈	—	316	2	no	—
205	C₈	—	316	2	no	—
206	C₈	—	316	2	no	—
207	C₈	—	316	2	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	4	2	no	—
211	C₂²	—	4	4	no	—
212	C₂²	#19098435	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	2	2	no	—
215	C₄	—	4	4	no	—
216	C₄	—	4	2	no	—
217	C₄	#19098677	1	2	yes	—
218	C₄	#19098682	1	2	yes	—
219	C₄	#19098730	1	1	yes	characteristic
220	C₄	#19098740	1	1	yes	characteristic
221	C₂	—	4	1	no	—
222	C₂	—	8	2	no	—
223	C₂	#23763609	1	1	yes	characteristic
224	C₂	#23763673	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23772467

C₂

yes

maximal, characteristic

#23762191

C₂

yes

maximal, complemented

#23762062

C₂

yes

maximal, characteristic

#23761233

C₂

yes

maximal, complemented

C₇₉×#1962

C₂

yes

maximal, characteristic

#19098641

—

D₄×C₇₉⋊C₈

—

#19098616

C₂²

yes

—

#19098616

C₂²

yes

—

—

—

—

C₄

yes

complemented

#19098307

C₄

yes

characteristic, complemented

#19098307

C₄

yes

characteristic, complemented

#532×C₇₉

C₂²

yes

characteristic

#423×C₇₉

C₂²

yes

characteristic

C₂×#104×C₇₉

—

C₄³×C₇₉

C₂²

yes

characteristic

C₂×#11929608

—

C₂²×C₇₉⋊C₈

—

C₄⋊(C₇₉⋊C₈)

C₂×C₄

yes

—

C₄⋊(C₇₉⋊C₈)

C₂×C₄

yes

—

C₄⋊(C₇₉⋊C₈)

D₄

yes

—

C₄⋊(C₇₉⋊C₈)

D₄

yes

—

C₂²⋊(C₇₉⋊C₈)

—

C₂²⋊(C₇₉⋊C₈)

—

C₂²⋊(C₇₉⋊C₈)

—

C₄×C₇₉⋊C₈

—

C₄×C₇₉⋊C₈

—

C₂×#41×C₇₉

D₄

yes

—

C₂×M₁₆×C₇₉

—

C₄⋊Q₈×C₇₉

C₂×C₄

yes

characteristic

#127×C₇₉

C₂×C₄

yes

characteristic

#124×C₇₉

Q₈

yes

—

#121×C₇₉

—

C₄×D₄×C₇₉

—

C₂×C₄²×C₇₉

—

C₂×C₄²×C₇₉

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄⋊C₈×C₇₉

—

#104×C₇₉

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

#41×C₇₉

—

#41×C₇₉

—

C₂×Q₈×C₇₉

C₄⋊C₄

yes

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

C₄⋊C₄

yes

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

#41

yes

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

—

C₄²×C₇₉

C₂×Q₈

yes

characteristic

C₂³×C₇₉

—

C₂³×C₇₉

#123

yes

characteristic

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×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⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄×D₄

yes

—

C₂×C₄×C₇₉

#120

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#385

yes

characteristic

100

C₄×C₇₉

—

101

C₄×C₇₉

—

102

C₄×C₇₉

—

103

C₄×C₇₉

—

104

C₄×C₇₉

#443

yes

—

105

C₄×C₇₉

#444

yes

—

106

C₄×C₇₉

#453

yes

characteristic

107

C₄×C₇₉

yes

108

—

109

C₂×C₇₉

—

110

C₂×C₇₉

—

111

C₂×C₇₉

#1856

yes

characteristic

112

C₂×C₇₉

#1866

yes

—

113

#2979

—

114

#1962

D₇₉

yes

characteristic

115

#1574

—

116

#1543

—

117

#1276

—

118

C₇₉

#8934

yes

characteristic, complemented, Sylow

119

#532

C₂×D₇₉

yes

characteristic

120

#436

—

158

—

121

C₈×D₄

—

158

—

122

#431

—

123

#431

—

124

#423

C₂×D₇₉

yes

characteristic

125

C₂×#104

—

126

#408

—

158

—

127

C₄³

C₂×D₇₉

yes

characteristic

128

#335

—

158

—

129

#334

—

158

—

130

#333

—

131

#331

—

132

#331

—

133

C₂×#41

C₂²⋊D₇₉

yes

—

134

C₂×M₁₆

—

135

C₂×M₁₆

—

158

—

136

C₂²×C₈

—

158

—

137

C₄⋊Q₈

C₄×D₇₉

yes

characteristic

138

#127

C₄×D₇₉

yes

characteristic

139

#124

Dic₁₅₈

yes

—

140

#121

—

141

C₄×D₄

—

142

C₂×C₄²

—

143

C₂×C₄²

—

144

C₂×C₄²

C₂²×D₇₉

yes

characteristic

145

C₄⋊C₈

—

146

C₄⋊C₈

—

147

C₄⋊C₈

—

148

C₄⋊C₈

—

149

C₄⋊C₈

—

150

#104

—

151

C₂²⋊C₈

—

158

—

152

C₂²⋊C₈

—

158

—

153

C₂²⋊C₈

—

158

—

154

C₄×C₈

—

158

—

155

C₄×C₈

—

158

—

156

#41

—

157

#41

—

158

C₂×Q₈

Dic₇₉⋊C₄

yes

—

159

C₂×D₄

—

160

C₂×D₄

—

161

C₂×D₄

Dic₇₉⋊C₄

yes

—

162

C₂²×C₄

—

163

C₂²×C₄

—

164

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

165

C₂²×C₄

#11929623

yes

—

166

M₁₆

—

167

M₁₆

—

632

—

168

C₂×C₈

—

169

C₂×C₈

—

632

—

170

C₂×C₈

—

158

—

171

C₂×C₈

—

158

—

172

C₂×C₈

—

158

—

173

C₂×C₈

—

158

—

174

C₄⋊C₄

—

175

C₂²⋊C₄

—

176

C₄²

—

177

C₄²

—

178

C₄²

—

179

C₄²

—

180

C₄²

C₂×C₄×D₇₉

yes

characteristic

181

C₄²

D₄×D₇₉

yes

—

182

C₄²

C₂×Dic₁₅₈

yes

characteristic

183

C₂³

—

184

C₂³

#14579142

yes

characteristic

185

Q₈

—

186

D₄

—

187

D₄

—

188

D₄

—

189

D₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

C₂×Dic₇₉⋊C₄

yes

characteristic

200

C₂×C₄

(C₂²⋊D₇₉)⋊C₄

yes

—

201

C₂×C₄

#14579129

yes

characteristic

202

C₂×C₄

#14579138

yes

—

203

C₈

—

204

C₈

—

316

—

205

C₈

—

316

—

206

C₈

—

316

—

207

C₈

—

316

—

208

C₂²

—

209

C₂²

—

210

C₂²

—

211

C₂²

—

212

C₂²

#19098435

yes

characteristic

213

C₄

—

214

C₄

—

215

C₄

—

216

C₄

—

217

C₄

#19098677

yes

—

218

C₄

#19098682

yes

—

219

C₄

#19098730

yes

characteristic

220

C₄

#19098740

yes

characteristic

221

C₂

—

222

C₂

—

223

C₂

#23763609

yes

characteristic

224

C₂

#23763673

yes

—

	subgroup	G/H
center	C₂×C₄	#14579129
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1962	C₂
socle	C₂²×C₇₉	#385
prime radical	#532×C₇₉	C₂²
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1962	D₇₉
hypocenter	C₇₉	#8934
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1962

subgroup

G/H

center

C₂×C₄

#14579129

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1962

C₂

socle

C₂²×C₇₉

#385

prime radical

#532×C₇₉

C₂²

cosocle

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1962

N	H ≅ G/N	type
2. #23762191	C₂	semidirect
4. #23761233	C₂	semidirect
13. #19098312	C₄	semidirect
14. #19098307	C₄	semidirect
15. #19098307	C₄	semidirect
118. C₇₉	#8934	semidirect

H ≅ G/N

type

2. #23762191

C₂

semidirect

4. #23761233

C₂

semidirect

13. #19098312

C₄

semidirect

14. #19098307

C₄

semidirect

15. #19098307

C₄

semidirect

118. C₇₉

#8934

semidirect

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

subgroup	index
#23772467	2
#23762191	2
#23762062	2
#23761233	2
C₇₉×#1962	2
#8934	79

subgroup

index

C₇₉×#1962

Subgroups — #34182771

Conjugacy classes of subgroups

Complements

Maximal subgroups