Group #34183596 — subgroups

Subgroups — #34183596

Order 20224.

426 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 111 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	#23772502	C₂	1	1	yes	maximal, characteristic, complemented
2	#23761995	C₂	1	1	yes	maximal, characteristic, complemented
3	#23761817	C₂	1	2	yes	maximal
4	#23761237	C₂	1	2	yes	maximal, complemented
5	C₇₉×#3131	C₂	1	1	yes	maximal, characteristic
6	#19098694	—	2	2	no	—
7	Q₈×C₇₉⋊C₈	—	2	2	no	—
8	#19098624	—	2	2	no	—
9	#19098616	C₂²	1	2	yes	—
10	C₄×#11929608	C₂²	1	2	yes	—
11	#19098305	—	2	2	no	—
12	D₄⋊(C₇₉⋊C₈)	—	2	2	no	—
13	D₄⋊(C₇₉⋊C₈)	—	2	2	no	—
14	#533×C₇₉	C₂²	1	1	yes	characteristic
15	#532×C₇₉	C₂²	1	1	yes	characteristic
16	#488×C₇₉	—	2	2	no	—
17	#487×C₇₉	C₄	1	2	yes	—
18	#486×C₇₉	C₄	1	2	yes	—
19	#416×C₇₉	—	2	2	no	—
20	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
21	C₂×#11929608	—	2	2	no	—
22	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
23	C₄⋊(C₇₉⋊C₈)	D₄	1	2	yes	—
24	C₄⋊(C₇₉⋊C₈)	D₄	1	2	yes	—
25	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
26	#14579072	—	2	2	no	—
27	#14579072	—	2	2	no	—
28	C₄×C₇₉⋊C₈	—	2	2	no	—
29	C₄×C₇₉⋊C₈	—	2	2	no	—
30	C₄×C₇₉⋊C₈	—	2	2	no	—
31	C₂×#41×C₇₉	—	2	1	no	—
32	C₂²×C₈×C₇₉	—	2	1	no	—
33	C₄⋊Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
34	C₄⋊Q₈×C₇₉	C₂×C₄	1	2	yes	—
35	#127×C₇₉	C₂×C₄	1	1	yes	characteristic
36	#125×C₇₉	D₄	1	2	yes	—
37	#124×C₇₉	—	2	1	no	—
38	C₂²⋊Q₈×C₇₉	—	2	2	no	—
39	#121×C₇₉	—	2	2	no	—
40	C₄×Q₈×C₇₉	—	2	2	no	—
41	C₄×D₄×C₇₉	—	2	2	no	—
42	#117×C₇₉	D₄	1	2	yes	—
43	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
44	#103×C₇₉	—	2	4	no	—
45	#102×C₇₉	—	2	4	no	—
46	C₄×C₈×C₇₉	—	2	1	no	—
47	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
48	#11929608	—	2	4	no	—
49	#11929608	—	4	2	no	—
50	C₂×C₇₉⋊C₈	—	2	2	no	—
51	C₂×C₇₉⋊C₈	—	2	2	no	—
52	C₂×C₇₉⋊C₈	—	4	2	no	—
53	#41×C₇₉	—	4	4	no	—
54	C₂×Q₈×C₇₉	—	2	2	no	—
55	C₂×Q₈×C₇₉	—	2	1	no	—
56	C₂×D₄×C₇₉	—	2	1	no	—
57	C₂×D₄×C₇₉	—	2	2	no	—
58	C₂×D₄×C₇₉	—	4	1	no	—
59	C₂²×C₄×C₇₉	—	4	1	no	—
60	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₇₉	#41	1	2	yes	—
62	C₂×C₈×C₇₉	—	2	2	no	—
63	C₂×C₈×C₇₉	—	4	2	no	—
64	C₄⋊C₄×C₇₉	—	2	2	no	—
65	C₄⋊C₄×C₇₉	—	2	4	no	—
66	C₄⋊C₄×C₇₉	—	2	2	no	—
67	C₄⋊C₄×C₇₉	C₂²⋊C₄	1	4	yes	—
68	C₂²⋊C₄×C₇₉	—	2	2	no	—
69	C₂²⋊C₄×C₇₉	—	4	2	no	—
70	C₄²×C₇₉	—	2	2	no	—
71	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₇₉	C₂×D₄	1	2	yes	—
74	C₂³×C₇₉	—	4	1	no	—
75	C₂³×C₇₉	#123	1	1	yes	characteristic
76	Q₈×C₇₉	—	4	2	no	—
77	Q₈×C₇₉	—	4	2	no	—
78	Q₈×C₇₉	—	4	2	no	—
79	D₄×C₇₉	—	4	2	no	—
80	D₄×C₇₉	—	4	2	no	—
81	D₄×C₇₉	—	4	2	no	—
82	D₄×C₇₉	—	8	2	no	—
83	C₂×C₄×C₇₉	—	2	2	no	—
84	C₂×C₄×C₇₉	—	2	2	no	—
85	C₂×C₄×C₇₉	—	2	2	no	—
86	C₂×C₄×C₇₉	—	2	4	no	—
87	C₂×C₄×C₇₉	—	4	1	no	—
88	C₂×C₄×C₇₉	—	8	2	no	—
89	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
91	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
92	C₂×C₄×C₇₉	#121	1	2	yes	—
93	C₈×C₇₉	—	4	2	no	—
94	C₇₉⋊C₈	—	4	4	no	—
95	C₇₉⋊C₈	—	4	2	no	—
96	C₇₉⋊C₈	—	4	2	no	—
97	C₂²×C₇₉	—	2	2	no	—
98	C₂²×C₇₉	—	2	1	no	—
99	C₂²×C₇₉	—	8	2	no	—
100	C₂²×C₇₉	—	8	1	no	—
101	C₂²×C₇₉	#386	1	1	yes	characteristic
102	C₄×C₇₉	—	2	2	no	—
103	C₄×C₇₉	—	2	1	no	—
104	C₄×C₇₉	—	4	4	no	—
105	C₄×C₇₉	—	8	1	no	—
106	C₄×C₇₉	#443	1	2	yes	—
107	C₄×C₇₉	#444	1	2	yes	—
108	C₄×C₇₉	#454	1	2	yes	—
109	#9070	—	79	1	no	maximal, Sylow
110	C₂×C₇₉	—	4	1	no	—
111	C₂×C₇₉	—	16	1	no	—
112	C₂×C₇₉	#1876	1	2	yes	—
113	C₂×C₇₉	#1888	1	1	yes	characteristic
114	#3131	D₇₉	1	1	yes	characteristic
115	#2986	—	79	1	no	—
116	#1526	—	79	1	no	—
117	#1484	—	79	2	no	—
118	#1278	—	79	2	no	—
119	C₇₉	#9070	1	1	yes	characteristic, complemented, Sylow
120	#533	C₂×D₇₉	1	1	yes	characteristic
121	#532	C₂×D₇₉	1	1	yes	characteristic
122	#488	—	2	2	no	—
123	#487	Dic₇₉	1	2	yes	—
124	#486	Dic₇₉	1	2	yes	—
125	#446	—	158	2	no	—
126	C₈×Q₈	—	158	2	no	—
127	#433	—	158	2	no	—
128	#431	—	79	2	no	—
129	#416	—	2	2	no	—
130	C₄×M₁₆	—	79	2	no	—
131	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
132	#330	—	158	2	no	—
133	D₄⋊C₈	—	158	2	no	—
134	D₄⋊C₈	—	158	2	no	—
135	C₂×#41	—	2	1	no	—
136	C₂×M₁₆	—	158	2	no	—
137	C₂²×C₈	—	2	1	no	—
138	C₄⋊Q₈	C₂×Dic₇₉	1	1	yes	characteristic
139	C₄⋊Q₈	C₂×Dic₇₉	1	2	yes	—
140	#127	C₂×Dic₇₉	1	1	yes	characteristic
141	#125	C₂²⋊D₇₉	1	2	yes	—
142	#124	—	2	1	no	—
143	C₂²⋊Q₈	—	2	2	no	—
144	#121	—	2	2	no	—
145	C₄×Q₈	—	2	2	no	—
146	C₄×D₄	—	2	2	no	—
147	#117	C₂²⋊D₇₉	1	2	yes	—
148	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
149	C₄⋊C₈	—	79	2	no	—
150	C₄⋊C₈	—	79	2	no	—
151	C₄⋊C₈	—	158	4	no	—
152	#103	—	2	4	no	—
153	#102	—	2	4	no	—
154	C₂²⋊C₈	—	158	2	no	—
155	#97	—	158	2	no	—
156	#97	—	158	2	no	—
157	C₄×C₈	—	2	1	no	—
158	C₄×C₈	—	158	2	no	—
159	C₄×C₈	—	158	2	no	—
160	C₄×C₈	—	158	2	no	—
161	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
162	#41	—	4	4	no	—
163	C₂×Q₈	—	2	2	no	—
164	C₂×Q₈	—	2	1	no	—
165	C₂×D₄	—	2	1	no	—
166	C₂×D₄	—	2	2	no	—
167	C₂×D₄	—	4	1	no	—
168	C₂²×C₄	—	4	1	no	—
169	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
170	C₂²×C₄	#11929623	1	2	yes	—
171	M₁₆	—	316	2	no	—
172	M₁₆	—	158	4	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	4	2	no	—
175	C₂×C₈	—	316	2	no	—
176	C₂×C₈	—	158	2	no	—
177	C₂×C₈	—	158	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	C₂²⋊Dic₇₉	1	4	yes	—
182	C₂²⋊C₄	—	2	2	no	—
183	C₂²⋊C₄	—	4	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
186	C₄²	D₄×D₇₉	1	2	yes	—
187	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
188	C₂³	—	4	1	no	—
189	C₂³	#14579142	1	1	yes	characteristic
190	Q₈	—	4	2	no	—
191	Q₈	—	4	2	no	—
192	Q₈	—	4	2	no	—
193	D₄	—	4	2	no	—
194	D₄	—	4	2	no	—
195	D₄	—	4	2	no	—
196	D₄	—	8	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	—	8	2	no	—
203	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
204	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
205	C₂×C₄	#14579129	1	1	yes	characteristic
206	C₂×C₄	#14579133	1	2	yes	—
207	C₈	—	4	2	no	—
208	C₈	—	316	4	no	—
209	C₈	—	316	2	no	—
210	C₈	—	316	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	1	no	—
213	C₂²	—	8	2	no	—
214	C₂²	—	8	1	no	—
215	C₂²	#19098443	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	2	1	no	—
218	C₄	—	4	4	no	—
219	C₄	—	8	1	no	—
220	C₄	#19098680	1	2	yes	—
221	C₄	#19098685	1	2	yes	—
222	C₄	#19098735	1	2	yes	—
223	C₂	—	4	1	no	—
224	C₂	—	16	1	no	—
225	C₂	#23763729	1	2	yes	—
226	C₂	#23763804	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23772502

C₂

yes

maximal, characteristic, complemented

#23761995

C₂

yes

maximal, characteristic, complemented

#23761817

C₂

yes

maximal

#23761237

C₂

yes

maximal, complemented

C₇₉×#3131

C₂

yes

maximal, characteristic

#19098694

—

Q₈×C₇₉⋊C₈

—

#19098624

—

#19098616

C₂²

yes

—

C₄×#11929608

C₂²

yes

—

#19098305

—

D₄⋊(C₇₉⋊C₈)

—

D₄⋊(C₇₉⋊C₈)

—

#533×C₇₉

C₂²

yes

characteristic

#532×C₇₉

C₂²

yes

characteristic

#488×C₇₉

—

#487×C₇₉

C₄

yes

—

#486×C₇₉

C₄

yes

—

#416×C₇₉

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×#11929608

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

D₄

yes

—

C₄⋊(C₇₉⋊C₈)

D₄

yes

—

C₂²⋊(C₇₉⋊C₈)

—

#14579072

—

#14579072

—

C₄×C₇₉⋊C₈

—

C₄×C₇₉⋊C₈

—

C₄×C₇₉⋊C₈

—

C₂×#41×C₇₉

—

C₂²×C₈×C₇₉

—

C₄⋊Q₈×C₇₉

C₂×C₄

yes

characteristic

C₄⋊Q₈×C₇₉

C₂×C₄

yes

—

#127×C₇₉

C₂×C₄

yes

characteristic

#125×C₇₉

D₄

yes

—

#124×C₇₉

—

C₂²⋊Q₈×C₇₉

—

#121×C₇₉

—

C₄×Q₈×C₇₉

—

C₄×D₄×C₇₉

—

#117×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

#103×C₇₉

—

#102×C₇₉

—

C₄×C₈×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

#41×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

#41

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

C₂²⋊C₄

yes

—

C₂²⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

—

C₂³×C₇₉

—

C₂³×C₇₉

#123

yes

characteristic

Q₈×C₇₉

—

Q₈×C₇₉

—

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×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄×D₄

yes

—

C₂×C₄×C₇₉

#120

yes

characteristic

C₂×C₄×C₇₉

#121

yes

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

100

C₂²×C₇₉

—

101

C₂²×C₇₉

#386

yes

characteristic

102

C₄×C₇₉

—

103

C₄×C₇₉

—

104

C₄×C₇₉

—

105

C₄×C₇₉

—

106

C₄×C₇₉

#443

yes

—

107

C₄×C₇₉

#444

yes

—

108

C₄×C₇₉

#454

yes

—

109

#9070

—

maximal, Sylow

110

C₂×C₇₉

—

111

C₂×C₇₉

—

112

C₂×C₇₉

#1876

yes

—

113

C₂×C₇₉

#1888

yes

characteristic

114

#3131

D₇₉

yes

characteristic

115

#2986

—

116

#1526

—

117

#1484

—

118

#1278

—

119

C₇₉

#9070

yes

characteristic, complemented, Sylow

120

#533

C₂×D₇₉

yes

characteristic

121

#532

C₂×D₇₉

yes

characteristic

122

#488

—

123

#487

Dic₇₉

yes

—

124

#486

Dic₇₉

yes

—

125

#446

—

158

—

126

C₈×Q₈

—

158

—

127

#433

—

158

—

128

#431

—

129

#416

—

130

C₄×M₁₆

—

131

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

132

#330

—

158

—

133

D₄⋊C₈

—

158

—

134

D₄⋊C₈

—

158

—

135

C₂×#41

—

136

C₂×M₁₆

—

158

—

137

C₂²×C₈

—

138

C₄⋊Q₈

C₂×Dic₇₉

yes

characteristic

139

C₄⋊Q₈

C₂×Dic₇₉

yes

—

140

#127

C₂×Dic₇₉

yes

characteristic

141

#125

C₂²⋊D₇₉

yes

—

142

#124

—

143

C₂²⋊Q₈

—

144

#121

—

145

C₄×Q₈

—

146

C₄×D₄

—

147

#117

C₂²⋊D₇₉

yes

—

148

C₂×C₄²

C₂²×D₇₉

yes

characteristic

149

C₄⋊C₈

—

150

C₄⋊C₈

—

151

C₄⋊C₈

—

158

—

152

#103

—

153

#102

—

154

C₂²⋊C₈

—

158

—

155

#97

—

158

—

156

#97

—

158

—

157

C₄×C₈

—

158

C₄×C₈

—

158

—

159

C₄×C₈

—

158

—

160

C₄×C₈

—

158

—

161

C₄×C₈

C₂×Dic₇₉

yes

—

162

#41

—

163

C₂×Q₈

—

164

C₂×Q₈

—

165

C₂×D₄

—

166

C₂×D₄

—

167

C₂×D₄

—

168

C₂²×C₄

—

169

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

170

C₂²×C₄

#11929623

yes

—

171

M₁₆

—

316

—

172

M₁₆

—

158

—

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

—

316

—

176

C₂×C₈

—

158

—

177

C₂×C₈

—

158

—

178

C₄⋊C₄

—

179

C₄⋊C₄

—

180

C₄⋊C₄

—

181

C₄⋊C₄

C₂²⋊Dic₇₉

yes

—

182

C₂²⋊C₄

—

183

C₂²⋊C₄

—

184

C₄²

—

185

C₄²

C₂²×Dic₇₉

yes

characteristic

186

C₄²

D₄×D₇₉

yes

—

187

C₄²

C₂×C₂²⋊D₇₉

yes

characteristic

188

C₂³

—

189

C₂³

#14579142

yes

characteristic

190

Q₈

—

191

Q₈

—

192

Q₈

—

193

D₄

—

194

D₄

—

195

D₄

—

196

D₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

C₂×C₂²⋊Dic₇₉

yes

characteristic

204

C₂×C₄

D₄×Dic₇₉

yes

—

205

C₂×C₄

#14579129

yes

characteristic

206

C₂×C₄

#14579133

yes

—

207

C₈

—

208

C₈

—

316

—

209

C₈

—

316

—

210

C₈

—

316

—

211

C₂²

—

212

C₂²

—

213

C₂²

—

214

C₂²

—

215

C₂²

#19098443

yes

characteristic

216

C₄

—

217

C₄

—

218

C₄

—

219

C₄

—

220

C₄

#19098680

yes

—

221

C₄

#19098685

yes

—

222

C₄

#19098735

yes

—

223

C₂

—

224

C₂

—

225

C₂

#23763729

yes

—

226

C₂

#23763804

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14579129
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#3131	C₂
socle	C₂²×C₇₉	#386
prime radical	#532×C₇₉	C₂²
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#3131	D₇₉
hypocenter	C₇₉	#9070
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3131

subgroup

G/H

center

C₂×C₄

#14579129

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#3131

C₂

socle

C₂²×C₇₉

#386

prime radical

#532×C₇₉

C₂²

cosocle

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3131

N	H ≅ G/N	type
1. #23772502	C₂	semidirect
2. #23761995	C₂	semidirect
4. #23761237	C₂	semidirect
119. C₇₉	#9070	semidirect

H ≅ G/N

type

1. #23772502

C₂

semidirect

2. #23761995

C₂

semidirect

4. #23761237

C₂

semidirect

119. C₇₉

#9070

semidirect

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

subgroup	index
#23772502	2
#23761995	2
#23761817	2
#23761237	2
C₇₉×#3131	2
#9070	79

subgroup

index

C₇₉×#3131

Subgroups — #34183596

Conjugacy classes of subgroups

Complements

Maximal subgroups