Group #34302166 — subgroups

Subgroups — #34302166

Order 20224.

826 conjugacy classes of subgroups; only the 243 normal subgroups are shown.

normal subgroups only

This group has 826 conjugacy classes of subgroups — too many to draw. Only its 243 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	#23772118	C₂	1	2	yes	maximal, complemented
2	#23764229	C₂	1	2	yes	maximal, complemented
3	#23763729	C₂	1	8	yes	maximal, complemented
4	#23763541	C₂	1	1	yes	maximal, characteristic, complemented
5	#23763148	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇₉×#3150	C₂	1	1	yes	maximal, characteristic
7	C₂×#14579168	C₂²	1	4	yes	—
8	#19098635	—	2	8	no	—
9	#19098587	C₂²	1	2	yes	—
10	C₂×#14579088	C₂²	1	8	yes	—
11	#19098530	C₂²	1	4	yes	—
12	C₂×#14579072	C₂²	1	1	yes	characteristic
13	C₄²×Dic₇₉	C₂²	1	1	yes	characteristic
14	#19098328	C₂²	1	4	yes	—
15	#19098317	C₂²	1	4	yes	—
16	C₂×#135×C₇₉	C₂²	1	4	yes	—
17	#532×C₇₉	C₂²	1	2	yes	—
18	#497×C₇₉	C₄	1	4	yes	complemented
19	#496×C₇₉	C₄	1	4	yes	complemented
20	C₂×#97×C₇₉	C₂²	1	1	yes	characteristic
21	#14579168	—	4	8	no	—
22	C₂×#11929608	—	2	4	no	—
23	C₂×#11929608	C₂³	1	4	yes	—
24	C₂²×C₇₉⋊C₈	—	2	4	no	—
25	C₂²×C₇₉⋊C₈	C₂³	1	2	yes	—
26	C₂×C₄×Dic₇₉	—	2	2	no	—
27	C₂×C₄×Dic₇₉	C₂³	1	2	yes	—
28	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
29	#14579088	—	4	8	no	—
30	C₂²⋊(C₇₉⋊C₈)	—	2	8	no	—
31	#14579072	D₄	1	4	yes	—
32	C₂×#41×C₇₉	C₂³	1	4	yes	—
33	#135×C₇₉	—	4	8	no	—
34	C₂×Q₁₆×C₇₉	C₂×C₄	1	4	yes	—
35	C₂×SD₁₆×C₇₉	C₂×C₄	1	8	yes	—
36	C₂×D₈×C₇₉	C₂×C₄	1	4	yes	—
37	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
38	C₄⋊Q₈×C₇₉	C₂×C₄	1	2	yes	—
39	#127×C₇₉	C₂×C₄	1	2	yes	—
40	#124×C₇₉	C₂×C₄	1	4	yes	—
41	#121×C₇₉	—	2	4	no	—
42	C₄×D₄×C₇₉	—	2	4	no	—
43	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
44	#97×C₇₉	C₂×C₄	1	4	yes	—
45	C₂²×Dic₇₉	—	2	2	no	—
46	#11929608	—	2	8	no	—
47	#11929608	—	4	8	no	—
48	C₂×C₇₉⋊C₈	—	2	4	no	—
49	C₂×C₇₉⋊C₈	—	2	4	no	—
50	C₂×C₇₉⋊C₈	—	4	8	no	—
51	C₂×C₇₉⋊C₈	C₂×D₄	1	4	yes	—
52	C₄×Dic₇₉	—	2	4	no	—
53	C₄×Dic₇₉	—	4	4	no	—
54	#41×C₇₉	—	2	8	no	—
55	#41×C₇₉	—	2	8	no	—
56	C₂×Q₈×C₇₉	C₂²×C₄	1	4	yes	—
57	C₂×D₄×C₇₉	—	2	4	no	—
58	C₂×D₄×C₇₉	—	2	4	no	—
59	C₂×D₄×C₇₉	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₇₉	—	2	4	no	—
61	C₂²×C₄×C₇₉	#41	1	2	yes	—
62	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
63	Q₁₆×C₇₉	—	4	4	no	—
64	SD₁₆×C₇₉	—	4	8	no	—
65	D₈×C₇₉	—	4	4	no	—
66	C₂×C₈×C₇₉	—	2	4	no	—
67	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
68	C₄⋊C₄×C₇₉	—	2	4	no	—
69	C₂²⋊C₄×C₇₉	—	2	8	no	—
70	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
71	C₄²×C₇₉	C₂×D₄	1	2	yes	—
72	C₂³×C₇₉	—	2	4	no	—
73	C₂³×C₇₉	C₂×#41	1	1	yes	characteristic
74	Q₈×C₇₉	—	2	8	no	—
75	D₄×C₇₉	—	2	8	no	—
76	D₄×C₇₉	—	4	8	no	—
77	D₄×C₇₉	—	8	4	no	—
78	C₂×C₄×C₇₉	—	2	4	no	—
79	C₂×C₄×C₇₉	—	4	2	no	—
80	C₂×C₄×C₇₉	—	4	4	no	—
81	C₂×C₄×C₇₉	—	4	4	no	—
82	C₂×C₄×C₇₉	C₄×D₄	1	4	yes	—
83	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
84	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
85	C₂×C₄×C₇₉	H(5,𝔽₂)	1	4	yes	—
86	C₂×Dic₇₉	—	2	4	no	—
87	C₂×Dic₇₉	—	4	2	no	—
88	C₂×Dic₇₉	—	4	2	no	—
89	C₈×C₇₉	—	2	4	no	—
90	C₇₉⋊C₈	—	2	4	no	—
91	C₇₉⋊C₈	—	4	4	no	—
92	C₂²×C₇₉	—	2	2	no	—
93	C₂²×C₇₉	—	4	4	no	—
94	C₂²×C₇₉	—	4	8	no	—
95	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
96	C₂²×C₇₉	#540	1	2	yes	—
97	C₄×C₇₉	—	2	4	no	—
98	C₄×C₇₉	—	4	4	no	—
99	C₄×C₇₉	#518	1	2	yes	—
100	C₄×C₇₉	#535	1	2	yes	—
101	Dic₇₉	—	4	4	no	—
102	#22283	—	79	1	no	maximal, Sylow
103	C₂×C₇₉	—	2	2	no	—
104	C₂×C₇₉	—	8	4	no	—
105	C₂×C₇₉	#2328	1	1	yes	characteristic
106	C₂×C₇₉	#2956	1	2	yes	—
107	#3150	D₇₉	1	1	yes	characteristic
108	#2932	—	79	2	no	—
109	#1962	—	79	2	no	—
110	#1876	—	79	8	no	—
111	#1838	—	79	1	no	—
112	#1745	—	79	1	no	—
113	C₇₉	#22283	1	1	yes	characteristic, complemented, Sylow
114	C₂×#135	C₂×D₇₉	1	4	yes	—
115	C₂×#131	—	79	4	no	—
116	#532	C₂×D₇₉	1	2	yes	—
117	#497	Dic₇₉	1	4	yes	complemented
118	#496	Dic₇₉	1	4	yes	complemented
119	#435	—	158	8	no	—
120	#423	—	79	2	no	—
121	C₂×#104	—	79	8	no	—
122	#408	—	79	4	no	—
123	C₂×#97	—	79	1	no	—
124	C₂×#97	C₂×D₇₉	1	1	yes	characteristic
125	C₄³	—	79	1	no	—
126	#341	—	79	4	no	—
127	#336	—	79	4	no	—
128	C₂×#41	C₂²×D₇₉	1	4	yes	—
129	#135	—	4	8	no	—
130	C₂×Q₁₆	C₂×Dic₇₉	1	4	yes	—
131	C₂×SD₁₆	C₂×Dic₇₉	1	8	yes	—
132	C₂×D₈	C₂×Dic₇₉	1	4	yes	—
133	#131	—	316	8	no	—
134	C₂×M₁₆	—	79	4	no	—
135	C₂×M₁₆	—	158	4	no	—
136	C₂²×C₈	—	79	2	no	—
137	C₂²×C₈	—	158	4	no	—
138	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
139	C₄⋊Q₈	C₂×Dic₇₉	1	2	yes	—
140	#127	C₂×Dic₇₉	1	2	yes	—
141	#124	C₂×Dic₇₉	1	4	yes	—
142	#121	—	2	4	no	—
143	C₄×D₄	—	2	4	no	—
144	C₂×C₄²	—	79	2	no	—
145	C₂×C₄²	—	158	2	no	—
146	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
147	C₄⋊C₈	—	158	4	no	—
148	#104	—	316	8	no	—
149	C₂²⋊C₈	—	158	8	no	—
150	#97	—	79	4	no	—
151	#97	C₂×Dic₇₉	1	4	yes	—
152	#41	—	2	8	no	—
153	#41	—	2	8	no	—
154	C₂×Q₈	C₂²×Dic₇₉	1	4	yes	—
155	C₂×D₄	—	2	4	no	—
156	C₂×D₄	—	2	4	no	—
157	C₂×D₄	C₂²×Dic₇₉	1	4	yes	—
158	C₂²×C₄	—	2	4	no	—
159	C₂²×C₄	—	158	2	no	—
160	C₂²×C₄	#11929623	1	2	yes	—
161	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
162	Q₁₆	—	4	4	no	—
163	SD₁₆	—	4	8	no	—
164	D₈	—	4	4	no	—
165	M₁₆	—	316	8	no	—
166	M₁₆	—	158	8	no	—
167	C₂×C₈	—	2	4	no	—
168	C₂×C₈	—	316	8	no	—
169	C₂×C₈	—	79	4	no	—
170	C₂×C₈	—	158	4	no	—
171	C₂×C₈	—	158	4	no	—
172	C₂×C₈	C₂²×Dic₇₉	1	4	yes	—
173	C₄⋊C₄	—	2	4	no	—
174	C₂²⋊C₄	—	2	8	no	—
175	C₄²	—	316	4	no	—
176	C₄²	—	158	4	no	—
177	C₄²	C₂²×Dic₇₉	1	2	yes	—
178	C₄²	D₄×D₇₉	1	2	yes	—
179	C₂³	—	2	4	no	—
180	C₂³	C₂×#11929623	1	1	yes	characteristic
181	Q₈	—	2	8	no	—
182	D₄	—	2	8	no	—
183	D₄	—	4	8	no	—
184	D₄	—	8	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	4	no	—
188	C₂×C₄	—	4	4	no	—
189	C₂×C₄	—	316	2	no	—
190	C₂×C₄	—	316	2	no	—
191	C₂×C₄	—	158	4	no	—
192	C₂×C₄	D₄×Dic₇₉	1	4	yes	—
193	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
194	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
195	C₂×C₄	#14579206	1	4	yes	—
196	C₈	—	2	4	no	—
197	C₈	—	316	4	no	—
198	C₈	—	158	4	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	4	4	no	—
201	C₂²	—	4	8	no	—
202	C₂²	C₂×D₄×Dic₇₉	1	1	yes	characteristic
203	C₂²	#19099167	1	2	yes	—
204	C₄	—	2	4	no	—
205	C₄	—	4	4	no	—
206	C₄	—	316	4	no	—
207	C₄	#19099038	1	2	yes	—
208	C₄	#19099126	1	2	yes	—
209	C₂	—	2	2	no	—
210	C₂	—	8	4	no	—
211	C₂	#23766126	1	1	yes	characteristic
212	C₂	#23772310	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23772118

C₂

yes

maximal, complemented

#23764229

C₂

yes

maximal, complemented

#23763729

C₂

yes

maximal, complemented

#23763541

C₂

yes

maximal, characteristic, complemented

#23763148

C₂

yes

maximal, characteristic, complemented

C₇₉×#3150

C₂

yes

maximal, characteristic

C₂×#14579168

C₂²

yes

—

#19098635

—

#19098587

C₂²

yes

—

C₂×#14579088

C₂²

yes

—

#19098530

C₂²

yes

—

C₂×#14579072

C₂²

yes

characteristic

C₄²×Dic₇₉

C₂²

yes

characteristic

#19098328

C₂²

yes

—

#19098317

C₂²

yes

—

C₂×#135×C₇₉

C₂²

yes

—

#532×C₇₉

C₂²

yes

—

#497×C₇₉

C₄

yes

complemented

#496×C₇₉

C₄

yes

complemented

C₂×#97×C₇₉

C₂²

yes

—

—

C₂³

yes

—

C₂²×C₇₉⋊C₈

—

C₂²×C₇₉⋊C₈

C₂³

yes

—

C₂×C₄×Dic₇₉

—

C₂×C₄×Dic₇₉

C₂³

yes

—

C₄⋊(C₇₉⋊C₈)

—

#14579088

—

C₂²⋊(C₇₉⋊C₈)

—

#14579072

D₄

yes

—

C₂×#41×C₇₉

C₂³

yes

—

#135×C₇₉

—

C₂×Q₁₆×C₇₉

C₂×C₄

yes

—

C₂×SD₁₆×C₇₉

C₂×C₄

yes

—

C₂×D₈×C₇₉

C₂×C₄

yes

—

C₂²×C₈×C₇₉

C₂³

yes

—

C₄⋊Q₈×C₇₉

C₂×C₄

yes

—

#127×C₇₉

C₂×C₄

yes

—

#124×C₇₉

C₂×C₄

yes

—

#121×C₇₉

—

C₄×D₄×C₇₉

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

#97×C₇₉

C₂×C₄

yes

—

C₂²×Dic₇₉

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₂×D₄

yes

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

#41×C₇₉

—

#41×C₇₉

—

C₂×Q₈×C₇₉

C₂²×C₄

yes

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

C₂²×C₄

yes

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

#41

yes

—

C₂²×C₄×C₇₉

C₂⁴

yes

characteristic

Q₁₆×C₇₉

—

SD₁₆×C₇₉

—

D₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂²×C₄

yes

—

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

C₂²×C₄

yes

—

C₄²×C₇₉

C₂×D₄

yes

—

C₂³×C₇₉

—

C₂³×C₇₉

C₂×#41

yes

characteristic

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₄×D₄

yes

—

C₂×C₄×C₇₉

C₂³×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²×D₄

yes

characteristic

C₂×C₄×C₇₉

H(5,𝔽₂)

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×C₄×D₄

yes

characteristic

C₂²×C₇₉

#540

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#518

yes

—

100

C₄×C₇₉

#535

yes

—

101

Dic₇₉

—

102

#22283

—

maximal, Sylow

103

C₂×C₇₉

—

104

C₂×C₇₉

—

105

C₂×C₇₉

#2328

yes

characteristic

106

C₂×C₇₉

#2956

yes

—

107

#3150

D₇₉

yes

characteristic

108

#2932

—

109

#1962

—

110

#1876

—

111

#1838

—

112

#1745

—

113

C₇₉

#22283

yes

characteristic, complemented, Sylow

114

C₂×#135

C₂×D₇₉

yes

—

115

C₂×#131

—

116

#532

C₂×D₇₉

yes

—

117

#497

Dic₇₉

yes

complemented

118

#496

Dic₇₉

yes

complemented

119

#435

—

158

—

120

#423

—

121

C₂×#104

—

122

#408

—

123

C₂×#97

—

124

C₂×#97

C₂×D₇₉

yes

characteristic

125

C₄³

—

126

#341

—

127

#336

—

128

C₂×#41

C₂²×D₇₉

yes

—

129

#135

—

130

C₂×Q₁₆

C₂×Dic₇₉

yes

—

131

C₂×SD₁₆

C₂×Dic₇₉

yes

—

132

C₂×D₈

C₂×Dic₇₉

yes

—

133

#131

—

316

—

134

C₂×M₁₆

—

135

C₂×M₁₆

—

158

—

136

C₂²×C₈

—

137

C₂²×C₈

—

158

—

138

C₂²×C₈

C₂²×D₇₉

yes

—

139

C₄⋊Q₈

C₂×Dic₇₉

yes

—

140

#127

C₂×Dic₇₉

yes

—

141

#124

C₂×Dic₇₉

yes

—

142

#121

—

143

C₄×D₄

—

144

C₂×C₄²

—

145

C₂×C₄²

—

158

—

146

C₂×C₄²

C₂²×D₇₉

yes

characteristic

147

C₄⋊C₈

—

158

—

148

#104

—

316

—

149

C₂²⋊C₈

—

158

—

150

#97

—

151

#97

C₂×Dic₇₉

yes

—

152

#41

—

153

#41

—

154

C₂×Q₈

C₂²×Dic₇₉

yes

—

155

C₂×D₄

—

156

C₂×D₄

—

157

C₂×D₄

C₂²×Dic₇₉

yes

—

158

C₂²×C₄

—

159

C₂²×C₄

—

158

—

160

C₂²×C₄

#11929623

yes

—

161

C₂²×C₄

C₂³×D₇₉

yes

characteristic

162

Q₁₆

—

163

SD₁₆

—

164

D₈

—

165

M₁₆

—

316

—

166

M₁₆

—

158

—

167

C₂×C₈

—

168

C₂×C₈

—

316

—

169

C₂×C₈

—

170

C₂×C₈

—

158

—

171

C₂×C₈

—

158

—

172

C₂×C₈

C₂²×Dic₇₉

yes

—

173

C₄⋊C₄

—

174

C₂²⋊C₄

—

175

C₄²

—

316

—

176

C₄²

—

158

—

177

C₄²

C₂²×Dic₇₉

yes

—

178

C₄²

D₄×D₇₉

yes

—

179

C₂³

—

180

C₂³

C₂×#11929623

yes

characteristic

181

Q₈

—

182

D₄

—

183

D₄

—

184

D₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

316

—

190

C₂×C₄

—

316

—

191

C₂×C₄

—

158

—

192

C₂×C₄

D₄×Dic₇₉

yes

—

193

C₂×C₄

C₂³×Dic₇₉

yes

characteristic

194

C₂×C₄

C₂×D₄×D₇₉

yes

characteristic

195

C₂×C₄

#14579206

yes

—

196

C₈

—

197

C₈

—

316

—

198

C₈

—

158

—

199

C₂²

—

200

C₂²

—

201

C₂²

—

202

C₂²

C₂×D₄×Dic₇₉

yes

characteristic

203

C₂²

#19099167

yes

—

204

C₄

—

205

C₄

—

206

C₄

—

316

—

207

C₄

#19099038

yes

—

208

C₄

#19099126

yes

—

209

C₂

—

210

C₂

—

211

C₂

#23766126

yes

characteristic

212

C₂

#23772310

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting, prime radical	C₇₉×#3150	C₂
socle	C₂²×C₇₉	C₂×C₄×D₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	#3150	D₇₉
hypocenter	C₇₉	#22283
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3150

subgroup

G/H

center

C₂×C₄

C₂×D₄×D₇₉

commutator

C₂×C₄×C₇₉

C₂³×C₄

Frattini

C₂²×C₄

C₂³×D₇₉

Fitting, prime radical

C₇₉×#3150

C₂

socle

C₂²×C₇₉

C₂×C₄×D₄

cosocle

C₂²×C₄×C₇₉

C₂⁴

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3150

N	H ≅ G/N	type
1. #23772118	C₂	semidirect
2. #23764229	C₂	semidirect
3. #23763729	C₂	semidirect
4. #23763541	C₂	semidirect
5. #23763148	C₂	semidirect
18. #497×C₇₉	C₄	semidirect
19. #496×C₇₉	C₄	semidirect
113. C₇₉	#22283	semidirect
117. #497	Dic₇₉	semidirect
118. #496	Dic₇₉	semidirect

H ≅ G/N

type

1. #23772118

C₂

semidirect

2. #23764229

C₂

semidirect

3. #23763729

C₂

semidirect

4. #23763541

C₂

semidirect

5. #23763148

C₂

semidirect

18. #497×C₇₉

C₄

semidirect

19. #496×C₇₉

C₄

semidirect

113. C₇₉

#22283

semidirect

117. #497

Dic₇₉

semidirect

118. #496

Dic₇₉

semidirect

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

subgroup	index
#23772118	2
#23764229	2
#23763729	2
#23763541	2
#23763148	2
C₇₉×#3150	2
#22283	79

subgroup

index

C₇₉×#3150

Subgroups — #34302166

Conjugacy classes of subgroups

Complements

Maximal subgroups