Group #34179272 — subgroups

Subgroups — #34179272

Order 20224.

452 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 143 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	#23762062	C₂	1	1	yes	maximal, characteristic
2	#23761992	C₂	1	1	yes	maximal, characteristic
3	C₈×#11929608	C₂	1	2	yes	maximal, complemented
4	#23761289	C₂	1	2	yes	maximal, complemented
5	C₇₉×#2931	C₂	1	1	yes	maximal, characteristic
6	#19098616	C₂²	1	2	yes	—
7	C₄×#11929608	C₂²	1	2	yes	—
8	#19098312	C₄	1	2	yes	—
9	#19098307	C₄	1	2	yes	—
10	#19098303	C₄	1	4	yes	—
11	#19098285	—	2	2	no	—
12	C₈×C₇₉⋊C₈	—	2	2	no	—
13	C₂×#131×C₇₉	C₄	1	2	yes	—
14	#532×C₇₉	C₂²	1	1	yes	characteristic
15	#436×C₇₉	—	2	2	no	—
16	C₈×D₄×C₇₉	—	2	2	no	—
17	#423×C₇₉	C₂²	1	1	yes	characteristic
18	#408×C₇₉	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
20	C₂×#11929608	—	2	2	no	—
21	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	4	yes	—
22	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
23	#14579072	C₂×C₄	1	4	yes	—
24	C₄×C₇₉⋊C₈	—	2	2	no	—
25	C₄×C₇₉⋊C₈	—	2	2	no	—
26	C₂×#41×C₇₉	C₂×C₄	1	2	yes	—
27	#131×C₇₉	—	4	4	no	—
28	C₂×M₁₆×C₇₉	—	2	2	no	—
29	C₂×M₁₆×C₇₉	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₇₉	—	2	2	no	—
31	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
33	C₄⋊Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
34	#127×C₇₉	C₂×C₄	1	1	yes	characteristic
35	#124×C₇₉	C₂×C₄	1	2	yes	—
36	#121×C₇₉	—	2	2	no	—
37	C₄×D₄×C₇₉	—	2	2	no	—
38	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₇₉	—	2	2	no	—
40	C₂²⋊C₈×C₇₉	—	2	4	no	—
41	C₄×C₈×C₇₉	D₄	1	2	yes	—
42	C₄×C₈×C₇₉	D₄	1	2	yes	—
43	#11929608	—	2	8	no	—
44	C₂×C₇₉⋊C₈	—	2	4	no	—
45	C₂×C₇₉⋊C₈	—	2	4	no	—
46	#41×C₇₉	—	4	2	no	—
47	#41×C₇₉	—	4	2	no	—
48	C₂×Q₈×C₇₉	C₄²	1	2	yes	—
49	C₂×D₄×C₇₉	—	2	2	no	—
50	C₂×D₄×C₇₉	—	2	2	no	—
51	C₂×D₄×C₇₉	C₄²	1	2	yes	—
52	C₂²×C₄×C₇₉	—	2	2	no	—
53	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₇₉	#41	1	2	yes	—
55	M₁₆×C₇₉	—	4	2	no	—
56	M₁₆×C₇₉	—	4	4	no	—
57	C₂×C₈×C₇₉	—	2	2	no	—
58	C₂×C₈×C₇₉	—	2	2	no	—
59	C₂×C₈×C₇₉	—	4	1	no	—
60	C₂×C₈×C₇₉	—	4	4	no	—
61	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₄⋊C₄×C₇₉	—	2	2	no	—
65	C₂²⋊C₄×C₇₉	—	2	4	no	—
66	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
67	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₇₉	C₂×D₄	1	2	yes	—
69	C₂³×C₇₉	—	2	2	no	—
70	C₂³×C₇₉	#117	1	1	yes	characteristic
71	Q₈×C₇₉	—	4	2	no	—
72	D₄×C₇₉	—	4	2	no	—
73	D₄×C₇₉	—	4	4	no	—
74	D₄×C₇₉	—	4	4	no	—
75	C₂×C₄×C₇₉	—	2	1	no	—
76	C₂×C₄×C₇₉	—	2	1	no	—
77	C₂×C₄×C₇₉	—	2	4	no	—
78	C₂×C₄×C₇₉	—	2	2	no	—
79	C₂×C₄×C₇₉	—	4	2	no	—
80	C₂×C₄×C₇₉	—	4	2	no	—
81	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
84	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
85	C₈×C₇₉	—	2	1	no	—
86	C₈×C₇₉	—	2	2	no	—
87	C₈×C₇₉	—	4	2	no	—
88	C₈×C₇₉	#100	1	2	yes	—
89	C₇₉⋊C₈	—	2	8	no	—
90	C₇₉⋊C₈	—	4	4	no	—
91	C₂²×C₇₉	—	2	1	no	—
92	C₂²×C₇₉	—	2	2	no	—
93	C₂²×C₇₉	—	4	2	no	—
94	C₂²×C₇₉	—	4	4	no	—
95	C₂²×C₇₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
96	C₄×C₇₉	—	2	1	no	—
97	C₄×C₇₉	—	2	2	no	—
98	C₄×C₇₉	—	4	2	no	—
99	C₄×C₇₉	C₂×#100	1	1	yes	characteristic
100	C₄×C₇₉	#413	1	1	yes	characteristic
101	C₄×C₇₉	#443	1	4	yes	—
102	#8241	—	79	1	no	maximal, Sylow
103	C₂×C₇₉	—	4	1	no	—
104	C₂×C₇₉	—	8	2	no	—
105	C₂×C₇₉	C₄×#100	1	1	yes	characteristic
106	C₂×C₇₉	#1764	1	2	yes	—
107	#2931	D₇₉	1	1	yes	characteristic
108	#1543	—	79	1	no	—
109	#1525	—	79	1	no	—
110	C₈×M₁₆	—	79	2	no	—
111	#1299	—	79	2	no	—
112	C₇₉	#8241	1	1	yes	characteristic, complemented, Sylow
113	C₂×#131	Dic₇₉	1	2	yes	—
114	#532	C₂×D₇₉	1	1	yes	characteristic
115	#436	—	2	2	no	—
116	C₈×D₄	—	2	2	no	—
117	#431	—	79	2	no	—
118	#423	C₂×D₇₉	1	1	yes	characteristic
119	#408	Dic₇₉	1	2	yes	—
120	C₄×M₁₆	—	79	2	no	—
121	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
122	#333	—	79	2	no	—
123	#331	—	79	2	no	—
124	#329	—	79	4	no	—
125	#322	—	158	2	no	—
126	C₈²	—	158	2	no	—
127	C₂×#41	C₂×Dic₇₉	1	2	yes	—
128	#131	—	4	4	no	—
129	C₂×M₁₆	—	2	2	no	—
130	C₂×M₁₆	—	158	2	no	—
131	C₂×M₁₆	C₂×Dic₇₉	1	2	yes	—
132	C₂²×C₈	—	2	2	no	—
133	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
134	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
135	C₄⋊Q₈	C₄×D₇₉	1	1	yes	characteristic
136	#127	C₄×D₇₉	1	1	yes	characteristic
137	#124	C₄×D₇₉	1	2	yes	—
138	#121	—	2	2	no	—
139	C₄×D₄	—	2	2	no	—
140	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
141	C₄⋊C₈	—	2	2	no	—
142	C₄⋊C₈	—	79	4	no	—
143	C₂²⋊C₈	—	2	4	no	—
144	C₂²⋊C₈	—	158	2	no	—
145	#97	—	79	4	no	—
146	C₄×C₈	—	158	2	no	—
147	C₄×C₈	—	158	2	no	—
148	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
149	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
150	#41	—	4	2	no	—
151	#41	—	4	2	no	—
152	C₂×Q₈	C₄×Dic₇₉	1	2	yes	—
153	C₂×D₄	—	2	2	no	—
154	C₂×D₄	—	2	2	no	—
155	C₂×D₄	C₄×Dic₇₉	1	2	yes	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
158	C₂²×C₄	#11929624	1	2	yes	—
159	M₁₆	—	4	4	no	—
160	M₁₆	—	4	2	no	—
161	M₁₆	—	158	8	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	4	1	no	—
165	C₂×C₈	—	4	4	no	—
166	C₂×C₈	—	158	4	no	—
167	C₂×C₈	—	158	4	no	—
168	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
169	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
170	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
171	C₄⋊C₄	—	2	2	no	—
172	C₂²⋊C₄	—	2	4	no	—
173	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
174	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
175	C₄²	C₂×C₂²⋊D₇₉	1	2	yes	—
176	C₂³	—	2	2	no	—
177	C₂³	#14579120	1	1	yes	characteristic
178	Q₈	—	4	2	no	—
179	D₄	—	4	2	no	—
180	D₄	—	4	4	no	—
181	D₄	—	4	4	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
189	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
190	C₂×C₄	C₄×C₂²⋊D₇₉	1	2	yes	—
191	C₂×C₄	C₄×C₂²⋊D₇₉	1	2	yes	—
192	C₈	—	2	1	no	—
193	C₈	—	2	2	no	—
194	C₈	—	4	2	no	—
195	C₈	—	316	4	no	—
196	C₈	—	158	8	no	—
197	C₈	#14579078	1	2	yes	—
198	C₂²	—	2	2	no	—
199	C₂²	—	2	1	no	—
200	C₂²	—	4	4	no	—
201	C₂²	—	4	2	no	—
202	C₂²	C₄×C₂²⋊Dic₇₉	1	1	yes	characteristic
203	C₄	—	2	2	no	—
204	C₄	—	2	1	no	—
205	C₄	—	4	2	no	—
206	C₄	C₂×#14579078	1	1	yes	characteristic
207	C₄	#19098547	1	1	yes	characteristic
208	C₄	#19098678	1	4	yes	—
209	C₂	—	4	1	no	—
210	C₂	—	8	2	no	—
211	C₂	C₄×#14579078	1	1	yes	characteristic
212	C₂	#23763235	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23762062

C₂

yes

maximal, characteristic

#23761992

C₂

yes

maximal, characteristic

C₈×#11929608

C₂

yes

maximal, complemented

#23761289

C₂

yes

maximal, complemented

C₇₉×#2931

C₂

yes

maximal, characteristic

#19098616

C₂²

yes

—

C₄×#11929608

C₂²

yes

—

#19098312

C₄

yes

—

#19098307

C₄

yes

—

#19098303

C₄

yes

—

#19098285

—

C₈×C₇₉⋊C₈

—

C₂×#131×C₇₉

C₄

yes

—

#532×C₇₉

C₂²

yes

characteristic

#436×C₇₉

—

C₈×D₄×C₇₉

—

#423×C₇₉

C₂²

yes

characteristic

#408×C₇₉

C₄

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×#11929608

—

C₄⋊(C₇₉⋊C₈)

C₂×C₄

yes

—

C₂²⋊(C₇₉⋊C₈)

—

#14579072

C₂×C₄

yes

—

C₄×C₇₉⋊C₈

—

C₄×C₇₉⋊C₈

—

C₂×#41×C₇₉

C₂×C₄

yes

—

#131×C₇₉

—

C₂×M₁₆×C₇₉

—

C₂×M₁₆×C₇₉

C₂×C₄

yes

—

C₂²×C₈×C₇₉

—

C₂²×C₈×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄⋊Q₈×C₇₉

C₂×C₄

yes

characteristic

#127×C₇₉

C₂×C₄

yes

characteristic

#124×C₇₉

C₂×C₄

yes

—

#121×C₇₉

—

C₄×D₄×C₇₉

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄⋊C₈×C₇₉

—

C₂²⋊C₈×C₇₉

—

C₄×C₈×C₇₉

D₄

yes

—

C₄×C₈×C₇₉

D₄

yes

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

#41×C₇₉

—

#41×C₇₉

—

C₂×Q₈×C₇₉

C₄²

yes

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

C₄²

yes

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₉

#41

yes

—

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₉

C₂²⋊C₄

yes

characteristic

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

—

C₂³×C₇₉

—

C₂³×C₇₉

#117

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

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄×D₄

yes

—

C₂×C₄×C₇₉

C₄×D₄

yes

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

#100

yes

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₄×C₂²⋊C₄

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×#100

yes

characteristic

100

C₄×C₇₉

#413

yes

characteristic

101

C₄×C₇₉

#443

yes

—

102

#8241

—

maximal, Sylow

103

C₂×C₇₉

—

104

C₂×C₇₉

—

105

C₂×C₇₉

C₄×#100

yes

characteristic

106

C₂×C₇₉

#1764

yes

—

107

#2931

D₇₉

yes

characteristic

108

#1543

—

109

#1525

—

110

C₈×M₁₆

—

111

#1299

—

112

C₇₉

#8241

yes

characteristic, complemented, Sylow

113

C₂×#131

Dic₇₉

yes

—

114

#532

C₂×D₇₉

yes

characteristic

115

#436

—

116

C₈×D₄

—

117

#431

—

118

#423

C₂×D₇₉

yes

characteristic

119

#408

Dic₇₉

yes

—

120

C₄×M₁₆

—

121

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

122

#333

—

123

#331

—

124

#329

—

125

#322

—

158

—

126

C₈²

—

158

—

127

C₂×#41

C₂×Dic₇₉

yes

—

128

#131

—

129

C₂×M₁₆

—

130

C₂×M₁₆

—

158

—

131

C₂×M₁₆

C₂×Dic₇₉

yes

—

132

C₂²×C₈

—

133

C₂²×C₈

C₂×Dic₇₉

yes

characteristic

134

C₂²×C₈

C₂×Dic₇₉

yes

characteristic

135

C₄⋊Q₈

C₄×D₇₉

yes

characteristic

136

#127

C₄×D₇₉

yes

characteristic

137

#124

C₄×D₇₉

yes

—

138

#121

—

139

C₄×D₄

—

140

C₂×C₄²

C₂²×D₇₉

yes

characteristic

141

C₄⋊C₈

—

142

C₄⋊C₈

—

143

C₂²⋊C₈

—

144

C₂²⋊C₈

—

158

—

145

#97

—

146

C₄×C₈

—

158

—

147

C₄×C₈

—

158

—

148

C₄×C₈

C₂²⋊D₇₉

yes

—

149

C₄×C₈

C₂²⋊D₇₉

yes

—

150

#41

—

151

#41

—

152

C₂×Q₈

C₄×Dic₇₉

yes

—

153

C₂×D₄

—

154

C₂×D₄

—

155

C₂×D₄

C₄×Dic₇₉

yes

—

156

C₂²×C₄

—

157

C₂²×C₄

C₂²×Dic₇₉

yes

characteristic

158

C₂²×C₄

#11929624

yes

—

159

M₁₆

—

160

M₁₆

—

161

M₁₆

—

158

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

158

—

167

C₂×C₈

—

158

—

168

C₂×C₈

C₂²⋊Dic₇₉

yes

—

169

C₂×C₈

C₂²⋊Dic₇₉

yes

characteristic

170

C₂×C₈

C₂²⋊Dic₇₉

yes

characteristic

171

C₄⋊C₄

—

172

C₂²⋊C₄

—

173

C₄²

C₂×C₄×D₇₉

yes

characteristic

174

C₄²

C₂×C₄×D₇₉

yes

characteristic

175

C₄²

C₂×C₂²⋊D₇₉

yes

—

176

C₂³

—

177

C₂³

#14579120

yes

characteristic

178

Q₈

—

179

D₄

—

180

D₄

—

181

D₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

C₂×C₄×Dic₇₉

yes

characteristic

189

C₂×C₄

C₂×C₂²⋊Dic₇₉

yes

characteristic

190

C₂×C₄

C₄×C₂²⋊D₇₉

yes

—

191

C₂×C₄

C₄×C₂²⋊D₇₉

yes

—

192

C₈

—

193

C₈

—

194

C₈

—

195

C₈

—

316

—

196

C₈

—

158

—

197

C₈

#14579078

yes

—

198

C₂²

—

199

C₂²

—

200

C₂²

—

201

C₂²

—

202

C₂²

C₄×C₂²⋊Dic₇₉

yes

characteristic

203

C₄

—

204

C₄

—

205

C₄

—

206

C₄

C₂×#14579078

yes

characteristic

207

C₄

#19098547

yes

characteristic

208

C₄

#19098678

yes

—

209

C₂

—

210

C₂

—

211

C₂

C₄×#14579078

yes

characteristic

212

C₂

#23763235

yes

—

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₇₉
commutator	C₂×C₄×C₇₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#2931	C₂
socle	C₂²×C₇₉	C₄×C₂²⋊C₄
prime radical	#532×C₇₉	C₂²
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#2931	D₇₉
hypocenter	C₇₉	#8241
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2931

subgroup

G/H

center

C₂×C₈

C₂²⋊Dic₇₉

commutator

C₂×C₄×C₇₉

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#2931

C₂

socle

C₂²×C₇₉

C₄×C₂²⋊C₄

prime radical

#532×C₇₉

C₂²

cosocle

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2931

N	H ≅ G/N	type
3. C₈×#11929608	C₂	semidirect
4. #23761289	C₂	semidirect
112. C₇₉	#8241	semidirect

H ≅ G/N

type

3. C₈×#11929608

C₂

semidirect

4. #23761289

C₂

semidirect

112. C₇₉

#8241

semidirect

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

subgroup	index
#23762062	2
#23761992	2
C₈×#11929608	2
#23761289	2
C₇₉×#2931	2
#8241	79

subgroup

index

C₇₉×#2931

Subgroups — #34179272

Conjugacy classes of subgroups

Complements

Maximal subgroups