Group #34174845 — subgroups

Subgroups — #34174845

Order 20224.

406 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 127 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	#23763380	C₂	1	2	yes	maximal
2	C₇₉×#1728	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
4	C₂×C₂²⋊(C₇₉⋊C₈)	C₄	1	4	yes	—
5	#19098318	—	2	2	no	—
6	C₂²×M₁₆×C₇₉	C₄	1	2	yes	—
7	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
8	#336×C₇₉	—	2	2	no	—
9	C₂²×C₇₉⋊C₈	—	2	4	no	—
10	C₄⋊(C₇₉⋊C₈)	—	4	4	no	—
11	C₂²⋊(C₇₉⋊C₈)	—	4	4	no	—
12	C₂³×C₄×C₇₉	C₂×C₄	1	2	yes	—
13	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
14	C₂×M₁₆×C₇₉	—	2	2	no	—
15	C₂×M₁₆×C₇₉	—	4	2	no	—
16	C₂×M₁₆×C₇₉	—	4	2	no	—
17	C₂×M₁₆×C₇₉	C₈	1	4	yes	complemented
18	C₂²×C₈×C₇₉	—	2	2	no	—
19	C₂×C₄²×C₇₉	—	4	2	no	—
20	C₂×C₄²×C₇₉	D₄	1	2	yes	—
21	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
22	C₂×C₄²×C₇₉	Q₈	1	1	yes	characteristic
23	C₂×C₇₉⋊C₈	—	4	4	no	—
24	C₂×C₇₉⋊C₈	—	4	4	no	—
25	C₂×C₇₉⋊C₈	—	4	4	no	—
26	C₂⁴×C₇₉	C₄²	1	1	yes	characteristic
27	C₂²×C₄×C₇₉	—	2	2	no	—
28	C₂²×C₄×C₇₉	—	2	1	no	—
29	C₂²×C₄×C₇₉	—	2	1	no	—
30	C₂²×C₄×C₇₉	—	2	2	no	—
31	C₂²×C₄×C₇₉	—	2	2	no	—
32	C₂²×C₄×C₇₉	—	4	2	no	—
33	C₂²×C₄×C₇₉	—	4	2	no	—
34	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
37	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₇₉	C₂×C₈	1	2	yes	—
39	C₂²×C₄×C₇₉	M₁₆	1	2	yes	—
40	M₁₆×C₇₉	—	4	4	no	—
41	M₁₆×C₇₉	—	4	4	no	—
42	C₂×C₈×C₇₉	—	2	4	no	—
43	C₂×C₈×C₇₉	—	4	2	no	—
44	C₂×C₈×C₇₉	—	4	2	no	—
45	C₄²×C₇₉	—	2	4	no	—
46	C₄²×C₇₉	—	4	1	no	—
47	C₄²×C₇₉	—	4	1	no	—
48	C₂³×C₇₉	—	2	1	no	—
49	C₂³×C₇₉	—	2	2	no	—
50	C₂³×C₇₉	—	2	1	no	—
51	C₂³×C₇₉	—	2	2	no	—
52	C₂³×C₇₉	#95	1	1	yes	characteristic
53	C₂³×C₇₉	C₄×C₈	1	1	yes	characteristic
54	C₂³×C₇₉	#97	1	1	yes	characteristic
55	C₂×C₄×C₇₉	—	2	2	no	—
56	C₂×C₄×C₇₉	—	2	1	no	—
57	C₂×C₄×C₇₉	—	2	1	no	—
58	C₂×C₄×C₇₉	—	2	2	no	—
59	C₂×C₄×C₇₉	—	2	2	no	—
60	C₂×C₄×C₇₉	—	2	2	no	—
61	C₂×C₄×C₇₉	—	2	2	no	—
62	C₂×C₄×C₇₉	—	2	2	no	—
63	C₂×C₄×C₇₉	—	2	2	no	—
64	C₂×C₄×C₇₉	—	2	2	no	—
65	C₂×C₄×C₇₉	—	2	2	no	—
66	C₂×C₄×C₇₉	—	4	4	no	—
67	C₂×C₄×C₇₉	—	4	4	no	—
68	C₂×C₄×C₇₉	C₂²⋊C₈	1	2	yes	—
69	C₂×C₄×C₇₉	#99	1	2	yes	—
70	C₂×C₄×C₇₉	#100	1	2	yes	—
71	C₂×C₄×C₇₉	#100	1	1	yes	characteristic
72	C₂×C₄×C₇₉	#101	1	2	yes	—
73	C₂×C₄×C₇₉	#101	1	1	yes	characteristic
74	C₂×C₄×C₇₉	C₄⋊C₈	1	2	yes	—
75	C₈×C₇₉	—	4	4	no	—
76	C₇₉⋊C₈	—	8	4	no	—
77	C₂²×C₇₉	—	2	1	no	—
78	C₂²×C₇₉	—	2	1	no	—
79	C₂²×C₇₉	—	2	2	no	—
80	C₂²×C₇₉	—	2	2	no	—
81	C₂²×C₇₉	—	4	2	no	—
82	C₂²×C₇₉	—	4	1	no	—
83	C₂²×C₇₉	—	4	1	no	—
84	C₂²×C₇₉	#336	1	1	yes	characteristic
85	C₂²×C₇₉	#342	1	1	yes	characteristic
86	C₂²×C₇₉	#343	1	2	yes	—
87	C₂²×C₇₉	#343	1	1	yes	characteristic
88	C₂²×C₇₉	#344	1	2	yes	—
89	C₄×C₇₉	—	2	2	no	—
90	C₄×C₇₉	—	2	2	no	—
91	C₄×C₇₉	—	2	2	no	—
92	C₄×C₇₉	—	4	2	no	—
93	C₄×C₇₉	#323	1	2	yes	—
94	C₄×C₇₉	#324	1	2	yes	—
95	#7034	—	79	1	no	maximal, Sylow
96	C₂×C₇₉	—	4	1	no	—
97	C₂×C₇₉	—	4	1	no	—
98	C₂×C₇₉	#1280	1	1	yes	characteristic
99	C₂×C₇₉	#1283	1	2	yes	—
100	C₂×C₇₉	#1305	1	2	yes	—
101	C₂×C₇₉	#1307	1	1	yes	characteristic
102	C₂×C₇₉	#1309	1	1	yes	characteristic
103	#1797	—	79	2	no	—
104	#1728	D₇₉	1	1	yes	characteristic
105	C₇₉	#7034	1	1	yes	characteristic, complemented, Sylow
106	C₂²×M₁₆	Dic₇₉	1	2	yes	—
107	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
108	C₂×C₄⋊C₈	—	158	2	no	—
109	C₂×C₂²⋊C₈	—	79	4	no	—
110	#336	—	2	2	no	—
111	#336	—	158	2	no	—
112	C₂³×C₄	C₄×D₇₉	1	2	yes	—
113	C₂³×C₄	C₂×Dic₇₉	1	1	yes	characteristic
114	C₂×M₁₆	—	2	2	no	—
115	C₂×M₁₆	—	4	2	no	—
116	C₂×M₁₆	—	4	2	no	—
117	C₂×M₁₆	C₇₉⋊C₈	1	4	yes	complemented
118	C₂²×C₈	—	2	2	no	—
119	C₂²×C₈	—	158	4	no	—
120	C₂×C₄²	—	4	2	no	—
121	C₂×C₄²	Dic₁₅₈	1	1	yes	characteristic
122	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
123	C₂×C₄²	C₂²⋊D₇₉	1	2	yes	—
124	C₄⋊C₈	—	316	4	no	—
125	C₂²⋊C₈	—	316	4	no	—
126	C₂⁴	C₄×Dic₇₉	1	1	yes	characteristic
127	C₂²×C₄	—	2	2	no	—
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	—	2	1	no	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	—	2	2	no	—
132	C₂²×C₄	—	4	2	no	—
133	C₂²×C₄	—	4	2	no	—
134	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
135	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
136	C₂²×C₄	Dic₇₉⋊C₄	1	2	yes	—
137	C₂²×C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
138	C₂²×C₄	C₂×C₇₉⋊C₈	1	2	yes	—
139	C₂²×C₄	#11929608	1	2	yes	—
140	M₁₆	—	4	4	no	—
141	M₁₆	—	4	4	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	4	2	no	—
144	C₂×C₈	—	4	2	no	—
145	C₂×C₈	—	316	4	no	—
146	C₂×C₈	—	316	4	no	—
147	C₂×C₈	—	316	4	no	—
148	C₄²	—	2	4	no	—
149	C₄²	—	4	1	no	—
150	C₄²	—	4	1	no	—
151	C₂³	—	2	1	no	—
152	C₂³	—	2	2	no	—
153	C₂³	—	2	1	no	—
154	C₂³	—	2	2	no	—
155	C₂³	C₇₉⋊#95	1	1	yes	characteristic
156	C₂³	C₄×C₇₉⋊C₈	1	1	yes	characteristic
157	C₂³	#14579072	1	1	yes	characteristic
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	2	yes	—
172	C₂×C₄	#14579076	1	2	yes	—
173	C₂×C₄	#14579077	1	2	yes	—
174	C₂×C₄	#14579078	1	1	yes	characteristic
175	C₂×C₄	#14579079	1	2	yes	—
176	C₂×C₄	#14579080	1	1	yes	characteristic
177	C₂×C₄	C₄⋊(C₇₉⋊C₈)	1	2	yes	—
178	C₈	—	4	4	no	—
179	C₈	—	632	4	no	—
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	—	2	2	no	—
183	C₂²	—	2	2	no	—
184	C₂²	—	4	2	no	—
185	C₂²	—	4	1	no	—
186	C₂²	—	4	1	no	—
187	C₂²	#19098318	1	1	yes	characteristic
188	C₂²	#19098330	1	1	yes	characteristic
189	C₂²	#19098331	1	2	yes	—
190	C₂²	#19098332	1	1	yes	characteristic
191	C₂²	#19098335	1	2	yes	—
192	C₄	—	2	2	no	—
193	C₄	—	2	2	no	—
194	C₄	—	2	2	no	—
195	C₄	—	4	2	no	—
196	C₄	#19098287	1	2	yes	—
197	C₄	#19098289	1	2	yes	—
198	C₂	—	4	1	no	—
199	C₂	—	4	1	no	—
200	C₂	#23761243	1	1	yes	characteristic
201	C₂	#23761251	1	2	yes	—
202	C₂	#23761305	1	2	yes	—
203	C₂	#23761311	1	1	yes	characteristic
204	C₂	C₇₉⋊#1309	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23763380

C₂

yes

maximal

C₇₉×#1728

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₇₉⋊C₈)

—

C₂×C₂²⋊(C₇₉⋊C₈)

C₄

yes

—

#19098318

—

C₂²×M₁₆×C₇₉

C₄

yes

—

C₂²×C₄²×C₇₉

C₂²

yes

characteristic

#336×C₇₉

—

C₂²×C₇₉⋊C₈

—

C₄⋊(C₇₉⋊C₈)

—

C₂²⋊(C₇₉⋊C₈)

—

C₂³×C₄×C₇₉

C₂×C₄

yes

—

C₂³×C₄×C₇₉

C₂×C₄

yes

characteristic

C₂×M₁₆×C₇₉

—

C₂×M₁₆×C₇₉

—

C₂×M₁₆×C₇₉

—

C₂×M₁₆×C₇₉

C₈

yes

complemented

C₂²×C₈×C₇₉

—

C₂×C₄²×C₇₉

—

C₂×C₄²×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₂×C₄²×C₇₉

Q₈

yes

characteristic

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂⁴×C₇₉

C₄²

yes

characteristic

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₉

C₄⋊C₄

yes

—

C₂²×C₄×C₇₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×C₈

yes

—

C₂²×C₄×C₇₉

M₁₆

yes

—

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

#95

yes

characteristic

C₂³×C₇₉

C₄×C₈

yes

characteristic

C₂³×C₇₉

#97

yes

characteristic

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×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊C₈

yes

—

C₂×C₄×C₇₉

#99

yes

—

C₂×C₄×C₇₉

#100

yes

—

C₂×C₄×C₇₉

#100

yes

characteristic

C₂×C₄×C₇₉

#101

yes

—

C₂×C₄×C₇₉

#101

yes

characteristic

C₂×C₄×C₇₉

C₄⋊C₈

yes

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

—

C₂²×C₇₉

#343

yes

characteristic

C₂²×C₇₉

#344

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#323

yes

—

C₄×C₇₉

#324

yes

—

#7034

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#1280

yes

characteristic

C₂×C₇₉

#1283

yes

—

100

C₂×C₇₉

#1305

yes

—

101

C₂×C₇₉

#1307

yes

characteristic

102

C₂×C₇₉

#1309

yes

characteristic

103

#1797

—

104

#1728

D₇₉

yes

characteristic

105

C₇₉

#7034

yes

characteristic, complemented, Sylow

106

C₂²×M₁₆

Dic₇₉

yes

—

107

C₂²×C₄²

C₂×D₇₉

yes

characteristic

108

C₂×C₄⋊C₈

—

158

—

109

C₂×C₂²⋊C₈

—

110

#336

—

111

#336

—

158

—

112

C₂³×C₄

C₄×D₇₉

yes

—

113

C₂³×C₄

C₂×Dic₇₉

yes

characteristic

114

C₂×M₁₆

—

115

C₂×M₁₆

—

116

C₂×M₁₆

—

117

C₂×M₁₆

C₇₉⋊C₈

yes

complemented

118

C₂²×C₈

—

119

C₂²×C₈

—

158

—

120

C₂×C₄²

—

121

C₂×C₄²

Dic₁₅₈

yes

characteristic

122

C₂×C₄²

D₃₁₆

yes

characteristic

123

C₂×C₄²

C₂²⋊D₇₉

yes

—

124

C₄⋊C₈

—

316

—

125

C₂²⋊C₈

—

316

—

126

C₂⁴

C₄×Dic₇₉

yes

characteristic

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂²×C₄

(C₂×D₇₉)⋊C₄

yes

—

135

C₂²×C₄

C₂²⋊Dic₇₉

yes

characteristic

136

C₂²×C₄

Dic₇₉⋊C₄

yes

—

137

C₂²×C₄

C₄⋊Dic₇₉

yes

characteristic

138

C₂²×C₄

C₂×C₇₉⋊C₈

yes

—

139

C₂²×C₄

#11929608

yes

—

140

M₁₆

—

141

M₁₆

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

316

—

146

C₂×C₈

—

316

—

147

C₂×C₈

—

316

—

148

C₄²

—

149

C₄²

—

150

C₄²

—

151

C₂³

—

152

C₂³

—

153

C₂³

—

154

C₂³

—

155

C₂³

C₇₉⋊#95

yes

characteristic

156

C₂³

C₄×C₇₉⋊C₈

yes

characteristic

157

C₂³

#14579072

yes

characteristic

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

C₂²⋊(C₇₉⋊C₈)

yes

—

172

C₂×C₄

#14579076

yes

—

173

C₂×C₄

#14579077

yes

—

174

C₂×C₄

#14579078

yes

characteristic

175

C₂×C₄

#14579079

yes

—

176

C₂×C₄

#14579080

yes

characteristic

177

C₂×C₄

C₄⋊(C₇₉⋊C₈)

yes

—

178

C₈

—

179

C₈

—

632

—

180

C₂²

—

181

C₂²

—

182

C₂²

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

#19098318

yes

characteristic

188

C₂²

#19098330

yes

characteristic

189

C₂²

#19098331

yes

—

190

C₂²

#19098332

yes

characteristic

191

C₂²

#19098335

yes

—

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

#19098287

yes

—

197

C₄

#19098289

yes

—

198

C₂

—

199

C₂

—

200

C₂

#23761243

yes

characteristic

201

C₂

#23761251

yes

—

202

C₂

#23761305

yes

—

203

C₂

#23761311

yes

characteristic

204

C₂

C₇₉⋊#1309

yes

characteristic

	subgroup	G/H
center	C₂³	C₇₉⋊#95
commutator	C₂³×C₇₉	C₄×C₈
Frattini	C₂²×C₄²	C₂×D₇₉
Fitting	C₇₉×#1728	C₂
socle	C₂³×C₇₉	#95
prime radical	C₂⁴×C₇₉	C₄²
cosocle	C₂²×C₄²×C₇₉	C₂²
hypercenter	#1728	D₇₉
hypocenter	C₇₉	#7034
LCS	G ▷ C₂³×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1728

subgroup

G/H

center

C₂³

C₇₉⋊#95

commutator

C₂³×C₇₉

C₄×C₈

Frattini

C₂²×C₄²

C₂×D₇₉

Fitting

C₇₉×#1728

C₂

socle

C₂³×C₇₉

#95

prime radical

C₂⁴×C₇₉

C₄²

cosocle

C₂²×C₄²×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂³×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1728

N	H ≅ G/N	type
17. C₂×M₁₆×C₇₉	C₈	semidirect
105. C₇₉	#7034	semidirect
117. C₂×M₁₆	C₇₉⋊C₈	semidirect

H ≅ G/N

type

17. C₂×M₁₆×C₇₉

C₈

semidirect

105. C₇₉

#7034

semidirect

117. C₂×M₁₆

C₇₉⋊C₈

semidirect

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

subgroup	index
#23763380	2
C₇₉×#1728	2
#7034	79

subgroup

index

#23763380

C₇₉×#1728

#7034

Subgroups — #34174845

Conjugacy classes of subgroups

Complements

Maximal subgroups