Group #34204631 — subgroups

Subgroups — #34204631

Order 20224.

302 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 123 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	#23765155	C₂	1	2	yes	maximal
2	#23763086	C₂	1	1	yes	maximal, characteristic
3	#23761482	C₂	1	1	yes	maximal, characteristic
4	#23761481	C₂	1	2	yes	maximal
5	C₇₉×#1376	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579101	C₂²	1	2	yes	—
7	C₂×#14579072	C₄	1	1	yes	characteristic
8	C₂×#14579071	C₄	1	1	yes	characteristic
9	C₂×#14579071	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
12	#19098409	C₄	1	1	yes	characteristic
13	Dic₇₉⋊C₁₆	—	2	4	no	—
14	C₂×M₃₂×C₇₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
16	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
20	#14579101	—	2	2	no	—
21	#14579101	—	2	2	no	—
22	C₂×C₇₉⋊C₁₆	—	2	2	no	—
23	#14579072	—	2	2	no	—
24	#14579071	—	2	2	no	—
25	#14579071	C₈	1	4	yes	—
26	C₈×Dic₇₉	—	2	2	no	—
27	C₈×Dic₇₉	D₄	1	2	yes	—
28	C₈×Dic₇₉	Q₈	1	2	yes	—
29	C₇₉⋊#95	—	2	2	no	—
30	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
33	M₃₂×C₇₉	—	2	2	no	—
34	M₃₂×C₇₉	—	2	2	no	—
35	C₂×C₁₆×C₇₉	—	2	2	no	—
36	C₄×C₈×C₇₉	—	2	2	no	—
37	C₂²×Dic₇₉	—	2	1	no	—
38	C₂²×Dic₇₉	M₁₆	1	2	yes	—
39	C₂×C₇₉⋊C₈	—	2	1	no	—
40	C₂×C₇₉⋊C₈	—	2	1	no	—
41	C₂×C₇₉⋊C₈	—	2	1	no	—
42	C₂×C₇₉⋊C₈	—	2	1	no	—
43	C₂×C₇₉⋊C₈	C₄⋊C₄	1	2	yes	—
44	C₂×C₇₉⋊C₈	C₂×C₈	1	2	yes	—
45	C₄×Dic₇₉	—	2	1	no	—
46	C₄×Dic₇₉	C₄⋊C₄	1	2	yes	—
47	C₄×Dic₇₉	C₂×C₈	1	2	yes	—
48	C₄×Dic₇₉	M₁₆	1	2	yes	—
49	C₇₉⋊C₁₆	—	4	2	no	—
50	C₂²×C₄×C₇₉	—	2	1	no	—
51	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₇₉	—	2	1	no	—
53	C₂×C₈×C₇₉	—	2	1	no	—
54	C₂×C₈×C₇₉	C₂×C₈	1	2	yes	—
55	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₇₉	#41	1	2	yes	—
58	C₂×C₈×C₇₉	#41	1	2	yes	—
59	C₄²×C₇₉	—	2	1	no	—
60	C₄²×C₇₉	M₁₆	1	2	yes	—
61	C₁₆×C₇₉	—	4	2	no	—
62	C₂³×C₇₉	C₂×M₁₆	1	1	yes	characteristic
63	C₂×C₄×C₇₉	—	2	2	no	—
64	C₂×C₄×C₇₉	—	4	2	no	—
65	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
67	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
68	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
71	C₂×Dic₇₉	—	2	2	no	—
72	C₂×Dic₇₉	—	4	2	no	—
73	C₂×Dic₇₉	—	4	2	no	—
74	C₂×Dic₇₉	C₄⋊C₈	1	4	yes	—
75	C₈×C₇₉	—	2	2	no	—
76	C₈×C₇₉	#124	1	2	yes	—
77	C₈×C₇₉	#125	1	2	yes	—
78	C₇₉⋊C₈	—	2	2	no	—
79	C₇₉⋊C₈	—	2	2	no	—
80	C₂²×C₇₉	—	2	2	no	—
81	C₂²×C₇₉	C₂×C₄⋊C₈	1	1	yes	characteristic
82	C₂²×C₇₉	#432	1	1	yes	characteristic
83	C₂²×C₇₉	#432	1	1	yes	characteristic
84	C₄×C₇₉	—	4	2	no	—
85	C₄×C₇₉	#382	1	1	yes	characteristic
86	C₄×C₇₉	#423	1	1	yes	characteristic
87	C₄×C₇₉	#431	1	1	yes	characteristic
88	C₄×C₇₉	#431	1	1	yes	characteristic
89	Dic₇₉	—	2	4	no	—
90	Dic₇₉	—	4	2	no	—
91	#12884	—	79	1	no	maximal, Sylow
92	C₂×C₇₉	—	2	1	no	—
93	C₂×C₇₉	—	2	1	no	—
94	C₂×C₇₉	#1831	1	1	yes	characteristic
95	C₂×C₇₉	#2153	1	2	yes	—
96	#2150	—	79	2	no	—
97	#1726	—	79	1	no	—
98	#1376	—	79	2	no	—
99	#1376	—	79	1	no	—
100	#1376	D₇₉	1	1	yes	characteristic
101	C₇₉	#12884	1	1	yes	characteristic, complemented, Sylow
102	C₂×M₃₂	—	79	2	no	—
103	C₂×M₃₂	C₂×D₇₉	1	2	yes	—
104	C₂×#97	—	79	1	no	—
105	C₂×#97	—	79	1	no	—
106	C₂×#97	—	79	1	no	—
107	C₂×C₄×C₈	—	79	1	no	—
108	C₂×C₄×C₈	—	79	1	no	—
109	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
110	#376	—	79	1	no	—
111	C₄⋊C₁₆	—	158	4	no	—
112	C₂²×C₈	—	79	1	no	—
113	C₂²×C₈	—	79	1	no	—
114	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
115	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
116	C₂×C₄²	—	79	1	no	—
117	C₂×C₄²	—	79	1	no	—
118	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
119	M₃₂	—	2	2	no	—
120	M₃₂	—	2	2	no	—
121	M₃₂	—	158	2	no	—
122	M₃₂	—	158	2	no	—
123	C₂×C₁₆	—	2	2	no	—
124	C₂×C₁₆	—	158	2	no	—
125	#97	—	79	4	no	—
126	#97	—	158	2	no	—
127	#97	—	158	2	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	—	79	2	no	—
130	C₄×C₈	—	79	2	no	—
131	C₄×C₈	—	158	2	no	—
132	#95	—	158	2	no	—
133	C₂²×C₄	—	2	1	no	—
134	C₂²×C₄	—	79	2	no	—
135	C₂²×C₄	—	158	1	no	—
136	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	79	2	no	—
140	C₂×C₈	—	79	2	no	—
141	C₂×C₈	—	158	1	no	—
142	C₂×C₈	—	158	1	no	—
143	C₂×C₈	—	158	1	no	—
144	C₂×C₈	—	158	1	no	—
145	C₂×C₈	C₈×D₇₉	1	2	yes	—
146	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
147	C₂×C₈	Q₈×D₇₉	1	1	yes	characteristic
148	C₂×C₈	#11929623	1	2	yes	—
149	C₂×C₈	#11929624	1	2	yes	—
150	C₄²	—	2	1	no	—
151	C₄²	—	79	2	no	—
152	C₄²	—	79	2	no	—
153	C₄²	—	79	2	no	—
154	C₄²	—	158	1	no	—
155	C₄²	#11929607	1	2	yes	—
156	C₁₆	—	4	2	no	—
157	C₁₆	—	316	2	no	—
158	C₂³	M₁₆×D₇₉	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	316	2	no	—
162	C₂×C₄	—	316	2	no	—
163	C₂×C₄	—	79	4	no	—
164	C₂×C₄	—	158	2	no	—
165	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
166	C₂×C₄	#14579117	1	1	yes	characteristic
167	C₂×C₄	#14579118	1	1	yes	characteristic
168	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
169	C₂×C₄	M₁₆×D₇₉	1	1	yes	characteristic
170	C₂×C₄	C₂×#11929607	1	1	yes	characteristic
171	C₈	—	2	2	no	—
172	C₈	—	158	2	no	—
173	C₈	—	158	2	no	—
174	C₈	#14579147	1	2	yes	—
175	C₈	#14579151	1	2	yes	—
176	C₂²	—	2	2	no	—
177	C₂²	C₄⋊C₈×D₇₉	1	1	yes	characteristic
178	C₂²	#19098617	1	1	yes	characteristic
179	C₂²	#19098618	1	1	yes	characteristic
180	C₄	—	4	2	no	—
181	C₄	—	316	2	no	—
182	C₄	—	158	4	no	—
183	C₄	#19098427	1	1	yes	characteristic
184	C₄	#19098585	1	1	yes	characteristic
185	C₄	#19098613	1	1	yes	characteristic
186	C₄	#19098614	1	1	yes	characteristic
187	C₂	—	2	1	no	—
188	C₂	—	2	1	no	—
189	C₂	#23763515	1	1	yes	characteristic
190	C₂	#23765164	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23765155

C₂

yes

maximal

#23763086

C₂

yes

maximal, characteristic

#23761482

C₂

yes

maximal, characteristic

#23761481

C₂

yes

maximal

C₇₉×#1376

C₂

yes

maximal, characteristic

C₂×#14579101

C₂²

yes

—

C₂×#14579072

C₄

yes

characteristic

C₂×#14579071

C₄

yes

characteristic

C₂×#14579071

C₄

yes

characteristic

C₂×C₈×Dic₇₉

C₂²

yes

characteristic

C₂×C₈×Dic₇₉

C₂²

yes

characteristic

#19098409

C₄

yes

characteristic

Dic₇₉⋊C₁₆

—

C₂×M₃₂×C₇₉

C₂²

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂²×C₇₉⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₇₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₄×Dic₇₉

C₂×C₄

yes

characteristic

C₂×C₄×Dic₇₉

C₂×C₄

yes

characteristic

#14579101

—

#14579101

—

C₂×C₇₉⋊C₁₆

—

#14579072

—

#14579071

—

#14579071

C₈

yes

—

C₈×Dic₇₉

—

C₈×Dic₇₉

D₄

yes

—

C₈×Dic₇₉

Q₈

yes

—

C₇₉⋊#95

—

C₂²×C₈×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₄²×C₇₉

C₂×C₄

yes

characteristic

M₃₂×C₇₉

—

M₃₂×C₇₉

—

C₂×C₁₆×C₇₉

—

C₄×C₈×C₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

M₁₆

yes

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₄⋊C₄

yes

—

C₂×C₇₉⋊C₈

C₂×C₈

yes

—

C₄×Dic₇₉

—

C₄×Dic₇₉

C₄⋊C₄

yes

—

C₄×Dic₇₉

C₂×C₈

yes

—

C₄×Dic₇₉

M₁₆

yes

—

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂×C₈

yes

—

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₉

#41

yes

—

C₂×C₈×C₇₉

#41

yes

—

C₄²×C₇₉

—

C₄²×C₇₉

M₁₆

yes

—

C₁₆×C₇₉

—

C₂³×C₇₉

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₉

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₉

C₂×M₁₆

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

C₄⋊C₈

yes

—

C₈×C₇₉

—

C₈×C₇₉

#124

yes

—

C₈×C₇₉

#125

yes

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×C₄⋊C₈

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

#12884

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#1831

yes

characteristic

C₂×C₇₉

#2153

yes

—

—

—

—

—

100

#1376

D₇₉

yes

characteristic

101

C₇₉

#12884

yes

characteristic, complemented, Sylow

102

C₂×M₃₂

—

103

C₂×M₃₂

C₂×D₇₉

yes

—

104

C₂×#97

—

105

C₂×#97

—

106

C₂×#97

—

107

C₂×C₄×C₈

—

108

C₂×C₄×C₈

—

109

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

110

#376

—

111

C₄⋊C₁₆

—

158

—

112

C₂²×C₈

—

113

C₂²×C₈

—

114

C₂²×C₈

C₄×D₇₉

yes

characteristic

115

C₂²×C₈

C₂²×D₇₉

yes

characteristic

116

C₂×C₄²

—

117

C₂×C₄²

—

118

C₂×C₄²

C₄×D₇₉

yes

characteristic

119

M₃₂

—

120

M₃₂

—

121

M₃₂

—

158

—

122

M₃₂

—

158

—

123

C₂×C₁₆

—

124

C₂×C₁₆

—

158

—

125

#97

—

126

#97

—

158

—

127

#97

—

158

—

128

C₄×C₈

—

129

C₄×C₈

—

130

C₄×C₈

—

131

C₄×C₈

—

158

—

132

#95

—

158

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

—

158

—

136

C₂²×C₄

C₂×C₄×D₇₉

yes

characteristic

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

158

—

142

C₂×C₈

—

158

—

143

C₂×C₈

—

158

—

144

C₂×C₈

—

158

—

145

C₂×C₈

C₈×D₇₉

yes

—

146

C₂×C₈

D₄×D₇₉

yes

characteristic

147

C₂×C₈

Q₈×D₇₉

yes

characteristic

148

C₂×C₈

#11929623

yes

—

149

C₂×C₈

#11929624

yes

—

150

C₄²

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

154

C₄²

—

158

—

155

C₄²

#11929607

yes

—

156

C₁₆

—

157

C₁₆

—

316

—

158

C₂³

M₁₆×D₇₉

yes

characteristic

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

316

—

162

C₂×C₄

—

316

—

163

C₂×C₄

—

164

C₂×C₄

—

158

—

165

C₂×C₄

C₄⋊C₄×D₇₉

yes

characteristic

166

C₂×C₄

#14579117

yes

characteristic

167

C₂×C₄

#14579118

yes

characteristic

168

C₂×C₄

C₂×C₈×D₇₉

yes

characteristic

169

C₂×C₄

M₁₆×D₇₉

yes

characteristic

170

C₂×C₄

C₂×#11929607

yes

characteristic

171

C₈

—

172

C₈

—

158

—

173

C₈

—

158

—

174

C₈

#14579147

yes

—

175

C₈

#14579151

yes

—

176

C₂²

—

177

C₂²

C₄⋊C₈×D₇₉

yes

characteristic

178

C₂²

#19098617

yes

characteristic

179

C₂²

#19098618

yes

characteristic

180

C₄

—

181

C₄

—

316

—

182

C₄

—

158

—

183

C₄

#19098427

yes

characteristic

184

C₄

#19098585

yes

characteristic

185

C₄

#19098613

yes

characteristic

186

C₄

#19098614

yes

characteristic

187

C₂

—

188

C₂

—

189

C₂

#23763515

yes

characteristic

190

C₂

#23765164

yes

—

	subgroup	G/H
center	C₂×C₈	D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1376	C₂
socle	C₂²×C₇₉	C₂×C₄⋊C₈
prime radical	C₂³×C₇₉	C₂×M₁₆
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#1376	D₇₉
hypocenter	C₇₉	#12884
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1376

subgroup

G/H

center

C₂×C₈

D₄×D₇₉

commutator

C₂×C₄×C₇₉

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₇₉×#1376

C₂

socle

C₂²×C₇₉

C₂×C₄⋊C₈

prime radical

C₂³×C₇₉

C₂×M₁₆

cosocle

C₂²×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1376

N	H ≅ G/N	type
101. C₇₉	#12884	semidirect

H ≅ G/N

type

101. C₇₉

#12884

semidirect

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

subgroup	index
#23765155	2
#23763086	2
#23761482	2
#23761481	2
C₇₉×#1376	2
#12884	79

subgroup

index

C₇₉×#1376

Subgroups — #34204631

Conjugacy classes of subgroups

Complements

Maximal subgroups