Group #34204608 — subgroups

Subgroups — #34204608

Order 20224.

366 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 119 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	#23772011	C₂	1	1	yes	maximal, characteristic
2	#23763446	C₂	1	1	yes	maximal, characteristic
3	#23761515	C₂	1	2	yes	maximal, complemented
4	#23761496	C₂	1	2	yes	maximal, complemented
5	C₇₉×#2113	C₂	1	1	yes	maximal, characteristic
6	#19099330	—	2	1	no	—
7	#19098612	—	2	1	no	—
8	#19098612	C₄	1	4	yes	complemented
9	C₂×#14579097	—	2	2	no	—
10	C₂×#14579097	C₂²	1	2	yes	—
11	#19098601	C₂²	1	2	yes	—
12	#19098537	—	2	1	no	—
13	#19098320	—	2	1	no	—
14	C₂×#131×C₇₉	C₂²	1	1	yes	characteristic
15	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
17	C₂²⋊C₁₆×C₇₉	—	2	2	no	—
18	#14579168	—	4	4	no	—
19	C₂×#11929608	—	2	4	no	—
20	C₂×#11929608	—	2	2	no	—
21	#14579120	—	2	2	no	—
22	#14579097	—	2	4	no	—
23	#14579097	—	2	2	no	—
24	#14579097	C₂×C₄	1	4	yes	—
25	#14579095	—	2	2	no	—
26	#14579093	—	2	2	no	—
27	#14579076	—	4	2	no	—
28	C₂×#41×C₇₉	D₄	1	1	yes	characteristic
29	#131×C₇₉	—	2	2	no	—
30	#131×C₇₉	—	2	2	no	—
31	C₂×M₁₆×C₇₉	C₂×C₄	1	2	yes	—
32	C₂×M₁₆×C₇₉	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₉	D₄	1	2	yes	—
34	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
35	M₃₂×C₇₉	—	2	2	no	—
36	C₂×C₁₆×C₇₉	—	2	1	no	—
37	C₂×C₁₆×C₇₉	—	2	1	no	—
38	C₂×C₁₆×C₇₉	—	2	2	no	—
39	#11929608	—	2	4	no	—
40	#11929608	—	4	4	no	—
41	#11929608	—	4	2	no	—
42	C₂×C₇₉⋊C₈	—	4	4	no	—
43	C₂×C₇₉⋊C₈	—	4	2	no	—
44	C₄⋊Dic₇₉	—	4	2	no	—
45	C₂²⋊Dic₇₉	—	4	2	no	—
46	C₄×Dic₇₉	—	4	2	no	—
47	#41×C₇₉	—	2	2	no	—
48	#41×C₇₉	—	2	2	no	—
49	C₂×Q₈×C₇₉	Q₁₆	1	1	yes	characteristic
50	C₂×D₄×C₇₉	SD₁₆	1	2	yes	—
51	C₂×D₄×C₇₉	Q₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
54	M₁₆×C₇₉	—	2	4	no	—
55	M₁₆×C₇₉	—	2	2	no	—
56	C₂×C₈×C₇₉	—	2	2	no	—
57	C₂×C₈×C₇₉	—	2	2	no	—
58	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
59	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₇₉	#41	1	2	yes	—
62	C₂×C₈×C₇₉	#41	1	2	yes	—
63	C₁₆×C₇₉	—	4	1	no	—
64	C₁₆×C₇₉	—	4	1	no	—
65	C₂³×C₇₉	#103	1	2	yes	—
66	C₂³×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
67	Q₈×C₇₉	—	2	2	no	—
68	D₄×C₇₉	—	2	4	no	—
69	D₄×C₇₉	—	2	2	no	—
70	C₂×C₄×C₇₉	—	2	2	no	—
71	C₂×C₄×C₇₉	—	2	2	no	—
72	C₂×C₄×C₇₉	#103	1	2	yes	—
73	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₇₉	#123	1	2	yes	—
75	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
77	C₂×Dic₇₉	—	8	2	no	—
78	C₈×C₇₉	—	2	2	no	—
79	C₈×C₇₉	#117	1	2	yes	—
80	C₈×C₇₉	#123	1	1	yes	characteristic
81	C₈×C₇₉	#123	1	1	yes	characteristic
82	C₇₉⋊C₈	—	4	4	no	—
83	C₂²×C₇₉	—	2	2	no	—
84	C₂²×C₇₉	—	4	1	no	—
85	C₂²×C₇₉	—	4	2	no	—
86	C₂²×C₇₉	C₂×#103	1	1	yes	characteristic
87	C₂²×C₇₉	#481	1	2	yes	—
88	C₄×C₇₉	—	2	2	no	—
89	C₄×C₇₉	#381	1	1	yes	characteristic
90	C₄×C₇₉	#418	1	1	yes	characteristic
91	C₄×C₇₉	#484	1	2	yes	—
92	Dic₇₉	—	16	2	no	—
93	#12877	—	79	1	no	maximal, Sylow
94	C₂×C₇₉	—	2	2	no	—
95	C₂×C₇₉	—	2	1	no	—
96	C₂×C₇₉	—	4	2	no	—
97	C₂×C₇₉	#1824	1	1	yes	characteristic
98	#2916	—	79	1	no	—
99	#2113	D₇₉	1	1	yes	characteristic
100	#1812	—	79	1	no	—
101	#1389	—	79	2	no	—
102	#1382	—	79	2	no	—
103	C₇₉	#12877	1	1	yes	characteristic, complemented, Sylow
104	#568	—	158	1	no	—
105	C₂×#131	C₂×D₇₉	1	1	yes	characteristic
106	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
107	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
108	#430	—	79	4	no	—
109	#430	—	158	1	no	—
110	C₂×#108	—	79	2	no	—
111	C₂×#108	—	158	2	no	—
112	#427	—	79	2	no	—
113	#410	—	158	1	no	—
114	C₂²⋊C₁₆	—	2	2	no	—
115	H(3,ℤ/4)	—	158	1	no	—
116	C₂×#41	D₃₁₆	1	1	yes	characteristic
117	#131	—	2	2	no	—
118	#131	—	2	2	no	—
119	#131	—	316	4	no	—
120	C₂×M₁₆	—	158	4	no	—
121	C₂×M₁₆	—	158	2	no	—
122	C₂×M₁₆	C₄×D₇₉	1	2	yes	—
123	C₂×M₁₆	D₃₁₆	1	1	yes	characteristic
124	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
125	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
126	#117	—	158	2	no	—
127	M₃₂	—	2	2	no	—
128	C₂×C₁₆	—	2	1	no	—
129	C₂×C₁₆	—	2	1	no	—
130	C₂×C₁₆	—	2	2	no	—
131	#108	—	79	4	no	—
132	#108	—	158	4	no	—
133	#108	—	158	2	no	—
134	#107	—	158	2	no	—
135	#106	—	158	2	no	—
136	#99	—	316	2	no	—
137	#41	—	2	2	no	—
138	#41	—	2	2	no	—
139	C₂×Q₈	Dic₃₁₆	1	1	yes	characteristic
140	C₂×D₄	#11929613	1	2	yes	—
141	C₂×D₄	Dic₃₁₆	1	1	yes	characteristic
142	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
143	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
144	M₁₆	—	2	2	no	—
145	M₁₆	—	2	4	no	—
146	M₁₆	—	316	4	no	—
147	M₁₆	—	316	2	no	—
148	M₁₆	—	158	4	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	316	4	no	—
152	C₂×C₈	—	316	2	no	—
153	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
154	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
155	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
156	C₂×C₈	#11929623	1	2	yes	—
157	C₂×C₈	#11929623	1	2	yes	—
158	C₄⋊C₄	—	316	2	no	—
159	C₂²⋊C₄	—	316	2	no	—
160	C₄²	—	316	2	no	—
161	C₁₆	—	4	1	no	—
162	C₁₆	—	4	1	no	—
163	C₂³	#14579086	1	2	yes	—
164	C₂³	C₂×#11929613	1	1	yes	characteristic
165	Q₈	—	2	2	no	—
166	D₄	—	2	4	no	—
167	D₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	632	2	no	—
171	C₂×C₄	#14579086	1	2	yes	—
172	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
173	C₂×C₄	#14579145	1	2	yes	—
174	C₂×C₄	C₂×Dic₃₁₆	1	1	yes	characteristic
175	C₂×C₄	#14579189	1	2	yes	—
176	C₈	—	2	2	no	—
177	C₈	—	316	4	no	—
178	C₈	#14579117	1	2	yes	—
179	C₈	#14579142	1	1	yes	characteristic
180	C₈	#14579142	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	—	4	1	no	—
183	C₂²	—	4	2	no	—
184	C₂²	C₂×#14579086	1	1	yes	characteristic
185	C₂²	#19098882	1	2	yes	—
186	C₄	—	2	2	no	—
187	C₄	—	1264	2	no	—
188	C₄	#19098424	1	1	yes	characteristic
189	C₄	#19098569	1	1	yes	characteristic
190	C₄	#19098899	1	2	yes	—
191	C₂	—	2	2	no	—
192	C₂	—	2	1	no	—
193	C₂	—	4	2	no	—
194	C₂	#23763491	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23772011

C₂

yes

maximal, characteristic

#23763446

C₂

yes

maximal, characteristic

#23761515

C₂

yes

maximal, complemented

#23761496

C₂

yes

maximal, complemented

C₇₉×#2113

C₂

yes

maximal, characteristic

#19099330

—

#19098612

—

#19098612

C₄

yes

complemented

C₂×#14579097

—

C₂×#14579097

C₂²

yes

—

#19098601

C₂²

yes

—

#19098537

—

#19098320

—

C₂×#131×C₇₉

C₂²

yes

characteristic

C₂×M₃₂×C₇₉

C₂²

yes

characteristic

C₂²×C₁₆×C₇₉

C₂²

yes

characteristic

C₂²⋊C₁₆×C₇₉

—

—

—

—

—

—

—

C₂×C₄

yes

—

#14579095

—

#14579093

—

#14579076

—

C₂×#41×C₇₉

D₄

yes

characteristic

#131×C₇₉

—

#131×C₇₉

—

C₂×M₁₆×C₇₉

C₂×C₄

yes

—

C₂×M₁₆×C₇₉

D₄

yes

characteristic

C₂²×C₈×C₇₉

D₄

yes

—

C₂²×C₈×C₇₉

C₂³

yes

characteristic

M₃₂×C₇₉

—

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

—

#11929608

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

#41×C₇₉

—

#41×C₇₉

—

C₂×Q₈×C₇₉

Q₁₆

yes

characteristic

C₂×D₄×C₇₉

SD₁₆

yes

—

C₂×D₄×C₇₉

Q₁₆

yes

characteristic

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

#41

yes

—

C₂×C₈×C₇₉

#41

yes

—

C₁₆×C₇₉

—

C₁₆×C₇₉

—

C₂³×C₇₉

#103

yes

—

C₂³×C₇₉

C₂×SD₁₆

yes

characteristic

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#103

yes

—

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

#123

yes

—

C₂×C₄×C₇₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₈×C₇₉

#117

yes

—

C₈×C₇₉

yes

C₈×C₇₉

yes

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×#103

yes

characteristic

C₂²×C₇₉

#481

yes

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

Dic₇₉

—

#12877

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

—

D₇₉

yes

characteristic

100

#1812

—

101

#1389

—

102

#1382

—

103

C₇₉

#12877

yes

characteristic, complemented, Sylow

104

#568

—

158

—

105

C₂×#131

C₂×D₇₉

yes

characteristic

106

C₂×M₃₂

C₂×D₇₉

yes

characteristic

107

C₂²×C₁₆

C₂×D₇₉

yes

characteristic

108

#430

—

109

#430

—

158

—

110

C₂×#108

—

111

C₂×#108

—

158

—

112

#427

—

113

#410

—

158

—

114

C₂²⋊C₁₆

—

115

H(3,ℤ/4)

—

158

—

116

C₂×#41

D₃₁₆

yes

characteristic

117

#131

—

118

#131

—

119

#131

—

316

—

120

C₂×M₁₆

—

158

—

121

C₂×M₁₆

—

158

—

122

C₂×M₁₆

C₄×D₇₉

yes

—

123

C₂×M₁₆

D₃₁₆

yes

characteristic

124

C₂²×C₈

C₂²⋊D₇₉

yes

—

125

C₂²×C₈

C₂²×D₇₉

yes

characteristic

126

#117

—

158

—

127

M₃₂

—

128

C₂×C₁₆

—

129

C₂×C₁₆

—

130

C₂×C₁₆

—

131

#108

—

132

#108

—

158

—

133

#108

—

158

—

134

#107

—

158

—

135

#106

—

158

—

136

#99

—

316

—

137

#41

—

138

#41

—

139

C₂×Q₈

Dic₃₁₆

yes

characteristic

140

C₂×D₄

#11929613

yes

—

141

C₂×D₄

Dic₃₁₆

yes

characteristic

142

C₂²×C₄

(C₂×D₇₉)⋊C₄

yes

—

143

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

144

M₁₆

—

145

M₁₆

—

146

M₁₆

—

316

—

147

M₁₆

—

316

—

148

M₁₆

—

158

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

316

—

152

C₂×C₈

—

316

—

153

C₂×C₈

(C₂×D₇₉)⋊C₄

yes

—

154

C₂×C₈

C₂×C₄×D₇₉

yes

characteristic

155

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

156

C₂×C₈

#11929623

yes

—

157

C₂×C₈

#11929623

yes

—

158

C₄⋊C₄

—

316

—

159

C₂²⋊C₄

—

316

—

160

C₄²

—

316

—

161

C₁₆

—

162

C₁₆

—

163

C₂³

#14579086

yes

—

164

C₂³

C₂×#11929613

yes

characteristic

165

Q₈

—

166

D₄

—

167

D₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

632

—

171

C₂×C₄

#14579086

yes

—

172

C₂×C₄

C₂×(C₂×D₇₉)⋊C₄

yes

characteristic

173

C₂×C₄

#14579145

yes

—

174

C₂×C₄

C₂×Dic₃₁₆

yes

characteristic

175

C₂×C₄

#14579189

yes

—

176

C₈

—

177

C₈

—

316

—

178

C₈

#14579117

yes

—

179

C₈

#14579142

yes

characteristic

180

C₈

#14579142

yes

characteristic

181

C₂²

—

182

C₂²

—

183

C₂²

—

184

C₂²

C₂×#14579086

yes

characteristic

185

C₂²

#19098882

yes

—

186

C₄

—

187

C₄

—

1264

—

188

C₄

#19098424

yes

characteristic

189

C₄

#19098569

yes

characteristic

190

C₄

#19098899

yes

—

191

C₂

—

192

C₂

—

193

C₂

—

194

C₂

#23763491

yes

characteristic

	subgroup	G/H
center	C₄	#19098569
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#2113	C₂
socle	C₂×C₇₉	#1824
prime radical	C₂×#41×C₇₉	D₄
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#2113	D₇₉
hypocenter	C₇₉	#12877
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #2113

subgroup

G/H

center

C₄

#19098569

commutator

C₂×C₈×C₇₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₇₉×#2113

C₂

socle

C₂×C₇₉

#1824

prime radical

C₂×#41×C₇₉

D₄

cosocle

C₂²×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #2113

N	H ≅ G/N	type
3. #23761515	C₂	semidirect
4. #23761496	C₂	semidirect
8. #19098612	C₄	semidirect
103. C₇₉	#12877	semidirect

H ≅ G/N

type

3. #23761515

C₂

semidirect

4. #23761496

C₂

semidirect

8. #19098612

C₄

semidirect

103. C₇₉

#12877

semidirect

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

subgroup	index
#23772011	2
#23763446	2
#23761515	2
#23761496	2
C₇₉×#2113	2
#12877	79

subgroup

index

C₇₉×#2113

Subgroups — #34204608

Conjugacy classes of subgroups

Complements

Maximal subgroups