Group #34188390 — subgroups

Subgroups — #34188390

Order 20224.

430 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 163 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	#23771763	C₂	1	1	yes	maximal, characteristic
2	#23763517	C₂	1	1	yes	maximal, characteristic
3	#23761284	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579153	—	2	1	no	—
6	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
7	#19098461	—	2	2	no	—
8	#19098461	C₄	1	4	yes	—
9	#19098433	C₂²	1	2	yes	—
10	#19098429	—	2	1	no	—
11	#19098318	—	2	2	no	—
12	C₂×C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₇₉	—	2	1	no	—
14	C₄³×C₇₉	C₂²	1	1	yes	characteristic
15	#336×C₇₉	—	2	1	no	—
16	C₂²×C₇₉⋊C₈	—	2	2	no	—
17	#14579153	—	4	4	no	—
18	C₂×C₄⋊Dic₇₉	—	2	2	no	—
19	C₂×C₄⋊Dic₇₉	—	2	2	no	—
20	C₂×C₄⋊Dic₇₉	C₂×C₄	1	4	yes	—
21	C₂×Dic₇₉⋊C₄	—	2	4	no	—
22	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
23	C₇₉⋊#95	—	2	4	no	—
24	C₂²×C₈×C₇₉	—	2	2	no	—
25	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₇₉	D₄	1	2	yes	—
29	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₇₉	—	2	2	no	—
31	C₄⋊C₈×C₇₉	Q₈	1	4	yes	—
32	C₄×C₈×C₇₉	—	4	2	no	—
33	C₂²×Dic₇₉	—	2	4	no	—
34	C₂×C₇₉⋊C₈	—	4	2	no	—
35	C₂×C₇₉⋊C₈	—	4	4	no	—
36	C₄⋊Dic₇₉	—	4	4	no	—
37	C₄⋊Dic₇₉	—	4	4	no	—
38	C₄⋊Dic₇₉	—	4	4	no	—
39	Dic₇₉⋊C₄	—	4	8	no	—
40	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₇₉	—	2	4	no	—
46	C₂×C₈×C₇₉	—	4	2	no	—
47	C₂×C₈×C₇₉	—	4	2	no	—
48	C₄²×C₇₉	—	2	4	no	—
49	C₄²×C₇₉	—	2	2	no	—
50	C₄²×C₇₉	—	4	2	no	—
51	C₄²×C₇₉	SD₁₆	1	2	yes	—
52	C₄²×C₇₉	Q₁₆	1	2	yes	—
53	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
54	C₄²×C₇₉	#41	1	2	yes	—
55	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₇₉	—	2	2	no	—
57	C₂×C₄×C₇₉	—	2	2	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	#103	1	4	yes	—
63	C₂×C₄×C₇₉	C₄×Q₈	1	2	yes	—
64	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
65	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
66	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
68	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄×C₇₉	#135	1	4	yes	—
70	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
71	C₂×Dic₇₉	—	4	4	no	—
72	C₂×Dic₇₉	—	4	4	no	—
73	C₂×Dic₇₉	—	4	4	no	—
74	C₈×C₇₉	—	4	4	no	—
75	C₇₉⋊C₈	—	8	2	no	—
76	C₂²×C₇₉	#391	1	1	yes	characteristic
77	C₂²×C₇₉	#413	1	2	yes	—
78	C₂²×C₇₉	C₂×#103	1	1	yes	characteristic
79	C₂²×C₇₉	#416	1	2	yes	—
80	C₂²×C₇₉	#418	1	1	yes	characteristic
81	C₄×C₇₉	—	2	4	no	—
82	C₄×C₇₉	—	2	2	no	—
83	C₄×C₇₉	—	4	2	no	—
84	C₄×C₇₉	#476	1	2	yes	—
85	C₄×C₇₉	#477	1	2	yes	—
86	C₄×C₇₉	#488	1	2	yes	—
87	C₄×C₇₉	#499	1	2	yes	—
88	Dic₇₉	—	8	4	no	—
89	#9897	—	79	1	no	maximal, Sylow
90	C₂×C₇₉	#1542	1	2	yes	—
91	C₂×C₇₉	#1553	1	2	yes	—
92	C₂×C₇₉	#1987	1	1	yes	characteristic
93	C₂×C₇₉	#1991	1	1	yes	characteristic
94	C₂×C₇₉	#1998	1	1	yes	characteristic
95	#2859	—	79	1	no	—
96	#1831	—	79	1	no	—
97	C₄×C₄⋊C₈	D₇₉	1	1	yes	characteristic
98	#1296	—	79	4	no	—
99	C₇₉	#9897	1	1	yes	characteristic, complemented, Sylow
100	C₂×#125	—	158	1	no	—
101	C₂×C₄⋊C₈	—	79	2	no	—
102	C₂×C₄⋊C₈	C₂×D₇₉	1	2	yes	—
103	C₂×C₄×C₈	—	2	1	no	—
104	#389	—	79	4	no	—
105	#389	—	158	2	no	—
106	#384	—	79	2	no	—
107	#382	—	158	1	no	—
108	C₄³	C₂×D₇₉	1	1	yes	characteristic
109	#336	—	2	1	no	—
110	#336	—	158	2	no	—
111	C₂²×C₈	—	2	2	no	—
112	C₂²×C₈	—	158	2	no	—
113	#125	—	316	4	no	—
114	C₂×C₄⋊C₄	—	79	4	no	—
115	C₂×C₄⋊C₄	—	158	4	no	—
116	C₂×C₄⋊C₄	—	158	2	no	—
117	C₂×C₄⋊C₄	—	158	2	no	—
118	C₂×C₄²	C₄×D₇₉	1	2	yes	—
119	C₂×C₄²	D₃₁₆	1	2	yes	—
120	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
121	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
122	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	158	4	no	—
125	C₄⋊C₈	Dic₁₅₈	1	4	yes	—
126	C₄×C₈	—	4	2	no	—
127	#95	—	158	4	no	—
128	C₂²×C₄	—	158	4	no	—
129	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
130	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
131	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
133	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
134	C₂×C₈	—	2	4	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	—	316	2	no	—
138	C₂×C₈	—	316	4	no	—
139	C₄⋊C₄	—	316	4	no	—
140	C₄⋊C₄	—	316	8	no	—
141	C₄⋊C₄	—	316	4	no	—
142	C₄⋊C₄	—	316	4	no	—
143	C₄²	—	2	4	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	4	2	no	—
146	C₄²	#11929612	1	2	yes	—
147	C₄²	#11929614	1	2	yes	—
148	C₄²	C₂×Dic₁₅₈	1	2	yes	—
149	C₄²	#11929624	1	2	yes	—
150	C₂³	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	316	4	no	—
158	C₂×C₄	—	316	4	no	—
159	C₂×C₄	—	316	4	no	—
160	C₂×C₄	#14579084	1	4	yes	—
161	C₂×C₄	C₄×Dic₁₅₈	1	2	yes	—
162	C₂×C₄	#14579140	1	2	yes	—
163	C₂×C₄	#14579149	1	1	yes	characteristic
164	C₂×C₄	#14579161	1	1	yes	characteristic
165	C₂×C₄	C₂×#11929612	1	1	yes	characteristic
166	C₂×C₄	C₂×#11929614	1	1	yes	characteristic
167	C₂×C₄	#14579183	1	4	yes	—
168	C₂×C₄	#14579186	1	2	yes	—
169	C₈	—	4	4	no	—
170	C₈	—	632	2	no	—
171	C₂²	#19098469	1	1	yes	characteristic
172	C₂²	#19098545	1	2	yes	—
173	C₂²	C₂×#14579084	1	1	yes	characteristic
174	C₂²	#19098560	1	2	yes	—
175	C₂²	#19098567	1	1	yes	characteristic
176	C₄	—	2	4	no	—
177	C₄	—	2	2	no	—
178	C₄	—	4	2	no	—
179	C₄	—	632	4	no	—
180	C₄	#19098853	1	2	yes	—
181	C₄	#19098858	1	2	yes	—
182	C₄	#19098918	1	2	yes	—
183	C₄	#19098965	1	2	yes	—
184	C₂	#23762058	1	2	yes	—
185	C₂	#23762103	1	2	yes	—
186	C₂	#23764348	1	1	yes	characteristic
187	C₂	#23764365	1	1	yes	characteristic
188	C₂	#23764393	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23771763

C₂

yes

maximal, characteristic

#23763517

C₂

yes

maximal, characteristic

#23761284

C₂

yes

maximal

C₄×C₄⋊C₈×C₇₉

C₂

yes

maximal, characteristic

C₂×#14579153

—

C₂×C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

#19098461

—

#19098461

C₄

yes

—

#19098433

C₂²

yes

—

#19098429

—

#19098318

—

C₂×C₄⋊C₈×C₇₉

C₂²

yes

—

C₂×C₄×C₈×C₇₉

—

C₄³×C₇₉

C₂²

yes

characteristic

#336×C₇₉

—

C₂²×C₇₉⋊C₈

—

#14579153

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

C₂×C₄

yes

—

C₂×Dic₇₉⋊C₄

—

C₄⋊(C₇₉⋊C₈)

—

C₇₉⋊#95

—

C₂²×C₈×C₇₉

—

C₂×C₄²×C₇₉

C₂×C₄

yes

—

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

Q₈

yes

—

C₄×C₈×C₇₉

—

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

SD₁₆

yes

—

C₄²×C₇₉

Q₁₆

yes

—

C₄²×C₇₉

C₂×Q₈

yes

—

C₄²×C₇₉

#41

yes

—

C₂³×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#103

yes

—

C₂×C₄×C₇₉

C₄×Q₈

yes

—

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₉

#124

yes

characteristic

C₂×C₄×C₇₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₉

#135

yes

—

C₂×C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

#391

yes

characteristic

C₂²×C₇₉

#413

yes

—

C₂²×C₇₉

C₂×#103

yes

characteristic

C₂²×C₇₉

#416

yes

—

C₂²×C₇₉

#418

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#476

yes

—

C₄×C₇₉

#477

yes

—

C₄×C₇₉

#488

yes

—

C₄×C₇₉

#499

yes

—

Dic₇₉

—

#9897

—

maximal, Sylow

C₂×C₇₉

#1542

yes

—

C₂×C₇₉

#1553

yes

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

—

C₄×C₄⋊C₈

D₇₉

yes

characteristic

#1296

—

C₇₉

#9897

yes

characteristic, complemented, Sylow

100

C₂×#125

—

158

—

101

C₂×C₄⋊C₈

—

102

C₂×C₄⋊C₈

C₂×D₇₉

yes

—

103

C₂×C₄×C₈

—

104

#389

—

105

#389

—

158

—

106

#384

—

107

#382

—

158

—

108

C₄³

C₂×D₇₉

yes

characteristic

109

#336

—

110

#336

—

158

—

111

C₂²×C₈

—

112

C₂²×C₈

—

158

—

113

#125

—

316

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

158

—

116

C₂×C₄⋊C₄

—

158

—

117

C₂×C₄⋊C₄

—

158

—

118

C₂×C₄²

C₄×D₇₉

yes

—

119

C₂×C₄²

D₃₁₆

yes

—

120

C₂×C₄²

C₂²⋊D₇₉

yes

characteristic

121

C₂×C₄²

C₂²⋊D₇₉

yes

characteristic

122

C₂×C₄²

C₂²×D₇₉

yes

characteristic

123

C₄⋊C₈

—

124

C₄⋊C₈

—

158

—

125

C₄⋊C₈

Dic₁₅₈

yes

—

126

C₄×C₈

—

127

#95

—

158

—

128

C₂²×C₄

—

158

—

129

C₂²×C₄

(C₂×D₇₉)⋊C₄

yes

—

130

C₂²×C₄

(C₂×D₇₉)⋊C₄

yes

—

131

C₂²×C₄

C₂×C₄×D₇₉

yes

characteristic

132

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

133

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

316

—

138

C₂×C₈

—

316

—

139

C₄⋊C₄

—

316

—

140

C₄⋊C₄

—

316

—

141

C₄⋊C₄

—

316

—

142

C₄⋊C₄

—

316

—

143

C₄²

—

144

C₄²

—

145

C₄²

—

146

C₄²

#11929612

yes

—

147

C₄²

#11929614

yes

—

148

C₄²

C₂×Dic₁₅₈

yes

—

149

C₄²

#11929624

yes

—

150

C₂³

C₂×(C₂×D₇₉)⋊C₄

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

316

—

158

C₂×C₄

—

316

—

159

C₂×C₄

—

316

—

160

C₂×C₄

#14579084

yes

—

161

C₂×C₄

C₄×Dic₁₅₈

yes

—

162

C₂×C₄

#14579140

yes

—

163

C₂×C₄

#14579149

yes

characteristic

164

C₂×C₄

#14579161

yes

characteristic

165

C₂×C₄

C₂×#11929612

yes

characteristic

166

C₂×C₄

C₂×#11929614

yes

characteristic

167

C₂×C₄

#14579183

yes

—

168

C₂×C₄

#14579186

yes

—

169

C₈

—

170

C₈

—

632

—

171

C₂²

#19098469

yes

characteristic

172

C₂²

#19098545

yes

—

173

C₂²

C₂×#14579084

yes

characteristic

174

C₂²

#19098560

yes

—

175

C₂²

#19098567

yes

characteristic

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

632

—

180

C₄

#19098853

yes

—

181

C₄

#19098858

yes

—

182

C₄

#19098918

yes

—

183

C₄

#19098965

yes

—

184

C₂

#23762058

yes

—

185

C₂

#23762103

yes

—

186

C₂

#23764348

yes

characteristic

187

C₂

#23764365

yes

characteristic

188

C₂

#23764393

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₇₉)⋊C₄
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₄×C₄⋊C₈×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₄×C₄⋊C₈	D₇₉
hypocenter	C₇₉	#9897
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×(C₂×D₇₉)⋊C₄

commutator

C₂²×C₄×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₄×C₄⋊C₈×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₇₉

C₂³

hypercenter

C₄×C₄⋊C₈

D₇₉

hypocenter

C₇₉

#9897

LCS

G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

N	H ≅ G/N	type
99. C₇₉	#9897	semidirect

H ≅ G/N

type

99. C₇₉

#9897

semidirect

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

subgroup	index
#23771763	2
#23763517	2
#23761284	2
C₄×C₄⋊C₈×C₇₉	2
#9897	79

subgroup

index

C₄×C₄⋊C₈×C₇₉

Subgroups — #34188390

Conjugacy classes of subgroups

Complements

Maximal subgroups