Group #34182677 — subgroups

Subgroups — #34182677

Order 20224.

336 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 99 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	#23764382	C₂	1	1	yes	maximal, characteristic
2	#23762168	C₂	1	2	yes	maximal
3	#23761995	C₂	1	1	yes	maximal, characteristic
4	#23761292	C₂	1	2	yes	maximal
5	C₇₉×#2987	C₂	1	1	yes	maximal, characteristic
6	#19098587	C₂²	1	1	yes	characteristic
7	C₄×#11929608	C₂²	1	2	yes	—
8	#19098409	C₂²	1	1	yes	characteristic
9	#19098322	—	2	2	no	—
10	#19098315	—	2	2	no	—
11	#19098313	—	2	2	no	—
12	#19098305	C₄	1	1	yes	characteristic, complemented
13	#19098305	C₄	1	2	yes	complemented
14	#19098305	C₄	1	1	yes	characteristic, complemented
15	#533×C₇₉	C₂²	1	1	yes	characteristic
16	#446×C₇₉	—	2	2	no	—
17	#446×C₇₉	—	2	2	no	—
18	#433×C₇₉	—	2	2	no	—
19	#432×C₇₉	C₂²	1	2	yes	—
20	C₂×#11929608	—	2	2	no	—
21	C₂×#11929608	—	2	1	no	—
22	C₂×C₄×Dic₇₉	—	2	1	no	—
23	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
24	#14579072	C₂×C₄	1	2	yes	—
25	#14579072	C₂×C₄	1	2	yes	—
26	C₄×C₇₉⋊C₈	—	2	2	no	—
27	C₇₉⋊#95	—	2	2	no	—
28	C₄⋊Q₈×C₇₉	Q₈	1	2	yes	—
29	#125×C₇₉	C₂×C₄	1	1	yes	characteristic
30	#125×C₇₉	C₂×C₄	1	1	yes	characteristic
31	C₂²⋊Q₈×C₇₉	—	2	2	no	—
32	C₄×Q₈×C₇₉	—	2	2	no	—
33	#117×C₇₉	D₄	1	2	yes	—
34	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₇₉	—	2	2	no	—
36	C₄⋊C₈×C₇₉	—	2	2	no	—
37	C₄⋊C₈×C₇₉	D₄	1	2	yes	—
38	C₄⋊C₈×C₇₉	D₄	1	2	yes	—
39	C₂²⋊C₈×C₇₉	—	2	2	no	—
40	#97×C₇₉	—	2	2	no	—
41	#97×C₇₉	—	2	2	no	—
42	C₄×C₈×C₇₉	—	2	2	no	—
43	C₂²×Dic₇₉	—	4	1	no	—
44	#11929608	—	4	2	no	—
45	#11929608	—	4	4	no	—
46	C₂×C₇₉⋊C₈	—	2	2	no	—
47	C₂×C₇₉⋊C₈	—	2	2	no	—
48	C₂×C₇₉⋊C₈	—	4	1	no	—
49	C₄×Dic₇₉	—	2	2	no	—
50	C₄×Dic₇₉	—	4	1	no	—
51	C₂×Q₈×C₇₉	—	2	2	no	—
52	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₇₉	#41	1	2	yes	—
54	C₂×C₈×C₇₉	—	2	2	no	—
55	C₂×C₈×C₇₉	—	2	2	no	—
56	C₄⋊C₄×C₇₉	—	2	2	no	—
57	C₄⋊C₄×C₇₉	—	2	2	no	—
58	C₄⋊C₄×C₇₉	—	2	2	no	—
59	C₄⋊C₄×C₇₉	C₄⋊C₄	1	2	yes	—
60	C₄⋊C₄×C₇₉	C₄⋊C₄	1	2	yes	—
61	C₂²⋊C₄×C₇₉	—	2	2	no	—
62	C₄²×C₇₉	—	2	2	no	—
63	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₇₉	C₂×D₄	1	2	yes	—
65	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
66	C₂³×C₇₉	#123	1	1	yes	characteristic
67	Q₈×C₇₉	—	8	2	no	—
68	C₂×C₄×C₇₉	—	2	1	no	—
69	C₂×C₄×C₇₉	—	2	1	no	—
70	C₂×C₄×C₇₉	—	2	2	no	—
71	C₂×C₄×C₇₉	—	2	2	no	—
72	C₂×C₄×C₇₉	—	4	2	no	—
73	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
75	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
76	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
77	C₂×Dic₇₉	—	4	2	no	—
78	C₂×Dic₇₉	—	8	2	no	—
79	C₈×C₇₉	—	4	2	no	—
80	C₈×C₇₉	—	4	2	no	—
81	C₇₉⋊C₈	—	2	4	no	—
82	C₇₉⋊C₈	—	4	2	no	—
83	C₇₉⋊C₈	—	8	1	no	—
84	C₂²×C₇₉	—	2	1	no	—
85	C₂²×C₇₉	—	2	2	no	—
86	C₂²×C₇₉	#385	1	1	yes	characteristic
87	C₄×C₇₉	—	2	1	no	—
88	C₄×C₇₉	—	2	2	no	—
89	C₄×C₇₉	—	2	2	no	—
90	C₄×C₇₉	—	4	2	no	—
91	C₄×C₇₉	—	4	2	no	—
92	C₄×C₇₉	#454	1	1	yes	characteristic
93	C₄×C₇₉	#456	1	1	yes	characteristic
94	Dic₇₉	—	8	2	no	—
95	#8919	—	79	1	no	maximal, Sylow
96	C₂×C₇₉	—	4	1	no	—
97	C₂×C₇₉	#1857	1	1	yes	characteristic
98	C₂×C₇₉	#1861	1	2	yes	—
99	#2987	D₇₉	1	1	yes	characteristic
100	#1995	—	79	1	no	—
101	#1569	—	79	2	no	—
102	#1526	—	79	1	no	—
103	#1300	—	79	2	no	—
104	C₇₉	#8919	1	1	yes	characteristic, complemented, Sylow
105	#533	C₂×D₇₉	1	1	yes	characteristic
106	#446	—	2	2	no	—
107	#446	—	2	2	no	—
108	#433	—	2	2	no	—
109	#432	C₂×D₇₉	1	2	yes	—
110	#423	—	79	1	no	—
111	C₄×M₁₆	—	79	2	no	—
112	#376	—	79	1	no	—
113	#339	—	158	2	no	—
114	#335	—	158	2	no	—
115	#334	—	158	2	no	—
116	#330	—	79	1	no	—
117	#330	—	79	2	no	—
118	#330	—	79	1	no	—
119	C₂×M₁₆	—	158	1	no	—
120	C₂×M₁₆	—	158	2	no	—
121	C₄⋊Q₈	Dic₁₅₈	1	2	yes	—
122	#125	C₄×D₇₉	1	1	yes	characteristic
123	#125	C₄×D₇₉	1	1	yes	characteristic
124	C₂²⋊Q₈	—	2	2	no	—
125	C₄×Q₈	—	2	2	no	—
126	#117	C₂²⋊D₇₉	1	2	yes	—
127	C₂×C₄²	—	158	1	no	—
128	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	2	2	no	—
131	C₄⋊C₈	—	158	2	no	—
132	C₄⋊C₈	C₂²⋊D₇₉	1	2	yes	—
133	C₄⋊C₈	C₂²⋊D₇₉	1	2	yes	—
134	C₂²⋊C₈	—	2	2	no	—
135	#97	—	2	2	no	—
136	#97	—	2	2	no	—
137	#97	—	79	2	no	—
138	#97	—	79	2	no	—
139	C₄×C₈	—	2	2	no	—
140	C₄×C₈	—	158	2	no	—
141	#95	—	158	2	no	—
142	C₂×Q₈	—	2	2	no	—
143	C₂²×C₄	—	316	1	no	—
144	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
145	C₂²×C₄	#11929624	1	2	yes	—
146	M₁₆	—	316	4	no	—
147	M₁₆	—	316	2	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	316	1	no	—
151	C₂×C₈	—	158	2	no	—
152	C₂×C₈	—	158	2	no	—
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	Dic₇₉⋊C₄	1	2	yes	—
157	C₄⋊C₄	Dic₇₉⋊C₄	1	2	yes	—
158	C₂²⋊C₄	—	2	2	no	—
159	C₄²	—	2	2	no	—
160	C₄²	—	316	1	no	—
161	C₄²	—	158	2	no	—
162	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
163	C₄²	C₂×C₂²⋊D₇₉	1	2	yes	—
164	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
165	C₂³	#14579144	1	1	yes	characteristic
166	Q₈	—	8	2	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	316	2	no	—
173	C₂×C₄	—	632	2	no	—
174	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
175	C₂×C₄	C₄×C₂²⋊D₇₉	1	2	yes	—
176	C₂×C₄	#14579130	1	1	yes	characteristic
177	C₂×C₄	#14579140	1	2	yes	—
178	C₈	—	4	2	no	—
179	C₈	—	4	2	no	—
180	C₈	—	316	2	no	—
181	C₈	—	632	1	no	—
182	C₈	—	158	4	no	—
183	C₂²	—	2	1	no	—
184	C₂²	—	2	2	no	—
185	C₂²	#19098436	1	1	yes	characteristic
186	C₄	—	2	1	no	—
187	C₄	—	2	2	no	—
188	C₄	—	2	2	no	—
189	C₄	—	4	2	no	—
190	C₄	—	4	2	no	—
191	C₄	—	632	2	no	—
192	C₄	#19098736	1	1	yes	characteristic
193	C₄	#19098746	1	1	yes	characteristic
194	C₂	—	4	1	no	—
195	C₂	#23763615	1	1	yes	characteristic
196	C₂	#23763640	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23764382

C₂

yes

maximal, characteristic

#23762168

C₂

yes

maximal

#23761995

C₂

yes

maximal, characteristic

#23761292

C₂

yes

maximal

C₇₉×#2987

C₂

yes

maximal, characteristic

#19098587

C₂²

yes

characteristic

C₄×#11929608

C₂²

yes

—

#19098409

C₂²

yes

—

—

—

C₄

yes

characteristic, complemented

#19098305

C₄

yes

complemented

#19098305

C₄

yes

characteristic, complemented

#533×C₇₉

C₂²

yes

characteristic

#446×C₇₉

—

#446×C₇₉

—

#433×C₇₉

—

#432×C₇₉

C₂²

yes

—

C₂×#11929608

—

C₂×#11929608

—

C₂×C₄×Dic₇₉

—

C₄⋊(C₇₉⋊C₈)

—

#14579072

C₂×C₄

yes

—

#14579072

C₂×C₄

yes

—

C₄×C₇₉⋊C₈

—

C₇₉⋊#95

—

C₄⋊Q₈×C₇₉

Q₈

yes

—

#125×C₇₉

C₂×C₄

yes

characteristic

#125×C₇₉

C₂×C₄

yes

characteristic

C₂²⋊Q₈×C₇₉

—

C₄×Q₈×C₇₉

—

#117×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

D₄

yes

—

C₄⋊C₈×C₇₉

D₄

yes

—

C₂²⋊C₈×C₇₉

—

#97×C₇₉

—

#97×C₇₉

—

C₄×C₈×C₇₉

—

C₂²×Dic₇₉

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₂×Q₈×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

#41

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

C₄⋊C₄

yes

—

C₄⋊C₄×C₇₉

C₄⋊C₄

yes

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

—

C₄²×C₇₉

C₂×Q₈

yes

characteristic

C₂³×C₇₉

#123

yes

characteristic

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄×D₄

yes

—

C₂×C₄×C₇₉

#120

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#385

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

#8919

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

#1857

yes

characteristic

C₂×C₇₉

#1861

yes

—

#2987

D₇₉

yes

characteristic

100

#1995

—

101

#1569

—

102

#1526

—

103

#1300

—

104

C₇₉

#8919

yes

characteristic, complemented, Sylow

105

#533

C₂×D₇₉

yes

characteristic

106

#446

—

107

#446

—

108

#433

—

109

#432

C₂×D₇₉

yes

—

110

#423

—

111

C₄×M₁₆

—

112

#376

—

113

#339

—

158

—

114

#335

—

158

—

115

#334

—

158

—

116

#330

—

117

#330

—

118

#330

—

119

C₂×M₁₆

—

158

—

120

C₂×M₁₆

—

158

—

121

C₄⋊Q₈

Dic₁₅₈

yes

—

122

#125

C₄×D₇₉

yes

characteristic

123

#125

C₄×D₇₉

yes

characteristic

124

C₂²⋊Q₈

—

125

C₄×Q₈

—

126

#117

C₂²⋊D₇₉

yes

—

127

C₂×C₄²

—

158

—

128

C₂×C₄²

C₂²×D₇₉

yes

characteristic

129

C₄⋊C₈

—

130

C₄⋊C₈

—

131

C₄⋊C₈

—

158

—

132

C₄⋊C₈

C₂²⋊D₇₉

yes

—

133

C₄⋊C₈

C₂²⋊D₇₉

yes

—

134

C₂²⋊C₈

—

135

#97

—

136

#97

—

137

#97

—

138

#97

—

139

C₄×C₈

—

140

C₄×C₈

—

158

—

141

#95

—

158

—

142

C₂×Q₈

—

143

C₂²×C₄

—

316

—

144

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

145

C₂²×C₄

#11929624

yes

—

146

M₁₆

—

316

—

147

M₁₆

—

316

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

316

—

151

C₂×C₈

—

158

—

152

C₂×C₈

—

158

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

Dic₇₉⋊C₄

yes

—

157

C₄⋊C₄

Dic₇₉⋊C₄

yes

—

158

C₂²⋊C₄

—

159

C₄²

—

160

C₄²

—

316

—

161

C₄²

—

158

—

162

C₄²

C₂×C₄×D₇₉

yes

characteristic

163

C₄²

C₂×C₂²⋊D₇₉

yes

—

164

C₄²

C₂×Dic₁₅₈

yes

characteristic

165

C₂³

#14579144

yes

characteristic

166

Q₈

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

316

—

173

C₂×C₄

—

632

—

174

C₂×C₄

C₂×Dic₇₉⋊C₄

yes

characteristic

175

C₂×C₄

C₄×C₂²⋊D₇₉

yes

—

176

C₂×C₄

#14579130

yes

characteristic

177

C₂×C₄

#14579140

yes

—

178

C₈

—

179

C₈

—

180

C₈

—

316

—

181

C₈

—

632

—

182

C₈

—

158

—

183

C₂²

—

184

C₂²

—

185

C₂²

#19098436

yes

characteristic

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

632

—

192

C₄

#19098736

yes

characteristic

193

C₄

#19098746

yes

characteristic

194

C₂

—

195

C₂

#23763615

yes

characteristic

196

C₂

#23763640

yes

—

	subgroup	G/H
center	C₂×C₄	#14579130
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#2987	C₂
socle	C₂²×C₇₉	#385
prime radical	C₂³×C₇₉	#123
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#2987	D₇₉
hypocenter	C₇₉	#8919
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2987

subgroup

G/H

center

C₂×C₄

#14579130

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#2987

C₂

C₂²×C₇₉

C₂³×C₇₉

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2987

N	H ≅ G/N	type
12. #19098305	C₄	semidirect
13. #19098305	C₄	semidirect
14. #19098305	C₄	semidirect
104. C₇₉	#8919	semidirect

H ≅ G/N

type

12. #19098305

C₄

semidirect

13. #19098305

C₄

semidirect

14. #19098305

C₄

semidirect

104. C₇₉

#8919

semidirect

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

subgroup	index
#23764382	2
#23762168	2
#23761995	2
#23761292	2
C₇₉×#2987	2
#8919	79

subgroup

index

C₇₉×#2987

Subgroups — #34182677

Conjugacy classes of subgroups

Complements

Maximal subgroups