Group #34193043 — subgroups

Subgroups — #34193043

Order 20224.

292 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 93 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	#23764381	C₂	1	1	yes	maximal, characteristic
2	#23762058	C₂	1	2	yes	maximal
3	#23761297	C₂	1	2	yes	maximal
4	#23761229	C₂	1	1	yes	maximal, characteristic
5	C₇₉×#1843	C₂	1	1	yes	maximal, characteristic
6	#19099120	C₂²	1	1	yes	characteristic
7	#19098616	C₂²	1	2	yes	—
8	#19098587	C₂²	1	1	yes	characteristic
9	#19098323	—	2	1	no	—
10	#19098323	—	2	1	no	—
11	#19098311	—	2	2	no	—
12	#19098308	—	2	2	no	—
13	C₂×C₄⋊C₈×C₇₉	—	2	1	no	—
14	C₂×C₄⋊C₈×C₇₉	—	2	1	no	—
15	C₂×C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
16	#376×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×#11929608	—	2	1	no	—
18	#14579161	D₄	1	2	yes	—
19	#14579153	—	2	1	no	—
20	#14579140	—	2	2	no	—
21	C₄×Dic₁₅₈	—	2	2	no	—
22	#14579120	—	2	1	no	—
23	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
24	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
25	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
26	C₄×C₇₉⋊C₈	—	2	2	no	—
27	C₂²×C₈×C₇₉	—	2	2	no	—
28	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₇₉	—	2	2	no	—
32	C₄⋊C₈×C₇₉	—	2	2	no	—
33	C₄⋊C₈×C₇₉	—	4	2	no	—
34	C₄⋊C₈×C₇₉	—	4	2	no	—
35	#95×C₇₉	—	2	2	no	—
36	C₂×Dic₁₅₈	—	2	2	no	—
37	#11929608	—	4	1	no	—
38	#11929608	—	4	1	no	—
39	C₂×C₇₉⋊C₈	—	4	2	no	—
40	C₂×C₇₉⋊C₈	—	4	1	no	—
41	C₄⋊Dic₇₉	—	2	2	no	—
42	C₄⋊Dic₇₉	—	2	2	no	—
43	Dic₇₉⋊C₄	—	4	2	no	—
44	C₂²⋊Dic₇₉	—	4	1	no	—
45	C₄×Dic₇₉	—	4	1	no	—
46	C₂²×C₄×C₇₉	—	2	1	no	—
47	C₂²×C₄×C₇₉	—	2	1	no	—
48	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₉	#41	1	2	yes	—
50	C₂×C₈×C₇₉	—	4	2	no	—
51	C₂×C₈×C₇₉	—	4	2	no	—
52	C₂×C₈×C₇₉	—	4	2	no	—
53	C₄²×C₇₉	—	2	1	no	—
54	C₄²×C₇₉	—	2	1	no	—
55	C₄²×C₇₉	SD₁₆	1	2	yes	—
56	C₄²×C₇₉	Q₁₆	1	2	yes	—
57	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₇₉	#41	1	1	yes	characteristic
59	C₄²×C₇₉	#41	1	2	yes	—
60	C₂³×C₇₉	#123	1	1	yes	characteristic
61	C₂×C₄×C₇₉	—	2	2	no	—
62	C₂×C₄×C₇₉	—	2	2	no	—
63	C₂×C₄×C₇₉	—	4	2	no	—
64	C₂×C₄×C₇₉	—	4	2	no	—
65	C₂×C₄×C₇₉	—	4	2	no	—
66	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
67	C₂×C₄×C₇₉	#123	1	2	yes	—
68	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
69	C₂×C₄×C₇₉	#126	1	2	yes	—
70	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
72	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
73	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
74	C₂×Dic₇₉	—	4	2	no	—
75	Dic₁₅₈	—	8	2	no	—
76	C₈×C₇₉	—	8	2	no	—
77	C₇₉⋊C₈	—	8	2	no	—
78	C₇₉⋊C₈	—	8	1	no	—
79	C₂²×C₇₉	—	2	2	no	—
80	C₂²×C₇₉	#397	1	1	yes	characteristic
81	C₂²×C₇₉	#483	1	1	yes	characteristic
82	C₂²×C₇₉	#484	1	1	yes	characteristic
83	C₄×C₇₉	—	2	2	no	—
84	C₄×C₇₉	—	2	2	no	—
85	C₄×C₇₉	—	4	2	no	—
86	C₄×C₇₉	—	4	2	no	—
87	C₄×C₇₉	#460	1	1	yes	characteristic
88	C₄×C₇₉	#462	1	1	yes	characteristic
89	C₄×C₇₉	#487	1	1	yes	characteristic
90	C₄×C₇₉	#489	1	1	yes	characteristic
91	Dic₇₉	—	8	2	no	—
92	#10628	—	79	1	no	maximal, Sylow
93	C₂×C₇₉	—	2	1	no	—
94	C₂×C₇₉	—	2	1	no	—
95	C₂×C₇₉	#2072	1	1	yes	characteristic
96	C₂×C₇₉	#2080	1	2	yes	—
97	#1995	—	79	1	no	—
98	#1843	D₇₉	1	1	yes	characteristic
99	#1542	—	79	2	no	—
100	#1301	—	79	2	no	—
101	#1274	—	79	1	no	—
102	C₇₉	#10628	1	1	yes	characteristic, complemented, Sylow
103	#533	—	79	1	no	—
104	#431	—	79	2	no	—
105	#423	—	79	1	no	—
106	C₂×C₄⋊C₈	—	2	1	no	—
107	C₂×C₄⋊C₈	—	2	1	no	—
108	C₂×C₄⋊C₈	C₂×D₇₉	1	2	yes	—
109	#376	C₂×D₇₉	1	1	yes	characteristic
110	#339	—	158	1	no	—
111	#339	—	158	1	no	—
112	#333	—	158	2	no	—
113	#332	—	158	2	no	—
114	C₂×M₁₆	—	158	1	no	—
115	C₂²×C₈	—	2	2	no	—
116	C₄⋊Q₈	—	79	2	no	—
117	#125	—	158	1	no	—
118	C₂²⋊Q₈	—	158	2	no	—
119	C₄×Q₈	—	158	2	no	—
120	#117	—	158	1	no	—
121	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
122	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
123	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	4	2	no	—
127	C₄⋊C₈	—	4	2	no	—
128	C₄⋊C₈	—	158	2	no	—
129	C₄⋊C₈	—	158	2	no	—
130	C₂²⋊C₈	—	158	2	no	—
131	C₄×C₈	—	158	2	no	—
132	#95	—	2	2	no	—
133	C₂×Q₈	—	158	2	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
137	C₂²×C₄	#11929624	1	2	yes	—
138	M₁₆	—	316	1	no	—
139	M₁₆	—	316	1	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	316	2	no	—
144	C₂×C₈	—	316	1	no	—
145	C₄⋊C₄	—	316	2	no	—
146	C₄⋊C₄	—	158	2	no	—
147	C₄⋊C₄	—	158	2	no	—
148	C₂²⋊C₄	—	316	1	no	—
149	C₄²	—	2	1	no	—
150	C₄²	—	2	1	no	—
151	C₄²	—	316	1	no	—
152	C₄²	#11929612	1	2	yes	—
153	C₄²	#11929614	1	2	yes	—
154	C₄²	D₄×D₇₉	1	1	yes	characteristic
155	C₄²	#11929623	1	1	yes	characteristic
156	C₄²	#11929625	1	2	yes	—
157	C₂³	#14579144	1	1	yes	characteristic
158	Q₈	—	632	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	316	2	no	—
165	C₂×C₄	#14579132	1	1	yes	characteristic
166	C₂×C₄	#14579146	1	2	yes	—
167	C₂×C₄	#14579150	1	1	yes	characteristic
168	C₂×C₄	#14579156	1	2	yes	—
169	C₂×C₄	C₂×#11929612	1	1	yes	characteristic
170	C₂×C₄	C₂×#11929614	1	1	yes	characteristic
171	C₂×C₄	#14579187	1	1	yes	characteristic
172	C₂×C₄	#14579192	1	1	yes	characteristic
173	C₈	—	8	2	no	—
174	C₈	—	632	2	no	—
175	C₈	—	632	1	no	—
176	C₂²	—	2	2	no	—
177	C₂²	#19098489	1	1	yes	characteristic
178	C₂²	#19098893	1	1	yes	characteristic
179	C₂²	#19098898	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	4	2	no	—
183	C₄	—	4	2	no	—
184	C₄	—	632	2	no	—
185	C₄	#19098760	1	1	yes	characteristic
186	C₄	#19098770	1	1	yes	characteristic
187	C₄	#19098915	1	1	yes	characteristic
188	C₄	#19098925	1	1	yes	characteristic
189	C₂	—	2	1	no	—
190	C₂	—	2	1	no	—
191	C₂	#23764809	1	1	yes	characteristic
192	C₂	#23764851	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23764381

C₂

yes

maximal, characteristic

#23762058

C₂

yes

maximal

#23761297

C₂

yes

maximal

#23761229

C₂

yes

maximal, characteristic

C₇₉×#1843

C₂

yes

maximal, characteristic

#19099120

C₂²

yes

characteristic

#19098616

C₂²

yes

—

#19098587

C₂²

yes

—

—

—

—

C₂×C₄⋊C₈×C₇₉

—

C₂×C₄⋊C₈×C₇₉

—

C₂×C₄⋊C₈×C₇₉

C₂²

yes

—

#376×C₇₉

C₂²

yes

characteristic

C₂×#11929608

—

#14579161

D₄

yes

—

#14579153

—

#14579140

—

C₄×Dic₁₅₈

—

#14579120

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₂²⋊(C₇₉⋊C₈)

—

C₄×C₇₉⋊C₈

—

C₂²×C₈×C₇₉

—

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

—

#95×C₇₉

—

C₂×Dic₁₅₈

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

#41

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

SD₁₆

yes

—

C₄²×C₇₉

Q₁₆

yes

—

C₄²×C₇₉

C₂×D₄

yes

C₄²×C₇₉

yes

C₄²×C₇₉

yes

—

C₂³×C₇₉

#123

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#121

yes

characteristic

C₂×C₄×C₇₉

#123

yes

—

C₂×C₄×C₇₉

#124

yes

characteristic

C₂×C₄×C₇₉

#126

yes

—

C₂×C₄×C₇₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₁₆

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×Dic₇₉

—

Dic₁₅₈

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

#10628

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#2072

yes

characteristic

C₂×C₇₉

#2080

yes

—

#1995

—

#1843

D₇₉

yes

characteristic

#1542

—

100

#1301

—

101

#1274

—

102

C₇₉

#10628

yes

characteristic, complemented, Sylow

103

#533

—

104

#431

—

105

#423

—

106

C₂×C₄⋊C₈

—

107

C₂×C₄⋊C₈

—

108

C₂×C₄⋊C₈

C₂×D₇₉

yes

—

109

#376

C₂×D₇₉

yes

characteristic

110

#339

—

158

—

111

#339

—

158

—

112

#333

—

158

—

113

#332

—

158

—

114

C₂×M₁₆

—

158

—

115

C₂²×C₈

—

116

C₄⋊Q₈

—

117

#125

—

158

—

118

C₂²⋊Q₈

—

158

—

119

C₄×Q₈

—

158

—

120

#117

—

158

—

121

C₂×C₄²

C₂²⋊D₇₉

yes

characteristic

122

C₂×C₄²

C₂²⋊D₇₉

yes

characteristic

123

C₂×C₄²

C₂²×D₇₉

yes

characteristic

124

C₄⋊C₈

—

125

C₄⋊C₈

—

126

C₄⋊C₈

—

127

C₄⋊C₈

—

128

C₄⋊C₈

—

158

—

129

C₄⋊C₈

—

158

—

130

C₂²⋊C₈

—

158

—

131

C₄×C₈

—

158

—

132

#95

—

133

C₂×Q₈

—

158

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

137

C₂²×C₄

#11929624

yes

—

138

M₁₆

—

316

—

139

M₁₆

—

316

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

316

—

144

C₂×C₈

—

316

—

145

C₄⋊C₄

—

316

—

146

C₄⋊C₄

—

158

—

147

C₄⋊C₄

—

158

—

148

C₂²⋊C₄

—

316

—

149

C₄²

—

150

C₄²

—

151

C₄²

—

316

—

152

C₄²

#11929612

yes

—

153

C₄²

#11929614

yes

—

154

C₄²

D₄×D₇₉

yes

characteristic

155

C₄²

#11929623

yes

characteristic

156

C₄²

#11929625

yes

—

157

C₂³

#14579144

yes

characteristic

158

Q₈

—

632

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

316

—

165

C₂×C₄

#14579132

yes

characteristic

166

C₂×C₄

#14579146

yes

—

167

C₂×C₄

#14579150

yes

characteristic

168

C₂×C₄

#14579156

yes

—

169

C₂×C₄

C₂×#11929612

yes

characteristic

170

C₂×C₄

C₂×#11929614

yes

characteristic

171

C₂×C₄

#14579187

yes

characteristic

172

C₂×C₄

#14579192

yes

characteristic

173

C₈

—

174

C₈

—

632

—

175

C₈

—

632

—

176

C₂²

—

177

C₂²

#19098489

yes

characteristic

178

C₂²

#19098893

yes

characteristic

179

C₂²

#19098898

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

632

—

185

C₄

#19098760

yes

characteristic

186

C₄

#19098770

yes

characteristic

187

C₄

#19098915

yes

characteristic

188

C₄

#19098925

yes

characteristic

189

C₂

—

190

C₂

—

191

C₂

#23764809

yes

characteristic

192

C₂

#23764851

yes

—

	subgroup	G/H
center	C₂×C₄	#14579150
commutator, cosocle	C₂×C₄²×C₇₉	C₂³
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1843	C₂
socle	C₂²×C₇₉	#397
prime radical	C₂³×C₇₉	#123
hypercenter	#1843	D₇₉
hypocenter	C₇₉	#10628
LCS	G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1843

subgroup

G/H

center

C₂×C₄

#14579150

commutator, cosocle

C₂×C₄²×C₇₉

C₂³

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1843

C₂

C₂²×C₇₉

C₂³×C₇₉

D₇₉

C₇₉

G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1843

N	H ≅ G/N	type
102. C₇₉	#10628	semidirect

H ≅ G/N

type

102. C₇₉

#10628

semidirect

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

subgroup	index
#23764381	2
#23762058	2
#23761297	2
#23761229	2
C₇₉×#1843	2
#10628	79

subgroup

index

C₇₉×#1843

Subgroups — #34193043

Conjugacy classes of subgroups

Complements

Maximal subgroups