Group #34193332 — subgroups

Subgroups — #34193332

Order 20224.

410 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 145 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	#23764360	C₂	1	2	yes	maximal
2	#23764202	C₂	1	1	yes	maximal, characteristic
3	#23763527	C₂	1	2	yes	maximal
4	C₇₉⋊#1273	C₂	1	1	yes	maximal, characteristic
5	C₇₉×#1845	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579161	C₂²	1	1	yes	characteristic
7	C₂×#14579086	—	2	2	no	—
8	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
9	#19098433	C₂²	1	1	yes	characteristic
10	#19098325	—	2	4	no	—
11	#19098325	—	2	4	no	—
12	C₂×#107×C₇₉	—	2	2	no	—
13	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
14	#384×C₇₉	C₂²	1	2	yes	—
15	C₂²×Dic₁₅₈	—	2	1	no	—
16	C₂²×C₇₉⋊C₈	D₄	1	4	yes	—
17	#14579161	—	2	2	no	—
18	#14579161	—	2	1	no	—
19	#14579161	Q₈	1	2	yes	—
20	C₂×C₄⋊Dic₇₉	—	2	2	no	—
21	C₂×C₄⋊Dic₇₉	—	2	1	no	—
22	C₂×C₄⋊Dic₇₉	—	2	2	no	—
23	#14579086	—	4	4	no	—
24	C₄×C₇₉⋊C₈	—	2	2	no	—
25	C₄×C₇₉⋊C₈	—	2	2	no	—
26	C₂²×C₈×C₇₉	D₄	1	2	yes	—
27	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
28	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
29	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
30	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
31	#107×C₇₉	—	4	4	no	—
32	C₄×C₈×C₇₉	—	2	2	no	—
33	C₂×Dic₁₅₈	—	2	2	no	—
34	C₂×Dic₁₅₈	—	4	1	no	—
35	C₂×Dic₁₅₈	—	4	2	no	—
36	C₂×Dic₁₅₈	—	4	2	no	—
37	C₂²×Dic₇₉	—	4	1	no	—
38	C₂²×Dic₇₉	—	4	1	no	—
39	C₂×C₇₉⋊C₈	—	2	4	no	—
40	C₂×C₇₉⋊C₈	—	2	4	no	—
41	C₂×C₇₉⋊C₈	—	2	4	no	—
42	C₄⋊Dic₇₉	—	2	4	no	—
43	C₄⋊Dic₇₉	—	4	2	no	—
44	C₄⋊Dic₇₉	—	4	2	no	—
45	C₄⋊Dic₇₉	—	4	4	no	—
46	C₂²×C₄×C₇₉	—	4	2	no	—
47	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
49	C₂×C₈×C₇₉	—	2	2	no	—
50	C₂×C₈×C₇₉	—	2	2	no	—
51	C₂×C₈×C₇₉	D₈	1	4	yes	—
52	C₄⋊C₄×C₇₉	—	4	4	no	—
53	C₄⋊C₄×C₇₉	—	4	4	no	—
54	C₄⋊C₄×C₇₉	—	4	4	no	—
55	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₇₉	#41	1	2	yes	—
57	C₄²×C₇₉	#41	1	1	yes	characteristic
58	C₂³×C₇₉	#127	1	1	yes	characteristic
59	C₂×C₄×C₇₉	—	8	2	no	—
60	C₂×C₄×C₇₉	—	8	2	no	—
61	C₂×C₄×C₇₉	—	8	2	no	—
62	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
64	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
65	C₂×C₄×C₇₉	#123	1	2	yes	—
66	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
67	C₂×C₄×C₇₉	#135	1	4	yes	—
68	C₂×C₄×C₇₉	#137	1	2	yes	—
69	C₂×C₄×C₇₉	#137	1	4	yes	—
70	C₂×Dic₇₉	—	4	2	no	—
71	C₂×Dic₇₉	—	8	1	no	—
72	C₂×Dic₇₉	—	8	1	no	—
73	C₂×Dic₇₉	—	8	1	no	—
74	C₂×Dic₇₉	—	8	2	no	—
75	Dic₁₅₈	—	4	4	no	—
76	Dic₁₅₈	—	8	2	no	—
77	C₈×C₇₉	—	2	4	no	—
78	C₇₉⋊C₈	—	4	4	no	—
79	C₂²×C₇₉	#399	1	1	yes	characteristic
80	C₂²×C₇₉	#493	1	1	yes	characteristic
81	C₂²×C₇₉	#495	1	2	yes	—
82	C₂²×C₇₉	#497	1	1	yes	characteristic
83	C₂²×C₇₉	#497	1	2	yes	—
84	C₄×C₇₉	—	16	2	no	—
85	C₄×C₇₉	#474	1	2	yes	—
86	C₄×C₇₉	#479	1	4	yes	—
87	C₄×C₇₉	#480	1	2	yes	—
88	C₄×C₇₉	#485	1	4	yes	—
89	Dic₇₉	—	8	2	no	—
90	Dic₇₉	—	16	1	no	—
91	#10676	—	79	1	no	maximal, Sylow
92	C₂×C₇₉	#1709	1	1	yes	characteristic
93	C₂×C₇₉	#1718	1	2	yes	—
94	C₂×C₇₉	#1722	1	1	yes	characteristic
95	C₂×C₇₉	#2093	1	1	yes	characteristic
96	C₂×C₇₉	#2097	1	2	yes	—
97	#1990	—	79	2	no	—
98	#1956	—	79	1	no	—
99	#1845	D₇₉	1	1	yes	characteristic
100	#1834	—	79	2	no	—
101	#1273	—	79	1	no	—
102	C₇₉	#10676	1	1	yes	characteristic, complemented, Sylow
103	C₂×C₄⋊Q₈	—	79	1	no	—
104	C₂×#107	—	2	2	no	—
105	C₂×#103	—	158	2	no	—
106	C₂×C₄×C₈	—	79	2	no	—
107	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
108	#384	—	79	1	no	—
109	#384	C₂×D₇₉	1	2	yes	—
110	#340	—	158	4	no	—
111	#340	—	158	4	no	—
112	C₂²×Q₈	—	158	1	no	—
113	C₂²×C₈	—	79	4	no	—
114	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
115	C₄⋊Q₈	—	79	2	no	—
116	C₄⋊Q₈	—	158	2	no	—
117	C₄⋊Q₈	—	158	1	no	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	158	2	no	—
122	C₂×C₄⋊C₄	—	158	1	no	—
123	C₂×C₄⋊C₄	—	158	2	no	—
124	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
125	#107	—	4	4	no	—
126	#103	—	316	4	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	158	2	no	—
129	C₄×C₈	—	158	2	no	—
130	C₂×Q₈	—	316	1	no	—
131	C₂×Q₈	—	316	2	no	—
132	C₂×Q₈	—	316	2	no	—
133	C₂×Q₈	—	158	2	no	—
134	C₂²×C₄	—	4	2	no	—
135	C₂²×C₄	—	316	1	no	—
136	C₂²×C₄	—	316	1	no	—
137	C₂²×C₄	D₄×D₇₉	1	2	yes	—
138	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	158	4	no	—
142	C₂×C₈	—	158	4	no	—
143	C₂×C₈	—	158	4	no	—
144	C₂×C₈	#11929609	1	4	yes	—
145	C₄⋊C₄	—	4	4	no	—
146	C₄⋊C₄	—	4	4	no	—
147	C₄⋊C₄	—	4	4	no	—
148	C₄⋊C₄	—	316	2	no	—
149	C₄⋊C₄	—	316	2	no	—
150	C₄⋊C₄	—	316	4	no	—
151	C₄⋊C₄	—	158	4	no	—
152	C₄²	Q₈×D₇₉	1	1	yes	characteristic
153	C₄²	#11929624	1	2	yes	—
154	C₄²	#11929625	1	1	yes	characteristic
155	C₂³	#14579157	1	1	yes	characteristic
156	Q₈	—	316	4	no	—
157	Q₈	—	632	2	no	—
158	C₂×C₄	—	8	2	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	—	316	2	no	—
162	C₂×C₄	—	632	1	no	—
163	C₂×C₄	—	632	1	no	—
164	C₂×C₄	—	632	1	no	—
165	C₂×C₄	—	632	2	no	—
166	C₂×C₄	#14579137	1	1	yes	characteristic
167	C₂×C₄	#14579139	1	2	yes	—
168	C₂×C₄	#14579144	1	1	yes	characteristic
169	C₂×C₄	#14579146	1	2	yes	—
170	C₂×C₄	C₂×#11929609	1	2	yes	—
171	C₂×C₄	#14579180	1	4	yes	—
172	C₂×C₄	#14579192	1	2	yes	—
173	C₂×C₄	#14579193	1	4	yes	—
174	C₈	—	2	4	no	—
175	C₈	—	316	4	no	—
176	C₂²	#19098498	1	1	yes	characteristic
177	C₂²	#19098939	1	1	yes	characteristic
178	C₂²	#19098946	1	2	yes	—
179	C₂²	#19098955	1	1	yes	characteristic
180	C₂²	#19098956	1	2	yes	—
181	C₄	—	16	2	no	—
182	C₄	—	632	2	no	—
183	C₄	—	1264	1	no	—
184	C₄	#19098840	1	2	yes	—
185	C₄	#19098869	1	4	yes	—
186	C₄	#19098876	1	2	yes	—
187	C₄	#19098906	1	4	yes	—
188	C₂	#23763005	1	1	yes	characteristic
189	C₂	#23763045	1	2	yes	—
190	C₂	#23763072	1	1	yes	characteristic
191	C₂	#23764929	1	1	yes	characteristic
192	C₂	#23764945	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23764360

C₂

yes

maximal

#23764202

C₂

yes

maximal, characteristic

#23763527

C₂

yes

maximal

C₇₉⋊#1273

C₂

yes

maximal, characteristic

C₇₉×#1845

C₂

yes

maximal, characteristic

C₂×#14579161

C₂²

yes

characteristic

C₂×#14579086

—

C₂×C₄×C₇₉⋊C₈

C₂²

yes

—

#19098433

C₂²

yes

characteristic

#19098325

—

#19098325

—

C₂×#107×C₇₉

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#384×C₇₉

C₂²

yes

—

C₂²×Dic₁₅₈

—

C₂²×C₇₉⋊C₈

D₄

yes

—

#14579161

—

#14579161

—

#14579161

Q₈

yes

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

#14579086

—

C₄×C₇₉⋊C₈

—

C₄×C₇₉⋊C₈

—

C₂²×C₈×C₇₉

D₄

yes

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

#107×C₇₉

—

C₄×C₈×C₇₉

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

D₈

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂×Q₈

yes

characteristic

C₄²×C₇₉

#41

yes

—

C₄²×C₇₉

yes

C₂³×C₇₉

yes

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₉

#123

yes

characteristic

C₂×C₄×C₇₉

#123

yes

—

C₂×C₄×C₇₉

C₂×D₈

yes

—

C₂×C₄×C₇₉

#135

yes

—

C₂×C₄×C₇₉

#137

yes

—

C₂×C₄×C₇₉

#137

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

—

C₂²×C₇₉

#497

yes

characteristic

C₂²×C₇₉

#497

yes

—

C₄×C₇₉

—

C₄×C₇₉

#474

yes

—

C₄×C₇₉

#479

yes

—

C₄×C₇₉

#480

yes

—

C₄×C₇₉

#485

yes

—

Dic₇₉

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

#1990

—

#1956

—

#1845

D₇₉

yes

characteristic

100

#1834

—

101

#1273

—

102

C₇₉

#10676

yes

characteristic, complemented, Sylow

103

C₂×C₄⋊Q₈

—

104

C₂×#107

—

105

C₂×#103

—

158

—

106

C₂×C₄×C₈

—

107

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

108

#384

—

109

#384

C₂×D₇₉

yes

—

110

#340

—

158

—

111

#340

—

158

—

112

C₂²×Q₈

—

158

—

113

C₂²×C₈

—

114

C₂²×C₈

C₂²⋊D₇₉

yes

—

115

C₄⋊Q₈

—

116

C₄⋊Q₈

—

158

—

117

C₄⋊Q₈

—

158

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

158

—

122

C₂×C₄⋊C₄

—

158

—

123

C₂×C₄⋊C₄

—

158

—

124

C₂×C₄²

C₂²×D₇₉

yes

characteristic

125

#107

—

126

#103

—

316

—

127

C₄×C₈

—

128

C₄×C₈

—

158

—

129

C₄×C₈

—

158

—

130

C₂×Q₈

—

316

—

131

C₂×Q₈

—

316

—

132

C₂×Q₈

—

316

—

133

C₂×Q₈

—

158

—

134

C₂²×C₄

—

135

C₂²×C₄

—

316

—

136

C₂²×C₄

—

316

—

137

C₂²×C₄

D₄×D₇₉

yes

—

138

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

158

—

142

C₂×C₈

—

158

—

143

C₂×C₈

—

158

—

144

C₂×C₈

#11929609

yes

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

316

—

149

C₄⋊C₄

—

316

—

150

C₄⋊C₄

—

316

—

151

C₄⋊C₄

—

158

—

152

C₄²

Q₈×D₇₉

yes

characteristic

153

C₄²

#11929624

yes

—

154

C₄²

#11929625

yes

characteristic

155

C₂³

#14579157

yes

characteristic

156

Q₈

—

316

—

157

Q₈

—

632

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

316

—

162

C₂×C₄

—

632

—

163

C₂×C₄

—

632

—

164

C₂×C₄

—

632

—

165

C₂×C₄

—

632

—

166

C₂×C₄

#14579137

yes

characteristic

167

C₂×C₄

#14579139

yes

—

168

C₂×C₄

#14579144

yes

characteristic

169

C₂×C₄

#14579146

yes

—

170

C₂×C₄

C₂×#11929609

yes

—

171

C₂×C₄

#14579180

yes

—

172

C₂×C₄

#14579192

yes

—

173

C₂×C₄

#14579193

yes

—

174

C₈

—

175

C₈

—

316

—

176

C₂²

#19098498

yes

characteristic

177

C₂²

#19098939

yes

characteristic

178

C₂²

#19098946

yes

—

179

C₂²

#19098955

yes

characteristic

180

C₂²

#19098956

yes

—

181

C₄

—

182

C₄

—

632

—

183

C₄

—

1264

—

184

C₄

#19098840

yes

—

185

C₄

#19098869

yes

—

186

C₄

#19098876

yes

—

187

C₄

#19098906

yes

—

188

C₂

#23763005

yes

characteristic

189

C₂

#23763045

yes

—

190

C₂

#23763072

yes

characteristic

191

C₂

#23764929

yes

characteristic

192

C₂

#23764945

yes

—

	subgroup	G/H
center	C₂³	#14579157
commutator, cosocle	C₂×C₄²×C₇₉	C₂³
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1845	C₂
socle, prime radical	C₂³×C₇₉	#127
hypercenter	#1845	D₇₉
hypocenter	C₇₉	#10676
LCS	G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

subgroup

G/H

center

C₂³

#14579157

commutator, cosocle

C₂×C₄²×C₇₉

C₂³

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1845

C₂

C₂³×C₇₉

D₇₉

C₇₉

G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

N	H ≅ G/N	type
102. C₇₉	#10676	semidirect

H ≅ G/N

type

102. C₇₉

#10676

semidirect

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

subgroup	index
#23764360	2
#23764202	2
#23763527	2
C₇₉⋊#1273	2
C₇₉×#1845	2
#10676	79

subgroup

index

#23764360

#23764202

#23763527

C₇₉⋊#1273

C₇₉×#1845

#10676

Subgroups — #34193332

Conjugacy classes of subgroups

Complements

Maximal subgroups