Group #23764971 — subgroups

Subgroups — #23764971

Order 10112.

198 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 79 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	#19098462	C₂	1	1	yes	maximal, characteristic
2	#19098428	C₂	1	1	yes	maximal, characteristic
3	#19098326	C₂	1	1	yes	maximal, characteristic
4	#19098325	C₂	1	1	yes	maximal, characteristic
5	#19098325	C₂	1	1	yes	maximal, characteristic
6	#19098317	C₂	1	1	yes	maximal, characteristic
7	#340×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₇₉	—	2	1	no	—
10	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
13	C₇₉⋊#95	—	2	1	no	—
14	C₇₉⋊#95	—	2	1	no	—
15	C₇₉⋊#95	—	2	1	no	—
16	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
19	C₂²×Dic₇₉	—	2	1	no	—
20	C₂²×Dic₇₉	—	2	1	no	—
21	C₂²×Dic₇₉	D₄	1	1	yes	characteristic
22	C₂²×Dic₇₉	D₄	1	1	yes	characteristic
23	C₂×C₇₉⋊C₈	—	2	1	no	—
24	C₂×C₇₉⋊C₈	—	2	1	no	—
25	C₂×C₇₉⋊C₈	—	2	1	no	—
26	C₄⋊Dic₇₉	—	2	1	no	—
27	C₄⋊Dic₇₉	—	2	1	no	—
28	C₄⋊Dic₇₉	—	2	1	no	—
29	C₄⋊Dic₇₉	—	2	1	no	—
30	C₄⋊Dic₇₉	—	4	2	no	—
31	C₄×Dic₇₉	—	2	1	no	—
32	C₄×Dic₇₉	—	2	1	no	—
33	C₂²×C₄×C₇₉	—	2	1	no	—
34	C₂²×C₄×C₇₉	—	2	1	no	—
35	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₇₉	—	2	1	no	—
37	C₂×C₈×C₇₉	—	2	1	no	—
38	C₂×C₈×C₇₉	—	2	1	no	—
39	C₄⋊C₄×C₇₉	—	2	1	no	—
40	C₄⋊C₄×C₇₉	—	2	1	no	—
41	C₄⋊C₄×C₇₉	—	2	1	no	—
42	C₄⋊C₄×C₇₉	Q₈	1	2	yes	—
43	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₇₉	—	2	2	no	—
45	C₂×C₄×C₇₉	—	4	1	no	—
46	C₂×C₄×C₇₉	—	4	1	no	—
47	C₂×C₄×C₇₉	—	4	1	no	—
48	C₂×C₄×C₇₉	—	4	1	no	—
49	C₂×C₄×C₇₉	—	4	1	no	—
50	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
52	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
53	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
54	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
55	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
56	C₂×Dic₇₉	—	2	1	no	—
57	C₂×Dic₇₉	—	2	1	no	—
58	C₂×Dic₇₉	—	2	1	no	—
59	C₂×Dic₇₉	—	2	1	no	—
60	C₂×Dic₇₉	—	2	1	no	—
61	C₂×Dic₇₉	—	2	1	no	—
62	C₂×Dic₇₉	—	4	1	no	—
63	C₂×Dic₇₉	—	4	1	no	—
64	C₂×Dic₇₉	—	4	1	no	—
65	C₂×Dic₇₉	—	4	1	no	—
66	C₂×Dic₇₉	—	4	1	no	—
67	C₂×Dic₇₉	—	4	1	no	—
68	C₈×C₇₉	—	4	1	no	—
69	C₇₉⋊C₈	—	4	1	no	—
70	C₂²×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂²×C₇₉	#123	1	1	yes	characteristic
72	C₂²×C₇₉	#124	1	1	yes	characteristic
73	C₂²×C₇₉	#135	1	1	yes	characteristic
74	C₂²×C₇₉	#135	1	1	yes	characteristic
75	C₂²×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
76	C₂²×C₇₉	#137	1	1	yes	characteristic
77	C₄×C₇₉	—	4	2	no	—
78	C₄×C₇₉	—	8	1	no	—
79	C₄×C₇₉	#125	1	1	yes	characteristic
80	C₄×C₇₉	#125	1	1	yes	characteristic
81	C₄×C₇₉	#126	1	1	yes	characteristic
82	C₄×C₇₉	#126	1	1	yes	characteristic
83	Dic₇₉	—	4	1	no	—
84	Dic₇₉	—	4	1	no	—
85	Dic₇₉	—	8	1	no	—
86	Dic₇₉	—	8	1	no	—
87	C₂×C₇₉	#400	1	1	yes	characteristic
88	C₂×C₇₉	#478	1	1	yes	characteristic
89	C₂×C₇₉	#479	1	1	yes	characteristic
90	C₂×C₇₉	#482	1	1	yes	characteristic
91	C₂×C₇₉	#485	1	1	yes	characteristic
92	C₂×C₇₉	#488	1	1	yes	characteristic
93	C₂×C₇₉	#489	1	1	yes	characteristic
94	#2101	—	79	1	no	maximal, Sylow
95	C₇₉	#2101	1	1	yes	characteristic, complemented, Sylow
96	#389	—	79	1	no	—
97	#382	—	79	1	no	—
98	#340	—	79	1	no	—
99	#340	—	79	1	no	—
100	#340	—	79	1	no	—
101	#340	D₇₉	1	1	yes	characteristic
102	#336	—	79	1	no	—
103	C₂²×C₈	—	79	1	no	—
104	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
105	C₂×C₄⋊C₄	—	79	1	no	—
106	C₂×C₄⋊C₄	—	79	1	no	—
107	C₂×C₄⋊C₄	—	158	1	no	—
108	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
109	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
110	C₂×C₄²	—	79	1	no	—
111	#95	—	158	1	no	—
112	#95	—	158	1	no	—
113	#95	—	158	1	no	—
114	C₂²×C₄	—	2	1	no	—
115	C₂²×C₄	—	2	1	no	—
116	C₂²×C₄	—	79	1	no	—
117	C₂²×C₄	—	79	1	no	—
118	C₂²×C₄	—	158	1	no	—
119	C₂²×C₄	—	158	1	no	—
120	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	158	1	no	—
125	C₂×C₈	—	158	1	no	—
126	C₂×C₈	—	158	1	no	—
127	C₄⋊C₄	—	2	1	no	—
128	C₄⋊C₄	—	2	1	no	—
129	C₄⋊C₄	—	2	1	no	—
130	C₄⋊C₄	—	316	2	no	—
131	C₄⋊C₄	—	158	1	no	—
132	C₄⋊C₄	—	158	1	no	—
133	C₄⋊C₄	—	158	1	no	—
134	C₄⋊C₄	—	158	1	no	—
135	C₄⋊C₄	Dic₁₅₈	1	2	yes	—
136	C₄²	—	158	1	no	—
137	C₄²	—	158	1	no	—
138	C₂³	D₄×D₇₉	1	1	yes	characteristic
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	4	1	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	—	316	1	no	—
146	C₂×C₄	—	316	1	no	—
147	C₂×C₄	—	316	1	no	—
148	C₂×C₄	—	316	1	no	—
149	C₂×C₄	—	316	1	no	—
150	C₂×C₄	—	316	1	no	—
151	C₂×C₄	—	158	1	no	—
152	C₂×C₄	—	158	1	no	—
153	C₂×C₄	—	158	1	no	—
154	C₂×C₄	—	158	1	no	—
155	C₂×C₄	—	158	1	no	—
156	C₂×C₄	—	158	1	no	—
157	C₂×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
158	C₂×C₄	#11929623	1	1	yes	characteristic
159	C₂×C₄	#11929623	1	1	yes	characteristic
160	C₂×C₄	#11929624	1	1	yes	characteristic
161	C₂×C₄	#11929624	1	1	yes	characteristic
162	C₂×C₄	#11929625	1	1	yes	characteristic
163	C₈	—	4	1	no	—
164	C₈	—	316	1	no	—
165	C₂²	#14579138	1	1	yes	characteristic
166	C₂²	#14579146	1	1	yes	characteristic
167	C₂²	#14579147	1	1	yes	characteristic
168	C₂²	#14579180	1	1	yes	characteristic
169	C₂²	#14579182	1	1	yes	characteristic
170	C₂²	#14579188	1	1	yes	characteristic
171	C₂²	#14579193	1	1	yes	characteristic
172	C₄	—	4	2	no	—
173	C₄	—	8	1	no	—
174	C₄	—	316	1	no	—
175	C₄	—	316	1	no	—
176	C₄	—	632	1	no	—
177	C₄	—	632	1	no	—
178	C₄	#14579152	1	1	yes	characteristic
179	C₄	#14579153	1	1	yes	characteristic
180	C₄	#14579154	1	1	yes	characteristic
181	C₄	#14579156	1	1	yes	characteristic
182	C₂	#19098505	1	1	yes	characteristic
183	C₂	#19098861	1	1	yes	characteristic
184	C₂	#19098868	1	1	yes	characteristic
185	C₂	#19098889	1	1	yes	characteristic
186	C₂	#19098906	1	1	yes	characteristic
187	C₂	#19098916	1	1	yes	characteristic
188	C₂	#19098924	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19098462

C₂

yes

maximal, characteristic

#19098428

C₂

yes

maximal, characteristic

#19098326

C₂

yes

maximal, characteristic

#19098325

C₂

yes

maximal, characteristic

#19098325

C₂

yes

maximal, characteristic

#19098317

C₂

yes

maximal, characteristic

#340×C₇₉

C₂

yes

maximal, characteristic

C₂²×C₇₉⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₉

C₂²

yes

characteristic

C₂×C₄×Dic₇₉

C₂²

yes

characteristic

C₇₉⋊#95

—

C₇₉⋊#95

—

C₇₉⋊#95

—

C₂²×C₈×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₉

C₂²

yes

characteristic

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

D₄

yes

characteristic

C₂²×Dic₇₉

D₄

yes

characteristic

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

Q₈

yes

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×Q₈

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

C₂²⋊Q₈

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

—

—

—

—

100

#340

—

101

#340

D₇₉

yes

characteristic

102

#336

—

103

C₂²×C₈

—

104

C₂²×C₈

C₂×D₇₉

yes

characteristic

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

158

—

108

C₂×C₄⋊C₄

C₂×D₇₉

yes

characteristic

109

C₂×C₄⋊C₄

C₂×D₇₉

yes

characteristic

110

C₂×C₄²

—

111

#95

—

158

—

112

#95

—

158

—

113

#95

—

158

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

158

—

119

C₂²×C₄

—

158

—

120

C₂²×C₄

C₂²×D₇₉

yes

characteristic

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

158

—

125

C₂×C₈

—

158

—

126

C₂×C₈

—

158

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

316

—

131

C₄⋊C₄

—

158

—

132

C₄⋊C₄

—

158

—

133

C₄⋊C₄

—

158

—

134

C₄⋊C₄

—

158

—

135

C₄⋊C₄

Dic₁₅₈

yes

—

136

C₄²

—

158

—

137

C₄²

—

158

—

138

C₂³

D₄×D₇₉

yes

characteristic

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

316

—

146

C₂×C₄

—

316

—

147

C₂×C₄

—

316

—

148

C₂×C₄

—

316

—

149

C₂×C₄

—

316

—

150

C₂×C₄

—

316

—

151

C₂×C₄

—

158

—

152

C₂×C₄

—

158

—

153

C₂×C₄

—

158

—

154

C₂×C₄

—

158

—

155

C₂×C₄

—

158

—

156

C₂×C₄

—

158

—

157

C₂×C₄

C₂×Dic₁₅₈

yes

characteristic

158

C₂×C₄

#11929623

yes

characteristic

159

C₂×C₄

#11929623

yes

characteristic

160

C₂×C₄

#11929624

yes

characteristic

161

C₂×C₄

#11929624

yes

characteristic

162

C₂×C₄

#11929625

yes

characteristic

163

C₈

—

164

C₈

—

316

—

165

C₂²

#14579138

yes

characteristic

166

C₂²

#14579146

yes

characteristic

167

C₂²

#14579147

yes

characteristic

168

C₂²

#14579180

yes

characteristic

169

C₂²

#14579182

yes

characteristic

170

C₂²

#14579188

yes

characteristic

171

C₂²

#14579193

yes

characteristic

172

C₄

—

173

C₄

—

174

C₄

—

316

—

175

C₄

—

316

—

176

C₄

—

632

—

177

C₄

—

632

—

178

C₄

#14579152

yes

characteristic

179

C₄

#14579153

yes

characteristic

180

C₄

#14579154

yes

characteristic

181

C₄

#14579156

yes

characteristic

182

C₂

#19098505

yes

characteristic

183

C₂

#19098861

yes

characteristic

184

C₂

#19098868

yes

characteristic

185

C₂

#19098889

yes

characteristic

186

C₂

#19098906

yes

characteristic

187

C₂

#19098916

yes

characteristic

188

C₂

#19098924

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₇₉
commutator, cosocle	C₂²×C₄×C₇₉	C₂³
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	#340×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₄
hypercenter	#340	D₇₉
hypocenter	C₇₉	#2101
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₇₉

commutator, cosocle

C₂²×C₄×C₇₉

C₂³

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

#340×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂×D₄

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
95. C₇₉	#2101	semidirect

H ≅ G/N

type

95. C₇₉

#2101

semidirect

Subgroups — #23764971

Conjugacy classes of subgroups

Complements