Group #23764536 — subgroups

Subgroups — #23764536

Order 10112.

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

normal subgroups only

This group has 326 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	C₂×#14579161	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14579082	C₂	1	2	yes	maximal
3	#19098325	C₂	1	2	yes	maximal, complemented
4	#19098318	C₂	1	1	yes	maximal, characteristic, complemented
5	#390×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₅₈	—	2	1	no	—
7	C₂²×C₇₉⋊C₈	C₂²	1	2	yes	—
8	#14579161	—	2	4	no	—
9	C₂×C₄⋊Dic₇₉	—	2	1	no	—
10	C₂×C₄⋊Dic₇₉	C₂²	1	2	yes	—
11	#14579082	—	2	4	no	—
12	C₂²×D₄×C₇₉	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
15	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
16	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₅₈	—	2	4	no	—
18	C₂×Dic₁₅₈	—	4	2	no	—
19	C₂×Dic₁₅₈	—	4	2	no	—
20	C₂²×Dic₇₉	—	2	2	no	—
21	C₂×C₇₉⋊C₈	—	2	2	no	—
22	C₂×C₇₉⋊C₈	—	2	2	no	—
23	C₂×C₇₉⋊C₈	—	2	2	no	—
24	C₄⋊Dic₇₉	—	2	4	no	—
25	C₄⋊Dic₇₉	—	2	2	no	—
26	C₄⋊Dic₇₉	D₄	1	4	yes	—
27	C₂⁴×C₇₉	—	2	1	no	—
28	C₂×D₄×C₇₉	—	2	2	no	—
29	C₂×D₄×C₇₉	—	2	2	no	—
30	C₂×D₄×C₇₉	—	2	1	no	—
31	C₂×D₄×C₇₉	Q₈	1	2	yes	complemented
32	C₂²×C₄×C₇₉	—	2	1	no	—
33	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₇₉	—	2	2	no	—
37	C₂²⋊C₄×C₇₉	—	4	2	no	—
38	C₂²⋊C₄×C₇₉	—	4	2	no	—
39	C₄²×C₇₉	—	2	2	no	—
40	C₂³×C₇₉	—	2	2	no	—
41	C₂³×C₇₉	—	4	1	no	—
42	C₂³×C₇₉	—	4	1	no	—
43	C₂³×C₇₉	—	4	1	no	—
44	C₂³×C₇₉	—	4	1	no	—
45	C₂³×C₇₉	—	4	1	no	—
46	C₂³×C₇₉	—	4	1	no	—
47	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
48	D₄×C₇₉	—	2	4	no	—
49	D₄×C₇₉	—	4	2	no	—
50	C₂×C₄×C₇₉	—	2	1	no	—
51	C₂×C₄×C₇₉	—	2	1	no	—
52	C₂×C₄×C₇₉	—	2	1	no	—
53	C₂×C₄×C₇₉	—	2	1	no	—
54	C₂×C₄×C₇₉	—	4	1	no	—
55	C₂×C₄×C₇₉	—	4	1	no	—
56	C₂×C₄×C₇₉	—	4	1	no	—
57	C₂×C₄×C₇₉	D₈	1	2	yes	—
58	C₂×C₄×C₇₉	SD₁₆	1	2	yes	—
59	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₄×C₇₉	#41	1	2	yes	—
62	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
63	C₂×Dic₇₉	—	2	4	no	—
64	C₂×Dic₇₉	—	4	2	no	—
65	C₂×Dic₇₉	—	4	2	no	—
66	Dic₁₅₈	—	4	8	no	—
67	C₇₉⋊C₈	—	4	2	no	—
68	C₂²×C₇₉	—	4	2	no	—
69	C₂²×C₇₉	—	4	2	no	—
70	C₂²×C₇₉	—	4	2	no	—
71	C₂²×C₇₉	—	8	1	no	—
72	C₂²×C₇₉	—	8	1	no	—
73	C₂²×C₇₉	—	8	1	no	—
74	C₂²×C₇₉	—	8	1	no	—
75	C₂²×C₇₉	#120	1	1	yes	characteristic
76	C₂²×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
77	C₂²×C₇₉	#124	1	1	yes	characteristic
78	C₂²×C₇₉	C₂×D₈	1	1	yes	characteristic
79	C₂²×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
80	C₂²×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₂²×C₇₉	#137	1	1	yes	characteristic
82	C₄×C₇₉	—	2	2	no	—
83	C₄×C₇₉	—	2	2	no	—
84	C₄×C₇₉	—	8	1	no	—
85	C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
86	C₄×C₇₉	#124	1	2	yes	—
87	Dic₇₉	—	4	4	no	—
88	C₂×C₇₉	—	8	2	no	—
89	C₂×C₇₉	#393	1	1	yes	characteristic
90	C₂×C₇₉	#447	1	1	yes	characteristic
91	C₂×C₇₉	#448	1	1	yes	characteristic
92	C₂×C₇₉	#474	1	1	yes	characteristic
93	C₂×C₇₉	#476	1	1	yes	characteristic
94	C₂×C₇₉	#486	1	1	yes	characteristic
95	C₂×C₇₉	#489	1	1	yes	characteristic
96	#2023	—	79	1	no	maximal, Sylow
97	C₇₉	#2023	1	1	yes	characteristic, complemented, Sylow
98	C₂×C₄⋊Q₈	—	79	1	no	—
99	C₂×#102	—	79	2	no	—
100	#390	D₇₉	1	1	yes	characteristic
101	#340	—	79	2	no	—
102	#336	—	79	1	no	—
103	C₂²×Q₈	—	158	1	no	—
104	C₂²×D₄	C₂×D₇₉	1	1	yes	characteristic
105	C₂²×C₈	—	79	2	no	—
106	C₄⋊Q₈	—	158	4	no	—
107	C₂×C₄⋊C₄	—	79	2	no	—
108	C₂×C₄⋊C₄	—	158	1	no	—
109	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
110	C₂×C₂²⋊C₄	—	2	1	no	—
111	C₂×C₂²⋊C₄	—	2	1	no	—
112	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
113	#102	—	158	4	no	—
114	C₂⁴	—	2	1	no	—
115	C₂×Q₈	—	316	2	no	—
116	C₂×Q₈	—	316	2	no	—
117	C₂×Q₈	—	158	4	no	—
118	C₂×D₄	—	2	2	no	—
119	C₂×D₄	—	2	2	no	—
120	C₂×D₄	—	2	1	no	—
121	C₂×D₄	Dic₁₅₈	1	2	yes	complemented
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	—	158	2	no	—
124	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
125	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
126	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
127	C₂×C₈	—	158	2	no	—
128	C₂×C₈	—	158	2	no	—
129	C₂×C₈	—	158	2	no	—
130	C₄⋊C₄	—	2	2	no	—
131	C₄⋊C₄	—	79	4	no	—
132	C₄⋊C₄	—	158	4	no	—
133	C₄⋊C₄	—	158	2	no	—
134	C₂²⋊C₄	—	4	2	no	—
135	C₂²⋊C₄	—	4	2	no	—
136	C₄²	—	2	2	no	—
137	C₂³	—	2	2	no	—
138	C₂³	—	4	1	no	—
139	C₂³	—	4	1	no	—
140	C₂³	—	4	1	no	—
141	C₂³	—	4	1	no	—
142	C₂³	—	4	1	no	—
143	C₂³	—	4	1	no	—
144	C₂³	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
145	Q₈	—	316	8	no	—
146	D₄	—	2	4	no	—
147	D₄	—	4	2	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	316	2	no	—
156	C₂×C₄	—	316	2	no	—
157	C₂×C₄	—	158	4	no	—
158	C₂×C₄	#11929609	1	2	yes	—
159	C₂×C₄	#11929612	1	2	yes	—
160	C₂×C₄	D₄×D₇₉	1	2	yes	—
161	C₂×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
162	C₂×C₄	#11929623	1	2	yes	—
163	C₂×C₄	#11929624	1	1	yes	characteristic
164	C₈	—	316	2	no	—
165	C₂²	—	4	2	no	—
166	C₂²	—	4	2	no	—
167	C₂²	—	4	2	no	—
168	C₂²	—	8	1	no	—
169	C₂²	—	8	1	no	—
170	C₂²	—	8	1	no	—
171	C₂²	—	8	1	no	—
172	C₂²	#14579129	1	1	yes	characteristic
173	C₂²	#14579140	1	1	yes	characteristic
174	C₂²	#14579148	1	1	yes	characteristic
175	C₂²	C₂×#11929609	1	1	yes	characteristic
176	C₂²	C₂×#11929612	1	1	yes	characteristic
177	C₂²	#14579186	1	1	yes	characteristic
178	C₂²	#14579191	1	1	yes	characteristic
179	C₄	—	2	2	no	—
180	C₄	—	2	2	no	—
181	C₄	—	8	1	no	—
182	C₄	—	316	4	no	—
183	C₄	#14579138	1	2	yes	—
184	C₄	#14579147	1	2	yes	—
185	C₂	—	8	2	no	—
186	C₂	#19098475	1	1	yes	characteristic
187	C₂	#19098696	1	1	yes	characteristic
188	C₂	#19098701	1	1	yes	characteristic
189	C₂	#19098843	1	1	yes	characteristic
190	C₂	#19098853	1	1	yes	characteristic
191	C₂	#19098909	1	1	yes	characteristic
192	C₂	#19098922	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579161

C₂

yes

maximal, characteristic, complemented

C₂×#14579082

C₂

yes

maximal

#19098325

C₂

yes

maximal, complemented

#19098318

C₂

yes

maximal, characteristic, complemented

#390×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

—

C₂²×C₇₉⋊C₈

C₂²

yes

—

#14579161

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

C₂²

yes

—

#14579082

—

C₂²×D₄×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₉

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₇₉

—

C₂×C₂²⋊C₄×C₇₉

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

D₄

yes

—

C₂⁴×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

Q₈

yes

complemented

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

D₈

yes

—

C₂×C₄×C₇₉

SD₁₆

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₉

#41

yes

—

C₂×C₄×C₇₉

#41

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#120

yes

characteristic

C₂²×C₇₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₇₉

#124

yes

characteristic

C₂²×C₇₉

C₂×D₈

yes

characteristic

C₂²×C₇₉

C₂×SD₁₆

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂²⋊Q₈

yes

—

C₄×C₇₉

#124

yes

—

Dic₇₉

—

C₂×C₇₉

—

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

C₂×C₄⋊Q₈

—

C₂×#102

—

100

#390

D₇₉

yes

characteristic

101

#340

—

102

#336

—

103

C₂²×Q₈

—

158

—

104

C₂²×D₄

C₂×D₇₉

yes

characteristic

105

C₂²×C₈

—

106

C₄⋊Q₈

—

158

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

158

—

109

C₂×C₄⋊C₄

C₂×D₇₉

yes

characteristic

110

C₂×C₂²⋊C₄

—

111

C₂×C₂²⋊C₄

—

112

C₂×C₄²

C₂×D₇₉

yes

characteristic

113

#102

—

158

—

114

C₂⁴

—

115

C₂×Q₈

—

316

—

116

C₂×Q₈

—

316

—

117

C₂×Q₈

—

158

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂×D₄

—

121

C₂×D₄

Dic₁₅₈

yes

complemented

122

C₂²×C₄

—

123

C₂²×C₄

—

158

—

124

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

125

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

126

C₂²×C₄

C₂²×D₇₉

yes

characteristic

127

C₂×C₈

—

158

—

128

C₂×C₈

—

158

—

129

C₂×C₈

—

158

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

158

—

133

C₄⋊C₄

—

158

—

134

C₂²⋊C₄

—

135

C₂²⋊C₄

—

136

C₄²

—

137

C₂³

—

138

C₂³

—

139

C₂³

—

140

C₂³

—

141

C₂³

—

142

C₂³

—

143

C₂³

—

144

C₂³

C₂×C₂²⋊D₇₉

yes

characteristic

145

Q₈

—

316

—

146

D₄

—

147

D₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

316

—

156

C₂×C₄

—

316

—

157

C₂×C₄

—

158

—

158

C₂×C₄

#11929609

yes

—

159

C₂×C₄

#11929612

yes

—

160

C₂×C₄

D₄×D₇₉

yes

—

161

C₂×C₄

C₂×Dic₁₅₈

yes

characteristic

162

C₂×C₄

#11929623

yes

—

163

C₂×C₄

#11929624

yes

characteristic

164

C₈

—

316

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

#14579129

yes

characteristic

173

C₂²

#14579140

yes

characteristic

174

C₂²

#14579148

yes

characteristic

175

C₂²

C₂×#11929609

yes

characteristic

176

C₂²

C₂×#11929612

yes

characteristic

177

C₂²

#14579186

yes

characteristic

178

C₂²

#14579191

yes

characteristic

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

316

—

183

C₄

#14579138

yes

—

184

C₄

#14579147

yes

—

185

C₂

—

186

C₂

#19098475

yes

characteristic

187

C₂

#19098696

yes

characteristic

188

C₂

#19098701

yes

characteristic

189

C₂

#19098843

yes

characteristic

190

C₂

#19098853

yes

characteristic

191

C₂

#19098909

yes

characteristic

192

C₂

#19098922

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₉
commutator, cosocle	C₂²×C₄×C₇₉	C₂³
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	#390×C₇₉	C₂
socle	C₂³×C₇₉	C₂×D₄
prime radical	C₂²×D₄×C₇₉	C₂²
hypercenter	#390	D₇₉
hypocenter	C₇₉	#2023
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₉

commutator, cosocle

C₂²×C₄×C₇₉

C₂³

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

#390×C₇₉

C₂

socle

C₂³×C₇₉

C₂×D₄

prime radical

C₂²×D₄×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

N	H ≅ G/N	type
1. C₂×#14579161	C₂	semidirect
3. #19098325	C₂	semidirect
4. #19098318	C₂	semidirect
31. C₂×D₄×C₇₉	Q₈	semidirect
97. C₇₉	#2023	semidirect
121. C₂×D₄	Dic₁₅₈	semidirect

H ≅ G/N

type

1. C₂×#14579161

C₂

semidirect

3. #19098325

C₂

semidirect

4. #19098318

C₂

semidirect

31. C₂×D₄×C₇₉

Q₈

semidirect

97. C₇₉

#2023

semidirect

121. C₂×D₄

Dic₁₅₈

semidirect

Subgroups — #23764536

Conjugacy classes of subgroups

Complements