Group #23774841 — subgroups

Subgroups — #23774841

Order 10112.

346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 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	#19099168	C₂	1	1	yes	maximal, characteristic
2	#19098870	C₂	1	2	yes	maximal, complemented
3	#19098860	C₂	1	2	yes	maximal, complemented
4	#19098811	C₂	1	2	yes	maximal, complemented
5	#19098794	C₂	1	2	yes	maximal, complemented
6	#19098765	C₂	1	2	yes	maximal, complemented
7	#19098763	C₂	1	2	yes	maximal, complemented
8	#19098590	C₂	1	1	yes	maximal, characteristic, complemented
9	#541×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11929611	C₂²	1	2	yes	—
11	C₂×#11929611	C₂²	1	2	yes	—
12	C₂×#11929608	C₂²	1	1	yes	characteristic
13	C₂²×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
14	#14579158	C₂²	1	1	yes	characteristic
15	#14579153	C₂²	1	1	yes	characteristic
16	#14579146	—	2	2	no	—
17	#14579135	C₂²	1	2	yes	—
18	#14579134	—	2	2	no	—
19	C₄×D₃₁₆	C₂²	1	2	yes	—
20	#14579094	C₂²	1	2	yes	—
21	#14579092	C₂²	1	2	yes	—
22	C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
23	C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
24	#14579085	C₂²	1	2	yes	—
25	#14579085	C₂²	1	2	yes	—
26	#14579081	C₂²	1	2	yes	—
27	#14579081	C₂²	1	2	yes	—
28	C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
29	#126×C₇₉	—	2	2	no	—
30	#125×C₇₉	—	2	2	no	—
31	#125×C₇₉	C₂²	1	1	yes	characteristic
32	C₂²⋊Q₈×C₇₉	C₂²	1	2	yes	—
33	C₄×Q₈×C₇₉	C₂²	1	2	yes	—
34	#117×C₇₉	C₂²	1	1	yes	characteristic
35	C₂×D₃₁₆	—	2	2	no	—
36	C₂×D₃₁₆	C₂³	1	2	yes	—
37	C₂×C₂²⋊D₇₉	—	2	2	no	—
38	C₂×C₄×D₇₉	—	2	2	no	—
39	#11929611	—	4	2	no	—
40	#11929611	—	4	2	no	—
41	#11929608	—	4	1	no	—
42	C₂×C₇₉⋊C₈	—	4	1	no	—
43	C₂×C₇₉⋊C₈	C₂³	1	2	yes	—
44	C₂×C₇₉⋊C₈	C₂³	1	2	yes	—
45	C₄⋊Dic₇₉	C₂³	1	2	yes	—
46	Dic₇₉⋊C₄	—	2	2	no	—
47	(C₂×D₇₉)⋊C₄	—	2	2	no	—
48	(C₂×D₇₉)⋊C₄	—	2	2	no	—
49	C₂×Q₈×C₇₉	C₂³	1	2	yes	—
50	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
51	C₄⋊C₄×C₇₉	—	2	2	no	—
52	C₄⋊C₄×C₇₉	—	2	2	no	—
53	C₄⋊C₄×C₇₉	—	2	2	no	—
54	C₄⋊C₄×C₇₉	—	2	2	no	—
55	C₄⋊C₄×C₇₉	D₄	1	2	yes	—
56	C₄⋊C₄×C₇₉	C₂³	1	2	yes	—
57	C₂²⋊C₄×C₇₉	—	2	2	no	—
58	C₂²⋊C₄×C₇₉	D₄	1	2	yes	—
59	C₄²×C₇₉	—	2	2	no	—
60	C₄²×C₇₉	C₂³	1	1	yes	characteristic
61	C₄²×C₇₉	C₂³	1	1	yes	characteristic
62	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×D₇₉	—	2	2	no	—
64	Q₈×C₇₉	—	2	2	no	—
65	Q₈×C₇₉	—	2	2	no	—
66	C₂×C₄×C₇₉	—	2	1	no	—
67	C₂×C₄×C₇₉	—	2	1	no	—
68	C₂×C₄×C₇₉	—	2	2	no	—
69	C₂×C₄×C₇₉	—	2	2	no	—
70	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
72	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
73	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
74	C₂²⋊D₇₉	—	8	2	no	—
75	C₂×Dic₇₉	—	2	2	no	—
76	D₃₁₆	—	2	2	no	—
77	D₃₁₆	—	2	2	no	—
78	D₃₁₆	—	8	2	no	—
79	C₄×D₇₉	—	8	2	no	—
80	C₇₉⋊C₈	—	2	2	no	—
81	C₇₉⋊C₈	—	2	2	no	—
82	C₂²×C₇₉	—	2	1	no	—
83	C₂²×C₇₉	—	2	2	no	—
84	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
85	C₂×D₇₉	—	4	2	no	—
86	C₂×D₇₉	—	4	2	no	—
87	C₂×D₇₉	—	4	2	no	—
88	C₄×C₇₉	—	2	1	no	—
89	C₄×C₇₉	—	2	2	no	—
90	C₄×C₇₉	—	2	2	no	—
91	C₄×C₇₉	—	4	2	no	—
92	C₄×C₇₉	—	4	2	no	—
93	C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
94	C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
95	Dic₇₉	—	4	2	no	—
96	C₂×C₇₉	—	4	1	no	—
97	C₂×C₇₉	#537	1	1	yes	characteristic
98	C₂×C₇₉	#577	1	2	yes	—
99	D₇₉	—	8	2	no	—
100	#3260	—	79	1	no	maximal, Sylow
101	C₇₉	#3260	1	1	yes	characteristic, complemented, Sylow
102	#541	D₇₉	1	1	yes	characteristic, complemented
103	#540	—	79	1	no	—
104	#479	—	79	2	no	—
105	#478	—	79	2	no	—
106	#468	—	79	2	no	—
107	#465	—	79	2	no	—
108	#461	—	79	2	no	—
109	#460	—	79	2	no	—
110	#424	—	79	1	no	—
111	C₂×SD₁₆	—	79	2	no	—
112	C₂×SD₁₆	—	79	2	no	—
113	C₂×M₁₆	—	79	1	no	—
114	C₂²×C₈	—	79	1	no	—
115	C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
116	#127	—	79	1	no	—
117	#126	—	2	2	no	—
118	#125	—	2	2	no	—
119	#125	—	79	1	no	—
120	#125	C₂×D₇₉	1	1	yes	characteristic
121	#123	—	158	2	no	—
122	C₂²⋊Q₈	C₂×D₇₉	1	2	yes	—
123	#121	—	79	2	no	—
124	#121	—	158	2	no	—
125	C₄×Q₈	C₂×D₇₉	1	2	yes	—
126	C₄×D₄	—	79	2	no	—
127	#117	C₂×D₇₉	1	1	yes	characteristic
128	#107	—	79	2	no	—
129	#106	—	79	2	no	—
130	C₄⋊C₈	—	79	2	no	—
131	C₄⋊C₈	—	79	2	no	—
132	#103	—	79	2	no	—
133	#103	—	79	2	no	—
134	#102	—	79	2	no	—
135	#102	—	79	2	no	—
136	C₂×Q₈	C₂²×D₇₉	1	2	yes	—
137	C₂×D₄	—	79	2	no	—
138	C₂×D₄	—	158	2	no	—
139	C₂×D₄	—	158	2	no	—
140	C₂²×C₄	—	158	2	no	—
141	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
142	SD₁₆	—	316	2	no	—
143	SD₁₆	—	316	2	no	—
144	M₁₆	—	316	1	no	—
145	C₂×C₈	—	316	1	no	—
146	C₂×C₈	—	79	2	no	—
147	C₂×C₈	—	79	2	no	—
148	C₄⋊C₄	—	2	2	no	—
149	C₄⋊C₄	—	2	2	no	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	79	2	no	—
153	C₄⋊C₄	—	158	2	no	—
154	C₄⋊C₄	C₂²⋊D₇₉	1	2	yes	—
155	C₄⋊C₄	C₂²×D₇₉	1	2	yes	—
156	C₂²⋊C₄	—	2	2	no	—
157	C₂²⋊C₄	—	158	2	no	—
158	C₂²⋊C₄	—	158	2	no	—
159	C₂²⋊C₄	C₂²⋊D₇₉	1	2	yes	—
160	C₄²	—	2	2	no	—
161	C₄²	C₂²×D₇₉	1	1	yes	characteristic
162	C₄²	C₂²×D₇₉	1	1	yes	characteristic
163	C₂³	—	158	2	no	—
164	C₂³	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
165	Q₈	—	2	2	no	—
166	Q₈	—	2	2	no	—
167	D₄	—	632	2	no	—
168	D₄	—	632	2	no	—
169	D₄	—	158	2	no	—
170	D₄	—	158	2	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	632	2	no	—
176	C₂×C₄	—	158	2	no	—
177	C₂×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
178	C₂×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
179	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
180	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
181	C₈	—	158	2	no	—
182	C₈	—	158	2	no	—
183	C₂²	—	2	1	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	316	2	no	—
186	C₂²	—	316	2	no	—
187	C₂²	—	316	2	no	—
188	C₂²	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
189	C₄	—	2	1	no	—
190	C₄	—	2	2	no	—
191	C₄	—	2	2	no	—
192	C₄	—	4	2	no	—
193	C₄	—	4	2	no	—
194	C₄	—	316	2	no	—
195	C₄	#14579207	1	1	yes	characteristic
196	C₄	#14579208	1	1	yes	characteristic
197	C₂	—	4	1	no	—
198	C₂	—	632	2	no	—
199	C₂	#19099141	1	1	yes	characteristic
200	C₂	#19099371	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19099168

C₂

yes

maximal, characteristic

#19098870

C₂

yes

maximal, complemented

#19098860

C₂

yes

maximal, complemented

#19098811

C₂

yes

maximal, complemented

#19098794

C₂

yes

maximal, complemented

#19098765

C₂

yes

maximal, complemented

#19098763

C₂

yes

maximal, complemented

#19098590

C₂

yes

maximal, characteristic, complemented

#541×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#11929611

C₂²

yes

—

C₂×#11929611

C₂²

yes

—

C₂×#11929608

C₂²

yes

characteristic

C₂²×C₇₉⋊C₈

C₂²

yes

characteristic

#14579158

C₂²

yes

characteristic

#14579153

C₂²

yes

characteristic

#14579146

—

#14579135

C₂²

yes

—

#14579134

—

C₄×D₃₁₆

C₂²

yes

—

#14579094

C₂²

yes

—

#14579092

C₂²

yes

—

C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

#14579085

C₂²

yes

—

#14579085

C₂²

yes

—

#14579081

C₂²

yes

—

#14579081

C₂²

yes

—

C₄⋊Q₈×C₇₉

C₂²

yes

characteristic

#126×C₇₉

—

#125×C₇₉

—

#125×C₇₉

C₂²

yes

characteristic

C₂²⋊Q₈×C₇₉

C₂²

yes

—

C₄×Q₈×C₇₉

C₂²

yes

—

#117×C₇₉

C₂²

yes

characteristic

C₂×D₃₁₆

—

C₂×D₃₁₆

C₂³

yes

—

C₂×C₂²⋊D₇₉

—

C₂×C₄×D₇₉

—

#11929611

—

#11929611

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₂³

yes

—

C₂×C₇₉⋊C₈

C₂³

yes

—

C₄⋊Dic₇₉

C₂³

yes

—

Dic₇₉⋊C₄

—

(C₂×D₇₉)⋊C₄

—

(C₂×D₇₉)⋊C₄

—

C₂×Q₈×C₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

D₄

yes

—

C₄⋊C₄×C₇₉

C₂³

yes

—

C₂²⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

D₄

yes

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂³

yes

characteristic

C₄²×C₇₉

C₂³

yes

characteristic

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂²×D₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂²⋊D₇₉

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

C₄×D₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

H(5,𝔽₂)

yes

characteristic

C₄×C₇₉

2₋¹⁺⁴

yes

characteristic

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#537

yes

characteristic

C₂×C₇₉

#577

yes

—

D₇₉

—

100

#3260

—

maximal, Sylow

101

C₇₉

#3260

yes

characteristic, complemented, Sylow

102

#541

D₇₉

yes

characteristic, complemented

103

#540

—

104

#479

—

105

#478

—

106

#468

—

107

#465

—

108

#461

—

109

#460

—

110

#424

—

111

C₂×SD₁₆

—

112

C₂×SD₁₆

—

113

C₂×M₁₆

—

114

C₂²×C₈

—

115

C₄⋊Q₈

C₂×D₇₉

yes

characteristic

116

#127

—

117

#126

—

118

#125

—

119

#125

—

120

#125

C₂×D₇₉

yes

characteristic

121

#123

—

158

—

122

C₂²⋊Q₈

C₂×D₇₉

yes

—

123

#121

—

124

#121

—

158

—

125

C₄×Q₈

C₂×D₇₉

yes

—

126

C₄×D₄

—

127

#117

C₂×D₇₉

yes

characteristic

128

#107

—

129

#106

—

130

C₄⋊C₈

—

131

C₄⋊C₈

—

132

#103

—

133

#103

—

134

#102

—

135

#102

—

136

C₂×Q₈

C₂²×D₇₉

yes

—

137

C₂×D₄

—

138

C₂×D₄

—

158

—

139

C₂×D₄

—

158

—

140

C₂²×C₄

—

158

—

141

C₂²×C₄

C₂²×D₇₉

yes

characteristic

142

SD₁₆

—

316

—

143

SD₁₆

—

316

—

144

M₁₆

—

316

—

145

C₂×C₈

—

316

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

158

—

154

C₄⋊C₄

C₂²⋊D₇₉

yes

—

155

C₄⋊C₄

C₂²×D₇₉

yes

—

156

C₂²⋊C₄

—

157

C₂²⋊C₄

—

158

—

158

C₂²⋊C₄

—

158

—

159

C₂²⋊C₄

C₂²⋊D₇₉

yes

—

160

C₄²

—

161

C₄²

C₂²×D₇₉

yes

characteristic

162

C₄²

C₂²×D₇₉

yes

characteristic

163

C₂³

—

158

—

164

C₂³

C₂×C₂²⋊D₇₉

yes

characteristic

165

Q₈

—

166

Q₈

—

167

D₄

—

632

—

168

D₄

—

632

—

169

D₄

—

158

—

170

D₄

—

158

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

632

—

176

C₂×C₄

—

158

—

177

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

178

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

179

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

180

C₂×C₄

C₂³×D₇₉

yes

characteristic

181

C₈

—

158

—

182

C₈

—

158

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

316

—

186

C₂²

—

316

—

187

C₂²

—

316

—

188

C₂²

C₂²×C₂²⋊D₇₉

yes

characteristic

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

—

316

—

195

C₄

#14579207

yes

characteristic

196

C₄

#14579208

yes

characteristic

197

C₂

—

198

C₂

—

632

—

199

C₂

#19099141

yes

characteristic

200

C₂

#19099371

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₇₉
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	#541×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₄
prime radical	#19099168	C₂
hypercenter	#541	D₇₉
hypocenter	C₇₉	#3260
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #541

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₇₉

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

#541×C₇₉

C₂

socle

C₂²×C₇₉

C₂²×D₄

C₂

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ #117 ◁ #541

N	H ≅ G/N	type
2. #19098870	C₂	semidirect
3. #19098860	C₂	semidirect
4. #19098811	C₂	semidirect
5. #19098794	C₂	semidirect
6. #19098765	C₂	semidirect
7. #19098763	C₂	semidirect
8. #19098590	C₂	semidirect
9. #541×C₇₉	C₂	semidirect
101. C₇₉	#3260	semidirect
102. #541	D₇₉	semidirect

H ≅ G/N

type

2. #19098870

C₂

semidirect

3. #19098860

C₂

semidirect

4. #19098811

C₂

semidirect

5. #19098794

C₂

semidirect

6. #19098765

C₂

semidirect

7. #19098763

C₂

semidirect

8. #19098590

C₂

semidirect

9. #541×C₇₉

C₂

semidirect

101. C₇₉

#3260

semidirect

102. #541

D₇₉

semidirect

Subgroups — #23774841

Conjugacy classes of subgroups

Complements