Group #23774598 — subgroups

Subgroups — #23774598

Order 10112.

366 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 121 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	#19099113	C₂	1	1	yes	maximal, characteristic
2	#19098973	C₂	1	2	yes	maximal, complemented
3	#19098929	C₂	1	2	yes	maximal, complemented
4	#19098887	C₂	1	4	yes	maximal, complemented
5	#19098850	C₂	1	2	yes	maximal, complemented
6	#19098840	C₂	1	2	yes	maximal, complemented
7	#19098620	C₂	1	1	yes	maximal, characteristic, complemented
8	#533×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11929624	—	2	1	no	—
10	#14579161	C₂²	1	1	yes	characteristic
11	#14579158	C₂²	1	1	yes	characteristic
12	#14579149	—	2	1	no	—
13	#14579135	—	2	1	no	—
14	#14579135	C₂²	1	2	yes	—
15	C₄×D₃₁₆	—	2	1	no	—
16	C₄×D₃₁₆	C₂²	1	2	yes	—
17	#14579094	C₂²	1	4	yes	—
18	#14579092	C₂²	1	4	yes	—
19	C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
20	#14579081	C₂²	1	4	yes	—
21	#14579081	C₂²	1	4	yes	—
22	C₂²⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
23	#14579072	C₂²	1	2	yes	—
24	C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
25	C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
26	#125×C₇₉	C₂²	1	2	yes	—
27	C₂²⋊Q₈×C₇₉	—	2	1	no	—
28	C₂²⋊Q₈×C₇₉	—	2	1	no	—
29	C₄×Q₈×C₇₉	—	2	1	no	—
30	C₄×Q₈×C₇₉	—	2	1	no	—
31	#117×C₇₉	C₂²	1	2	yes	—
32	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
33	#11929624	—	4	4	no	—
34	C₂×Dic₁₅₈	—	2	1	no	—
35	C₂×D₃₁₆	—	2	1	no	—
36	C₂×D₃₁₆	—	2	1	no	—
37	C₂×D₃₁₆	C₂³	1	2	yes	—
38	C₂×C₂²⋊D₇₉	—	2	2	no	—
39	C₂×C₄×D₇₉	—	2	2	no	—
40	C₂×C₇₉⋊C₈	C₂³	1	4	yes	—
41	C₄⋊Dic₇₉	—	2	1	no	—
42	C₄⋊Dic₇₉	C₂³	1	2	yes	—
43	(C₂×D₇₉)⋊C₄	—	2	2	no	—
44	(C₂×D₇₉)⋊C₄	—	2	2	no	—
45	C₂×Q₈×C₇₉	—	2	1	no	—
46	C₂×Q₈×C₇₉	—	2	1	no	—
47	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
48	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
50	C₄⋊C₄×C₇₉	—	2	2	no	—
51	C₄⋊C₄×C₇₉	—	2	2	no	—
52	C₄⋊C₄×C₇₉	—	2	1	no	—
53	C₄⋊C₄×C₇₉	—	2	1	no	—
54	C₄⋊C₄×C₇₉	C₂³	1	2	yes	—
55	C₄⋊C₄×C₇₉	C₂³	1	2	yes	—
56	C₂²⋊C₄×C₇₉	—	2	2	no	—
57	C₄²×C₇₉	—	2	2	no	—
58	C₄²×C₇₉	D₄	1	1	yes	characteristic
59	C₄²×C₇₉	D₄	1	1	yes	characteristic
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	—	4	2	no	—
65	Q₈×C₇₉	—	4	2	no	—
66	C₂×C₄×C₇₉	—	2	2	no	—
67	C₂×C₄×C₇₉	—	2	2	no	—
68	C₂×C₄×C₇₉	—	2	2	no	—
69	C₂×C₄×C₇₉	—	2	2	no	—
70	C₂×C₄×C₇₉	—	4	1	no	—
71	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
77	C₂²⋊D₇₉	—	4	4	no	—
78	C₂×Dic₇₉	—	2	2	no	—
79	D₃₁₆	—	2	4	no	—
80	D₃₁₆	—	4	2	no	—
81	D₃₁₆	—	4	2	no	—
82	C₄×D₇₉	—	4	4	no	—
83	Dic₁₅₈	—	4	2	no	—
84	C₇₉⋊C₈	—	2	4	no	—
85	C₂²×C₇₉	—	2	1	no	—
86	C₂²×C₇₉	—	2	1	no	—
87	C₂²×C₇₉	—	2	1	no	—
88	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
89	C₂×D₇₉	—	4	2	no	—
90	C₂×D₇₉	—	4	2	no	—
91	C₂×D₇₉	—	4	2	no	—
92	C₄×C₇₉	—	2	1	no	—
93	C₄×C₇₉	—	2	1	no	—
94	C₄×C₇₉	—	2	1	no	—
95	C₄×C₇₉	—	2	1	no	—
96	C₄×C₇₉	—	4	2	no	—
97	C₄×C₇₉	—	4	2	no	—
98	C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
99	C₄×C₇₉	2₋¹⁺⁴	1	2	yes	—
100	Dic₇₉	—	4	2	no	—
101	C₂×C₇₉	—	4	1	no	—
102	C₂×C₇₉	#536	1	1	yes	characteristic
103	C₂×C₇₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
104	C₂×C₇₉	#575	1	1	yes	characteristic
105	D₇₉	—	8	2	no	—
106	#3235	—	79	1	no	maximal, Sylow
107	C₇₉	#3235	1	1	yes	characteristic, complemented, Sylow
108	#533	D₇₉	1	1	yes	characteristic, complemented
109	#532	—	79	1	no	—
110	#501	—	79	2	no	—
111	#490	—	79	2	no	—
112	#482	—	79	4	no	—
113	#476	—	79	2	no	—
114	#474	—	79	2	no	—
115	#432	—	79	1	no	—
116	C₂×#41	—	158	1	no	—
117	C₄⋊Q₈	—	79	1	no	—
118	C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
119	C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
120	#127	—	79	1	no	—
121	#125	C₂×D₇₉	1	2	yes	—
122	#124	—	158	1	no	—
123	C₂²⋊Q₈	—	2	1	no	—
124	C₂²⋊Q₈	—	2	1	no	—
125	#121	—	79	2	no	—
126	#121	—	158	1	no	—
127	C₄×Q₈	—	2	1	no	—
128	C₄×Q₈	—	2	1	no	—
129	C₄×D₄	—	79	2	no	—
130	C₄×D₄	—	158	1	no	—
131	#117	C₂×D₇₉	1	2	yes	—
132	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
133	#107	—	79	4	no	—
134	#106	—	79	4	no	—
135	C₄⋊C₈	—	79	2	no	—
136	#102	—	79	4	no	—
137	#102	—	79	4	no	—
138	C₂²⋊C₈	—	79	2	no	—
139	#97	—	79	2	no	—
140	#41	—	316	4	no	—
141	C₂×Q₈	—	2	1	no	—
142	C₂×Q₈	—	2	1	no	—
143	C₂×Q₈	—	158	1	no	—
144	C₂×D₄	—	79	2	no	—
145	C₂×D₄	—	158	2	no	—
146	C₂×D₄	—	158	1	no	—
147	C₂×D₄	—	158	1	no	—
148	C₂²×C₄	—	158	2	no	—
149	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
150	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
151	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
152	C₂×C₈	—	79	4	no	—
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	2	1	no	—
156	C₄⋊C₄	—	2	1	no	—
157	C₄⋊C₄	—	79	2	no	—
158	C₄⋊C₄	—	158	1	no	—
159	C₄⋊C₄	C₂²×D₇₉	1	2	yes	—
160	C₄⋊C₄	C₂²×D₇₉	1	2	yes	—
161	C₂²⋊C₄	—	2	2	no	—
162	C₂²⋊C₄	—	158	2	no	—
163	C₂²⋊C₄	—	158	2	no	—
164	C₄²	—	2	2	no	—
165	C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
166	C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
167	C₄²	C₂²×D₇₉	1	1	yes	characteristic
168	C₄²	C₂²×D₇₉	1	1	yes	characteristic
169	C₂³	—	158	2	no	—
170	C₂³	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
171	Q₈	—	4	2	no	—
172	Q₈	—	4	2	no	—
173	Q₈	—	316	2	no	—
174	D₄	—	316	2	no	—
175	D₄	—	316	2	no	—
176	D₄	—	316	4	no	—
177	D₄	—	158	4	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	4	1	no	—
183	C₂×C₄	—	316	4	no	—
184	C₂×C₄	—	158	2	no	—
185	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
186	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
187	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
188	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
189	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
190	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
191	C₈	—	158	4	no	—
192	C₂²	—	2	1	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	316	2	no	—
196	C₂²	—	316	2	no	—
197	C₂²	—	316	2	no	—
198	C₂²	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
199	C₄	—	2	1	no	—
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	4	2	no	—
204	C₄	—	4	2	no	—
205	C₄	—	316	2	no	—
206	C₄	#14579186	1	2	yes	—
207	C₄	#14579209	1	2	yes	—
208	C₂	—	4	1	no	—
209	C₂	—	632	2	no	—
210	C₂	#19099137	1	1	yes	characteristic
211	C₂	C₂×#14579186	1	1	yes	characteristic
212	C₂	#19099361	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19099113

C₂

yes

maximal, characteristic

#19098973

C₂

yes

maximal, complemented

#19098929

C₂

yes

maximal, complemented

#19098887

C₂

yes

maximal, complemented

#19098850

C₂

yes

maximal, complemented

#19098840

C₂

yes

maximal, complemented

#19098620

C₂

yes

maximal, characteristic, complemented

#533×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#11929624

—

#14579161

C₂²

yes

characteristic

#14579158

C₂²

yes

—

—

C₂²

yes

—

C₄×D₃₁₆

—

C₄×D₃₁₆

C₂²

yes

—

#14579094

C₂²

yes

—

#14579092

C₂²

yes

—

C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

#14579081

C₂²

yes

—

#14579081

C₂²

yes

—

C₂²⋊(C₇₉⋊C₈)

C₂²

yes

—

#14579072

C₂²

yes

—

C₄⋊Q₈×C₇₉

C₂²

yes

characteristic

C₄⋊Q₈×C₇₉

C₂²

yes

characteristic

#125×C₇₉

C₂²

yes

—

C₂²⋊Q₈×C₇₉

—

C₂²⋊Q₈×C₇₉

—

C₄×Q₈×C₇₉

—

C₄×Q₈×C₇₉

—

#117×C₇₉

C₂²

yes

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

#11929624

—

C₂×Dic₁₅₈

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

C₂³

yes

—

C₂×C₂²⋊D₇₉

—

C₂×C₄×D₇₉

—

C₂×C₇₉⋊C₈

C₂³

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

C₂³

yes

—

(C₂×D₇₉)⋊C₄

—

(C₂×D₇₉)⋊C₄

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

C₂³

yes

—

C₄⋊C₄×C₇₉

C₂³

yes

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

D₄

yes

characteristic

C₄²×C₇₉

D₄

yes

characteristic

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×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂²⋊D₇₉

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

C₄×D₇₉

—

Dic₁₅₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

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

—

C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₄×C₇₉

2₋¹⁺⁴

yes

—

100

Dic₇₉

—

101

C₂×C₇₉

—

102

C₂×C₇₉

#536

yes

characteristic

103

C₂×C₇₉

C₂×AGL(1,ℤ/8)

yes

characteristic

104

C₂×C₇₉

#575

yes

characteristic

105

D₇₉

—

106

#3235

—

maximal, Sylow

107

C₇₉

#3235

yes

characteristic, complemented, Sylow

108

#533

D₇₉

yes

characteristic, complemented

109

#532

—

110

#501

—

111

#490

—

112

#482

—

113

#476

—

114

#474

—

115

#432

—

116

C₂×#41

—

158

—

117

C₄⋊Q₈

—

118

C₄⋊Q₈

C₂×D₇₉

yes

characteristic

119

C₄⋊Q₈

C₂×D₇₉

yes

characteristic

120

#127

—

121

#125

C₂×D₇₉

yes

—

122

#124

—

158

—

123

C₂²⋊Q₈

—

124

C₂²⋊Q₈

—

125

#121

—

126

#121

—

158

—

127

C₄×Q₈

—

128

C₄×Q₈

—

129

C₄×D₄

—

130

C₄×D₄

—

158

—

131

#117

C₂×D₇₉

yes

—

132

C₂×C₄²

C₂×D₇₉

yes

characteristic

133

#107

—

134

#106

—

135

C₄⋊C₈

—

136

#102

—

137

#102

—

138

C₂²⋊C₈

—

139

#97

—

140

#41

—

316

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂×Q₈

—

158

—

144

C₂×D₄

—

145

C₂×D₄

—

158

—

146

C₂×D₄

—

158

—

147

C₂×D₄

—

158

—

148

C₂²×C₄

—

158

—

149

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

150

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

151

C₂²×C₄

C₂²×D₇₉

yes

characteristic

152

C₂×C₈

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

158

—

159

C₄⋊C₄

C₂²×D₇₉

yes

—

160

C₄⋊C₄

C₂²×D₇₉

yes

—

161

C₂²⋊C₄

—

162

C₂²⋊C₄

—

158

—

163

C₂²⋊C₄

—

158

—

164

C₄²

—

165

C₄²

C₂²⋊D₇₉

yes

characteristic

166

C₄²

C₂²⋊D₇₉

yes

characteristic

167

C₄²

C₂²×D₇₉

yes

characteristic

168

C₄²

C₂²×D₇₉

yes

characteristic

169

C₂³

—

158

—

170

C₂³

C₂×C₂²⋊D₇₉

yes

characteristic

171

Q₈

—

172

Q₈

—

173

Q₈

—

316

—

174

D₄

—

316

—

175

D₄

—

316

—

176

D₄

—

316

—

177

D₄

—

158

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

316

—

184

C₂×C₄

—

158

—

185

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

186

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

187

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

188

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

189

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

190

C₂×C₄

C₂³×D₇₉

yes

characteristic

191

C₈

—

158

—

192

C₂²

—

193

C₂²

—

194

C₂²

—

195

C₂²

—

316

—

196

C₂²

—

316

—

197

C₂²

—

316

—

198

C₂²

C₂²×C₂²⋊D₇₉

yes

characteristic

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

—

316

—

206

C₄

#14579186

yes

—

207

C₄

#14579209

yes

—

208

C₂

—

209

C₂

—

632

—

210

C₂

#19099137

yes

characteristic

211

C₂

C₂×#14579186

yes

characteristic

212

C₂

#19099361

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₇₉
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	#533×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₄
prime radical	#19099113	C₂
hypercenter	#533	D₇₉
hypocenter	C₇₉	#3235
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₇₉

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

#533×C₇₉

C₂

socle

C₂²×C₇₉

C₂²×D₄

C₂

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #19098973	C₂	semidirect
3. #19098929	C₂	semidirect
4. #19098887	C₂	semidirect
5. #19098850	C₂	semidirect
6. #19098840	C₂	semidirect
7. #19098620	C₂	semidirect
8. #533×C₇₉	C₂	semidirect
107. C₇₉	#3235	semidirect
108. #533	D₇₉	semidirect

H ≅ G/N

type

2. #19098973

C₂

semidirect

3. #19098929

C₂

semidirect

4. #19098887

C₂

semidirect

5. #19098850

C₂

semidirect

6. #19098840

C₂

semidirect

7. #19098620

C₂

semidirect

8. #533×C₇₉

C₂

semidirect

107. C₇₉

#3235

semidirect

108. #533

D₇₉

semidirect

Subgroups — #23774598

Conjugacy classes of subgroups

Complements