Group #23768610 — subgroups

Subgroups — #23768610

Order 10112.

500 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 185 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×#14579160	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×Dic₇₉	C₂	1	1	yes	maximal, characteristic
3	#19098484	C₂	1	4	yes	maximal
4	#19098470	C₂	1	1	yes	maximal, characteristic
5	#19098460	C₂	1	4	yes	maximal
6	#19098433	C₂	1	1	yes	maximal, characteristic
7	#19098432	C₂	1	1	yes	maximal, characteristic
8	C₄×Dic₇₉⋊C₄	C₂	1	1	yes	maximal, characteristic
9	#391×C₇₉	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₁₅₈	C₂²	1	2	yes	—
11	#14579160	—	2	4	no	—
12	Q₈×Dic₇₉	—	2	4	no	—
13	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₇₉	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₇₉	C₂²	1	4	yes	—
17	C₂×Dic₇₉⋊C₄	C₂²	1	2	yes	—
18	C₂×Dic₇₉⋊C₄	C₂²	1	4	yes	—
19	C₂×Dic₇₉⋊C₄	C₂²	1	2	yes	—
20	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₇₉	C₂²	1	2	yes	—
22	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
23	C₇₉⋊#95	C₂²	1	2	yes	—
24	C₇₉⋊#95	C₂²	1	4	yes	—
25	C₂²×Q₈×C₇₉	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
28	#95×C₇₉	C₂²	1	4	yes	—
29	C₂×Dic₁₅₈	—	2	4	no	—
30	C₂×Dic₁₅₈	—	4	4	no	—
31	C₂²×Dic₇₉	C₂³	1	2	yes	—
32	C₂²×Dic₇₉	C₂³	1	2	yes	—
33	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
34	C₂²×Dic₇₉	C₂³	1	2	yes	—
35	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
36	C₄⋊Dic₇₉	—	2	4	no	—
37	C₄⋊Dic₇₉	—	2	2	no	—
38	C₄⋊Dic₇₉	—	2	2	no	—
39	C₄⋊Dic₇₉	—	4	4	no	—
40	Dic₇₉⋊C₄	—	2	4	no	—
41	Dic₇₉⋊C₄	—	4	4	no	—
42	Dic₇₉⋊C₄	Q₈	1	8	yes	complemented
43	C₄×Dic₇₉	—	2	4	no	—
44	C₄×Dic₇₉	—	2	2	no	—
45	C₄×Dic₇₉	D₄	1	4	yes	—
46	C₂×Q₈×C₇₉	—	2	2	no	—
47	C₂×Q₈×C₇₉	—	2	2	no	—
48	C₂×Q₈×C₇₉	Q₈	1	4	yes	complemented
49	C₂²×C₄×C₇₉	C₂³	1	2	yes	—
50	C₂²×C₄×C₇₉	C₂³	1	2	yes	—
51	C₂²×C₄×C₇₉	C₂³	1	2	yes	—
52	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
53	C₄⋊C₄×C₇₉	—	2	2	no	—
54	C₄²×C₇₉	—	2	2	no	—
55	C₂³×C₇₉	C₂⁴	1	1	yes	characteristic
56	Q₈×C₇₉	—	2	8	no	—
57	C₂×C₄×C₇₉	—	2	2	no	—
58	C₂×C₄×C₇₉	—	2	2	no	—
59	C₂×C₄×C₇₉	—	2	2	no	—
60	C₂×C₄×C₇₉	—	2	2	no	—
61	C₂×C₄×C₇₉	—	2	2	no	—
62	C₂×C₄×C₇₉	—	2	4	no	—
63	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
64	C₂×C₄×C₇₉	C₂×Q₈	1	4	yes	—
65	C₂×C₄×C₇₉	C₂×Q₈	1	4	yes	—
66	C₂×C₄×C₇₉	C₂×Q₈	1	2	yes	—
67	C₂×C₄×C₇₉	#41	1	2	yes	—
68	C₂×Dic₇₉	—	2	2	no	—
69	C₂×Dic₇₉	—	2	2	no	—
70	C₂×Dic₇₉	—	2	2	no	—
71	C₂×Dic₇₉	—	2	2	no	—
72	C₂×Dic₇₉	—	2	2	no	—
73	C₂×Dic₇₉	—	2	4	no	—
74	C₂×Dic₇₉	—	2	4	no	—
75	C₂×Dic₇₉	C₂×D₄	1	2	yes	—
76	C₂×Dic₇₉	C₂×D₄	1	2	yes	—
77	C₂×Dic₇₉	C₂×Q₈	1	4	yes	—
78	C₂×Dic₇₉	C₂×Q₈	1	4	yes	—
79	Dic₁₅₈	—	4	8	no	—
80	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
81	C₂²×C₇₉	C₂²×Q₈	1	2	yes	—
82	C₂²×C₇₉	C₂²×Q₈	1	1	yes	characteristic
83	C₂²×C₇₉	C₂×#41	1	1	yes	characteristic
84	C₂²×C₇₉	H(5,𝔽₂)	1	2	yes	—
85	C₄×C₇₉	—	2	4	no	—
86	C₄×C₇₉	—	2	4	no	—
87	C₄×C₇₉	—	4	2	no	—
88	C₄×C₇₉	C₂²⋊Q₈	1	4	yes	—
89	Dic₇₉	—	2	2	no	—
90	Dic₇₉	—	2	4	no	—
91	Dic₇₉	—	4	2	no	—
92	Dic₇₉	—	4	2	no	—
93	Dic₇₉	C₄⋊Q₈	1	4	yes	—
94	C₂×C₇₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
96	C₂×C₇₉	#535	1	1	yes	characteristic
97	C₂×C₇₉	#554	1	2	yes	—
98	C₂×C₇₉	Q₈²	1	2	yes	—
99	#2567	—	79	1	no	maximal, Sylow
100	C₇₉	#2567	1	1	yes	characteristic, complemented, Sylow
101	C₂×C₄⋊Q₈	—	79	1	no	—
102	C₂×C₄×Q₈	—	79	1	no	—
103	#395	—	79	4	no	—
104	#391	—	79	1	no	—
105	#391	D₇₉	1	1	yes	characteristic
106	#389	—	79	4	no	—
107	#384	—	79	1	no	—
108	#384	—	79	1	no	—
109	C₄×C₄⋊C₄	—	79	1	no	—
110	C₂²×Q₈	—	79	2	no	—
111	C₂²×Q₈	C₂×D₇₉	1	1	yes	characteristic
112	C₄⋊Q₈	—	158	4	no	—
113	C₄×Q₈	—	158	4	no	—
114	C₂×C₄⋊C₄	—	79	4	no	—
115	C₂×C₄⋊C₄	—	79	4	no	—
116	C₂×C₄⋊C₄	—	79	2	no	—
117	C₂×C₄⋊C₄	—	79	1	no	—
118	C₂×C₄⋊C₄	—	79	2	no	—
119	C₂×C₄⋊C₄	—	79	1	no	—
120	C₂×C₄⋊C₄	—	79	2	no	—
121	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
122	C₂×C₄²	—	79	1	no	—
123	C₂×C₄²	—	79	2	no	—
124	C₂×C₄²	—	79	1	no	—
125	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
126	#95	—	79	2	no	—
127	#95	—	79	4	no	—
128	#95	C₂×D₇₉	1	4	yes	—
129	C₂×Q₈	—	2	2	no	—
130	C₂×Q₈	—	2	2	no	—
131	C₂×Q₈	—	316	4	no	—
132	C₂×Q₈	—	158	4	no	—
133	C₂×Q₈	Dic₁₅₈	1	4	yes	complemented
134	C₂²×C₄	—	79	1	no	—
135	C₂²×C₄	—	79	2	no	—
136	C₂²×C₄	—	79	1	no	—
137	C₂²×C₄	—	79	2	no	—
138	C₂²×C₄	—	79	2	no	—
139	C₂²×C₄	C₂²×D₇₉	1	2	yes	—
140	C₂²×C₄	C₂²×D₇₉	1	2	yes	—
141	C₂²×C₄	C₂²×D₇₉	1	2	yes	—
142	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
143	C₄⋊C₄	—	2	2	no	—
144	C₄⋊C₄	—	316	4	no	—
145	C₄⋊C₄	—	316	4	no	—
146	C₄⋊C₄	—	79	8	no	—
147	C₄⋊C₄	—	158	4	no	—
148	C₄⋊C₄	—	158	4	no	—
149	C₄⋊C₄	—	158	2	no	—
150	C₄⋊C₄	—	158	2	no	—
151	C₄²	—	2	2	no	—
152	C₄²	—	79	4	no	—
153	C₄²	—	158	4	no	—
154	C₄²	—	158	2	no	—
155	C₂³	C₂³×D₇₉	1	1	yes	characteristic
156	Q₈	—	2	8	no	—
157	Q₈	—	316	8	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	79	4	no	—
165	C₂×C₄	—	79	2	no	—
166	C₂×C₄	—	79	4	no	—
167	C₂×C₄	—	79	2	no	—
168	C₂×C₄	—	158	4	no	—
169	C₂×C₄	—	158	4	no	—
170	C₂×C₄	—	158	2	no	—
171	C₂×C₄	—	158	2	no	—
172	C₂×C₄	—	158	2	no	—
173	C₂×C₄	—	158	2	no	—
174	C₂×C₄	—	158	2	no	—
175	C₂×C₄	D₄×D₇₉	1	2	yes	—
176	C₂×C₄	Q₈×D₇₉	1	4	yes	—
177	C₂×C₄	C₂×Dic₁₅₈	1	4	yes	—
178	C₂×C₄	C₂×Dic₁₅₈	1	2	yes	—
179	C₂×C₄	#11929623	1	2	yes	—
180	C₂²	C₂×D₄×D₇₉	1	1	yes	characteristic
181	C₂²	C₂×Q₈×D₇₉	1	2	yes	—
182	C₂²	C₂²×Dic₁₅₈	1	1	yes	characteristic
183	C₂²	C₂×#11929623	1	1	yes	characteristic
184	C₂²	#14579207	1	2	yes	—
185	C₄	—	2	4	no	—
186	C₄	—	2	4	no	—
187	C₄	—	4	2	no	—
188	C₄	—	316	2	no	—
189	C₄	—	316	2	no	—
190	C₄	—	79	4	no	—
191	C₄	—	158	4	no	—
192	C₄	—	158	2	no	—
193	C₄	#14579138	1	4	yes	—
194	C₂	C₂×#14579138	1	1	yes	characteristic
195	C₂	C₄⋊Q₈×D₇₉	1	1	yes	characteristic
196	C₂	#19099131	1	1	yes	characteristic
197	C₂	#19099271	1	2	yes	—
198	C₂	Q₈×Dic₁₅₈	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579160

C₂

yes

maximal, characteristic

C₂×Q₈×Dic₇₉

C₂

yes

maximal, characteristic

#19098484

C₂

yes

maximal

#19098470

C₂

yes

maximal, characteristic

#19098460

C₂

yes

maximal

#19098433

C₂

yes

maximal, characteristic

#19098432

C₂

yes

maximal, characteristic

C₄×Dic₇₉⋊C₄

C₂

yes

maximal, characteristic

#391×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

C₂²

yes

—

#14579160

—

Q₈×Dic₇₉

—

C₂×C₄⋊Dic₇₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₂×Dic₇₉⋊C₄

C₂²

yes

—

C₂×Dic₇₉⋊C₄

C₂²

yes

—

C₂×Dic₇₉⋊C₄

C₂²

yes

—

C₂×C₄×Dic₇₉

C₂²

yes

characteristic

C₂×C₄×Dic₇₉

C₂²

yes

—

C₂×C₄×Dic₇₉

C₂²

yes

characteristic

C₇₉⋊#95

C₂²

yes

—

C₇₉⋊#95

C₂²

yes

—

C₂²×Q₈×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₉

C₂²

yes

characteristic

C₂×C₄²×C₇₉

C₂²

yes

characteristic

#95×C₇₉

C₂²

yes

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂²×Dic₇₉

C₂³

yes

—

C₂²×Dic₇₉

C₂³

yes

—

C₂²×Dic₇₉

C₂³

yes

characteristic

C₂²×Dic₇₉

C₂³

yes

—

C₂²×Dic₇₉

C₂³

yes

characteristic

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

Q₈

yes

complemented

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

D₄

yes

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

Q₈

yes

complemented

C₂²×C₄×C₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₂³×C₇₉

C₂⁴

yes

characteristic

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×Q₈

yes

—

C₂×C₄×C₇₉

C₂×Q₈

yes

—

C₂×C₄×C₇₉

C₂×Q₈

yes

—

C₂×C₄×C₇₉

#41

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

C₂×D₄

yes

—

C₂×Dic₇₉

C₂×D₄

yes

—

C₂×Dic₇₉

C₂×Q₈

yes

—

C₂×Dic₇₉

C₂×Q₈

yes

—

Dic₁₅₈

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₂²×C₇₉

C₂²×Q₈

yes

—

C₂²×C₇₉

C₂²×Q₈

yes

characteristic

C₂²×C₇₉

C₂×#41

yes

characteristic

C₂²×C₇₉

H(5,𝔽₂)

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂²⋊Q₈

yes

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

C₄⋊Q₈

yes

—

C₂×C₇₉

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₇₉

C₂×C₄⋊Q₈

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₂×C₇₉

Q₈²

yes

—

#2567

—

maximal, Sylow

100

C₇₉

#2567

yes

characteristic, complemented, Sylow

101

C₂×C₄⋊Q₈

—

102

C₂×C₄×Q₈

—

103

#395

—

104

#391

—

105

#391

D₇₉

yes

characteristic

106

#389

—

107

#384

—

108

#384

—

109

C₄×C₄⋊C₄

—

110

C₂²×Q₈

—

111

C₂²×Q₈

C₂×D₇₉

yes

characteristic

112

C₄⋊Q₈

—

158

—

113

C₄×Q₈

—

158

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

C₂×D₇₉

yes

characteristic

122

C₂×C₄²

—

123

C₂×C₄²

—

124

C₂×C₄²

—

125

C₂×C₄²

C₂×D₇₉

yes

characteristic

126

#95

—

127

#95

—

128

#95

C₂×D₇₉

yes

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

316

—

132

C₂×Q₈

—

158

—

133

C₂×Q₈

Dic₁₅₈

yes

complemented

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

C₂²×D₇₉

yes

—

140

C₂²×C₄

C₂²×D₇₉

yes

—

141

C₂²×C₄

C₂²×D₇₉

yes

—

142

C₂²×C₄

C₂²×D₇₉

yes

characteristic

143

C₄⋊C₄

—

144

C₄⋊C₄

—

316

—

145

C₄⋊C₄

—

316

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

158

—

148

C₄⋊C₄

—

158

—

149

C₄⋊C₄

—

158

—

150

C₄⋊C₄

—

158

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

158

—

154

C₄²

—

158

—

155

C₂³

C₂³×D₇₉

yes

characteristic

156

Q₈

—

157

Q₈

—

316

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

158

—

169

C₂×C₄

—

158

—

170

C₂×C₄

—

158

—

171

C₂×C₄

—

158

—

172

C₂×C₄

—

158

—

173

C₂×C₄

—

158

—

174

C₂×C₄

—

158

—

175

C₂×C₄

D₄×D₇₉

yes

—

176

C₂×C₄

Q₈×D₇₉

yes

—

177

C₂×C₄

C₂×Dic₁₅₈

yes

—

178

C₂×C₄

C₂×Dic₁₅₈

yes

—

179

C₂×C₄

#11929623

yes

—

180

C₂²

C₂×D₄×D₇₉

yes

characteristic

181

C₂²

C₂×Q₈×D₇₉

yes

—

182

C₂²

C₂²×Dic₁₅₈

yes

characteristic

183

C₂²

C₂×#11929623

yes

characteristic

184

C₂²

#14579207

yes

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

316

—

189

C₄

—

316

—

190

C₄

—

191

C₄

—

158

—

192

C₄

—

158

—

193

C₄

#14579138

yes

—

194

C₂

C₂×#14579138

yes

characteristic

195

C₂

C₄⋊Q₈×D₇₉

yes

characteristic

196

C₂

#19099131

yes

characteristic

197

C₂

#19099271

yes

—

198

C₂

Q₈×Dic₁₅₈

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇₉
commutator, socle, prime radical, cosocle	C₂³×C₇₉	C₂⁴
Fitting	#391×C₇₉	C₂
hypercenter	#391	D₇₉
hypocenter	C₇₉	#2567
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ #391

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇₉

commutator, socle, prime radical, cosocle

C₂³×C₇₉

C₂⁴

Fitting

#391×C₇₉

C₂

D₇₉

C₇₉

G ▷ C₂³×C₇₉ ▷ C₇₉

C₁ ◁ C₂³ ◁ #391

N	H ≅ G/N	type
42. Dic₇₉⋊C₄	Q₈	semidirect
48. C₂×Q₈×C₇₉	Q₈	semidirect
100. C₇₉	#2567	semidirect
133. C₂×Q₈	Dic₁₅₈	semidirect

H ≅ G/N

type

42. Dic₇₉⋊C₄

Q₈

semidirect

48. C₂×Q₈×C₇₉

Q₈

semidirect

100. C₇₉

#2567

semidirect

133. C₂×Q₈

Dic₁₅₈

semidirect

Subgroups — #23768610

Conjugacy classes of subgroups

Complements