Group #23771752 — subgroups

Subgroups — #23771752

Order 10112.

644 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 237 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×#14579140	C₂	1	2	yes	maximal
3	#19098469	C₂	1	2	yes	maximal, complemented
4	#19098461	C₂	1	4	yes	maximal, complemented
5	#19098433	C₂	1	1	yes	maximal, characteristic, complemented
6	#19098423	C₂	1	4	yes	maximal
7	C₂²×C₄²×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₅₈	C₂²	1	2	yes	—
9	#14579161	—	2	4	no	—
10	#14579140	—	2	8	no	—
11	C₂×C₄⋊Dic₇₉	C₂²	1	4	yes	—
12	C₂×C₄⋊Dic₇₉	C₂²	1	4	yes	—
13	C₂×C₄⋊Dic₇₉	C₂²	1	2	yes	—
14	C₂×Dic₇₉⋊C₄	C₂²	1	4	yes	—
15	C₂×C₂²⋊Dic₇₉	C₂²	1	4	yes	—
16	C₇₉⋊#95	C₂²	1	8	yes	—
17	C₂³×C₄×C₇₉	C₂²	1	1	yes	characteristic
18	C₂³×C₄×C₇₉	C₂²	1	2	yes	—
19	C₂×C₄²×C₇₉	—	2	4	no	—
20	C₂×C₄²×C₇₉	C₂²	1	2	yes	—
21	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₅₈	—	2	4	no	—
24	C₂×Dic₁₅₈	—	2	4	no	—
25	C₂×Dic₁₅₈	—	2	4	no	—
26	C₂²×Dic₇₉	C₂³	1	4	yes	—
27	C₂²×Dic₇₉	C₂³	1	4	yes	—
28	C₄⋊Dic₇₉	—	2	4	no	—
29	C₄⋊Dic₇₉	—	2	8	no	—
30	C₄⋊Dic₇₉	—	2	8	no	—
31	Dic₇₉⋊C₄	—	2	8	no	—
32	C₂²⋊Dic₇₉	—	2	8	no	—
33	C₂⁴×C₇₉	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₇₉	—	2	2	no	—
35	C₂²×C₄×C₇₉	—	2	4	no	—
36	C₂²×C₄×C₇₉	—	2	2	no	—
37	C₂²×C₄×C₇₉	—	2	4	no	—
38	C₂²×C₄×C₇₉	D₄	1	4	yes	—
39	C₂²×C₄×C₇₉	Q₈	1	8	yes	complemented
40	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₇₉	C₂³	1	2	yes	—
42	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₇₉	C₂³	1	2	yes	—
44	C₄²×C₇₉	—	2	2	no	—
45	C₄²×C₇₉	—	2	4	no	—
46	C₄²×C₇₉	D₄	1	4	yes	—
47	C₂³×C₇₉	—	2	1	no	—
48	C₂³×C₇₉	—	2	1	no	—
49	C₂³×C₇₉	—	2	2	no	—
50	C₂³×C₇₉	C₂×D₄	1	2	yes	—
51	C₂³×C₇₉	C₂×Q₈	1	4	yes	—
52	C₂³×C₇₉	C₂⁴	1	1	yes	characteristic
53	C₂×C₄×C₇₉	—	2	1	no	—
54	C₂×C₄×C₇₉	—	2	2	no	—
55	C₂×C₄×C₇₉	—	2	1	no	—
56	C₂×C₄×C₇₉	—	2	2	no	—
57	C₂×C₄×C₇₉	—	2	8	no	—
58	C₂×C₄×C₇₉	—	2	16	no	—
59	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₇₉	C₂×D₄	1	4	yes	—
61	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₇₉	C₂×Q₈	1	4	yes	—
64	C₂×C₄×C₇₉	C₂×Q₈	1	4	yes	—
65	C₂×C₄×C₇₉	#41	1	2	yes	—
66	C₂×C₄×C₇₉	#41	1	4	yes	—
67	C₂×Dic₇₉	—	2	4	no	—
68	C₂×Dic₇₉	—	2	4	no	—
69	C₂×Dic₇₉	—	2	4	no	—
70	C₂×Dic₇₉	—	2	8	no	—
71	C₂×Dic₇₉	—	2	4	no	—
72	Dic₁₅₈	—	4	8	no	—
73	C₂²×C₇₉	—	2	2	no	—
74	C₂²×C₇₉	—	2	4	no	—
75	C₂²×C₇₉	—	2	2	no	—
76	C₂²×C₇₉	—	2	4	no	—
77	C₂²×C₇₉	C₄⋊Q₈	1	4	yes	complemented
78	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
79	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
80	C₂²×C₇₉	C₂²×Q₈	1	2	yes	—
81	C₂²×C₇₉	C₂×#41	1	1	yes	characteristic
82	C₂²×C₇₉	C₂×#41	1	2	yes	—
83	C₄×C₇₉	—	2	2	no	—
84	C₄×C₇₉	—	2	4	no	—
85	C₄×C₇₉	#121	1	4	yes	—
86	C₄×C₇₉	C₂²⋊Q₈	1	8	yes	complemented
87	Dic₇₉	—	4	4	no	—
88	Dic₇₉	—	4	4	no	—
89	C₂×C₇₉	—	2	4	no	—
90	C₂×C₇₉	C₂×#121	1	1	yes	characteristic
91	C₂×C₇₉	C₂×C₂²⋊Q₈	1	2	yes	—
92	C₂×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
93	C₂×C₇₉	#532	1	1	yes	characteristic
94	C₂×C₇₉	#533	1	2	yes	—
95	#2857	—	79	1	no	maximal, Sylow
96	C₇₉	#2857	1	1	yes	characteristic, complemented, Sylow
97	C₂×C₄⋊Q₈	—	79	1	no	—
98	C₂×C₂²⋊Q₈	—	79	2	no	—
99	C₂²×C₄²	D₇₉	1	1	yes	characteristic
100	#391	—	79	2	no	—
101	#389	—	79	4	no	—
102	#384	—	79	1	no	—
103	#380	—	79	4	no	—
104	C₂²×Q₈	—	79	2	no	—
105	C₂³×C₄	C₂×D₇₉	1	2	yes	—
106	C₂³×C₄	C₂×D₇₉	1	1	yes	characteristic
107	C₄⋊Q₈	—	158	4	no	—
108	C₂²⋊Q₈	—	158	8	no	—
109	C₂×C₄⋊C₄	—	79	2	no	—
110	C₂×C₄⋊C₄	—	79	4	no	—
111	C₂×C₄⋊C₄	—	79	4	no	—
112	C₂×C₄⋊C₄	—	79	4	no	—
113	C₂×C₂²⋊C₄	—	79	4	no	—
114	C₂×C₄²	—	2	4	no	—
115	C₂×C₄²	C₂×D₇₉	1	2	yes	—
116	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
117	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
118	#95	—	79	8	no	—
119	C₂⁴	C₂²×D₇₉	1	1	yes	characteristic
120	C₂×Q₈	—	158	4	no	—
121	C₂×Q₈	—	158	4	no	—
122	C₂×Q₈	—	158	4	no	—
123	C₂²×C₄	—	2	2	no	—
124	C₂²×C₄	—	2	4	no	—
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	—	2	4	no	—
127	C₂²×C₄	—	79	4	no	—
128	C₂²×C₄	—	79	4	no	—
129	C₂²×C₄	Dic₁₅₈	1	8	yes	complemented
130	C₂²×C₄	D₃₁₆	1	4	yes	—
131	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
132	C₂²×C₄	C₂²×D₇₉	1	2	yes	—
133	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
134	C₂²×C₄	C₂²×D₇₉	1	2	yes	—
135	C₄⋊C₄	—	158	8	no	—
136	C₄⋊C₄	—	158	8	no	—
137	C₄⋊C₄	—	158	4	no	—
138	C₄⋊C₄	—	158	8	no	—
139	C₂²⋊C₄	—	158	8	no	—
140	C₄²	—	2	2	no	—
141	C₄²	—	2	4	no	—
142	C₄²	C₂²⋊D₇₉	1	4	yes	—
143	C₂³	—	2	1	no	—
144	C₂³	—	2	2	no	—
145	C₂³	—	2	1	no	—
146	C₂³	C₂×D₃₁₆	1	2	yes	—
147	C₂³	C₂×Dic₁₅₈	1	4	yes	—
148	C₂³	C₂³×D₇₉	1	1	yes	characteristic
149	Q₈	—	316	8	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	8	no	—
155	C₂×C₄	—	2	16	no	—
156	C₂×C₄	—	158	8	no	—
157	C₂×C₄	—	158	4	no	—
158	C₂×C₄	—	158	4	no	—
159	C₂×C₄	—	158	4	no	—
160	C₂×C₄	—	158	4	no	—
161	C₂×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
162	C₂×C₄	C₂×C₂²⋊D₇₉	1	4	yes	—
163	C₂×C₄	C₂×D₃₁₆	1	2	yes	—
164	C₂×C₄	C₂×D₃₁₆	1	2	yes	—
165	C₂×C₄	C₂×Dic₁₅₈	1	4	yes	—
166	C₂×C₄	C₂×Dic₁₅₈	1	4	yes	—
167	C₂×C₄	#11929624	1	2	yes	—
168	C₂×C₄	#11929624	1	4	yes	—
169	C₂²	—	2	2	no	—
170	C₂²	—	2	4	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	2	4	no	—
173	C₂²	#14579161	1	4	yes	complemented
174	C₂²	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
175	C₂²	C₂²×D₃₁₆	1	1	yes	characteristic
176	C₂²	C₂²×Dic₁₅₈	1	2	yes	—
177	C₂²	C₂×#11929624	1	1	yes	characteristic
178	C₂²	C₂×#11929624	1	2	yes	—
179	C₄	—	2	2	no	—
180	C₄	—	2	4	no	—
181	C₄	—	316	4	no	—
182	C₄	—	316	4	no	—
183	C₄	#14579135	1	4	yes	—
184	C₄	#14579140	1	8	yes	complemented
185	C₂	—	2	4	no	—
186	C₂	C₂×#14579135	1	1	yes	characteristic
187	C₂	C₂×#14579140	1	2	yes	—
188	C₂	C₂×#14579161	1	1	yes	characteristic
189	C₂	#19099113	1	1	yes	characteristic
190	C₂	#19099120	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579161

C₂

yes

maximal, characteristic, complemented

C₂×#14579140

C₂

yes

maximal

#19098469

C₂

yes

maximal, complemented

#19098461

C₂

yes

maximal, complemented

#19098433

C₂

yes

maximal, characteristic, complemented

#19098423

C₂

yes

maximal

C₂²×C₄²×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

C₂²

yes

—

#14579161

—

#14579140

—

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₂×Dic₇₉⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₇₉

C₂²

yes

—

C₇₉⋊#95

C₂²

yes

—

C₂³×C₄×C₇₉

C₂²

yes

characteristic

C₂³×C₄×C₇₉

C₂²

yes

—

C₂×C₄²×C₇₉

—

C₂×C₄²×C₇₉

C₂²

yes

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂²×Dic₇₉

C₂³

yes

—

C₂²×Dic₇₉

C₂³

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

C₂⁴×C₇₉

C₂³

yes

characteristic

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

D₄

yes

—

C₂²×C₄×C₇₉

Q₈

yes

complemented

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

D₄

yes

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

C₂×D₄

yes

—

C₂³×C₇₉

C₂×Q₈

yes

—

C₂³×C₇₉

C₂⁴

yes

characteristic

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×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×Q₈

yes

—

C₂×C₄×C₇₉

C₂×Q₈

yes

—

C₂×C₄×C₇₉

#41

yes

—

C₂×C₄×C₇₉

#41

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₄⋊Q₈

yes

complemented

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₂²×C₇₉

C₂²×Q₈

yes

—

C₂²×C₇₉

C₂×#41

yes

characteristic

C₂²×C₇₉

C₂×#41

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#121

yes

—

C₄×C₇₉

C₂²⋊Q₈

yes

complemented

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

C₂×#121

yes

characteristic

C₂×C₇₉

C₂×C₂²⋊Q₈

yes

—

C₂×C₇₉

C₂×C₄⋊Q₈

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

#2857

—

maximal, Sylow

C₇₉

#2857

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×C₂²⋊Q₈

—

C₂²×C₄²

D₇₉

yes

characteristic

100

#391

—

101

#389

—

102

#384

—

103

#380

—

104

C₂²×Q₈

—

105

C₂³×C₄

C₂×D₇₉

yes

—

106

C₂³×C₄

C₂×D₇₉

yes

characteristic

107

C₄⋊Q₈

—

158

—

108

C₂²⋊Q₈

—

158

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₂²⋊C₄

—

114

C₂×C₄²

—

115

C₂×C₄²

C₂×D₇₉

yes

—

116

C₂×C₄²

C₂×D₇₉

yes

characteristic

117

C₂×C₄²

C₂×D₇₉

yes

characteristic

118

#95

—

119

C₂⁴

C₂²×D₇₉

yes

characteristic

120

C₂×Q₈

—

158

—

121

C₂×Q₈

—

158

—

122

C₂×Q₈

—

158

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

Dic₁₅₈

yes

complemented

130

C₂²×C₄

D₃₁₆

yes

—

131

C₂²×C₄

C₂²×D₇₉

yes

characteristic

132

C₂²×C₄

C₂²×D₇₉

yes

—

133

C₂²×C₄

C₂²×D₇₉

yes

characteristic

134

C₂²×C₄

C₂²×D₇₉

yes

—

135

C₄⋊C₄

—

158

—

136

C₄⋊C₄

—

158

—

137

C₄⋊C₄

—

158

—

138

C₄⋊C₄

—

158

—

139

C₂²⋊C₄

—

158

—

140

C₄²

—

141

C₄²

—

142

C₄²

C₂²⋊D₇₉

yes

—

143

C₂³

—

144

C₂³

—

145

C₂³

—

146

C₂³

C₂×D₃₁₆

yes

—

147

C₂³

C₂×Dic₁₅₈

yes

—

148

C₂³

C₂³×D₇₉

yes

characteristic

149

Q₈

—

316

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

158

—

157

C₂×C₄

—

158

—

158

C₂×C₄

—

158

—

159

C₂×C₄

—

158

—

160

C₂×C₄

—

158

—

161

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

162

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

163

C₂×C₄

C₂×D₃₁₆

yes

—

164

C₂×C₄

C₂×D₃₁₆

yes

—

165

C₂×C₄

C₂×Dic₁₅₈

yes

—

166

C₂×C₄

C₂×Dic₁₅₈

yes

—

167

C₂×C₄

#11929624

yes

—

168

C₂×C₄

#11929624

yes

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

#14579161

yes

complemented

174

C₂²

C₂²×C₂²⋊D₇₉

yes

characteristic

175

C₂²

C₂²×D₃₁₆

yes

characteristic

176

C₂²

C₂²×Dic₁₅₈

yes

—

177

C₂²

C₂×#11929624

yes

characteristic

178

C₂²

C₂×#11929624

yes

—

179

C₄

—

180

C₄

—

181

C₄

—

316

—

182

C₄

—

316

—

183

C₄

#14579135

yes

—

184

C₄

#14579140

yes

complemented

185

C₂

—

186

C₂

C₂×#14579135

yes

characteristic

187

C₂

C₂×#14579140

yes

—

188

C₂

C₂×#14579161

yes

characteristic

189

C₂

#19099113

yes

characteristic

190

C₂

#19099120

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇₉
commutator, socle, cosocle	C₂³×C₇₉	C₂⁴
Fitting	C₂²×C₄²×C₇₉	C₂
prime radical	C₂⁴×C₇₉	C₂³
hypercenter	C₂²×C₄²	D₇₉
hypocenter	C₇₉	#2857
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄²

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇₉

commutator, socle, cosocle

C₂³×C₇₉

C₂⁴

Fitting

C₂²×C₄²×C₇₉

C₂

prime radical

C₂⁴×C₇₉

C₂³

hypercenter

C₂²×C₄²

D₇₉

C₇₉

G ▷ C₂³×C₇₉ ▷ C₇₉

C₁ ◁ C₂³ ◁ C₂²×C₄²

N	H ≅ G/N	type
1. C₂×#14579161	C₂	semidirect
3. #19098469	C₂	semidirect
4. #19098461	C₂	semidirect
5. #19098433	C₂	semidirect
39. C₂²×C₄×C₇₉	Q₈	semidirect
77. C₂²×C₇₉	C₄⋊Q₈	semidirect
86. C₄×C₇₉	C₂²⋊Q₈	semidirect
96. C₇₉	#2857	semidirect
129. C₂²×C₄	Dic₁₅₈	semidirect
173. C₂²	#14579161	semidirect
184. C₄	#14579140	semidirect

H ≅ G/N

type

1. C₂×#14579161

C₂

semidirect

3. #19098469

C₂

semidirect

4. #19098461

C₂

semidirect

5. #19098433

C₂

semidirect

39. C₂²×C₄×C₇₉

Q₈

semidirect

77. C₂²×C₇₉

C₄⋊Q₈

semidirect

86. C₄×C₇₉

C₂²⋊Q₈

semidirect

96. C₇₉

#2857

semidirect

129. C₂²×C₄

Dic₁₅₈

semidirect

173. C₂²

#14579161

semidirect

184. C₄

#14579140

semidirect

Subgroups — #23771752

Conjugacy classes of subgroups

Complements