Group #23761977 — subgroups

Subgroups — #23761977

Order 10112.

218 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 77 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	#19098616	C₂	1	1	yes	maximal, characteristic, complemented
2	#19098431	C₂	1	1	yes	maximal, characteristic, complemented
3	#19098300	C₂	1	2	yes	maximal, complemented
4	#19098297	C₂	1	2	yes	maximal
5	#529×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
7	C₂²⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
8	C₄×C₇₉⋊C₈	—	2	1	no	—
9	C₇₉⋊#95	—	2	1	no	—
10	C₇₉⋊#95	—	2	1	no	—
11	C₇₉⋊#95	C₂²	1	2	yes	—
12	#126×C₇₉	—	2	1	no	—
13	#125×C₇₉	C₄	1	1	yes	characteristic, complemented
14	#124×C₇₉	C₄	1	1	yes	characteristic, complemented
15	#123×C₇₉	—	2	1	no	—
16	C₂²⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
17	#121×C₇₉	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₇₉	C₄	1	1	yes	characteristic, complemented
19	C₄×D₄×C₇₉	—	2	1	no	—
20	C₄×D₄×C₇₉	C₄	1	1	yes	characteristic, complemented
21	#117×C₇₉	—	2	1	no	—
22	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
23	C₂²×Dic₇₉	—	2	2	no	—
24	C₂×C₇₉⋊C₈	—	2	2	no	—
25	C₂×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
26	C₂×D₄×C₇₉	—	2	1	no	—
27	C₂×D₄×C₇₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₇₉	—	2	1	no	—
29	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₇₉	—	2	1	no	—
33	C₄⋊C₄×C₇₉	—	2	1	no	—
34	C₄⋊C₄×C₇₉	—	2	1	no	—
35	C₄⋊C₄×C₇₉	—	2	1	no	—
36	C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
37	C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
38	C₂²⋊C₄×C₇₉	—	2	1	no	—
39	C₂²⋊C₄×C₇₉	—	2	1	no	—
40	C₂²⋊C₄×C₇₉	—	2	1	no	—
41	C₂²⋊C₄×C₇₉	—	2	1	no	—
42	C₂²⋊C₄×C₇₉	—	2	1	no	—
43	C₄²×C₇₉	—	2	1	no	—
44	C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
45	C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
46	C₄²×C₇₉	D₄	1	1	yes	characteristic
47	C₄²×C₇₉	D₄	1	1	yes	characteristic
48	C₂³×C₇₉	—	2	1	no	—
49	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
50	Q₈×C₇₉	—	4	1	no	—
51	D₄×C₇₉	—	4	1	no	—
52	D₄×C₇₉	—	4	2	no	—
53	C₂×C₄×C₇₉	—	2	2	no	—
54	C₂×C₄×C₇₉	—	2	1	no	—
55	C₂×C₄×C₇₉	—	2	1	no	—
56	C₂×C₄×C₇₉	—	2	1	no	—
57	C₂×C₄×C₇₉	—	4	1	no	—
58	C₂×C₄×C₇₉	—	4	1	no	—
59	C₂×C₄×C₇₉	—	4	2	no	—
60	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
66	C₂×Dic₇₉	—	4	2	no	—
67	C₂×Dic₇₉	—	4	2	no	—
68	C₂×Dic₇₉	—	4	2	no	—
69	C₇₉⋊C₈	—	4	2	no	—
70	C₂²×C₇₉	—	2	1	no	—
71	C₂²×C₇₉	—	2	1	no	—
72	C₂²×C₇₉	—	4	1	no	—
73	C₂²×C₇₉	—	4	1	no	—
74	C₂²×C₇₉	—	4	1	no	—
75	C₂²×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₇₉	#135	1	2	yes	—
77	C₄×C₇₉	—	2	1	no	—
78	C₄×C₇₉	—	2	1	no	—
79	C₄×C₇₉	—	2	1	no	—
80	C₄×C₇₉	—	2	1	no	—
81	C₄×C₇₉	—	2	1	no	—
82	C₄×C₇₉	—	2	1	no	—
83	C₄×C₇₉	—	4	1	no	—
84	C₄×C₇₉	—	4	1	no	—
85	C₄×C₇₉	—	4	1	no	—
86	Dic₇₉	—	8	2	no	—
87	C₂×C₇₉	—	2	2	no	—
88	C₂×C₇₉	—	8	1	no	—
89	C₂×C₇₉	#410	1	1	yes	characteristic
90	C₂×C₇₉	#416	1	1	yes	characteristic
91	C₂×C₇₉	#421	1	1	yes	characteristic
92	#1521	—	79	1	no	maximal, Sylow
93	C₇₉	#1521	1	1	yes	characteristic, complemented, Sylow
94	#529	D₇₉	1	1	yes	characteristic
95	#431	—	79	1	no	—
96	#383	—	79	1	no	—
97	#328	—	79	2	no	—
98	#327	—	79	2	no	—
99	#126	—	2	1	no	—
100	#125	Dic₇₉	1	1	yes	characteristic, complemented
101	#124	Dic₇₉	1	1	yes	characteristic, complemented
102	#123	—	2	1	no	—
103	C₂²⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
104	#121	C₂×D₇₉	1	1	yes	characteristic
105	C₄×Q₈	Dic₇₉	1	1	yes	characteristic, complemented
106	C₄×D₄	—	2	1	no	—
107	C₄×D₄	Dic₇₉	1	1	yes	characteristic, complemented
108	#117	—	2	1	no	—
109	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
110	C₄⋊C₈	—	158	1	no	—
111	C₂²⋊C₈	—	79	2	no	—
112	C₄×C₈	—	158	1	no	—
113	#95	—	79	2	no	—
114	#95	—	158	1	no	—
115	#95	—	158	1	no	—
116	C₂×Q₈	C₂×Dic₇₉	1	1	yes	characteristic
117	C₂×D₄	—	2	1	no	—
118	C₂×D₄	C₂×Dic₇₉	1	1	yes	characteristic
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	—	158	2	no	—
121	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
122	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
123	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
124	C₂×C₈	—	158	2	no	—
125	C₄⋊C₄	—	2	1	no	—
126	C₄⋊C₄	—	2	1	no	—
127	C₄⋊C₄	—	2	1	no	—
128	C₄⋊C₄	—	2	1	no	—
129	C₄⋊C₄	C₂×Dic₇₉	1	1	yes	characteristic
130	C₄⋊C₄	C₂×Dic₇₉	1	1	yes	characteristic
131	C₂²⋊C₄	—	2	1	no	—
132	C₂²⋊C₄	—	2	1	no	—
133	C₂²⋊C₄	—	2	1	no	—
134	C₂²⋊C₄	—	2	1	no	—
135	C₂²⋊C₄	—	2	1	no	—
136	C₄²	—	2	1	no	—
137	C₄²	C₂×Dic₇₉	1	1	yes	characteristic
138	C₄²	C₂×Dic₇₉	1	1	yes	characteristic
139	C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
140	C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
141	C₂³	—	2	1	no	—
142	C₂³	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
143	Q₈	—	4	1	no	—
144	D₄	—	4	1	no	—
145	D₄	—	4	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	316	2	no	—
154	C₂×C₄	—	316	2	no	—
155	C₂×C₄	—	316	2	no	—
156	C₂×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
157	C₂×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
158	C₂×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
159	C₂×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
160	C₂×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
161	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
162	C₈	—	316	2	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	4	1	no	—
166	C₂²	—	4	1	no	—
167	C₂²	—	4	1	no	—
168	C₂²	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
169	C₂²	#14579181	1	2	yes	—
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	—	4	1	no	—
177	C₄	—	4	1	no	—
178	C₄	—	4	1	no	—
179	C₄	—	632	2	no	—
180	C₂	—	2	2	no	—
181	C₂	—	8	1	no	—
182	C₂	#19098537	1	1	yes	characteristic
183	C₂	#19098559	1	1	yes	characteristic
184	C₂	#19098580	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19098616

C₂

yes

maximal, characteristic, complemented

#19098431

C₂

yes

maximal, characteristic, complemented

#19098300

C₂

yes

maximal, complemented

#19098297

C₂

yes

maximal

#529×C₇₉

C₂

yes

maximal, characteristic

C₄⋊(C₇₉⋊C₈)

—

C₂²⋊(C₇₉⋊C₈)

C₂²

yes

—

C₄×C₇₉⋊C₈

—

C₇₉⋊#95

—

C₇₉⋊#95

—

C₇₉⋊#95

C₂²

yes

—

#126×C₇₉

—

#125×C₇₉

C₄

yes

characteristic, complemented

#124×C₇₉

C₄

yes

characteristic, complemented

#123×C₇₉

—

C₂²⋊Q₈×C₇₉

C₂²

yes

characteristic

#121×C₇₉

C₂²

yes

characteristic

C₄×Q₈×C₇₉

C₄

yes

characteristic, complemented

C₄×D₄×C₇₉

—

C₄×D₄×C₇₉

C₄

yes

characteristic, complemented

#117×C₇₉

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

—

C₂×Q₈×C₇₉

C₂×C₄

yes

characteristic

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₄×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

C₂×C₄

yes

characteristic

C₄⋊C₄×C₇₉

C₂×C₄

yes

characteristic

C₂²⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂×C₄

yes

characteristic

C₄²×C₇₉

C₂×C₄

yes

characteristic

C₄²×C₇₉

D₄

yes

characteristic

C₄²×C₇₉

D₄

yes

characteristic

C₂³×C₇₉

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

Q₈×C₇₉

—

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

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇₉

#135

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

#529

D₇₉

yes

—

—

—

—

—

100

#125

Dic₇₉

yes

characteristic, complemented

101

#124

Dic₇₉

yes

characteristic, complemented

102

#123

—

103

C₂²⋊Q₈

C₂×D₇₉

yes

characteristic

104

#121

C₂×D₇₉

yes

characteristic

105

C₄×Q₈

Dic₇₉

yes

characteristic, complemented

106

C₄×D₄

—

107

C₄×D₄

Dic₇₉

yes

characteristic, complemented

108

#117

—

109

C₂×C₄²

C₂×D₇₉

yes

characteristic

110

C₄⋊C₈

—

158

—

111

C₂²⋊C₈

—

112

C₄×C₈

—

158

—

113

#95

—

114

#95

—

158

—

115

#95

—

158

—

116

C₂×Q₈

C₂×Dic₇₉

yes

characteristic

117

C₂×D₄

—

118

C₂×D₄

C₂×Dic₇₉

yes

characteristic

119

C₂²×C₄

—

120

C₂²×C₄

—

158

—

121

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

122

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

123

C₂²×C₄

C₂²×D₇₉

yes

characteristic

124

C₂×C₈

—

158

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

C₂×Dic₇₉

yes

characteristic

130

C₄⋊C₄

C₂×Dic₇₉

yes

characteristic

131

C₂²⋊C₄

—

132

C₂²⋊C₄

—

133

C₂²⋊C₄

—

134

C₂²⋊C₄

—

135

C₂²⋊C₄

—

136

C₄²

—

137

C₄²

C₂×Dic₇₉

yes

characteristic

138

C₄²

C₂×Dic₇₉

yes

characteristic

139

C₄²

C₂²⋊D₇₉

yes

characteristic

140

C₄²

C₂²⋊D₇₉

yes

characteristic

141

C₂³

—

142

C₂³

C₂×C₂²⋊D₇₉

yes

characteristic

143

Q₈

—

144

D₄

—

145

D₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

316

—

154

C₂×C₄

—

316

—

155

C₂×C₄

—

316

—

156

C₂×C₄

C₂²⋊Dic₇₉

yes

characteristic

157

C₂×C₄

C₂²⋊Dic₇₉

yes

characteristic

158

C₂×C₄

C₂²⋊Dic₇₉

yes

characteristic

159

C₂×C₄

C₂²⋊Dic₇₉

yes

characteristic

160

C₂×C₄

C₂²×Dic₇₉

yes

characteristic

161

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

162

C₈

—

316

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

C₂×C₂²⋊Dic₇₉

yes

characteristic

169

C₂²

#14579181

yes

—

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

632

—

180

C₂

—

181

C₂

—

182

C₂

#19098537

yes

characteristic

183

C₂

#19098559

yes

characteristic

184

C₂

#19098580

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	#529×C₇₉	C₂
socle	C₂²×C₇₉	C₂×C₂²⋊C₄
prime radical	#121×C₇₉	C₂²
cosocle	C₂²×C₄×C₇₉	C₂³
hypercenter	#529	D₇₉
hypocenter	C₇₉	#1521
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #529

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₇₉

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

#529×C₇₉

C₂

socle

C₂²×C₇₉

C₂×C₂²⋊C₄

prime radical

#121×C₇₉

C₂²

cosocle

C₂²×C₄×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #529

N	H ≅ G/N	type
1. #19098616	C₂	semidirect
2. #19098431	C₂	semidirect
3. #19098300	C₂	semidirect
13. #125×C₇₉	C₄	semidirect
14. #124×C₇₉	C₄	semidirect
18. C₄×Q₈×C₇₉	C₄	semidirect
20. C₄×D₄×C₇₉	C₄	semidirect
93. C₇₉	#1521	semidirect
100. #125	Dic₇₉	semidirect
101. #124	Dic₇₉	semidirect
105. C₄×Q₈	Dic₇₉	semidirect
107. C₄×D₄	Dic₇₉	semidirect

H ≅ G/N

type

1. #19098616

C₂

semidirect

2. #19098431

C₂

semidirect

3. #19098300

C₂

semidirect

13. #125×C₇₉

C₄

semidirect

14. #124×C₇₉

C₄

semidirect

18. C₄×Q₈×C₇₉

C₄

semidirect

20. C₄×D₄×C₇₉

C₄

semidirect

93. C₇₉

#1521

semidirect

100. #125

Dic₇₉

semidirect

101. #124

Dic₇₉

semidirect

105. C₄×Q₈

Dic₇₉

semidirect

107. C₄×D₄

Dic₇₉

semidirect

Subgroups — #23761977

Conjugacy classes of subgroups

Complements