Group #23775158 — subgroups

Subgroups — #23775158

Order 10112.

504 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 125 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×#14579180	C₂	1	2	yes	maximal, complemented
2	#19099233	C₂	1	2	yes	maximal, complemented
3	#19098796	C₂	1	2	yes	maximal, complemented
4	#19098760	C₂	1	2	yes	maximal, complemented
5	#19098705	C₂	1	4	yes	maximal, complemented
6	C₄×Dic₃₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×D₃₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	#493×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11929623	C₂²	1	4	yes	—
10	#14579180	—	2	8	no	—
11	C₂×Dic₃₁₆	C₂²	1	1	yes	characteristic
12	C₂×#11929612	C₂²	1	4	yes	—
13	C₂×C₈×D₇₉	C₂²	1	2	yes	—
14	#14579157	—	2	2	no	—
15	#14579133	—	2	4	no	—
16	#14579132	C₂²	1	4	yes	—
17	C₄×Dic₁₅₈	C₂²	1	2	yes	—
18	C₄×D₃₁₆	C₂²	1	1	yes	characteristic
19	D₄×Dic₇₉	—	2	2	no	—
20	#14579095	C₂²	1	1	yes	characteristic
21	C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
22	#14579086	C₂²	1	2	yes	—
23	#14579082	C₂²	1	4	yes	—
24	(C₂×D₇₉)⋊C₈	C₂²	1	2	yes	—
25	C₂×D₈×C₇₉	C₂²	1	2	yes	—
26	C₂×D₈×C₇₉	C₂²	1	2	yes	—
27	#127×C₇₉	C₂²	1	2	yes	—
28	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
29	#11929623	—	2	8	no	—
30	#11929623	—	4	4	no	—
31	C₂×Dic₁₅₈	C₂³	1	2	yes	—
32	C₂×D₃₁₆	D₄	1	1	yes	characteristic
33	C₂×C₂²⋊D₇₉	—	2	4	no	—
34	C₂×C₂²⋊D₇₉	—	2	4	no	—
35	C₂²×Dic₇₉	—	2	4	no	—
36	C₂×C₄×D₇₉	C₂³	1	2	yes	—
37	Dic₃₁₆	D₄	1	4	yes	complemented
38	#11929612	—	2	8	no	—
39	C₂×C₇₉⋊C₈	C₂³	1	2	yes	—
40	C₈×D₇₉	—	2	4	no	—
41	C₄⋊Dic₇₉	D₄	1	1	yes	characteristic
42	C₄⋊Dic₇₉	C₂³	1	2	yes	—
43	Dic₇₉⋊C₄	—	2	2	no	—
44	C₂²⋊Dic₇₉	—	2	4	no	—
45	(C₂×D₇₉)⋊C₄	D₄	1	2	yes	—
46	C₄×Dic₇₉	—	2	2	no	—
47	C₂×D₄×C₇₉	—	2	2	no	—
48	C₂×D₄×C₇₉	—	2	2	no	—
49	C₂×D₄×C₇₉	C₂³	1	4	yes	—
50	D₈×C₇₉	—	2	4	no	—
51	D₈×C₇₉	—	4	2	no	—
52	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
54	C₄²×C₇₉	C₂³	1	1	yes	characteristic
55	C₂³×C₇₉	—	2	4	no	—
56	C₂²×D₇₉	C₂×D₄	1	2	yes	—
57	D₄×C₇₉	—	2	4	no	—
58	D₄×C₇₉	—	2	4	no	—
59	D₄×C₇₉	—	4	4	no	—
60	D₄×C₇₉	—	8	2	no	—
61	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
64	C₂²⋊D₇₉	—	4	8	no	—
65	C₂²⋊D₇₉	—	8	4	no	—
66	C₂×Dic₇₉	—	2	2	no	—
67	C₂×Dic₇₉	—	2	2	no	—
68	C₂×Dic₇₉	—	4	8	no	—
69	C₂×Dic₇₉	C₂×D₄	1	2	yes	—
70	D₃₁₆	—	2	2	no	—
71	C₄×D₇₉	—	2	2	no	—
72	C₄×D₇₉	—	2	2	no	—
73	Dic₁₅₈	—	2	2	no	—
74	Dic₁₅₈	C₂×D₄	1	4	yes	—
75	C₈×C₇₉	—	2	1	no	—
76	C₈×C₇₉	C₂×D₄	1	2	yes	—
77	C₇₉⋊C₈	—	2	2	no	—
78	C₂²×C₇₉	—	4	4	no	—
79	C₂²×C₇₉	—	4	4	no	—
80	C₂²×C₇₉	—	4	4	no	—
81	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
82	C₂×D₇₉	—	2	2	no	—
83	C₂×D₇₉	—	2	2	no	—
84	C₂×D₇₉	—	2	2	no	—
85	C₄×C₇₉	—	2	1	no	—
86	C₄×C₇₉	#135	1	2	yes	complemented
87	C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
88	C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
89	Dic₇₉	—	2	4	no	—
90	Dic₇₉	—	2	2	no	—
91	Dic₇₉	—	4	2	no	—
92	C₂×C₇₉	—	8	4	no	—
93	C₂×C₇₉	D₄²	1	1	yes	characteristic
94	C₂×C₇₉	C₂×#135	1	1	yes	characteristic
95	C₂×C₇₉	#576	1	1	yes	characteristic
96	D₇₉	—	4	2	no	—
97	#3287	—	79	1	no	maximal, Sylow
98	C₇₉	#3287	1	1	yes	characteristic, complemented, Sylow
99	C₂×#135	—	79	2	no	—
100	#550	—	79	2	no	—
101	#493	D₇₉	1	1	yes	characteristic, complemented
102	#466	—	79	2	no	—
103	#460	—	79	2	no	—
104	#449	—	79	4	no	—
105	C₄×Q₁₆	—	79	1	no	—
106	C₈×D₄	—	79	1	no	—
107	C₂×#41	—	79	4	no	—
108	#135	—	158	8	no	—
109	C₂×Q₁₆	—	79	1	no	—
110	C₂×SD₁₆	—	79	4	no	—
111	C₂×D₈	C₂×D₇₉	1	2	yes	—
112	C₂×D₈	C₂×D₇₉	1	2	yes	—
113	C₂²×C₈	—	79	2	no	—
114	#127	—	158	2	no	—
115	#127	C₂×D₇₉	1	2	yes	—
116	#121	—	79	4	no	—
117	#121	—	158	4	no	—
118	C₄×Q₈	—	79	2	no	—
119	C₄×D₄	—	79	1	no	—
120	C₄×D₄	—	158	2	no	—
121	#107	—	79	1	no	—
122	C₄⋊C₈	—	79	1	no	—
123	#103	—	79	2	no	—
124	#102	—	79	4	no	—
125	C₂²⋊C₈	—	79	2	no	—
126	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
127	#41	—	316	4	no	—
128	#41	—	158	8	no	—
129	C₂×Q₈	—	79	2	no	—
130	C₂×D₄	—	2	2	no	—
131	C₂×D₄	—	2	2	no	—
132	C₂×D₄	—	79	1	no	—
133	C₂×D₄	—	158	4	no	—
134	C₂×D₄	—	158	4	no	—
135	C₂×D₄	C₂²×D₇₉	1	4	yes	—
136	C₂²×C₄	—	79	2	no	—
137	C₂²×C₄	—	158	4	no	—
138	Q₁₆	—	79	4	no	—
139	SD₁₆	—	158	8	no	—
140	D₈	—	2	4	no	—
141	D₈	—	4	2	no	—
142	C₂×C₈	—	79	2	no	—
143	C₂×C₈	—	158	4	no	—
144	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
145	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
146	C₄⋊C₄	—	79	1	no	—
147	C₄⋊C₄	—	79	2	no	—
148	C₄⋊C₄	—	158	2	no	—
149	C₂²⋊C₄	—	79	2	no	—
150	C₂²⋊C₄	—	158	4	no	—
151	C₄²	—	158	2	no	—
152	C₄²	C₂²×D₇₉	1	1	yes	characteristic
153	C₂³	—	2	4	no	—
154	C₂³	—	79	2	no	—
155	Q₈	—	79	4	no	—
156	Q₈	—	158	2	no	—
157	D₄	—	2	4	no	—
158	D₄	—	2	4	no	—
159	D₄	—	4	4	no	—
160	D₄	—	8	2	no	—
161	D₄	—	316	8	no	—
162	D₄	—	632	4	no	—
163	D₄	—	158	2	no	—
164	C₂×C₄	—	316	8	no	—
165	C₂×C₄	—	79	2	no	—
166	C₂×C₄	—	158	2	no	—
167	C₂×C₄	—	158	2	no	—
168	C₂×C₄	—	158	2	no	—
169	C₂×C₄	—	158	2	no	—
170	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
171	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
172	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
173	C₈	—	2	1	no	—
174	C₈	—	158	2	no	—
175	C₈	D₄×D₇₉	1	2	yes	—
176	C₂²	—	4	4	no	—
177	C₂²	—	4	4	no	—
178	C₂²	—	4	4	no	—
179	C₂²	—	158	2	no	—
180	C₂²	—	158	2	no	—
181	C₂²	—	158	2	no	—
182	C₂²	C₂×D₄×D₇₉	1	1	yes	characteristic
183	C₄	—	2	1	no	—
184	C₄	—	316	2	no	—
185	C₄	—	158	2	no	—
186	C₄	—	158	4	no	—
187	C₄	#14579180	1	2	yes	complemented
188	C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
189	C₄	#14579206	1	1	yes	characteristic
190	C₂	—	8	4	no	—
191	C₂	—	316	2	no	—
192	C₂	#19099198	1	1	yes	characteristic
193	C₂	C₂×#14579180	1	1	yes	characteristic
194	C₂	#19099364	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579180

C₂

yes

maximal, complemented

#19099233

C₂

yes

maximal, complemented

#19098796

C₂

yes

maximal, complemented

#19098760

C₂

yes

maximal, complemented

#19098705

C₂

yes

maximal, complemented

C₄×Dic₃₁₆

C₂

yes

maximal, characteristic, complemented

C₈×D₃₁₆

C₂

yes

maximal, characteristic, complemented

#493×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#11929623

C₂²

yes

—

#14579180

—

C₂×Dic₃₁₆

C₂²

yes

characteristic

C₂×#11929612

C₂²

yes

—

C₂×C₈×D₇₉

C₂²

yes

—

#14579157

—

#14579133

—

#14579132

C₂²

yes

—

C₄×Dic₁₅₈

C₂²

yes

—

C₄×D₃₁₆

C₂²

yes

characteristic

D₄×Dic₇₉

—

#14579095

C₂²

yes

characteristic

C₄⋊(C₇₉⋊C₈)

C₂²

yes

characteristic

#14579086

C₂²

yes

—

#14579082

C₂²

yes

—

(C₂×D₇₉)⋊C₈

C₂²

yes

—

C₂×D₈×C₇₉

C₂²

yes

—

C₂×D₈×C₇₉

C₂²

yes

—

#127×C₇₉

C₂²

yes

—

C₄×C₈×C₇₉

C₂²

yes

characteristic

#11929623

—

#11929623

—

C₂×Dic₁₅₈

C₂³

yes

—

C₂×D₃₁₆

D₄

yes

characteristic

C₂×C₂²⋊D₇₉

—

C₂×C₂²⋊D₇₉

—

C₂²×Dic₇₉

—

C₂×C₄×D₇₉

C₂³

yes

—

Dic₃₁₆

D₄

yes

complemented

#11929612

—

C₂×C₇₉⋊C₈

C₂³

yes

—

C₈×D₇₉

—

C₄⋊Dic₇₉

D₄

yes

characteristic

C₄⋊Dic₇₉

C₂³

yes

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

(C₂×D₇₉)⋊C₄

D₄

yes

—

C₄×Dic₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

C₂³

yes

—

D₈×C₇₉

—

D₈×C₇₉

—

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₄²×C₇₉

C₂³

yes

characteristic

C₂³×C₇₉

—

C₂²×D₇₉

C₂×D₄

yes

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂²⋊D₇₉

—

C₂²⋊D₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

C₂×D₄

yes

—

D₃₁₆

—

C₄×D₇₉

—

C₄×D₇₉

—

Dic₁₅₈

—

Dic₁₅₈

C₂×D₄

yes

—

C₈×C₇₉

—

C₈×C₇₉

C₂×D₄

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#135

yes

complemented

C₄×C₇₉

C₂²×D₄

yes

characteristic

C₄×C₇₉

H(5,𝔽₂)

yes

characteristic

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

D₄²

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

D₇₉

—

#3287

—

maximal, Sylow

C₇₉

#3287

yes

characteristic, complemented, Sylow

C₂×#135

—

100

#550

—

101

#493

D₇₉

yes

characteristic, complemented

102

#466

—

103

#460

—

104

#449

—

105

C₄×Q₁₆

—

106

C₈×D₄

—

107

C₂×#41

—

108

#135

—

158

—

109

C₂×Q₁₆

—

110

C₂×SD₁₆

—

111

C₂×D₈

C₂×D₇₉

yes

—

112

C₂×D₈

C₂×D₇₉

yes

—

113

C₂²×C₈

—

114

#127

—

158

—

115

#127

C₂×D₇₉

yes

—

116

#121

—

117

#121

—

158

—

118

C₄×Q₈

—

119

C₄×D₄

—

120

C₄×D₄

—

158

—

121

#107

—

122

C₄⋊C₈

—

123

#103

—

124

#102

—

125

C₂²⋊C₈

—

126

C₄×C₈

C₂×D₇₉

yes

characteristic

127

#41

—

316

—

128

#41

—

158

—

129

C₂×Q₈

—

130

C₂×D₄

—

131

C₂×D₄

—

132

C₂×D₄

—

133

C₂×D₄

—

158

—

134

C₂×D₄

—

158

—

135

C₂×D₄

C₂²×D₇₉

yes

—

136

C₂²×C₄

—

137

C₂²×C₄

—

158

—

138

Q₁₆

—

139

SD₁₆

—

158

—

140

D₈

—

141

D₈

—

142

C₂×C₈

—

143

C₂×C₈

—

158

—

144

C₂×C₈

C₂²×D₇₉

yes

characteristic

145

C₂×C₈

C₂²×D₇₉

yes

characteristic

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

158

—

149

C₂²⋊C₄

—

150

C₂²⋊C₄

—

158

—

151

C₄²

—

158

—

152

C₄²

C₂²×D₇₉

yes

characteristic

153

C₂³

—

154

C₂³

—

155

Q₈

—

156

Q₈

—

158

—

157

D₄

—

158

D₄

—

159

D₄

—

160

D₄

—

161

D₄

—

316

—

162

D₄

—

632

—

163

D₄

—

158

—

164

C₂×C₄

—

316

—

165

C₂×C₄

—

166

C₂×C₄

—

158

—

167

C₂×C₄

—

158

—

168

C₂×C₄

—

158

—

169

C₂×C₄

—

158

—

170

C₂×C₄

D₄×D₇₉

yes

characteristic

171

C₂×C₄

D₄×D₇₉

yes

characteristic

172

C₂×C₄

C₂³×D₇₉

yes

characteristic

173

C₈

—

174

C₈

—

158

—

175

C₈

D₄×D₇₉

yes

—

176

C₂²

—

177

C₂²

—

178

C₂²

—

179

C₂²

—

158

—

180

C₂²

—

158

—

181

C₂²

—

158

—

182

C₂²

C₂×D₄×D₇₉

yes

characteristic

183

C₄

—

184

C₄

—

316

—

185

C₄

—

158

—

186

C₄

—

158

—

187

C₄

#14579180

yes

complemented

188

C₄

C₂×D₄×D₇₉

yes

characteristic

189

C₄

#14579206

yes

characteristic

190

C₂

—

191

C₂

—

316

—

192

C₂

#19099198

yes

characteristic

193

C₂

C₂×#14579180

yes

characteristic

194

C₂

#19099364

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₉
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	#493×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₄
hypercenter	#493	D₇₉
hypocenter	C₇₉	#3287
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

C₂×D₄×D₇₉

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

#493×C₇₉

C₂

socle

C₂²×C₇₉

C₂²×D₄

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. C₂×#14579180	C₂	semidirect
2. #19099233	C₂	semidirect
3. #19098796	C₂	semidirect
4. #19098760	C₂	semidirect
5. #19098705	C₂	semidirect
6. C₄×Dic₃₁₆	C₂	semidirect
7. C₈×D₃₁₆	C₂	semidirect
8. #493×C₇₉	C₂	semidirect
37. Dic₃₁₆	D₄	semidirect
86. C₄×C₇₉	#135	semidirect
98. C₇₉	#3287	semidirect
101. #493	D₇₉	semidirect
187. C₄	#14579180	semidirect

H ≅ G/N

type

1. C₂×#14579180

C₂

semidirect

2. #19099233

C₂

semidirect

3. #19098796

C₂

semidirect

4. #19098760

C₂

semidirect

5. #19098705

C₂

semidirect

6. C₄×Dic₃₁₆

C₂

semidirect

7. C₈×D₃₁₆

C₂

semidirect

8. #493×C₇₉

C₂

semidirect

37. Dic₃₁₆

D₄

semidirect

86. C₄×C₇₉

#135

semidirect

98. C₇₉

#3287

semidirect

101. #493

D₇₉

semidirect

187. C₄

#14579180

semidirect

Subgroups — #23775158

Conjugacy classes of subgroups

Complements