Group #23762665 — subgroups

Subgroups — #23762665

Order 10112.

234 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 59 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	D₄×Dic₁₅₈	C₂	1	1	yes	maximal, characteristic
2	#19098961	C₂	1	1	yes	maximal, characteristic, complemented
3	#19098849	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098768	C₂	1	1	yes	maximal, characteristic
5	#19098310	C₂	1	1	yes	maximal, characteristic, complemented
6	#19098295	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄⋊C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₅₈	—	2	1	no	—
9	C₂×#11929612	—	2	1	no	—
10	#14579161	C₂²	1	1	yes	characteristic
11	#14579161	C₂²	1	1	yes	characteristic
12	#14579160	—	2	1	no	—
13	#14579140	—	2	1	no	—
14	#14579138	—	2	2	no	—
15	C₄×Dic₁₅₈	C₂²	1	1	yes	characteristic
16	D₄×Dic₇₉	—	2	1	no	—
17	#14579093	—	2	2	no	—
18	#14579093	—	2	1	no	—
19	C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
20	#14579086	—	2	1	no	—
21	#14579084	—	2	1	no	—
22	C₄×D₄×C₇₉	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
24	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
25	C₂×Dic₁₅₈	—	2	2	no	—
26	C₂×Dic₁₅₈	—	2	1	no	—
27	C₂×Dic₁₅₈	—	2	1	no	—
28	C₂×Dic₁₅₈	—	4	2	no	—
29	C₂×Dic₁₅₈	—	4	2	no	—
30	C₂×Dic₁₅₈	D₄	1	1	yes	characteristic
31	C₂²×Dic₇₉	—	2	2	no	—
32	#11929612	—	4	2	no	—
33	C₂×C₇₉⋊C₈	—	2	1	no	—
34	C₄⋊Dic₇₉	—	2	1	no	—
35	C₄⋊Dic₇₉	—	2	1	no	—
36	C₄⋊Dic₇₉	—	2	1	no	—
37	C₄⋊Dic₇₉	D₄	1	1	yes	characteristic
38	C₄⋊Dic₇₉	D₄	1	2	yes	—
39	Dic₇₉⋊C₄	—	2	2	no	—
40	Dic₇₉⋊C₄	—	2	1	no	—
41	C₂²⋊Dic₇₉	—	2	2	no	—
42	C₄×Dic₇₉	—	2	1	no	—
43	C₂×D₄×C₇₉	D₄	1	1	yes	characteristic
44	C₂²×C₄×C₇₉	—	2	1	no	—
45	C₂×C₈×C₇₉	—	2	1	no	—
46	C₂×C₈×C₇₉	—	2	1	no	—
47	C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
48	C₂²⋊C₄×C₇₉	—	2	1	no	—
49	C₄²×C₇₉	C₂³	1	1	yes	characteristic
50	C₂³×C₇₉	—	2	1	no	—
51	D₄×C₇₉	—	2	1	no	—
52	D₄×C₇₉	SD₁₆	1	2	yes	—
53	C₂×C₄×C₇₉	—	2	1	no	—
54	C₂×C₄×C₇₉	—	4	1	no	—
55	C₂×C₄×C₇₉	—	4	1	no	—
56	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
59	C₂×Dic₇₉	—	2	1	no	—
60	C₂×Dic₇₉	—	2	1	no	—
61	C₂×Dic₇₉	—	2	2	no	—
62	C₂×Dic₇₉	—	4	4	no	—
63	C₂×Dic₇₉	—	4	1	no	—
64	Dic₁₅₈	—	2	1	no	—
65	Dic₁₅₈	—	4	4	no	—
66	Dic₁₅₈	—	4	1	no	—
67	Dic₁₅₈	—	4	1	no	—
68	Dic₁₅₈	—	4	1	no	—
69	Dic₁₅₈	—	4	1	no	—
70	Dic₁₅₈	SD₁₆	1	2	yes	complemented
71	C₈×C₇₉	—	2	2	no	—
72	C₈×C₇₉	—	4	1	no	—
73	C₇₉⋊C₈	—	4	1	no	—
74	C₂²×C₇₉	—	2	2	no	—
75	C₂²×C₇₉	—	4	1	no	—
76	C₂²×C₇₉	—	4	1	no	—
77	C₂²×C₇₉	#120	1	1	yes	characteristic
78	C₄×C₇₉	—	2	1	no	—
79	C₄×C₇₉	—	4	1	no	—
80	C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
81	C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
82	C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
83	C₄×C₇₉	#137	1	1	yes	characteristic
84	Dic₇₉	—	2	2	no	—
85	Dic₇₉	—	4	2	no	—
86	Dic₇₉	—	4	1	no	—
87	Dic₇₉	—	8	1	no	—
88	C₂×C₇₉	—	4	2	no	—
89	C₂×C₇₉	#448	1	1	yes	characteristic
90	C₂×C₇₉	#450	1	1	yes	characteristic
91	C₂×C₇₉	#456	1	1	yes	characteristic
92	#1651	—	79	1	no	maximal, Sylow
93	C₇₉	#1651	1	1	yes	characteristic, complemented, Sylow
94	D₄×Q₈	—	79	1	no	—
95	#498	—	79	1	no	—
96	#475	—	79	1	no	—
97	#461	—	79	1	no	—
98	#332	—	79	1	no	—
99	Q₈⋊C₈	—	79	1	no	—
100	D₄⋊C₈	D₇₉	1	1	yes	characteristic
101	C₂²×Q₈	—	158	1	no	—
102	C₂×SD₁₆	—	158	1	no	—
103	C₄⋊Q₈	—	79	1	no	—
104	C₄⋊Q₈	—	79	1	no	—
105	C₄⋊Q₈	—	158	1	no	—
106	C₂²⋊Q₈	—	158	2	no	—
107	C₂²⋊Q₈	—	158	1	no	—
108	C₄×Q₈	—	79	1	no	—
109	C₄×D₄	—	158	1	no	—
110	C₄×D₄	C₂×D₇₉	1	1	yes	characteristic
111	#106	—	158	1	no	—
112	#106	—	158	2	no	—
113	C₄⋊C₈	—	79	1	no	—
114	C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
115	#103	—	158	1	no	—
116	#103	—	158	1	no	—
117	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
118	C₂×Q₈	—	316	2	no	—
119	C₂×Q₈	—	316	2	no	—
120	C₂×Q₈	—	79	1	no	—
121	C₂×Q₈	—	158	2	no	—
122	C₂×Q₈	—	158	1	no	—
123	C₂×Q₈	—	158	1	no	—
124	C₂×D₄	D₃₁₆	1	1	yes	characteristic
125	C₂²×C₄	—	2	1	no	—
126	C₂²×C₄	—	158	2	no	—
127	SD₁₆	—	316	2	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	158	1	no	—
131	C₄⋊C₄	—	79	1	no	—
132	C₄⋊C₄	—	79	2	no	—
133	C₄⋊C₄	—	158	2	no	—
134	C₄⋊C₄	—	158	1	no	—
135	C₄⋊C₄	—	158	1	no	—
136	C₄⋊C₄	—	158	1	no	—
137	C₄⋊C₄	—	158	1	no	—
138	C₄⋊C₄	D₃₁₆	1	1	yes	characteristic
139	C₂²⋊C₄	—	2	1	no	—
140	C₂²⋊C₄	—	158	2	no	—
141	C₄²	—	158	1	no	—
142	C₄²	C₂²×D₇₉	1	1	yes	characteristic
143	C₂³	—	2	1	no	—
144	Q₈	—	316	4	no	—
145	Q₈	—	316	1	no	—
146	Q₈	—	316	1	no	—
147	Q₈	—	316	1	no	—
148	Q₈	—	316	1	no	—
149	Q₈	—	79	2	no	—
150	Q₈	—	158	1	no	—
151	D₄	—	2	1	no	—
152	D₄	#11929613	1	2	yes	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	316	4	no	—
157	C₂×C₄	—	316	1	no	—
158	C₂×C₄	—	158	1	no	—
159	C₂×C₄	—	158	2	no	—
160	C₂×C₄	—	158	1	no	—
161	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
162	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
163	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
164	C₈	—	2	2	no	—
165	C₈	—	4	1	no	—
166	C₈	—	316	1	no	—
167	C₂²	—	2	2	no	—
168	C₂²	—	4	1	no	—
169	C₂²	—	4	1	no	—
170	C₂²	#14579131	1	1	yes	characteristic
171	C₄	—	2	1	no	—
172	C₄	—	4	1	no	—
173	C₄	—	316	2	no	—
174	C₄	—	316	1	no	—
175	C₄	—	632	1	no	—
176	C₄	—	158	2	no	—
177	C₄	SD₁₆×D₇₉	1	1	yes	characteristic
178	C₄	C₂×#11929613	1	1	yes	characteristic
179	C₄	#14579185	1	1	yes	characteristic
180	C₄	#14579194	1	1	yes	characteristic
181	C₂	—	4	2	no	—
182	C₂	#19098704	1	1	yes	characteristic
183	C₂	#19098715	1	1	yes	characteristic
184	C₂	#19098747	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₄×Dic₁₅₈

C₂

yes

maximal, characteristic

#19098961

C₂

yes

maximal, characteristic, complemented

#19098849

C₂

yes

maximal, characteristic, complemented

#19098768

C₂

yes

maximal, characteristic

#19098310

C₂

yes

maximal, characteristic, complemented

#19098295

C₂

yes

maximal, characteristic, complemented

D₄⋊C₈×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

—

C₂×#11929612

—

#14579161

C₂²

yes

characteristic

#14579161

C₂²

yes

—

—

—

C₄×Dic₁₅₈

C₂²

yes

characteristic

D₄×Dic₇₉

—

#14579093

—

#14579093

—

C₄⋊(C₇₉⋊C₈)

C₂²

yes

characteristic

#14579086

—

#14579084

—

C₄×D₄×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

C₂²

yes

characteristic

C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

D₄

yes

characteristic

C₂²×Dic₇₉

—

#11929612

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

D₄

yes

characteristic

C₄⋊Dic₇₉

D₄

yes

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₂×D₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

D₄

yes

characteristic

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

C₂³

yes

characteristic

C₂³×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

SD₁₆

yes

—

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×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

SD₁₆

yes

complemented

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#120

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×SD₁₆

yes

characteristic

C₄×C₇₉

C₂×SD₁₆

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

D₄×Q₈

—

—

—

—

—

Q₈⋊C₈

—

100

D₄⋊C₈

D₇₉

yes

characteristic

101

C₂²×Q₈

—

158

—

102

C₂×SD₁₆

—

158

—

103

C₄⋊Q₈

—

104

C₄⋊Q₈

—

105

C₄⋊Q₈

—

158

—

106

C₂²⋊Q₈

—

158

—

107

C₂²⋊Q₈

—

158

—

108

C₄×Q₈

—

109

C₄×D₄

—

158

—

110

C₄×D₄

C₂×D₇₉

yes

characteristic

111

#106

—

158

—

112

#106

—

158

—

113

C₄⋊C₈

—

114

C₄⋊C₈

C₂×D₇₉

yes

characteristic

115

#103

—

158

—

116

#103

—

158

—

117

C₄×C₈

C₂×D₇₉

yes

characteristic

118

C₂×Q₈

—

316

—

119

C₂×Q₈

—

316

—

120

C₂×Q₈

—

121

C₂×Q₈

—

158

—

122

C₂×Q₈

—

158

—

123

C₂×Q₈

—

158

—

124

C₂×D₄

D₃₁₆

yes

characteristic

125

C₂²×C₄

—

126

C₂²×C₄

—

158

—

127

SD₁₆

—

316

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

158

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

158

—

134

C₄⋊C₄

—

158

—

135

C₄⋊C₄

—

158

—

136

C₄⋊C₄

—

158

—

137

C₄⋊C₄

—

158

—

138

C₄⋊C₄

D₃₁₆

yes

characteristic

139

C₂²⋊C₄

—

140

C₂²⋊C₄

—

158

—

141

C₄²

—

158

—

142

C₄²

C₂²×D₇₉

yes

characteristic

143

C₂³

—

144

Q₈

—

316

—

145

Q₈

—

316

—

146

Q₈

—

316

—

147

Q₈

—

316

—

148

Q₈

—

316

—

149

Q₈

—

150

Q₈

—

158

—

151

D₄

—

152

D₄

#11929613

yes

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

316

—

157

C₂×C₄

—

316

—

158

C₂×C₄

—

158

—

159

C₂×C₄

—

158

—

160

C₂×C₄

—

158

—

161

C₂×C₄

D₄×D₇₉

yes

characteristic

162

C₂×C₄

D₄×D₇₉

yes

characteristic

163

C₂×C₄

C₂×D₃₁₆

yes

characteristic

164

C₈

—

165

C₈

—

166

C₈

—

316

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

#14579131

yes

characteristic

171

C₄

—

172

C₄

—

173

C₄

—

316

—

174

C₄

—

316

—

175

C₄

—

632

—

176

C₄

—

158

—

177

C₄

SD₁₆×D₇₉

yes

characteristic

178

C₄

C₂×#11929613

yes

characteristic

179

C₄

#14579185

yes

characteristic

180

C₄

#14579194

yes

characteristic

181

C₂

—

182

C₂

#19098704

yes

characteristic

183

C₂

#19098715

yes

characteristic

184

C₂

#19098747

yes

characteristic

	subgroup	G/H
center	C₂²	#14579131
commutator, cosocle	C₄²×C₇₉	C₂³
Frattini	C₄²	C₂²×D₇₉
Fitting	D₄⋊C₈×C₇₉	C₂
socle	C₂²×C₇₉	#120
prime radical	C₂×D₄×C₇₉	D₄
hypercenter	D₄⋊C₈	D₇₉
hypocenter	C₇₉	#1651
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ D₄⋊C₈

subgroup

G/H

center

C₂²

#14579131

commutator, cosocle

C₄²×C₇₉

C₂³

Frattini

C₄²

C₂²×D₇₉

Fitting

D₄⋊C₈×C₇₉

C₂

socle

C₂²×C₇₉

#120

prime radical

C₂×D₄×C₇₉

D₄

hypercenter

D₄⋊C₈

D₇₉

hypocenter

C₇₉

#1651

LCS

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ D₄⋊C₈

N	H ≅ G/N	type
2. #19098961	C₂	semidirect
3. #19098849	C₂	semidirect
5. #19098310	C₂	semidirect
6. #19098295	C₂	semidirect
70. Dic₁₅₈	SD₁₆	semidirect
93. C₇₉	#1651	semidirect

H ≅ G/N

type

2. #19098961

C₂

semidirect

3. #19098849

C₂

semidirect

5. #19098310

C₂

semidirect

6. #19098295

C₂

semidirect

70. Dic₁₅₈

SD₁₆

semidirect

93. C₇₉

#1651

semidirect

Subgroups — #23762665

Conjugacy classes of subgroups

Complements