Group #23762663 — subgroups

Subgroups — #23762663

Order 10112.

234 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 65 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	#19098959	C₂	1	1	yes	maximal, characteristic, complemented
2	#19098848	C₂	1	1	yes	maximal, characteristic, complemented
3	#19098768	C₂	1	1	yes	maximal, characteristic
4	#19098309	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₈⋊(C₇₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄⋊(C₇₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
7	D₄×Q₈×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#11929612	—	2	1	no	—
9	#14579161	C₂²	1	1	yes	characteristic
10	#14579092	—	2	2	no	—
11	#14579092	—	2	1	no	—
12	C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
13	C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
14	#14579085	—	2	1	no	—
15	#14579084	—	2	1	no	—
16	C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₂²×Q₈×C₇₉	—	2	1	no	—
18	C₄⋊Q₈×C₇₉	—	2	1	no	—
19	C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
20	C₂²⋊Q₈×C₇₉	—	2	1	no	—
21	C₂²⋊Q₈×C₇₉	—	2	2	no	—
22	C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₇₉	—	2	1	no	—
24	C₄×D₄×C₇₉	C₂²	1	1	yes	characteristic
25	C₂×Dic₁₅₈	—	2	1	no	—
26	#11929612	—	4	2	no	—
27	C₂×C₇₉⋊C₈	—	2	1	no	—
28	C₂×C₇₉⋊C₈	—	2	1	no	—
29	C₂×C₇₉⋊C₈	—	2	1	no	—
30	C₄⋊Dic₇₉	—	2	1	no	—
31	C₄⋊Dic₇₉	—	2	1	no	—
32	C₂×Q₈×C₇₉	—	2	2	no	—
33	C₂×Q₈×C₇₉	—	2	1	no	—
34	C₂×Q₈×C₇₉	—	4	2	no	—
35	C₂×Q₈×C₇₉	—	4	2	no	—
36	C₂×Q₈×C₇₉	D₄	1	1	yes	characteristic
37	C₂×D₄×C₇₉	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₇₉	—	2	2	no	—
39	C₂²×C₄×C₇₉	—	2	1	no	—
40	C₄⋊C₄×C₇₉	—	2	2	no	—
41	C₄⋊C₄×C₇₉	—	2	1	no	—
42	C₄⋊C₄×C₇₉	—	2	1	no	—
43	C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
44	C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
45	C₄⋊C₄×C₇₉	D₄	1	2	yes	—
46	C₂²⋊C₄×C₇₉	—	2	2	no	—
47	C₂²⋊C₄×C₇₉	—	2	1	no	—
48	C₄²×C₇₉	—	2	1	no	—
49	C₄²×C₇₉	C₂³	1	1	yes	characteristic
50	C₂³×C₇₉	—	2	1	no	—
51	Q₈×C₇₉	—	2	1	no	—
52	Q₈×C₇₉	—	4	4	no	—
53	Q₈×C₇₉	—	4	1	no	—
54	Q₈×C₇₉	—	4	1	no	—
55	Q₈×C₇₉	SD₁₆	1	2	yes	complemented
56	D₄×C₇₉	—	2	1	no	—
57	D₄×C₇₉	SD₁₆	1	2	yes	—
58	C₂×C₄×C₇₉	—	2	1	no	—
59	C₂×C₄×C₇₉	—	2	1	no	—
60	C₂×C₄×C₇₉	—	2	2	no	—
61	C₂×C₄×C₇₉	—	2	1	no	—
62	C₂×C₄×C₇₉	—	4	4	no	—
63	C₂×C₄×C₇₉	—	4	1	no	—
64	C₂×C₄×C₇₉	—	4	1	no	—
65	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
68	C₂×Dic₇₉	—	4	1	no	—
69	Dic₁₅₈	—	4	1	no	—
70	Dic₁₅₈	—	4	1	no	—
71	C₇₉⋊C₈	—	2	2	no	—
72	C₇₉⋊C₈	—	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	2	no	—
79	C₄×C₇₉	—	2	1	no	—
80	C₄×C₇₉	—	4	2	no	—
81	C₄×C₇₉	—	4	1	no	—
82	C₄×C₇₉	—	4	1	no	—
83	C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
84	C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
85	C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
86	C₄×C₇₉	#137	1	1	yes	characteristic
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	D₇₉	1	1	yes	characteristic
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	—	79	1	no	—
101	C₂²×Q₈	—	2	1	no	—
102	C₂×SD₁₆	—	158	1	no	—
103	C₄⋊Q₈	—	2	1	no	—
104	C₄⋊Q₈	—	79	1	no	—
105	C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
106	C₂²⋊Q₈	—	2	1	no	—
107	C₂²⋊Q₈	—	2	2	no	—
108	C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
109	C₄×D₄	—	2	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	—	79	1	no	—
115	#103	—	158	1	no	—
116	#103	—	158	1	no	—
117	C₄×C₈	—	79	1	no	—
118	C₂×Q₈	—	2	2	no	—
119	C₂×Q₈	—	2	1	no	—
120	C₂×Q₈	—	4	2	no	—
121	C₂×Q₈	—	4	2	no	—
122	C₂×Q₈	—	158	1	no	—
123	C₂×Q₈	C₂²⋊D₇₉	1	1	yes	characteristic
124	C₂×D₄	C₂²⋊D₇₉	1	1	yes	characteristic
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	—	2	1	no	—
127	SD₁₆	—	316	2	no	—
128	C₂×C₈	—	158	1	no	—
129	C₂×C₈	—	158	1	no	—
130	C₂×C₈	—	158	1	no	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	—	2	1	no	—
134	C₄⋊C₄	—	158	1	no	—
135	C₄⋊C₄	—	158	1	no	—
136	C₄⋊C₄	C₂²⋊D₇₉	1	1	yes	characteristic
137	C₄⋊C₄	C₂²⋊D₇₉	1	1	yes	characteristic
138	C₄⋊C₄	C₂²⋊D₇₉	1	2	yes	—
139	C₂²⋊C₄	—	2	2	no	—
140	C₂²⋊C₄	—	2	1	no	—
141	C₄²	—	2	1	no	—
142	C₄²	C₂²×D₇₉	1	1	yes	characteristic
143	C₂³	—	2	1	no	—
144	Q₈	—	2	1	no	—
145	Q₈	—	4	4	no	—
146	Q₈	—	4	1	no	—
147	Q₈	—	4	1	no	—
148	Q₈	—	316	1	no	—
149	Q₈	—	316	1	no	—
150	Q₈	#11929612	1	2	yes	complemented
151	D₄	—	2	1	no	—
152	D₄	#11929611	1	2	yes	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	—	316	1	no	—
161	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
162	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
163	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
164	C₈	—	316	1	no	—
165	C₈	—	316	1	no	—
166	C₈	—	158	2	no	—
167	C₂²	—	2	2	no	—
168	C₂²	—	4	1	no	—
169	C₂²	—	4	1	no	—
170	C₂²	#14579130	1	1	yes	characteristic
171	C₄	—	2	2	no	—
172	C₄	—	2	1	no	—
173	C₄	—	4	2	no	—
174	C₄	—	4	1	no	—
175	C₄	—	4	1	no	—
176	C₄	—	632	1	no	—
177	C₄	C₂×#11929611	1	1	yes	characteristic
178	C₄	C₂×#11929612	1	1	yes	characteristic
179	C₄	#14579187	1	1	yes	characteristic
180	C₄	#14579192	1	1	yes	characteristic
181	C₂	—	4	2	no	—
182	C₂	#19098702	1	1	yes	characteristic
183	C₂	#19098714	1	1	yes	characteristic
184	C₂	#19098746	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19098959

C₂

yes

maximal, characteristic, complemented

#19098848

C₂

yes

maximal, characteristic, complemented

#19098768

C₂

yes

maximal, characteristic

#19098309

C₂

yes

maximal, characteristic, complemented

Q₈⋊(C₇₉⋊C₈)

C₂

yes

maximal, characteristic, complemented

D₄⋊(C₇₉⋊C₈)

C₂

yes

maximal, characteristic

D₄×Q₈×C₇₉

C₂

yes

maximal, characteristic

C₂×#11929612

—

#14579161

C₂²

yes

characteristic

#14579092

—

#14579092

—

C₄⋊(C₇₉⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₇₉⋊C₈)

C₂²

yes

characteristic

#14579085

—

#14579084

—

C₄×C₇₉⋊C₈

C₂²

yes

characteristic

C₂²×Q₈×C₇₉

—

C₄⋊Q₈×C₇₉

—

C₄⋊Q₈×C₇₉

C₂²

yes

characteristic

C₂²⋊Q₈×C₇₉

—

C₂²⋊Q₈×C₇₉

—

C₄×Q₈×C₇₉

C₂²

yes

characteristic

C₄×D₄×C₇₉

—

C₄×D₄×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

—

#11929612

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

D₄

yes

characteristic

C₂×D₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

D₄

yes

characteristic

C₄⋊C₄×C₇₉

D₄

yes

characteristic

C₄⋊C₄×C₇₉

D₄

yes

—

C₂²⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂³

yes

characteristic

C₂³×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

SD₁₆

yes

complemented

D₄×C₇₉

—

D₄×C₇₉

SD₁₆

yes

—

C₂×C₄×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

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#120

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×SD₁₆

yes

characteristic

C₄×C₇₉

C₂×SD₁₆

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

D₄×Q₈

D₇₉

yes

—

—

—

—

Q₈⋊C₈

—

100

D₄⋊C₈

—

101

C₂²×Q₈

—

102

C₂×SD₁₆

—

158

—

103

C₄⋊Q₈

—

104

C₄⋊Q₈

—

105

C₄⋊Q₈

C₂×D₇₉

yes

characteristic

106

C₂²⋊Q₈

—

107

C₂²⋊Q₈

—

108

C₄×Q₈

C₂×D₇₉

yes

characteristic

109

C₄×D₄

—

110

C₄×D₄

C₂×D₇₉

yes

characteristic

111

#106

—

158

—

112

#106

—

158

—

113

C₄⋊C₈

—

114

C₄⋊C₈

—

115

#103

—

158

—

116

#103

—

158

—

117

C₄×C₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂×Q₈

—

121

C₂×Q₈

—

122

C₂×Q₈

—

158

—

123

C₂×Q₈

C₂²⋊D₇₉

yes

characteristic

124

C₂×D₄

C₂²⋊D₇₉

yes

characteristic

125

C₂²×C₄

—

126

C₂²×C₄

—

127

SD₁₆

—

316

—

128

C₂×C₈

—

158

—

129

C₂×C₈

—

158

—

130

C₂×C₈

—

158

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

158

—

135

C₄⋊C₄

—

158

—

136

C₄⋊C₄

C₂²⋊D₇₉

yes

characteristic

137

C₄⋊C₄

C₂²⋊D₇₉

yes

characteristic

138

C₄⋊C₄

C₂²⋊D₇₉

yes

—

139

C₂²⋊C₄

—

140

C₂²⋊C₄

—

141

C₄²

—

142

C₄²

C₂²×D₇₉

yes

characteristic

143

C₂³

—

144

Q₈

—

145

Q₈

—

146

Q₈

—

147

Q₈

—

148

Q₈

—

316

—

149

Q₈

—

316

—

150

Q₈

#11929612

yes

complemented

151

D₄

—

152

D₄

#11929611

yes

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

316

—

161

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

162

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

163

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

164

C₈

—

316

—

165

C₈

—

316

—

166

C₈

—

158

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

#14579130

yes

characteristic

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

632

—

177

C₄

C₂×#11929611

yes

characteristic

178

C₄

C₂×#11929612

yes

characteristic

179

C₄

#14579187

yes

characteristic

180

C₄

#14579192

yes

characteristic

181

C₂

—

182

C₂

#19098702

yes

characteristic

183

C₂

#19098714

yes

characteristic

184

C₂

#19098746

yes

characteristic

	subgroup	G/H
center	C₂²	#14579130
commutator, cosocle	C₄²×C₇₉	C₂³
Frattini	C₄²	C₂²×D₇₉
Fitting	D₄×Q₈×C₇₉	C₂
socle	C₂²×C₇₉	#120
prime radical	C₂×D₄×C₇₉	D₄
hypercenter	D₄×Q₈	D₇₉
hypocenter	C₇₉	#1651
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ D₄×Q₈

subgroup

G/H

center

C₂²

#14579130

commutator, cosocle

C₄²×C₇₉

C₂³

Frattini

C₄²

C₂²×D₇₉

Fitting

D₄×Q₈×C₇₉

C₂

socle

C₂²×C₇₉

#120

prime radical

C₂×D₄×C₇₉

D₄

hypercenter

D₄×Q₈

D₇₉

hypocenter

C₇₉

#1651

LCS

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ D₄×Q₈

N	H ≅ G/N	type
1. #19098959	C₂	semidirect
2. #19098848	C₂	semidirect
4. #19098309	C₂	semidirect
5. Q₈⋊(C₇₉⋊C₈)	C₂	semidirect
55. Q₈×C₇₉	SD₁₆	semidirect
93. C₇₉	#1651	semidirect
150. Q₈	#11929612	semidirect

H ≅ G/N

type

1. #19098959

C₂

semidirect

2. #19098848

C₂

semidirect

4. #19098309

C₂

semidirect

5. Q₈⋊(C₇₉⋊C₈)

C₂

semidirect

55. Q₈×C₇₉

SD₁₆

semidirect

93. C₇₉

#1651

semidirect

150. Q₈

#11929612

semidirect

Subgroups — #23762663

Conjugacy classes of subgroups

Complements