Group #23775120 — subgroups

Subgroups — #23775120

Order 10112.

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

normal subgroups only

This group has 484 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×#14579184	C₂	1	2	yes	maximal, complemented
2	#19099216	C₂	1	2	yes	maximal, complemented
3	#19098801	C₂	1	2	yes	maximal, complemented
4	#19098765	C₂	1	2	yes	maximal, complemented
5	#19098724	C₂	1	4	yes	maximal, complemented
6	C₄×D₆₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×D₃₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	#494×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11929625	C₂²	1	4	yes	—
10	#14579184	—	2	8	no	—
11	C₂×#11929611	C₂²	1	4	yes	—
12	C₂×D₆₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₈×D₇₉	C₂²	1	2	yes	—
14	#14579146	—	2	4	no	—
15	#14579137	C₂²	1	4	yes	—
16	#14579136	—	2	2	no	—
17	C₄×D₃₁₆	C₂²	1	2	yes	—
18	C₄×D₃₁₆	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×D₇₉	—	2	2	no	—
20	#14579095	C₂²	1	1	yes	characteristic
21	C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
22	#14579085	C₂²	1	4	yes	—
23	#14579083	C₂²	1	2	yes	—
24	(C₂×D₇₉)⋊C₈	C₂²	1	2	yes	—
25	C₂×Q₁₆×C₇₉	C₂²	1	2	yes	—
26	C₂×Q₁₆×C₇₉	C₂²	1	2	yes	—
27	C₄⋊Q₈×C₇₉	C₂²	1	2	yes	—
28	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
29	#11929625	—	2	8	no	—
30	#11929625	—	4	4	no	—
31	C₂×D₃₁₆	—	2	4	no	—
32	C₂×D₃₁₆	D₄	1	1	yes	characteristic
33	C₂×D₃₁₆	C₂³	1	2	yes	—
34	C₂×C₄×D₇₉	—	2	4	no	—
35	C₂×C₄×D₇₉	—	2	2	no	—
36	C₂×C₄×D₇₉	C₂³	1	2	yes	—
37	#11929611	—	2	8	no	—
38	D₆₃₂	D₄	1	4	yes	complemented
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	4	no	—
44	(C₂×D₇₉)⋊C₄	—	2	4	no	—
45	(C₂×D₇₉)⋊C₄	—	2	2	no	—
46	(C₂×D₇₉)⋊C₄	D₄	1	2	yes	—
47	C₂×Q₈×C₇₉	C₂³	1	4	yes	—
48	Q₁₆×C₇₉	—	2	4	no	—
49	Q₁₆×C₇₉	—	4	2	no	—
50	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
52	C₄⋊C₄×C₇₉	—	2	2	no	—
53	C₄⋊C₄×C₇₉	—	2	2	no	—
54	C₄²×C₇₉	C₂³	1	1	yes	characteristic
55	C₂²×D₇₉	—	2	2	no	—
56	C₂²×D₇₉	C₂×D₄	1	2	yes	—
57	Q₈×C₇₉	—	2	4	no	—
58	Q₈×C₇₉	—	2	4	no	—
59	C₂×C₄×C₇₉	—	2	4	no	—
60	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
63	C₂×Dic₇₉	—	2	2	no	—
64	C₂×Dic₇₉	C₂×D₄	1	2	yes	—
65	D₃₁₆	—	2	2	no	—
66	D₃₁₆	—	2	2	no	—
67	D₃₁₆	—	4	8	no	—
68	D₃₁₆	C₂×D₄	1	4	yes	—
69	C₄×D₇₉	—	2	2	no	—
70	C₄×D₇₉	—	2	2	no	—
71	C₄×D₇₉	—	4	8	no	—
72	C₄×D₇₉	—	4	4	no	—
73	C₈×C₇₉	—	2	1	no	—
74	C₈×C₇₉	C₂×D₄	1	2	yes	—
75	C₇₉⋊C₈	—	2	2	no	—
76	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
77	C₂×D₇₉	—	2	4	no	—
78	C₂×D₇₉	—	2	2	no	—
79	C₂×D₇₉	—	2	2	no	—
80	C₂×D₇₉	—	2	2	no	—
81	C₂×D₇₉	—	4	2	no	—
82	C₂×D₇₉	—	4	2	no	—
83	C₄×C₇₉	—	2	1	no	—
84	C₄×C₇₉	—	4	4	no	—
85	C₄×C₇₉	#135	1	2	yes	complemented
86	C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
87	C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
88	Dic₇₉	—	2	2	no	—
89	Dic₇₉	—	4	2	no	—
90	C₂×C₇₉	D₄²	1	1	yes	characteristic
91	C₂×C₇₉	C₂×#135	1	1	yes	characteristic
92	C₂×C₇₉	#578	1	1	yes	characteristic
93	D₇₉	—	4	2	no	—
94	D₇₉	—	4	4	no	—
95	#3283	—	79	1	no	maximal, Sylow
96	C₇₉	#3283	1	1	yes	characteristic, complemented, Sylow
97	C₂×#135	—	79	2	no	—
98	#547	—	79	2	no	—
99	#494	D₇₉	1	1	yes	characteristic, complemented
100	#467	—	79	2	no	—
101	#461	—	79	2	no	—
102	#452	—	79	4	no	—
103	C₄×D₈	—	79	1	no	—
104	C₈×D₄	—	79	1	no	—
105	C₂×#41	—	79	4	no	—
106	#135	—	158	8	no	—
107	C₂×Q₁₆	C₂×D₇₉	1	2	yes	—
108	C₂×Q₁₆	C₂×D₇₉	1	2	yes	—
109	C₂×SD₁₆	—	79	4	no	—
110	C₂×D₈	—	79	1	no	—
111	C₂²×C₈	—	79	2	no	—
112	C₄⋊Q₈	C₂×D₇₉	1	2	yes	—
113	#123	—	158	4	no	—
114	C₂²⋊Q₈	—	79	4	no	—
115	#121	—	158	2	no	—
116	C₄×D₄	—	79	1	no	—
117	C₄×D₄	—	79	2	no	—
118	C₂×C₄⋊C₄	—	158	2	no	—
119	#107	—	79	1	no	—
120	C₄⋊C₈	—	79	1	no	—
121	#103	—	79	4	no	—
122	#102	—	79	2	no	—
123	C₂²⋊C₈	—	79	2	no	—
124	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
125	#41	—	316	4	no	—
126	#41	—	158	8	no	—
127	C₂×Q₈	C₂²×D₇₉	1	4	yes	—
128	C₂×D₄	—	79	1	no	—
129	C₂×D₄	—	79	2	no	—
130	C₂×D₄	—	158	4	no	—
131	C₂²×C₄	—	79	2	no	—
132	C₂²×C₄	—	158	4	no	—
133	C₂²×C₄	—	158	2	no	—
134	Q₁₆	—	2	4	no	—
135	Q₁₆	—	4	2	no	—
136	SD₁₆	—	158	8	no	—
137	D₈	—	79	4	no	—
138	C₂×C₈	—	79	2	no	—
139	C₂×C₈	—	158	4	no	—
140	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
141	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
142	C₄⋊C₄	—	2	2	no	—
143	C₄⋊C₄	—	2	2	no	—
144	C₄⋊C₄	—	79	1	no	—
145	C₄⋊C₄	—	79	2	no	—
146	C₄⋊C₄	—	158	4	no	—
147	C₂²⋊C₄	—	79	2	no	—
148	C₂²⋊C₄	—	158	4	no	—
149	C₂²⋊C₄	—	158	2	no	—
150	C₄²	C₂²×D₇₉	1	1	yes	characteristic
151	C₂³	—	79	2	no	—
152	C₂³	—	158	2	no	—
153	Q₈	—	2	4	no	—
154	Q₈	—	2	4	no	—
155	D₄	—	316	8	no	—
156	D₄	—	79	4	no	—
157	D₄	—	158	2	no	—
158	D₄	—	158	2	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	316	8	no	—
161	C₂×C₄	—	316	4	no	—
162	C₂×C₄	—	79	2	no	—
163	C₂×C₄	—	158	2	no	—
164	C₂×C₄	—	158	2	no	—
165	C₂×C₄	—	158	2	no	—
166	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
167	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
168	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
169	C₈	—	2	1	no	—
170	C₈	—	158	2	no	—
171	C₈	D₄×D₇₉	1	2	yes	—
172	C₂²	—	316	2	no	—
173	C₂²	—	316	2	no	—
174	C₂²	—	158	4	no	—
175	C₂²	—	158	2	no	—
176	C₂²	—	158	2	no	—
177	C₂²	—	158	2	no	—
178	C₂²	C₂×D₄×D₇₉	1	1	yes	characteristic
179	C₄	—	2	1	no	—
180	C₄	—	4	4	no	—
181	C₄	—	316	2	no	—
182	C₄	—	158	2	no	—
183	C₄	#14579184	1	2	yes	complemented
184	C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
185	C₄	#14579206	1	1	yes	characteristic
186	C₂	—	316	4	no	—
187	C₂	—	316	2	no	—
188	C₂	#19099198	1	1	yes	characteristic
189	C₂	C₂×#14579184	1	1	yes	characteristic
190	C₂	#19099376	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579184

C₂

yes

maximal, complemented

#19099216

C₂

yes

maximal, complemented

#19098801

C₂

yes

maximal, complemented

#19098765

C₂

yes

maximal, complemented

#19098724

C₂

yes

maximal, complemented

C₄×D₆₃₂

C₂

yes

maximal, characteristic, complemented

C₈×D₃₁₆

C₂

yes

maximal, characteristic, complemented

#494×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#11929625

C₂²

yes

—

#14579184

—

C₂×#11929611

C₂²

yes

—

C₂×D₆₃₂

C₂²

yes

characteristic

C₂×C₈×D₇₉

C₂²

yes

—

#14579146

—

#14579137

C₂²

yes

—

#14579136

—

C₄×D₃₁₆

C₂²

yes

—

C₄×D₃₁₆

C₂²

yes

characteristic

C₄⋊C₄×D₇₉

—

#14579095

C₂²

yes

characteristic

C₄⋊(C₇₉⋊C₈)

C₂²

yes

characteristic

#14579085

C₂²

yes

—

#14579083

C₂²

yes

—

(C₂×D₇₉)⋊C₈

C₂²

yes

—

C₂×Q₁₆×C₇₉

C₂²

yes

—

C₂×Q₁₆×C₇₉

C₂²

yes

—

C₄⋊Q₈×C₇₉

C₂²

yes

—

C₄×C₈×C₇₉

C₂²

yes

characteristic

#11929625

—

#11929625

—

C₂×D₃₁₆

—

C₂×D₃₁₆

D₄

yes

characteristic

C₂×D₃₁₆

C₂³

yes

—

C₂×C₄×D₇₉

—

C₂×C₄×D₇₉

—

C₂×C₄×D₇₉

C₂³

yes

—

#11929611

—

D₆₃₂

D₄

yes

complemented

C₂×C₇₉⋊C₈

C₂³

yes

—

C₈×D₇₉

—

C₄⋊Dic₇₉

D₄

yes

characteristic

C₄⋊Dic₇₉

C₂³

yes

—

Dic₇₉⋊C₄

—

(C₂×D₇₉)⋊C₄

—

(C₂×D₇₉)⋊C₄

—

(C₂×D₇₉)⋊C₄

D₄

yes

—

C₂×Q₈×C₇₉

C₂³

yes

—

Q₁₆×C₇₉

—

Q₁₆×C₇₉

—

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂³

yes

characteristic

C₂²×D₇₉

—

C₂²×D₇₉

C₂×D₄

yes

—

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

C₂×D₄

yes

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

C₂×D₄

yes

—

C₄×D₇₉

—

C₄×D₇₉

—

C₄×D₇₉

—

C₄×D₇₉

—

C₈×C₇₉

—

C₈×C₇₉

C₂×D₄

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#135

yes

complemented

C₄×C₇₉

C₂²×D₄

yes

characteristic

C₄×C₇₉

H(5,𝔽₂)

yes

characteristic

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

D₄²

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

D₇₉

—

D₇₉

—

#3283

—

maximal, Sylow

C₇₉

#3283

yes

characteristic, complemented, Sylow

C₂×#135

—

#547

—

#494

D₇₉

yes

characteristic, complemented

100

#467

—

101

#461

—

102

#452

—

103

C₄×D₈

—

104

C₈×D₄

—

105

C₂×#41

—

106

#135

—

158

—

107

C₂×Q₁₆

C₂×D₇₉

yes

—

108

C₂×Q₁₆

C₂×D₇₉

yes

—

109

C₂×SD₁₆

—

110

C₂×D₈

—

111

C₂²×C₈

—

112

C₄⋊Q₈

C₂×D₇₉

yes

—

113

#123

—

158

—

114

C₂²⋊Q₈

—

115

#121

—

158

—

116

C₄×D₄

—

117

C₄×D₄

—

118

C₂×C₄⋊C₄

—

158

—

119

#107

—

120

C₄⋊C₈

—

121

#103

—

122

#102

—

123

C₂²⋊C₈

—

124

C₄×C₈

C₂×D₇₉

yes

characteristic

125

#41

—

316

—

126

#41

—

158

—

127

C₂×Q₈

C₂²×D₇₉

yes

—

128

C₂×D₄

—

129

C₂×D₄

—

130

C₂×D₄

—

158

—

131

C₂²×C₄

—

132

C₂²×C₄

—

158

—

133

C₂²×C₄

—

158

—

134

Q₁₆

—

135

Q₁₆

—

136

SD₁₆

—

158

—

137

D₈

—

138

C₂×C₈

—

139

C₂×C₈

—

158

—

140

C₂×C₈

C₂²×D₇₉

yes

characteristic

141

C₂×C₈

C₂²×D₇₉

yes

characteristic

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

158

—

147

C₂²⋊C₄

—

148

C₂²⋊C₄

—

158

—

149

C₂²⋊C₄

—

158

—

150

C₄²

C₂²×D₇₉

yes

characteristic

151

C₂³

—

152

C₂³

—

158

—

153

Q₈

—

154

Q₈

—

155

D₄

—

316

—

156

D₄

—

157

D₄

—

158

—

158

D₄

—

158

—

159

C₂×C₄

—

160

C₂×C₄

—

316

—

161

C₂×C₄

—

316

—

162

C₂×C₄

—

163

C₂×C₄

—

158

—

164

C₂×C₄

—

158

—

165

C₂×C₄

—

158

—

166

C₂×C₄

D₄×D₇₉

yes

characteristic

167

C₂×C₄

D₄×D₇₉

yes

characteristic

168

C₂×C₄

C₂³×D₇₉

yes

characteristic

169

C₈

—

170

C₈

—

158

—

171

C₈

D₄×D₇₉

yes

—

172

C₂²

—

316

—

173

C₂²

—

316

—

174

C₂²

—

158

—

175

C₂²

—

158

—

176

C₂²

—

158

—

177

C₂²

—

158

—

178

C₂²

C₂×D₄×D₇₉

yes

characteristic

179

C₄

—

180

C₄

—

181

C₄

—

316

—

182

C₄

—

158

—

183

C₄

#14579184

yes

complemented

184

C₄

C₂×D₄×D₇₉

yes

characteristic

185

C₄

#14579206

yes

characteristic

186

C₂

—

316

—

187

C₂

—

316

—

188

C₂

#19099198

yes

characteristic

189

C₂

C₂×#14579184

yes

characteristic

190

C₂

#19099376

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₉
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	#494×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₄
hypercenter	#494	D₇₉
hypocenter	C₇₉	#3283
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

C₂×D₄×D₇₉

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

#494×C₇₉

C₂

socle

C₂²×C₇₉

C₂²×D₄

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
1. C₂×#14579184	C₂	semidirect
2. #19099216	C₂	semidirect
3. #19098801	C₂	semidirect
4. #19098765	C₂	semidirect
5. #19098724	C₂	semidirect
6. C₄×D₆₃₂	C₂	semidirect
7. C₈×D₃₁₆	C₂	semidirect
8. #494×C₇₉	C₂	semidirect
38. D₆₃₂	D₄	semidirect
85. C₄×C₇₉	#135	semidirect
96. C₇₉	#3283	semidirect
99. #494	D₇₉	semidirect
183. C₄	#14579184	semidirect

H ≅ G/N

type

1. C₂×#14579184

C₂

semidirect

2. #19099216

C₂

semidirect

3. #19098801

C₂

semidirect

4. #19098765

C₂

semidirect

5. #19098724

C₂

semidirect

6. C₄×D₆₃₂

C₂

semidirect

7. C₈×D₃₁₆

C₂

semidirect

8. #494×C₇₉

C₂

semidirect

38. D₆₃₂

D₄

semidirect

85. C₄×C₇₉

#135

semidirect

96. C₇₉

#3283

semidirect

99. #494

D₇₉

semidirect

183. C₄

#14579184

semidirect

Subgroups — #23775120

Conjugacy classes of subgroups

Complements