Group #34275976 — subgroups

Subgroups — #34275976

Order 20224.

706 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 187 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	#23773985	C₂	1	4	yes	maximal
2	#23773911	C₂	1	2	yes	maximal
3	#23763017	C₂	1	2	yes	maximal, complemented
4	#23763010	C₂	1	1	yes	maximal, characteristic, complemented
5	#23762996	C₂	1	4	yes	maximal, complemented
6	C₈²×D₇₉	C₂	1	1	yes	maximal, characteristic
7	C₇₉×#1710	C₂	1	1	yes	maximal, characteristic, complemented
8	#19099118	C₂²	1	4	yes	—
9	#19098968	C₂²	1	2	yes	—
10	#19098966	C₂²	1	4	yes	—
11	#19098964	—	2	4	no	—
12	#19098959	C₂²	1	4	yes	—
13	#19098942	C₂²	1	4	yes	—
14	#19098914	C₂²	1	4	yes	—
15	#19098803	—	2	8	no	—
16	#19098655	—	2	8	no	—
17	#19098600	—	2	4	no	—
18	C₄×C₈×D₇₉	C₂²	1	2	yes	—
19	C₄×C₈×D₇₉	C₂²	1	1	yes	characteristic
20	C₈×C₇₉⋊C₈	C₂²	1	2	yes	—
21	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
22	#500×C₇₉	C₂²	1	2	yes	—
23	#487×C₇₉	C₂²	1	4	yes	—
24	C₈²×C₇₉	C₂²	1	1	yes	characteristic
25	C₂×#11929614	—	2	8	no	—
26	C₂×C₈×D₇₉	D₄	1	4	yes	—
27	C₂×C₈×D₇₉	D₄	1	2	yes	—
28	#14579161	C₂³	1	4	yes	—
29	#14579152	—	2	4	no	—
30	#14579141	—	2	4	no	—
31	#14579137	—	2	4	no	—
32	Q₈×Dic₇₉	—	2	4	no	—
33	#14579126	—	2	4	no	—
34	#14579119	—	2	4	no	—
35	C₄²×D₇₉	C₂³	1	1	yes	characteristic
36	#14579095	—	2	4	no	—
37	#14579094	—	2	8	no	—
38	#14579092	—	2	8	no	—
39	#14579086	—	2	8	no	—
40	C₄×C₇₉⋊C₈	C₂³	1	2	yes	—
41	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
42	C₈×Dic₇₉	D₄	1	4	yes	—
43	C₈×Dic₇₉	D₄	1	2	yes	—
44	C₄⋊Q₈×C₇₉	C₂³	1	4	yes	—
45	#107×C₇₉	—	2	4	no	—
46	#103×C₇₉	—	2	8	no	—
47	C₄×C₈×C₇₉	C₂³	1	2	yes	—
48	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
49	C₂×Dic₁₅₈	—	2	4	no	—
50	C₂×C₄×D₇₉	C₂×D₄	1	2	yes	—
51	C₂×C₄×D₇₉	C₂×D₄	1	1	yes	characteristic
52	#11929614	—	4	16	no	—
53	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
54	C₂×C₇₉⋊C₈	C₂×D₄	1	4	yes	—
55	C₈×D₇₉	—	2	8	no	—
56	C₈×D₇₉	Q₁₆	1	8	yes	complemented
57	C₄⋊Dic₇₉	—	2	4	no	—
58	C₄⋊Dic₇₉	—	2	4	no	—
59	C₄⋊Dic₇₉	—	4	4	no	—
60	Dic₇₉⋊C₄	—	4	4	no	—
61	(C₂×D₇₉)⋊C₄	—	4	4	no	—
62	C₄×Dic₇₉	—	4	4	no	—
63	C₄×Dic₇₉	C₂×D₄	1	2	yes	—
64	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
65	C₂×Q₈×C₇₉	—	2	4	no	—
66	C₂×C₈×C₇₉	C₂×D₄	1	4	yes	—
67	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
68	C₄⋊C₄×C₇₉	—	2	4	no	—
69	C₄⋊C₄×C₇₉	—	2	4	no	—
70	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
71	C₂²×D₇₉	#127	1	1	yes	characteristic
72	Q₈×C₇₉	—	4	8	no	—
73	C₂×C₄×C₇₉	—	4	4	no	—
74	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
75	C₂×C₄×C₇₉	C₂²×D₄	1	2	yes	—
76	C₂×Dic₇₉	—	4	4	no	—
77	C₂×Dic₇₉	#127	1	2	yes	—
78	C₂×Dic₇₉	#127	1	1	yes	characteristic
79	C₄×D₇₉	—	2	4	no	—
80	C₄×D₇₉	C₂×Q₁₆	1	4	yes	—
81	Dic₁₅₈	—	4	8	no	—
82	C₈×C₇₉	—	2	4	no	—
83	C₈×C₇₉	C₂×Q₁₆	1	4	yes	—
84	C₇₉⋊C₈	C₂×Q₁₆	1	4	yes	—
85	C₇₉⋊C₈	#135	1	8	yes	—
86	C₂²×C₇₉	C₂×#127	1	1	yes	characteristic
87	C₂×D₇₉	—	2	2	no	—
88	C₂×D₇₉	#494	1	2	yes	—
89	C₄×C₇₉	—	8	4	no	—
90	C₄×C₇₉	C₂²×Q₁₆	1	2	yes	—
91	C₄×C₇₉	C₂×#135	1	4	yes	—
92	Dic₇₉	—	2	2	no	—
93	Dic₇₉	—	8	4	no	—
94	Dic₇₉	#494	1	2	yes	—
95	#19604	—	79	1	no	maximal, Sylow
96	C₂×C₇₉	C₂×#494	1	1	yes	characteristic
97	C₂×C₇₉	#3147	1	2	yes	—
98	D₇₉	—	2	2	no	—
99	#3170	—	79	4	no	—
100	#3160	—	79	2	no	—
101	#1713	—	79	2	no	—
102	#1710	—	79	1	no	—
103	#1710	D₇₉	1	1	yes	characteristic, complemented
104	#1707	—	79	4	no	—
105	C₂×C₈²	—	79	1	no	—
106	C₇₉	#19604	1	1	yes	characteristic, complemented, Sylow
107	#533	—	79	4	no	—
108	#500	—	79	4	no	—
109	#500	—	79	2	no	—
110	#500	C₂×D₇₉	1	2	yes	—
111	#499	—	158	4	no	—
112	#498	—	79	4	no	—
113	#494	—	79	4	no	—
114	#487	—	79	4	no	—
115	#487	C₂×D₇₉	1	4	yes	—
116	#467	—	158	8	no	—
117	C₄×Q₁₆	—	158	8	no	—
118	#427	—	158	4	no	—
119	C₂×C₄×C₈	—	79	1	no	—
120	C₂×C₄×C₈	—	79	2	no	—
121	C₈²	—	79	2	no	—
122	C₈²	—	79	1	no	—
123	C₈²	C₂×D₇₉	1	1	yes	characteristic
124	C₂×Q₁₆	—	158	8	no	—
125	C₂²×C₈	—	79	2	no	—
126	C₂²×C₈	—	79	4	no	—
127	C₄⋊Q₈	—	79	4	no	—
128	C₄⋊Q₈	C₂²×D₇₉	1	4	yes	—
129	#125	—	158	4	no	—
130	C₂²⋊Q₈	—	158	4	no	—
131	C₂²⋊Q₈	—	158	4	no	—
132	C₄×Q₈	—	158	4	no	—
133	C₄×Q₈	—	158	4	no	—
134	#117	—	158	4	no	—
135	C₂×C₄²	—	79	1	no	—
136	#107	—	2	4	no	—
137	#107	—	158	8	no	—
138	#107	—	158	4	no	—
139	#106	—	158	8	no	—
140	#103	—	2	8	no	—
141	#103	—	158	8	no	—
142	C₄×C₈	—	79	4	no	—
143	C₄×C₈	—	79	2	no	—
144	C₄×C₈	—	79	2	no	—
145	C₄×C₈	—	79	1	no	—
146	C₄×C₈	C₂²×D₇₉	1	2	yes	—
147	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
148	C₂×Q₈	—	2	4	no	—
149	C₂×Q₈	—	158	4	no	—
150	C₂²×C₄	—	79	1	no	—
151	C₂²×C₄	—	79	2	no	—
152	Q₁₆	—	316	16	no	—
153	C₂×C₈	—	79	2	no	—
154	C₂×C₈	—	79	4	no	—
155	C₂×C₈	—	79	8	no	—
156	C₂×C₈	—	158	8	no	—
157	C₂×C₈	D₄×D₇₉	1	4	yes	—
158	C₂×C₈	D₄×D₇₉	1	2	yes	—
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	316	4	no	—
162	C₄⋊C₄	—	316	4	no	—
163	C₄⋊C₄	—	158	4	no	—
164	C₄⋊C₄	—	158	4	no	—
165	C₂²⋊C₄	—	316	4	no	—
166	C₄²	—	316	4	no	—
167	C₄²	—	79	2	no	—
168	C₄²	—	79	1	no	—
169	C₄²	C₂³×D₇₉	1	1	yes	characteristic
170	C₂³	—	79	1	no	—
171	Q₈	—	4	8	no	—
172	Q₈	—	316	8	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	—	316	4	no	—
175	C₂×C₄	—	79	1	no	—
176	C₂×C₄	—	79	4	no	—
177	C₂×C₄	—	79	2	no	—
178	C₂×C₄	—	158	4	no	—
179	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
180	C₂×C₄	C₂×D₄×D₇₉	1	2	yes	—
181	C₈	—	2	4	no	—
182	C₈	—	79	4	no	—
183	C₈	—	79	8	no	—
184	C₈	Q₁₆×D₇₉	1	4	yes	—
185	C₂²	—	79	2	no	—
186	C₂²	—	158	2	no	—
187	C₂²	#127×D₇₉	1	1	yes	characteristic
188	C₄	—	8	4	no	—
189	C₄	—	79	2	no	—
190	C₄	—	632	4	no	—
191	C₄	—	158	2	no	—
192	C₄	C₂×Q₁₆×D₇₉	1	2	yes	—
193	C₄	#135×D₇₉	1	4	yes	—
194	C₂	—	158	2	no	—
195	C₂	#494×D₇₉	1	1	yes	characteristic
196	C₂	#23773825	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23773985

C₂

yes

maximal

#23773911

C₂

yes

maximal

#23763017

C₂

yes

maximal, complemented

#23763010

C₂

yes

maximal, characteristic, complemented

#23762996

C₂

yes

maximal, complemented

C₈²×D₇₉

C₂

yes

maximal, characteristic

C₇₉×#1710

C₂

yes

maximal, characteristic, complemented

#19099118

C₂²

yes

—

#19098968

C₂²

yes

—

#19098966

C₂²

yes

—

#19098964

—

#19098959

C₂²

yes

—

#19098942

C₂²

yes

—

#19098914

C₂²

yes

—

#19098803

—

#19098655

—

#19098600

—

C₄×C₈×D₇₉

C₂²

yes

—

C₄×C₈×D₇₉

C₂²

yes

characteristic

C₈×C₇₉⋊C₈

C₂²

yes

—

C₈×C₇₉⋊C₈

C₂²

yes

characteristic

#500×C₇₉

C₂²

yes

—

#487×C₇₉

C₂²

yes

—

C₈²×C₇₉

C₂²

yes

characteristic

C₂×#11929614

—

C₂×C₈×D₇₉

D₄

yes

—

C₂×C₈×D₇₉

D₄

yes

—

#14579161

C₂³

yes

—

#14579152

—

#14579141

—

#14579137

—

Q₈×Dic₇₉

—

#14579126

—

#14579119

—

C₄²×D₇₉

C₂³

yes

—

—

—

—

C₄×C₇₉⋊C₈

C₂³

yes

—

C₄×C₇₉⋊C₈

C₂³

yes

characteristic

C₈×Dic₇₉

D₄

yes

—

C₈×Dic₇₉

D₄

yes

—

C₄⋊Q₈×C₇₉

C₂³

yes

—

#107×C₇₉

—

#103×C₇₉

—

C₄×C₈×C₇₉

C₂³

yes

—

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×Dic₁₅₈

—

C₂×C₄×D₇₉

C₂×D₄

yes

—

C₂×C₄×D₇₉

C₂×D₄

yes

characteristic

#11929614

—

C₂×C₇₉⋊C₈

C₂×D₄

yes

—

C₂×C₇₉⋊C₈

C₂×D₄

yes

—

C₈×D₇₉

—

C₈×D₇₉

Q₁₆

yes

complemented

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

(C₂×D₇₉)⋊C₄

—

C₄×Dic₇₉

—

C₄×Dic₇₉

C₂×D₄

yes

—

C₄×Dic₇₉

C₂×D₄

yes

characteristic

C₂×Q₈×C₇₉

—

C₂×C₈×C₇₉

C₂×D₄

yes

—

C₂×C₈×C₇₉

C₂×D₄

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂⁴

yes

characteristic

C₂²×D₇₉

#127

yes

characteristic

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂²×D₄

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

#127

yes

—

C₂×Dic₇₉

#127

yes

characteristic

C₄×D₇₉

—

C₄×D₇₉

C₂×Q₁₆

yes

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

C₂×Q₁₆

yes

—

C₇₉⋊C₈

C₂×Q₁₆

yes

—

C₇₉⋊C₈

#135

yes

—

C₂²×C₇₉

C₂×#127

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

#494

yes

—

C₄×C₇₉

—

C₄×C₇₉

C₂²×Q₁₆

yes

—

C₄×C₇₉

C₂×#135

yes

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

#494

yes

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

D₇₉

—

#3170

—

100

#3160

—

101

#1713

—

102

#1710

—

103

#1710

D₇₉

yes

characteristic, complemented

104

#1707

—

105

C₂×C₈²

—

106

C₇₉

#19604

yes

characteristic, complemented, Sylow

107

#533

—

108

#500

—

109

#500

—

110

#500

C₂×D₇₉

yes

—

111

#499

—

158

—

112

#498

—

113

#494

—

114

#487

—

115

#487

C₂×D₇₉

yes

—

116

#467

—

158

—

117

C₄×Q₁₆

—

158

—

118

#427

—

158

—

119

C₂×C₄×C₈

—

120

C₂×C₄×C₈

—

121

C₈²

—

122

C₈²

—

123

C₈²

C₂×D₇₉

yes

characteristic

124

C₂×Q₁₆

—

158

—

125

C₂²×C₈

—

126

C₂²×C₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

C₂²×D₇₉

yes

—

129

#125

—

158

—

130

C₂²⋊Q₈

—

158

—

131

C₂²⋊Q₈

—

158

—

132

C₄×Q₈

—

158

—

133

C₄×Q₈

—

158

—

134

#117

—

158

—

135

C₂×C₄²

—

136

#107

—

137

#107

—

158

—

138

#107

—

158

—

139

#106

—

158

—

140

#103

—

141

#103

—

158

—

142

C₄×C₈

—

143

C₄×C₈

—

144

C₄×C₈

—

145

C₄×C₈

—

146

C₄×C₈

C₂²×D₇₉

yes

—

147

C₄×C₈

C₂²×D₇₉

yes

characteristic

148

C₂×Q₈

—

149

C₂×Q₈

—

158

—

150

C₂²×C₄

—

151

C₂²×C₄

—

152

Q₁₆

—

316

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

158

—

157

C₂×C₈

D₄×D₇₉

yes

—

158

C₂×C₈

D₄×D₇₉

yes

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

316

—

162

C₄⋊C₄

—

316

—

163

C₄⋊C₄

—

158

—

164

C₄⋊C₄

—

158

—

165

C₂²⋊C₄

—

316

—

166

C₄²

—

316

—

167

C₄²

—

168

C₄²

—

169

C₄²

C₂³×D₇₉

yes

characteristic

170

C₂³

—

171

Q₈

—

172

Q₈

—

316

—

173

C₂×C₄

—

174

C₂×C₄

—

316

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

158

—

179

C₂×C₄

C₂×D₄×D₇₉

yes

characteristic

180

C₂×C₄

C₂×D₄×D₇₉

yes

—

181

C₈

—

182

C₈

—

183

C₈

—

184

C₈

Q₁₆×D₇₉

yes

—

185

C₂²

—

186

C₂²

—

158

—

187

C₂²

#127×D₇₉

yes

characteristic

188

C₄

—

189

C₄

—

190

C₄

—

632

—

191

C₄

—

158

—

192

C₄

C₂×Q₁₆×D₇₉

yes

—

193

C₄

#135×D₇₉

yes

—

194

C₂

—

158

—

195

C₂

#494×D₇₉

yes

characteristic

196

C₂

#23773825

yes

—

	subgroup	G/H
center	C₂²	#127×D₇₉
commutator, cosocle	C₄²×C₇₉	C₂⁴
Frattini	C₄²	C₂³×D₇₉
Fitting	C₇₉×#1710	C₂
socle	C₂²×C₇₉	C₂×#127
prime radical	C₂²×D₇₉	#127
hypercenter	#1710	D₇₉
hypocenter	C₇₉	#19604
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1710

subgroup

G/H

center

C₂²

#127×D₇₉

commutator, cosocle

C₄²×C₇₉

C₂⁴

Frattini

C₄²

C₂³×D₇₉

Fitting

C₇₉×#1710

C₂

C₂²×C₇₉

C₂²×D₇₉

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #1710

N	H ≅ G/N	type
3. #23763017	C₂	semidirect
4. #23763010	C₂	semidirect
5. #23762996	C₂	semidirect
7. C₇₉×#1710	C₂	semidirect
56. C₈×D₇₉	Q₁₆	semidirect
103. #1710	D₇₉	semidirect
106. C₇₉	#19604	semidirect

H ≅ G/N

type

3. #23763017

C₂

semidirect

4. #23763010

C₂

semidirect

5. #23762996

C₂

semidirect

7. C₇₉×#1710

C₂

semidirect

56. C₈×D₇₉

Q₁₆

semidirect

103. #1710

D₇₉

semidirect

106. C₇₉

#19604

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#23773985	2
#23773911	2
#23763017	2
#23763010	2
#23762996	2
C₈²×D₇₉	2
C₇₉×#1710	2
#19604	79

subgroup

index

C₈²×D₇₉

C₇₉×#1710

#19604

Subgroups — #34275976

Conjugacy classes of subgroups

Complements

Maximal subgroups