Group #34208161 — subgroups

Subgroups — #34208161

Order 20224.

292 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 83 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	#23765195	C₂	1	1	yes	maximal, characteristic, complemented
2	#23765010	C₂	1	1	yes	maximal, characteristic
3	#23763435	C₂	1	2	yes	maximal
4	#23761513	C₂	1	2	yes	maximal, complemented
5	C₇₉×#1385	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₇₉	C₂²	1	1	yes	characteristic
7	C₂×#14579101	C₂²	1	1	yes	characteristic
8	C₂×#14579097	C₂²	1	2	yes	—
9	#19098526	—	2	2	no	—
10	#19098393	—	2	2	no	—
11	(C₂×D₇₉)⋊C₁₆	—	2	2	no	—
12	#19098327	—	2	2	no	—
13	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
14	C₂×#108×C₇₉	C₂²	1	2	yes	—
15	C₂²×C₄×D₇₉	D₄	1	1	yes	characteristic
16	C₂×#11929608	—	2	2	no	—
17	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
18	C₂×C₈×D₇₉	—	2	1	no	—
19	C₂×C₈×D₇₉	—	2	2	no	—
20	C₂×C₈×D₇₉	—	2	2	no	—
21	C₂×C₈×D₇₉	—	2	1	no	—
22	#14579101	—	2	2	no	—
23	C₂×C₇₉⋊C₁₆	—	2	1	no	—
24	#14579097	—	2	2	no	—
25	#14579097	—	2	2	no	—
26	(C₂×D₇₉)⋊C₈	—	4	4	no	—
27	C₂×M₁₆×C₇₉	—	2	2	no	—
28	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
29	M₃₂×C₇₉	—	2	2	no	—
30	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
31	#108×C₇₉	—	2	2	no	—
32	#108×C₇₉	—	2	2	no	—
33	C₂³×D₇₉	Q₁₆	1	1	yes	characteristic
34	C₂²×Dic₇₉	Q₁₆	1	1	yes	characteristic
35	C₂×C₄×D₇₉	—	2	1	no	—
36	C₂×C₄×D₇₉	—	2	2	no	—
37	C₂×C₄×D₇₉	—	2	1	no	—
38	C₂×C₄×D₇₉	—	2	2	no	—
39	#11929608	—	4	2	no	—
40	#11929608	—	4	2	no	—
41	C₂×C₇₉⋊C₈	—	2	1	no	—
42	C₂×C₇₉⋊C₈	—	2	2	no	—
43	C₂×C₇₉⋊C₈	—	4	2	no	—
44	C₈×D₇₉	—	4	1	no	—
45	C₈×D₇₉	—	4	1	no	—
46	C₈×D₇₉	—	4	2	no	—
47	C₇₉⋊C₁₆	—	4	1	no	—
48	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
49	M₁₆×C₇₉	—	4	2	no	—
50	M₁₆×C₇₉	—	4	2	no	—
51	C₂×C₈×C₇₉	—	4	2	no	—
52	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
54	C₂×C₈×C₇₉	#41	1	2	yes	—
55	C₂×C₈×C₇₉	#41	1	2	yes	—
56	C₁₆×C₇₉	—	2	2	no	—
57	C₂³×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
58	C₂²×D₇₉	—	2	1	no	—
59	C₂²×D₇₉	—	2	2	no	—
60	C₂²×D₇₉	—	4	2	no	—
61	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
62	C₂×C₄×C₇₉	#123	1	2	yes	—
63	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
64	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
65	C₂×Dic₇₉	—	2	1	no	—
66	C₂×Dic₇₉	—	2	2	no	—
67	C₄×D₇₉	—	4	1	no	—
68	C₄×D₇₉	—	4	1	no	—
69	C₄×D₇₉	—	4	2	no	—
70	C₈×C₇₉	—	8	2	no	—
71	C₈×C₇₉	#123	1	1	yes	characteristic
72	C₈×C₇₉	#124	1	1	yes	characteristic
73	C₈×C₇₉	#126	1	2	yes	—
74	C₇₉⋊C₈	—	4	1	no	—
75	C₇₉⋊C₈	—	8	2	no	—
76	C₂²×C₇₉	—	4	1	no	—
77	C₂²×C₇₉	#467	1	1	yes	characteristic
78	C₂²×C₇₉	#484	1	2	yes	—
79	C₂×D₇₉	—	2	2	no	—
80	C₂×D₇₉	—	2	4	no	—
81	C₂×D₇₉	—	4	1	no	—
82	C₂×D₇₉	—	4	2	no	—
83	C₂×D₇₉	—	4	1	no	—
84	C₄×C₇₉	#397	1	1	yes	characteristic
85	C₄×C₇₉	#469	1	1	yes	characteristic
86	C₄×C₇₉	#482	1	2	yes	—
87	Dic₇₉	—	4	1	no	—
88	#13498	—	79	1	no	maximal, Sylow
89	C₂×C₇₉	—	2	1	no	—
90	C₂×C₇₉	—	2	2	no	—
91	C₂×C₇₉	#2076	1	1	yes	characteristic
92	D₇₉	—	4	2	no	—
93	#2161	—	79	1	no	—
94	#2112	—	79	1	no	—
95	#1810	—	79	2	no	—
96	#1388	—	79	2	no	—
97	#1385	D₇₉	1	1	yes	characteristic, complemented
98	C₇₉	#13498	1	1	yes	characteristic, complemented, Sylow
99	C₂³×C₈	—	79	1	no	—
100	C₂×M₃₂	—	79	1	no	—
101	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
102	C₂×#108	—	79	2	no	—
103	C₂×#108	C₂×D₇₉	1	2	yes	—
104	#407	—	158	2	no	—
105	#368	—	158	2	no	—
106	C₂²⋊C₁₆	—	158	2	no	—
107	#341	—	158	2	no	—
108	C₂³×C₄	—	79	1	no	—
109	C₂×M₁₆	—	2	2	no	—
110	C₂×M₁₆	—	158	2	no	—
111	C₂²×C₈	—	79	1	no	—
112	C₂²×C₈	—	158	1	no	—
113	C₂²×C₈	—	158	2	no	—
114	C₂²×C₈	—	158	2	no	—
115	C₂²×C₈	—	158	1	no	—
116	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
117	M₃₂	—	2	2	no	—
118	M₃₂	—	158	2	no	—
119	C₂×C₁₆	—	158	1	no	—
120	C₂×C₁₆	D₃₁₆	1	2	yes	—
121	#108	—	2	2	no	—
122	#108	—	2	2	no	—
123	#108	—	158	2	no	—
124	#108	—	158	2	no	—
125	C₂²⋊C₈	—	316	4	no	—
126	C₂⁴	—	79	1	no	—
127	C₂²×C₄	—	79	1	no	—
128	C₂²×C₄	—	158	1	no	—
129	C₂²×C₄	—	158	2	no	—
130	C₂²×C₄	—	158	1	no	—
131	C₂²×C₄	—	158	2	no	—
132	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
133	M₁₆	—	4	2	no	—
134	M₁₆	—	4	2	no	—
135	M₁₆	—	316	2	no	—
136	M₁₆	—	316	2	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	—	316	1	no	—
139	C₂×C₈	—	316	1	no	—
140	C₂×C₈	—	316	2	no	—
141	C₂×C₈	—	316	2	no	—
142	C₂×C₈	—	158	1	no	—
143	C₂×C₈	—	158	2	no	—
144	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
145	C₂×C₈	#11929623	1	2	yes	—
146	C₂×C₈	#11929624	1	2	yes	—
147	C₂×C₈	#11929625	1	1	yes	characteristic
148	C₁₆	—	2	2	no	—
149	C₁₆	—	316	1	no	—
150	C₂³	—	316	2	no	—
151	C₂³	—	158	2	no	—
152	C₂³	—	158	1	no	—
153	C₂³	Q₁₆×D₇₉	1	1	yes	characteristic
154	C₂×C₄	—	316	2	no	—
155	C₂×C₄	—	316	1	no	—
156	C₂×C₄	—	316	1	no	—
157	C₂×C₄	—	158	2	no	—
158	C₂×C₄	—	158	1	no	—
159	C₂×C₄	#14579136	1	1	yes	characteristic
160	C₂×C₄	#14579143	1	2	yes	—
161	C₂×C₄	#14579180	1	1	yes	characteristic
162	C₂×C₄	#14579188	1	2	yes	—
163	C₈	—	8	2	no	—
164	C₈	—	316	1	no	—
165	C₈	—	632	2	no	—
166	C₈	#14579145	1	1	yes	characteristic
167	C₈	#14579149	1	1	yes	characteristic
168	C₈	#14579156	1	2	yes	—
169	C₂²	—	4	1	no	—
170	C₂²	—	316	2	no	—
171	C₂²	—	316	1	no	—
172	C₂²	—	316	1	no	—
173	C₂²	—	158	4	no	—
174	C₂²	—	158	2	no	—
175	C₂²	#19098805	1	1	yes	characteristic
176	C₂²	#19098897	1	2	yes	—
177	C₄	—	316	1	no	—
178	C₄	#19098490	1	1	yes	characteristic
179	C₄	#19098817	1	1	yes	characteristic
180	C₄	#19098885	1	2	yes	—
181	C₂	—	2	2	no	—
182	C₂	—	2	1	no	—
183	C₂	—	316	2	no	—
184	C₂	#23764830	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765195

C₂

yes

maximal, characteristic, complemented

#23765010

C₂

yes

maximal, characteristic

#23763435

C₂

yes

maximal

#23761513

C₂

yes

maximal, complemented

C₇₉×#1385

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×D₇₉

C₂²

yes

characteristic

C₂×#14579101

C₂²

yes

characteristic

C₂×#14579097

C₂²

yes

—

#19098526

—

#19098393

—

(C₂×D₇₉)⋊C₁₆

—

#19098327

—

C₂×M₃₂×C₇₉

C₂²

yes

characteristic

C₂×#108×C₇₉

C₂²

yes

—

C₂²×C₄×D₇₉

D₄

yes

characteristic

C₂×#11929608

—

C₂²×C₇₉⋊C₈

D₄

yes

characteristic

C₂×C₈×D₇₉

—

C₂×C₈×D₇₉

—

C₂×C₈×D₇₉

—

C₂×C₈×D₇₉

—

#14579101

—

C₂×C₇₉⋊C₁₆

—

#14579097

—

#14579097

—

(C₂×D₇₉)⋊C₈

—

C₂×M₁₆×C₇₉

—

C₂²×C₈×C₇₉

C₂³

yes

characteristic

M₃₂×C₇₉

—

C₂×C₁₆×C₇₉

D₄

yes

—

#108×C₇₉

—

#108×C₇₉

—

C₂³×D₇₉

Q₁₆

yes

characteristic

C₂²×Dic₇₉

Q₁₆

yes

characteristic

C₂×C₄×D₇₉

—

C₂×C₄×D₇₉

—

C₂×C₄×D₇₉

—

C₂×C₄×D₇₉

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₈×D₇₉

—

C₈×D₇₉

—

C₈×D₇₉

—

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂×D₄

yes

C₂×C₈×C₇₉

yes

C₂×C₈×C₇₉

yes

—

C₂×C₈×C₇₉

#41

yes

—

C₁₆×C₇₉

—

C₂³×C₇₉

C₂×Q₁₆

yes

characteristic

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂×C₄×C₇₉

#121

yes

characteristic

C₂×C₄×C₇₉

#123

yes

—

C₂×C₄×C₇₉

#135

yes

characteristic

C₂×C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₄×D₇₉

—

C₄×D₇₉

—

C₄×D₇₉

—

C₈×C₇₉

—

C₈×C₇₉

yes

C₈×C₇₉

yes

C₈×C₇₉

yes

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

#467

yes

characteristic

C₂²×C₇₉

#484

yes

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

Dic₇₉

—

#13498

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#2076

yes

characteristic

D₇₉

—

—

—

—

—

D₇₉

yes

characteristic, complemented

C₇₉

#13498

yes

characteristic, complemented, Sylow

C₂³×C₈

—

100

C₂×M₃₂

—

101

C₂×M₃₂

C₂×D₇₉

yes

characteristic

102

C₂×#108

—

103

C₂×#108

C₂×D₇₉

yes

—

104

#407

—

158

—

105

#368

—

158

—

106

C₂²⋊C₁₆

—

158

—

107

#341

—

158

—

108

C₂³×C₄

—

109

C₂×M₁₆

—

110

C₂×M₁₆

—

158

—

111

C₂²×C₈

—

112

C₂²×C₈

—

158

—

113

C₂²×C₈

—

158

—

114

C₂²×C₈

—

158

—

115

C₂²×C₈

—

158

—

116

C₂²×C₈

C₂²×D₇₉

yes

characteristic

117

M₃₂

—

118

M₃₂

—

158

—

119

C₂×C₁₆

—

158

—

120

C₂×C₁₆

D₃₁₆

yes

—

121

#108

—

122

#108

—

123

#108

—

158

—

124

#108

—

158

—

125

C₂²⋊C₈

—

316

—

126

C₂⁴

—

127

C₂²×C₄

—

128

C₂²×C₄

—

158

—

129

C₂²×C₄

—

158

—

130

C₂²×C₄

—

158

—

131

C₂²×C₄

—

158

—

132

C₂²×C₄

D₄×D₇₉

yes

characteristic

133

M₁₆

—

134

M₁₆

—

135

M₁₆

—

316

—

136

M₁₆

—

316

—

137

C₂×C₈

—

138

C₂×C₈

—

316

—

139

C₂×C₈

—

316

—

140

C₂×C₈

—

316

—

141

C₂×C₈

—

316

—

142

C₂×C₈

—

158

—

143

C₂×C₈

—

158

—

144

C₂×C₈

C₂×D₃₁₆

yes

characteristic

145

C₂×C₈

#11929623

yes

—

146

C₂×C₈

#11929624

yes

—

147

C₂×C₈

#11929625

yes

characteristic

148

C₁₆

—

149

C₁₆

—

316

—

150

C₂³

—

316

—

151

C₂³

—

158

—

152

C₂³

—

158

—

153

C₂³

Q₁₆×D₇₉

yes

characteristic

154

C₂×C₄

—

316

—

155

C₂×C₄

—

316

—

156

C₂×C₄

—

316

—

157

C₂×C₄

—

158

—

158

C₂×C₄

—

158

—

159

C₂×C₄

#14579136

yes

characteristic

160

C₂×C₄

#14579143

yes

—

161

C₂×C₄

#14579180

yes

characteristic

162

C₂×C₄

#14579188

yes

—

163

C₈

—

164

C₈

—

316

—

165

C₈

—

632

—

166

C₈

#14579145

yes

characteristic

167

C₈

#14579149

yes

characteristic

168

C₈

#14579156

yes

—

169

C₂²

—

170

C₂²

—

316

—

171

C₂²

—

316

—

172

C₂²

—

316

—

173

C₂²

—

158

—

174

C₂²

—

158

—

175

C₂²

#19098805

yes

characteristic

176

C₂²

#19098897

yes

—

177

C₄

—

316

—

178

C₄

#19098490

yes

characteristic

179

C₄

#19098817

yes

characteristic

180

C₄

#19098885

yes

—

181

C₂

—

182

C₂

—

183

C₂

—

316

—

184

C₂

#23764830

yes

characteristic

	subgroup	G/H
center	C₄	#19098817
commutator, cosocle	C₂²×C₈×C₇₉	C₂³
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1385	C₂
socle	C₂×C₇₉	#2076
prime radical	C₂³×D₇₉	Q₁₆
hypercenter	#1385	D₇₉
hypocenter	C₇₉	#13498
LCS	G ▷ C₂²×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

subgroup

G/H

center

C₄

#19098817

commutator, cosocle

C₂²×C₈×C₇₉

C₂³

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₇₉×#1385

C₂

socle

C₂×C₇₉

#2076

prime radical

C₂³×D₇₉

Q₁₆

D₇₉

C₇₉

G ▷ C₂²×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

N	H ≅ G/N	type
1. #23765195	C₂	semidirect
4. #23761513	C₂	semidirect
5. C₇₉×#1385	C₂	semidirect
97. #1385	D₇₉	semidirect
98. C₇₉	#13498	semidirect

H ≅ G/N

type

1. #23765195

C₂

semidirect

4. #23761513

C₂

semidirect

5. C₇₉×#1385

C₂

semidirect

97. #1385

D₇₉

semidirect

98. C₇₉

#13498

semidirect

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

subgroup	index
#23765195	2
#23765010	2
#23763435	2
#23761513	2
C₇₉×#1385	2
#13498	79

subgroup

index

C₇₉×#1385

Subgroups — #34208161

Conjugacy classes of subgroups

Complements

Maximal subgroups