Group #34208147 — subgroups

Subgroups — #34208147

Order 20224.

284 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 91 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×#19098391	C₂	1	1	yes	maximal, characteristic, complemented
2	#23763464	C₂	1	2	yes	maximal
3	#23761509	C₂	1	2	yes	maximal, complemented
4	#23761494	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₉×#2113	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579097	C₂²	1	2	yes	—
7	C₂×#14579095	C₂²	1	2	yes	—
8	#19098564	—	2	2	no	—
9	#19098391	—	2	2	no	—
10	#19098331	—	2	2	no	—
11	C₂×#131×C₇₉	C₂²	1	1	yes	characteristic
12	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
14	C₂²⋊C₁₆×C₇₉	—	2	2	no	—
15	C₂×#11929608	—	2	2	no	—
16	C₂×C₄⋊Dic₇₉	—	2	2	no	—
17	#14579097	—	2	2	no	—
18	#14579097	—	2	2	no	—
19	#14579095	—	2	2	no	—
20	#14579095	—	2	2	no	—
21	#14579085	—	4	2	no	—
22	#14579082	—	4	2	no	—
23	C₂×#41×C₇₉	D₄	1	1	yes	characteristic
24	#131×C₇₉	—	2	2	no	—
25	#131×C₇₉	—	2	2	no	—
26	C₂×M₁₆×C₇₉	—	2	1	no	—
27	C₂×M₁₆×C₇₉	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₇₉	—	2	1	no	—
29	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
30	M₃₂×C₇₉	—	2	2	no	—
31	C₂×C₁₆×C₇₉	—	2	2	no	—
32	C₂×C₁₆×C₇₉	—	2	1	no	—
33	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
34	C₂²×Dic₇₉	—	4	2	no	—
35	#11929608	—	4	2	no	—
36	#11929608	—	4	2	no	—
37	C₂×C₇₉⋊C₈	—	4	2	no	—
38	C₄⋊Dic₇₉	—	4	2	no	—
39	C₄⋊Dic₇₉	—	4	2	no	—
40	#41×C₇₉	—	2	2	no	—
41	#41×C₇₉	—	2	2	no	—
42	C₂×Q₈×C₇₉	D₈	1	1	yes	characteristic
43	C₂×D₄×C₇₉	—	2	1	no	—
44	C₂×D₄×C₇₉	D₈	1	1	yes	characteristic
45	C₂²×C₄×C₇₉	—	2	1	no	—
46	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
47	M₁₆×C₇₉	—	2	2	no	—
48	M₁₆×C₇₉	—	4	2	no	—
49	C₂×C₈×C₇₉	—	2	1	no	—
50	C₂×C₈×C₇₉	—	4	1	no	—
51	C₂×C₈×C₇₉	—	4	1	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	C₁₆×C₇₉	—	4	1	no	—
58	C₂³×C₇₉	—	2	1	no	—
59	C₂³×C₇₉	#135	1	1	yes	characteristic
60	Q₈×C₇₉	—	2	2	no	—
61	D₄×C₇₉	—	2	2	no	—
62	D₄×C₇₉	—	4	2	no	—
63	C₂×C₄×C₇₉	—	2	1	no	—
64	C₂×C₄×C₇₉	—	4	1	no	—
65	C₂×C₄×C₇₉	—	4	1	no	—
66	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
67	C₂×C₄×C₇₉	#123	1	2	yes	—
68	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₇₉	#137	1	2	yes	—
70	C₂×Dic₇₉	—	8	2	no	—
71	C₂×Dic₇₉	—	16	2	no	—
72	C₈×C₇₉	—	4	1	no	—
73	C₈×C₇₉	#123	1	1	yes	characteristic
74	C₈×C₇₉	#124	1	1	yes	characteristic
75	C₈×C₇₉	#126	1	2	yes	—
76	C₇₉⋊C₈	—	8	2	no	—
77	C₂²×C₇₉	—	2	2	no	—
78	C₂²×C₇₉	—	4	1	no	—
79	C₂²×C₇₉	—	4	2	no	—
80	C₂²×C₇₉	#466	1	1	yes	characteristic
81	C₂²×C₇₉	#485	1	2	yes	—
82	C₄×C₇₉	—	4	1	no	—
83	C₄×C₇₉	#397	1	1	yes	characteristic
84	C₄×C₇₉	#470	1	1	yes	characteristic
85	C₄×C₇₉	#480	1	2	yes	—
86	Dic₇₉	—	16	2	no	—
87	#13496	—	79	1	no	maximal, Sylow
88	C₂×C₇₉	—	2	2	no	—
89	C₂×C₇₉	—	8	1	no	—
90	C₂×C₇₉	#2075	1	1	yes	characteristic
91	C₂×C₇₉	#2223	1	2	yes	—
92	C₂×#367	—	79	1	no	—
93	#2113	D₇₉	1	1	yes	characteristic
94	#1816	—	79	2	no	—
95	#1387	—	79	2	no	—
96	#1381	—	79	1	no	—
97	C₇₉	#13496	1	1	yes	characteristic, complemented, Sylow
98	C₂×#131	C₂×D₇₉	1	1	yes	characteristic
99	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
100	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
101	C₂×#108	—	79	2	no	—
102	C₂×#107	—	79	2	no	—
103	#417	—	158	2	no	—
104	#367	—	158	2	no	—
105	C₂²⋊C₁₆	—	2	2	no	—
106	#343	—	158	2	no	—
107	C₂×#41	D₃₁₆	1	1	yes	characteristic
108	#131	—	2	2	no	—
109	#131	—	2	2	no	—
110	C₂×M₁₆	—	2	1	no	—
111	C₂×M₁₆	—	158	2	no	—
112	C₂×M₁₆	D₃₁₆	1	1	yes	characteristic
113	C₂²×C₈	—	2	1	no	—
114	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
115	C₂×C₄⋊C₄	—	158	2	no	—
116	M₃₂	—	2	2	no	—
117	C₂×C₁₆	—	2	2	no	—
118	C₂×C₁₆	—	2	1	no	—
119	C₂×C₁₆	C₂²⋊D₇₉	1	2	yes	—
120	#108	—	158	2	no	—
121	#108	—	158	2	no	—
122	#107	—	158	2	no	—
123	#107	—	158	2	no	—
124	#103	—	316	2	no	—
125	#102	—	316	2	no	—
126	#41	—	2	2	no	—
127	#41	—	2	2	no	—
128	C₂×Q₈	D₆₃₂	1	1	yes	characteristic
129	C₂×D₄	—	2	1	no	—
130	C₂×D₄	D₆₃₂	1	1	yes	characteristic
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	—	316	2	no	—
133	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
134	M₁₆	—	2	2	no	—
135	M₁₆	—	4	2	no	—
136	M₁₆	—	316	2	no	—
137	M₁₆	—	316	2	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	4	1	no	—
140	C₂×C₈	—	4	1	no	—
141	C₂×C₈	—	316	2	no	—
142	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
143	C₂×C₈	#11929623	1	2	yes	—
144	C₂×C₈	#11929623	1	2	yes	—
145	C₂×C₈	#11929624	1	1	yes	characteristic
146	C₄⋊C₄	—	316	2	no	—
147	C₄⋊C₄	—	316	2	no	—
148	C₁₆	—	2	2	no	—
149	C₁₆	—	4	1	no	—
150	C₂³	—	2	1	no	—
151	C₂³	#14579183	1	1	yes	characteristic
152	Q₈	—	2	2	no	—
153	D₄	—	2	2	no	—
154	D₄	—	4	2	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	—	632	2	no	—
159	C₂×C₄	—	1264	2	no	—
160	C₂×C₄	#14579135	1	1	yes	characteristic
161	C₂×C₄	#14579145	1	2	yes	—
162	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
163	C₂×C₄	#14579194	1	2	yes	—
164	C₈	—	4	1	no	—
165	C₈	—	632	2	no	—
166	C₈	#14579142	1	1	yes	characteristic
167	C₈	#14579148	1	1	yes	characteristic
168	C₈	#14579154	1	2	yes	—
169	C₂²	—	2	2	no	—
170	C₂²	—	4	2	no	—
171	C₂²	—	4	1	no	—
172	C₂²	#19098799	1	1	yes	characteristic
173	C₂²	#19098905	1	2	yes	—
174	C₄	—	4	1	no	—
175	C₄	—	1264	2	no	—
176	C₄	#19098488	1	1	yes	characteristic
177	C₄	#19098821	1	1	yes	characteristic
178	C₄	#19098877	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	—	8	1	no	—
181	C₂	#23764823	1	1	yes	characteristic
182	C₂	#23765530	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19098391

C₂

yes

maximal, characteristic, complemented

#23763464

C₂

yes

maximal

#23761509

C₂

yes

maximal, complemented

#23761494

C₂

yes

maximal, characteristic, complemented

C₇₉×#2113

C₂

yes

maximal, characteristic

C₂×#14579097

C₂²

yes

—

C₂×#14579095

C₂²

yes

—

#19098564

—

#19098391

—

#19098331

—

C₂×#131×C₇₉

C₂²

yes

characteristic

C₂×M₃₂×C₇₉

C₂²

yes

characteristic

C₂²×C₁₆×C₇₉

C₂²

yes

characteristic

C₂²⋊C₁₆×C₇₉

—

C₂×#11929608

—

C₂×C₄⋊Dic₇₉

—

—

—

—

—

—

—

C₂×#41×C₇₉

D₄

yes

characteristic

#131×C₇₉

—

#131×C₇₉

—

C₂×M₁₆×C₇₉

—

C₂×M₁₆×C₇₉

D₄

yes

characteristic

C₂²×C₈×C₇₉

—

C₂²×C₈×C₇₉

C₂³

yes

characteristic

M₃₂×C₇₉

—

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

D₄

yes

—

C₂²×Dic₇₉

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

#41×C₇₉

—

#41×C₇₉

—

C₂×Q₈×C₇₉

D₈

yes

characteristic

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

D₈

yes

characteristic

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×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

—

C₁₆×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

#135

yes

characteristic

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#121

yes

characteristic

C₂×C₄×C₇₉

#123

yes

—

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

#137

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₈×C₇₉

yes

C₈×C₇₉

yes

C₈×C₇₉

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#466

yes

characteristic

C₂²×C₇₉

#485

yes

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

Dic₇₉

—

#13496

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#2075

yes

characteristic

C₂×C₇₉

#2223

yes

—

C₂×#367

—

#2113

D₇₉

yes

—

—

—

C₇₉

#13496

yes

characteristic, complemented, Sylow

C₂×#131

C₂×D₇₉

yes

characteristic

C₂×M₃₂

C₂×D₇₉

yes

characteristic

100

C₂²×C₁₆

C₂×D₇₉

yes

characteristic

101

C₂×#108

—

102

C₂×#107

—

103

#417

—

158

—

104

#367

—

158

—

105

C₂²⋊C₁₆

—

106

#343

—

158

—

107

C₂×#41

D₃₁₆

yes

characteristic

108

#131

—

109

#131

—

110

C₂×M₁₆

—

111

C₂×M₁₆

—

158

—

112

C₂×M₁₆

D₃₁₆

yes

characteristic

113

C₂²×C₈

—

114

C₂²×C₈

C₂²×D₇₉

yes

characteristic

115

C₂×C₄⋊C₄

—

158

—

116

M₃₂

—

117

C₂×C₁₆

—

118

C₂×C₁₆

—

119

C₂×C₁₆

C₂²⋊D₇₉

yes

—

120

#108

—

158

—

121

#108

—

158

—

122

#107

—

158

—

123

#107

—

158

—

124

#103

—

316

—

125

#102

—

316

—

126

#41

—

127

#41

—

128

C₂×Q₈

D₆₃₂

yes

characteristic

129

C₂×D₄

—

130

C₂×D₄

D₆₃₂

yes

characteristic

131

C₂²×C₄

—

132

C₂²×C₄

—

316

—

133

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

134

M₁₆

—

135

M₁₆

—

136

M₁₆

—

316

—

137

M₁₆

—

316

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

316

—

142

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

143

C₂×C₈

#11929623

yes

—

144

C₂×C₈

#11929623

yes

—

145

C₂×C₈

#11929624

yes

characteristic

146

C₄⋊C₄

—

316

—

147

C₄⋊C₄

—

316

—

148

C₁₆

—

149

C₁₆

—

150

C₂³

—

151

C₂³

#14579183

yes

characteristic

152

Q₈

—

153

D₄

—

154

D₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

632

—

159

C₂×C₄

—

1264

—

160

C₂×C₄

#14579135

yes

characteristic

161

C₂×C₄

#14579145

yes

—

162

C₂×C₄

C₂×D₆₃₂

yes

characteristic

163

C₂×C₄

#14579194

yes

—

164

C₈

—

165

C₈

—

632

—

166

C₈

#14579142

yes

characteristic

167

C₈

#14579148

yes

characteristic

168

C₈

#14579154

yes

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

#19098799

yes

characteristic

173

C₂²

#19098905

yes

—

174

C₄

—

175

C₄

—

1264

—

176

C₄

#19098488

yes

characteristic

177

C₄

#19098821

yes

characteristic

178

C₄

#19098877

yes

—

179

C₂

—

180

C₂

—

181

C₂

#23764823

yes

characteristic

182

C₂

#23765530

yes

—

	subgroup	G/H
center	C₂²	#19098799
commutator, cosocle	C₂²×C₈×C₇₉	C₂³
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#2113	C₂
socle	C₂²×C₇₉	#466
prime radical	C₂×#41×C₇₉	D₄
hypercenter	#2113	D₇₉
hypocenter	C₇₉	#13496
LCS	G ▷ C₂²×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ #2113

subgroup

G/H

center

C₂²

#19098799

commutator, cosocle

C₂²×C₈×C₇₉

C₂³

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₇₉×#2113

C₂

socle

C₂²×C₇₉

#466

prime radical

C₂×#41×C₇₉

D₄

D₇₉

C₇₉

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ #2113

N	H ≅ G/N	type
1. C₂×#19098391	C₂	semidirect
3. #23761509	C₂	semidirect
4. #23761494	C₂	semidirect
97. C₇₉	#13496	semidirect

H ≅ G/N

type

1. C₂×#19098391

C₂

semidirect

3. #23761509

C₂

semidirect

4. #23761494

C₂

semidirect

97. C₇₉

#13496

semidirect

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

subgroup	index
C₂×#19098391	2
#23763464	2
#23761509	2
#23761494	2
C₇₉×#2113	2
#13496	79

subgroup

index

C₇₉×#2113

Subgroups — #34208147

Conjugacy classes of subgroups

Complements

Maximal subgroups