Group #34187737 — subgroups

Subgroups — #34187737

Order 20224.

582 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 159 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	#23763572	C₂	1	1	yes	maximal, characteristic, complemented
2	#23762068	C₂	1	4	yes	maximal, complemented
3	#23761683	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#493×C₇₉	C₂	1	1	yes	maximal, characteristic
5	#19098616	C₂²	1	2	yes	—
6	C₂×#14579097	—	2	2	no	—
7	#19098587	C₂²	1	2	yes	—
8	#19098307	—	2	8	no	—
9	#19098284	—	2	1	no	—
10	C₈×C₇₉⋊C₈	—	2	1	no	—
11	C₂²×D₈×C₇₉	—	2	2	no	—
12	C₂×#127×C₇₉	C₂²	1	2	yes	—
13	#493×C₇₉	—	2	1	no	—
14	#493×C₇₉	—	2	1	no	—
15	#493×C₇₉	C₄	1	4	yes	—
16	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×#11929608	—	2	2	no	—
18	#14579097	—	4	4	no	—
19	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
20	C₄⋊(C₇₉⋊C₈)	D₄	1	4	yes	—
21	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
22	C₄×C₇₉⋊C₈	—	2	2	no	—
23	C₂²×D₄×C₇₉	—	2	2	no	—
24	C₂²×D₄×C₇₉	—	2	4	no	—
25	C₂×D₈×C₇₉	—	2	4	no	—
26	C₂×D₈×C₇₉	—	2	4	no	—
27	C₂×D₈×C₇₉	—	4	4	no	—
28	C₂×D₈×C₇₉	—	4	4	no	—
29	C₂²×C₈×C₇₉	D₄	1	2	yes	—
30	#127×C₇₉	—	2	2	no	—
31	#127×C₇₉	—	2	4	no	—
32	#127×C₇₉	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
34	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
36	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
37	#11929608	—	4	4	no	—
38	C₂×C₇₉⋊C₈	—	2	4	no	—
39	C₂×C₇₉⋊C₈	—	4	2	no	—
40	C₂⁴×C₇₉	—	4	2	no	—
41	C₂×D₄×C₇₉	—	2	4	no	—
42	C₂×D₄×C₇₉	—	2	8	no	—
43	C₂×D₄×C₇₉	—	4	4	no	—
44	C₂×D₄×C₇₉	—	4	2	no	—
45	C₂×D₄×C₇₉	—	4	4	no	—
46	C₂×D₄×C₇₉	—	4	4	no	—
47	C₂×D₄×C₇₉	—	8	8	no	—
48	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₉	#41	1	2	yes	—
50	D₈×C₇₉	—	4	8	no	—
51	D₈×C₇₉	—	4	8	no	—
52	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
53	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₇₉	D₈	1	4	yes	—
55	C₂×C₈×C₇₉	SD₁₆	1	4	yes	—
56	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₇₉	C₂×D₄	1	2	yes	—
59	C₂³×C₇₉	—	4	4	no	—
60	C₂³×C₇₉	—	8	8	no	—
61	C₂³×C₇₉	—	8	4	no	—
62	C₂³×C₇₉	#124	1	1	yes	characteristic
63	D₄×C₇₉	—	4	8	no	—
64	D₄×C₇₉	—	8	8	no	—
65	D₄×C₇₉	—	8	8	no	—
66	D₄×C₇₉	—	8	2	no	—
67	D₄×C₇₉	—	8	2	no	—
68	C₂×C₄×C₇₉	—	2	4	no	—
69	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
71	C₂×C₄×C₇₉	#121	1	2	yes	—
72	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
73	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
74	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
75	C₈×C₇₉	#102	1	4	yes	—
76	C₈×C₇₉	#102	1	4	yes	—
77	C₇₉⋊C₈	—	4	4	no	—
78	C₇₉⋊C₈	—	8	2	no	—
79	C₂²×C₇₉	—	2	2	no	—
80	C₂²×C₇₉	—	8	4	no	—
81	C₂²×C₇₉	—	8	8	no	—
82	C₂²×C₇₉	—	16	8	no	—
83	C₂²×C₇₉	#390	1	1	yes	characteristic
84	C₂²×C₇₉	#486	1	2	yes	—
85	C₄×C₇₉	—	2	2	no	—
86	C₄×C₇₉	C₂×#102	1	2	yes	—
87	C₄×C₇₉	#472	1	4	yes	—
88	C₄×C₇₉	#492	1	1	yes	characteristic
89	C₄×C₇₉	#493	1	1	yes	characteristic
90	#9773	—	79	1	no	maximal, Sylow
91	C₂×C₇₉	—	2	2	no	—
92	C₂×C₇₉	—	16	4	no	—
93	C₂×C₇₉	#1957	1	1	yes	characteristic
94	C₂×C₇₉	#1978	1	2	yes	—
95	C₂×#493	D₇₉	1	1	yes	characteristic
96	#1848	—	79	1	no	—
97	#1545	—	79	4	no	—
98	#1450	—	79	1	no	—
99	C₇₉	#9773	1	1	yes	characteristic, complemented, Sylow
100	C₂²×D₈	—	2	2	no	—
101	C₂×#127	C₂×D₇₉	1	2	yes	—
102	#493	—	2	1	no	—
103	#493	—	2	1	no	—
104	#493	Dic₇₉	1	4	yes	—
105	#431	—	79	2	no	—
106	C₂×#108	—	158	2	no	—
107	#423	—	79	2	no	—
108	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
109	#331	—	158	8	no	—
110	#322	—	158	1	no	—
111	C₈²	—	158	1	no	—
112	C₂²×D₄	—	2	2	no	—
113	C₂²×D₄	—	2	4	no	—
114	C₂×D₈	—	2	4	no	—
115	C₂×D₈	—	2	4	no	—
116	C₂×D₈	—	4	4	no	—
117	C₂×D₈	—	4	4	no	—
118	C₂×M₁₆	—	158	2	no	—
119	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
120	#127	—	2	4	no	—
121	#127	—	2	2	no	—
122	#127	C₂×Dic₇₉	1	4	yes	—
123	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
124	#108	—	316	4	no	—
125	C₄⋊C₈	—	79	4	no	—
126	C₄⋊C₈	—	158	4	no	—
127	C₂²⋊C₈	—	158	2	no	—
128	C₄×C₈	—	158	2	no	—
129	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
130	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
131	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
132	C₂⁴	—	4	2	no	—
133	C₂×D₄	—	2	4	no	—
134	C₂×D₄	—	2	8	no	—
135	C₂×D₄	—	4	4	no	—
136	C₂×D₄	—	4	2	no	—
137	C₂×D₄	—	4	4	no	—
138	C₂×D₄	—	4	4	no	—
139	C₂×D₄	—	8	8	no	—
140	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
141	C₂²×C₄	#11929623	1	2	yes	—
142	D₈	—	4	8	no	—
143	D₈	—	4	8	no	—
144	M₁₆	—	316	4	no	—
145	C₂×C₈	—	316	2	no	—
146	C₂×C₈	—	158	4	no	—
147	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
148	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
149	C₂×C₈	#11929609	1	4	yes	—
150	C₂×C₈	#11929612	1	4	yes	—
151	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
152	C₄²	D₄×D₇₉	1	2	yes	—
153	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
154	C₂³	—	4	4	no	—
155	C₂³	—	8	8	no	—
156	C₂³	—	8	4	no	—
157	C₂³	#14579148	1	1	yes	characteristic
158	D₄	—	4	8	no	—
159	D₄	—	8	8	no	—
160	D₄	—	8	2	no	—
161	D₄	—	8	2	no	—
162	D₄	—	8	8	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
165	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
166	C₂×C₄	#14579132	1	2	yes	—
167	C₂×C₄	#14579157	1	1	yes	characteristic
168	C₂×C₄	C₂×#11929609	1	2	yes	—
169	C₂×C₄	C₂×#11929612	1	2	yes	—
170	C₈	—	316	4	no	—
171	C₈	—	632	2	no	—
172	C₈	#14579082	1	4	yes	—
173	C₈	#14579082	1	4	yes	—
174	C₂²	—	2	2	no	—
175	C₂²	—	8	8	no	—
176	C₂²	—	8	4	no	—
177	C₂²	—	16	8	no	—
178	C₂²	#19098466	1	1	yes	characteristic
179	C₂²	#19098909	1	2	yes	—
180	C₄	—	2	2	no	—
181	C₄	C₂×#14579082	1	2	yes	—
182	C₄	#19098830	1	4	yes	—
183	C₄	#19098935	1	1	yes	characteristic
184	C₄	#19098939	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	—	16	4	no	—
187	C₂	#23764207	1	1	yes	characteristic
188	C₂	#23764311	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23763572

C₂

yes

maximal, characteristic, complemented

#23762068

C₂

yes

maximal, complemented

#23761683

C₂

yes

maximal, characteristic, complemented

C₂×#493×C₇₉

C₂

yes

maximal, characteristic

#19098616

C₂²

yes

—

C₂×#14579097

—

#19098587

C₂²

yes

—

#19098307

—

#19098284

—

C₈×C₇₉⋊C₈

—

C₂²×D₈×C₇₉

—

C₂×#127×C₇₉

C₂²

yes

—

#493×C₇₉

—

#493×C₇₉

—

#493×C₇₉

C₄

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×#11929608

—

#14579097

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

D₄

yes

—

C₂²⋊(C₇₉⋊C₈)

—

C₄×C₇₉⋊C₈

—

C₂²×D₄×C₇₉

—

C₂²×D₄×C₇₉

—

C₂×D₈×C₇₉

—

C₂×D₈×C₇₉

—

C₂×D₈×C₇₉

—

C₂×D₈×C₇₉

—

C₂²×C₈×C₇₉

D₄

yes

—

#127×C₇₉

—

#127×C₇₉

—

#127×C₇₉

C₂×C₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂⁴×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

#41

yes

—

D₈×C₇₉

—

D₈×C₇₉

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

D₈

yes

—

C₂×C₈×C₇₉

SD₁₆

yes

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

#124

yes

characteristic

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄×D₄

yes

—

C₂×C₄×C₇₉

#121

yes

—

C₂×C₄×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

—

C₂×C₄×C₇₉

C₂×SD₁₆

yes

—

C₈×C₇₉

#102

yes

—

C₈×C₇₉

#102

yes

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#390

yes

characteristic

C₂²×C₇₉

#486

yes

—

C₄×C₇₉

—

C₄×C₇₉

C₂×#102

yes

—

C₄×C₇₉

#472

yes

—

C₄×C₇₉

yes

C₄×C₇₉

yes

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#1957

yes

characteristic

C₂×C₇₉

#1978

yes

—

C₂×#493

D₇₉

yes

—

—

—

C₇₉

#9773

yes

characteristic, complemented, Sylow

100

C₂²×D₈

—

101

C₂×#127

C₂×D₇₉

yes

—

102

#493

—

103

#493

—

104

#493

Dic₇₉

yes

—

105

#431

—

106

C₂×#108

—

158

—

107

#423

—

108

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

109

#331

—

158

—

110

#322

—

158

—

111

C₈²

—

158

—

112

C₂²×D₄

—

113

C₂²×D₄

—

114

C₂×D₈

—

115

C₂×D₈

—

116

C₂×D₈

—

117

C₂×D₈

—

118

C₂×M₁₆

—

158

—

119

C₂²×C₈

C₂²⋊D₇₉

yes

—

120

#127

—

121

#127

—

122

#127

C₂×Dic₇₉

yes

—

123

C₂×C₄²

C₂²×D₇₉

yes

characteristic

124

#108

—

316

—

125

C₄⋊C₈

—

126

C₄⋊C₈

—

158

—

127

C₂²⋊C₈

—

158

—

128

C₄×C₈

—

158

—

129

C₄×C₈

C₂×Dic₇₉

yes

—

130

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

131

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

132

C₂⁴

—

133

C₂×D₄

—

134

C₂×D₄

—

135

C₂×D₄

—

136

C₂×D₄

—

137

C₂×D₄

—

138

C₂×D₄

—

139

C₂×D₄

—

140

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

141

C₂²×C₄

#11929623

yes

—

142

D₈

—

143

D₈

—

144

M₁₆

—

316

—

145

C₂×C₈

—

316

—

146

C₂×C₈

—

158

—

147

C₂×C₈

C₂²⋊Dic₇₉

yes

—

148

C₂×C₈

C₂²⋊Dic₇₉

yes

—

149

C₂×C₈

#11929609

yes

—

150

C₂×C₈

#11929612

yes

—

151

C₄²

C₂²×Dic₇₉

yes

characteristic

152

C₄²

D₄×D₇₉

yes

—

153

C₄²

C₂×C₂²⋊D₇₉

yes

characteristic

154

C₂³

—

155

C₂³

—

156

C₂³

—

157

C₂³

#14579148

yes

characteristic

158

D₄

—

159

D₄

—

160

D₄

—

161

D₄

—

162

D₄

—

163

C₂×C₄

—

164

C₂×C₄

C₂×C₂²⋊Dic₇₉

yes

characteristic

165

C₂×C₄

D₄×Dic₇₉

yes

—

166

C₂×C₄

#14579132

yes

—

167

C₂×C₄

#14579157

yes

characteristic

168

C₂×C₄

C₂×#11929609

yes

—

169

C₂×C₄

C₂×#11929612

yes

—

170

C₈

—

316

—

171

C₈

—

632

—

172

C₈

#14579082

yes

—

173

C₈

#14579082

yes

—

174

C₂²

—

175

C₂²

—

176

C₂²

—

177

C₂²

—

178

C₂²

#19098466

yes

characteristic

179

C₂²

#19098909

yes

—

180

C₄

—

181

C₄

C₂×#14579082

yes

—

182

C₄

#19098830

yes

—

183

C₄

#19098935

yes

characteristic

184

C₄

#19098939

yes

characteristic

185

C₂

—

186

C₂

—

187

C₂

#23764207

yes

characteristic

188

C₂

#23764311

yes

—

	subgroup	G/H
center	C₂²	#19098466
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting, prime radical	C₂×#493×C₇₉	C₂
socle	C₂²×C₇₉	#390
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×#493	D₇₉
hypocenter	C₇₉	#9773
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#493

subgroup

G/H

center

C₂²

#19098466

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting, prime radical

C₂×#493×C₇₉

C₂

socle

C₂²×C₇₉

#390

cosocle

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#493

N	H ≅ G/N	type
1. #23763572	C₂	semidirect
2. #23762068	C₂	semidirect
3. #23761683	C₂	semidirect
99. C₇₉	#9773	semidirect

H ≅ G/N

type

1. #23763572

C₂

semidirect

2. #23762068

C₂

semidirect

3. #23761683

C₂

semidirect

99. C₇₉

#9773

semidirect

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

subgroup	index
#23763572	2
#23762068	2
#23761683	2
C₂×#493×C₇₉	2
#9773	79

subgroup

index

C₂×#493×C₇₉

Subgroups — #34187737

Conjugacy classes of subgroups

Complements

Maximal subgroups