Group #34204582 — subgroups

Subgroups — #34204582

Order 20224.

398 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 131 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	#23765011	C₂	1	1	yes	maximal, characteristic
2	#23763438	C₂	1	1	yes	maximal, characteristic
3	#23761508	C₂	1	4	yes	maximal, complemented
4	C₂×#427×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579101	C₂²	1	2	yes	—
6	C₂×#14579097	C₂²	1	2	yes	—
7	#19098527	—	2	1	no	—
8	C₂²⋊(C₇₉⋊C₁₆)	—	2	2	no	—
9	#19098328	—	2	1	no	—
10	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
11	C₂×#117×C₇₉	—	2	1	no	—
12	#427×C₇₉	—	2	2	no	—
13	#427×C₇₉	C₄	1	4	yes	—
14	C₂×#107×C₇₉	C₂²	1	2	yes	—
15	C₂×#106×C₇₉	—	2	1	no	—
16	C₂×#11929608	—	2	2	no	—
17	#14579101	—	2	4	no	—
18	C₂×C₇₉⋊C₁₆	—	2	2	no	—
19	#14579097	—	2	4	no	—
20	C₂²⋊(C₇₉⋊C₈)	—	4	2	no	—
21	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₇₉	—	2	4	no	—
23	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₇₉	D₄	1	2	yes	—
26	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
27	#117×C₇₉	—	2	4	no	—
28	#117×C₇₉	—	4	2	no	—
29	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
30	C₂×C₂²⋊C₄×C₇₉	—	4	1	no	—
31	C₂×C₄²×C₇₉	—	4	1	no	—
32	#107×C₇₉	—	2	2	no	—
33	#107×C₇₉	C₂×C₄	1	4	yes	—
34	#106×C₇₉	—	2	4	no	—
35	#11929608	—	4	2	no	—
36	#11929608	—	4	2	no	—
37	C₂×C₇₉⋊C₈	—	4	2	no	—
38	C₇₉⋊C₁₆	—	4	2	no	—
39	C₂⁴×C₇₉	SD₁₆	1	1	yes	characteristic
40	C₂²×C₄×C₇₉	—	2	2	no	—
41	C₂²×C₄×C₇₉	—	2	2	no	—
42	C₂²×C₄×C₇₉	—	4	2	no	—
43	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₇₉	D₈	1	2	yes	—
45	C₂²×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₇₉	—	2	4	no	—
48	C₂×C₈×C₇₉	—	2	4	no	—
49	C₂×C₈×C₇₉	—	2	2	no	—
50	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₇₉	#41	1	4	yes	—
54	C₄⋊C₄×C₇₉	—	2	4	no	—
55	C₄⋊C₄×C₇₉	—	4	2	no	—
56	C₂²⋊C₄×C₇₉	—	4	4	no	—
57	C₄²×C₇₉	—	4	4	no	—
58	C₂³×C₇₉	—	2	2	no	—
59	C₂³×C₇₉	—	4	2	no	—
60	C₂³×C₇₉	#102	1	2	yes	—
61	C₂³×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₇₉	—	2	2	no	—
63	C₂×C₄×C₇₉	—	2	2	no	—
64	C₂×C₄×C₇₉	—	2	2	no	—
65	C₂×C₄×C₇₉	—	2	4	no	—
66	C₂×C₄×C₇₉	—	4	4	no	—
67	C₂×C₄×C₇₉	—	8	4	no	—
68	C₂×C₄×C₇₉	#102	1	2	yes	—
69	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₇₉	#123	1	2	yes	—
71	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
72	C₂×C₄×C₇₉	#137	1	2	yes	—
73	C₈×C₇₉	—	2	2	no	—
74	C₈×C₇₉	#117	1	2	yes	—
75	C₈×C₇₉	#123	1	2	yes	—
76	C₇₉⋊C₈	—	8	2	no	—
77	C₂²×C₇₉	—	2	2	no	—
78	C₂²×C₇₉	—	2	2	no	—
79	C₂²×C₇₉	—	2	4	no	—
80	C₂²×C₇₉	—	4	4	no	—
81	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
82	C₂²×C₇₉	#483	1	2	yes	—
83	C₄×C₇₉	—	2	2	no	—
84	C₄×C₇₉	—	8	4	no	—
85	C₄×C₇₉	#381	1	1	yes	characteristic
86	C₄×C₇₉	#419	1	1	yes	characteristic
87	C₄×C₇₉	#480	1	2	yes	—
88	#12872	—	79	1	no	maximal, Sylow
89	C₂×C₇₉	—	2	2	no	—
90	C₂×C₇₉	—	4	2	no	—
91	C₂×C₇₉	#1823	1	1	yes	characteristic
92	C₂×C₇₉	#2148	1	2	yes	—
93	C₂×#427	D₇₉	1	1	yes	characteristic
94	#2112	—	79	1	no	—
95	#1810	—	79	1	no	—
96	#1387	—	79	4	no	—
97	C₇₉	#12872	1	1	yes	characteristic, complemented, Sylow
98	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
99	C₂×#117	—	2	1	no	—
100	C₂×M₃₂	—	79	2	no	—
101	C₂×#108	—	79	2	no	—
102	#427	—	2	2	no	—
103	#427	Dic₇₉	1	4	yes	—
104	C₂×#107	C₂×D₇₉	1	2	yes	—
105	C₂×#106	—	2	1	no	—
106	#407	—	158	1	no	—
107	C₂²⋊C₁₆	—	158	2	no	—
108	#341	—	158	1	no	—
109	C₂³×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
110	C₂×M₁₆	—	158	2	no	—
111	C₂²×C₈	—	2	4	no	—
112	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
113	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
114	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
115	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
116	#117	—	2	4	no	—
117	#117	—	4	2	no	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₂²⋊C₄	—	4	1	no	—
120	C₂×C₄²	—	4	1	no	—
121	M₃₂	—	158	4	no	—
122	C₂×C₁₆	—	158	2	no	—
123	#108	—	158	4	no	—
124	#107	—	2	2	no	—
125	#107	C₂×Dic₇₉	1	4	yes	—
126	#106	—	2	4	no	—
127	C₂²⋊C₈	—	316	2	no	—
128	C₂⁴	#11929612	1	1	yes	characteristic
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	C₂²⋊Dic₇₉	1	2	yes	—
133	C₂²×C₄	#11929609	1	2	yes	—
134	C₂²×C₄	#11929612	1	1	yes	characteristic
135	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
136	M₁₆	—	316	2	no	—
137	M₁₆	—	316	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	316	2	no	—
142	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
143	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
144	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
145	C₂×C₈	#11929624	1	4	yes	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₂²⋊C₄	—	4	4	no	—
149	C₄²	—	4	4	no	—
150	C₁₆	—	316	2	no	—
151	C₂³	—	2	2	no	—
152	C₂³	—	4	2	no	—
153	C₂³	#14579082	1	2	yes	—
154	C₂³	C₂×#11929612	1	1	yes	characteristic
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	4	4	no	—
160	C₂×C₄	—	8	4	no	—
161	C₂×C₄	#14579082	1	2	yes	—
162	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
163	C₂×C₄	#14579144	1	2	yes	—
164	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
165	C₂×C₄	#14579192	1	2	yes	—
166	C₈	—	2	2	no	—
167	C₈	—	632	2	no	—
168	C₈	#14579120	1	2	yes	—
169	C₈	#14579144	1	2	yes	—
170	C₂²	—	2	2	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	2	4	no	—
173	C₂²	—	4	4	no	—
174	C₂²	C₂×#14579082	1	1	yes	characteristic
175	C₂²	#19098893	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	8	4	no	—
178	C₄	#19098426	1	1	yes	characteristic
179	C₄	#19098572	1	1	yes	characteristic
180	C₄	#19098876	1	2	yes	—
181	C₂	—	2	2	no	—
182	C₂	—	4	2	no	—
183	C₂	#23763486	1	1	yes	characteristic
184	C₂	#23765149	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23765011

C₂

yes

maximal, characteristic

#23763438

C₂

yes

maximal, characteristic

#23761508

C₂

yes

maximal, complemented

C₂×#427×C₇₉

C₂

yes

maximal, characteristic

C₂×#14579101

C₂²

yes

—

C₂×#14579097

C₂²

yes

—

#19098527

—

C₂²⋊(C₇₉⋊C₁₆)

—

#19098328

—

C₂³×C₈×C₇₉

C₂²

yes

characteristic

C₂×#117×C₇₉

—

#427×C₇₉

—

#427×C₇₉

C₄

yes

—

C₂×#107×C₇₉

C₂²

yes

—

C₂×#106×C₇₉

—

C₂×#11929608

—

#14579101

—

C₂×C₇₉⋊C₁₆

—

#14579097

—

C₂²⋊(C₇₉⋊C₈)

—

C₂³×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₈×C₇₉

—

C₂²×C₈×C₇₉

C₂×C₄

yes

—

C₂²×C₈×C₇₉

D₄

yes

characteristic

C₂²×C₈×C₇₉

D₄

yes

—

C₂²×C₈×C₇₉

C₂³

yes

characteristic

#117×C₇₉

—

#117×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₂²⋊C₄×C₇₉

—

C₂×C₄²×C₇₉

—

#107×C₇₉

—

#107×C₇₉

C₂×C₄

yes

—

#106×C₇₉

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₇₉⋊C₁₆

—

C₂⁴×C₇₉

SD₁₆

yes

characteristic

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₉

D₈

yes

—

C₂²×C₄×C₇₉

SD₁₆

yes

characteristic

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

#41

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

#102

yes

—

C₂³×C₇₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#102

yes

—

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

#123

yes

—

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

#137

yes

—

C₈×C₇₉

—

C₈×C₇₉

#117

yes

—

C₈×C₇₉

#123

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×#102

yes

characteristic

C₂²×C₇₉

#483

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

#12872

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#1823

yes

characteristic

C₂×C₇₉

#2148

yes

—

C₂×#427

D₇₉

yes

—

—

—

C₇₉

#12872

yes

characteristic, complemented, Sylow

C₂³×C₈

C₂×D₇₉

yes

characteristic

C₂×#117

—

100

C₂×M₃₂

—

101

C₂×#108

—

102

#427

—

103

#427

Dic₇₉

yes

—

104

C₂×#107

C₂×D₇₉

yes

—

105

C₂×#106

—

106

#407

—

158

—

107

C₂²⋊C₁₆

—

158

—

108

#341

—

158

—

109

C₂³×C₄

C₂²⋊D₇₉

yes

characteristic

110

C₂×M₁₆

—

158

—

111

C₂²×C₈

—

112

C₂²×C₈

C₂×Dic₇₉

yes

—

113

C₂²×C₈

C₂²⋊D₇₉

yes

characteristic

114

C₂²×C₈

C₂²⋊D₇₉

yes

—

115

C₂²×C₈

C₂²×D₇₉

yes

characteristic

116

#117

—

117

#117

—

118

C₂×C₄⋊C₄

—

119

C₂×C₂²⋊C₄

—

120

C₂×C₄²

—

121

M₃₂

—

158

—

122

C₂×C₁₆

—

158

—

123

#108

—

158

—

124

#107

—

125

#107

C₂×Dic₇₉

yes

—

126

#106

—

127

C₂²⋊C₈

—

316

—

128

C₂⁴

#11929612

yes

characteristic

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

C₂²⋊Dic₇₉

yes

—

133

C₂²×C₄

#11929609

yes

—

134

C₂²×C₄

#11929612

yes

characteristic

135

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

136

M₁₆

—

316

—

137

M₁₆

—

316

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

316

—

142

C₂×C₈

C₂²⋊Dic₇₉

yes

—

143

C₂×C₈

C₂²×Dic₇₉

yes

characteristic

144

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

145

C₂×C₈

#11929624

yes

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₂²⋊C₄

—

149

C₄²

—

150

C₁₆

—

316

—

151

C₂³

—

152

C₂³

—

153

C₂³

#14579082

yes

—

154

C₂³

C₂×#11929612

yes

characteristic

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

#14579082

yes

—

162

C₂×C₄

C₂×C₂²⋊Dic₇₉

yes

characteristic

163

C₂×C₄

#14579144

yes

—

164

C₂×C₄

C₂×#11929609

yes

characteristic

165

C₂×C₄

#14579192

yes

—

166

C₈

—

167

C₈

—

632

—

168

C₈

#14579120

yes

—

169

C₈

#14579144

yes

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

C₂×#14579082

yes

characteristic

175

C₂²

#19098893

yes

—

176

C₄

—

177

C₄

—

178

C₄

#19098426

yes

characteristic

179

C₄

#19098572

yes

characteristic

180

C₄

#19098876

yes

—

181

C₂

—

182

C₂

—

183

C₂

#23763486

yes

characteristic

184

C₂

#23765149

yes

—

	subgroup	G/H
center	C₂²	C₂×#14579082
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#427×C₇₉	C₂
socle	C₂²×C₇₉	C₂×#102
prime radical	C₂⁴×C₇₉	SD₁₆
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#427	D₇₉
hypocenter	C₇₉	#12872
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#427

subgroup

G/H

center

C₂²

C₂×#14579082

commutator

C₂×C₈×C₇₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₂×#427×C₇₉

C₂

socle

C₂²×C₇₉

C₂×#102

prime radical

C₂⁴×C₇₉

SD₁₆

cosocle

C₂²×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#427

N	H ≅ G/N	type
3. #23761508	C₂	semidirect
97. C₇₉	#12872	semidirect

H ≅ G/N

type

3. #23761508

C₂

semidirect

97. C₇₉

#12872

semidirect

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

subgroup	index
#23765011	2
#23763438	2
#23761508	2
C₂×#427×C₇₉	2
#12872	79

subgroup

index

C₂×#427×C₇₉

Subgroups — #34204582

Conjugacy classes of subgroups

Complements

Maximal subgroups