Group #34187766 — subgroups

Subgroups — #34187766

Order 20224.

390 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 127 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	#23763538	C₂	1	1	yes	maximal, characteristic
2	#23762085	C₂	1	2	yes	maximal
3	#23762081	C₂	1	2	yes	maximal
4	#23761692	C₂	1	1	yes	maximal, characteristic
5	C₂×#500×C₇₉	C₂	1	1	yes	maximal, characteristic
6	#19098616	C₂²	1	1	yes	characteristic
7	#19098616	C₂²	1	1	yes	characteristic
8	#19098587	C₂²	1	2	yes	—
9	#19098328	—	2	2	no	—
10	#19098312	—	2	4	no	—
11	#19098309	—	2	4	no	—
12	#19098285	—	2	2	no	—
13	C₂×C₄⋊Q₈×C₇₉	C₂²	1	2	yes	—
14	#500×C₇₉	—	2	2	no	—
15	#500×C₇₉	C₄	1	4	yes	—
16	C₂×#107×C₇₉	—	2	2	no	—
17	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
18	C₂×#11929608	—	2	2	no	—
19	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
20	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
21	C₄⋊(C₇₉⋊C₈)	D₄	1	2	yes	—
22	C₄⋊(C₇₉⋊C₈)	D₄	1	2	yes	—
23	C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
24	C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
25	C₄×C₇₉⋊C₈	—	2	1	no	—
26	C₄×C₇₉⋊C₈	—	2	1	no	—
27	C₂²×Q₈×C₇₉	—	2	2	no	—
28	C₂²×C₈×C₇₉	D₄	1	2	yes	—
29	C₄⋊Q₈×C₇₉	—	2	2	no	—
30	C₄⋊Q₈×C₇₉	—	2	2	no	—
31	C₄⋊Q₈×C₇₉	—	2	2	no	—
32	C₄⋊Q₈×C₇₉	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
34	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
35	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
36	#107×C₇₉	—	2	8	no	—
37	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₇₉	D₄	1	2	yes	—
39	#11929608	—	4	4	no	—
40	C₂×C₇₉⋊C₈	—	2	2	no	—
41	C₂×C₇₉⋊C₈	—	2	2	no	—
42	C₂×C₇₉⋊C₈	—	4	2	no	—
43	C₂×Q₈×C₇₉	—	2	4	no	—
44	C₂×Q₈×C₇₉	—	4	2	no	—
45	C₂×Q₈×C₇₉	—	8	4	no	—
46	C₂²×C₄×C₇₉	—	4	2	no	—
47	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
49	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
50	C₂×C₈×C₇₉	—	2	4	no	—
51	C₂×C₈×C₇₉	C₂²⋊C₄	1	4	yes	—
52	C₄⋊C₄×C₇₉	—	2	4	no	—
53	C₄⋊C₄×C₇₉	—	2	4	no	—
54	C₄⋊C₄×C₇₉	—	4	2	no	—
55	C₄⋊C₄×C₇₉	—	4	2	no	—
56	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
60	C₂³×C₇₉	#124	1	1	yes	characteristic
61	Q₈×C₇₉	—	8	4	no	—
62	Q₈×C₇₉	—	8	4	no	—
63	C₂×C₄×C₇₉	—	2	2	no	—
64	C₂×C₄×C₇₉	—	2	2	no	—
65	C₂×C₄×C₇₉	—	4	4	no	—
66	C₂×C₄×C₇₉	—	8	4	no	—
67	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
71	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
72	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
73	C₂×C₄×C₇₉	#137	1	4	yes	—
74	C₈×C₇₉	—	2	4	no	—
75	C₇₉⋊C₈	—	4	2	no	—
76	C₇₉⋊C₈	—	4	2	no	—
77	C₇₉⋊C₈	—	8	2	no	—
78	C₂²×C₇₉	—	2	1	no	—
79	C₂²×C₇₉	—	2	1	no	—
80	C₂²×C₇₉	#390	1	1	yes	characteristic
81	C₂²×C₇₉	#491	1	2	yes	—
82	C₄×C₇₉	—	2	1	no	—
83	C₄×C₇₉	—	2	1	no	—
84	C₄×C₇₉	—	8	4	no	—
85	C₄×C₇₉	#419	1	2	yes	—
86	C₄×C₇₉	#472	1	2	yes	—
87	C₄×C₇₉	#473	1	2	yes	—
88	C₄×C₇₉	#497	1	2	yes	—
89	#9778	—	79	1	no	maximal, Sylow
90	C₂×C₇₉	—	2	2	no	—
91	C₂×C₇₉	#1960	1	1	yes	characteristic
92	C₂×C₇₉	#1978	1	1	yes	characteristic
93	C₂×C₇₉	#1979	1	1	yes	characteristic
94	C₂×#500	D₇₉	1	1	yes	characteristic
95	#1837	—	79	1	no	—
96	#1549	—	79	2	no	—
97	#1548	—	79	2	no	—
98	#1452	—	79	1	no	—
99	C₇₉	#9778	1	1	yes	characteristic, complemented, Sylow
100	C₂×C₄⋊Q₈	C₂×D₇₉	1	2	yes	—
101	#500	—	2	2	no	—
102	#500	Dic₇₉	1	4	yes	—
103	#431	—	79	1	no	—
104	#431	—	79	1	no	—
105	C₂×#107	—	2	2	no	—
106	#423	—	79	2	no	—
107	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
108	#341	—	158	2	no	—
109	#333	—	158	4	no	—
110	#332	—	158	4	no	—
111	#322	—	158	2	no	—
112	C₂²×Q₈	—	2	2	no	—
113	C₂×M₁₆	—	158	2	no	—
114	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
115	C₄⋊Q₈	—	2	2	no	—
116	C₄⋊Q₈	—	2	2	no	—
117	C₄⋊Q₈	—	2	2	no	—
118	C₄⋊Q₈	C₂×Dic₇₉	1	4	yes	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
122	#107	—	2	8	no	—
123	C₄⋊C₈	—	79	2	no	—
124	C₄⋊C₈	—	79	2	no	—
125	C₄⋊C₈	—	158	2	no	—
126	C₄⋊C₈	—	158	2	no	—
127	C₂²⋊C₈	—	158	1	no	—
128	C₂²⋊C₈	—	158	1	no	—
129	C₄×C₈	—	158	1	no	—
130	C₄×C₈	—	158	1	no	—
131	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
132	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
133	C₂×Q₈	—	2	4	no	—
134	C₂×Q₈	—	4	2	no	—
135	C₂×Q₈	—	8	4	no	—
136	C₂²×C₄	—	4	2	no	—
137	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
138	C₂²×C₄	#11929623	1	1	yes	characteristic
139	C₂²×C₄	#11929623	1	1	yes	characteristic
140	M₁₆	—	316	4	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	316	2	no	—
143	C₂×C₈	—	158	2	no	—
144	C₂×C₈	—	158	2	no	—
145	C₂×C₈	C₂²⋊Dic₇₉	1	4	yes	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	2	4	no	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
151	C₄²	D₄×D₇₉	1	1	yes	characteristic
152	C₄²	D₄×D₇₉	1	1	yes	characteristic
153	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
154	C₂³	#14579148	1	1	yes	characteristic
155	Q₈	—	8	4	no	—
156	Q₈	—	8	4	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	4	4	no	—
160	C₂×C₄	—	8	4	no	—
161	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
162	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
163	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
164	C₂×C₄	#14579132	1	1	yes	characteristic
165	C₂×C₄	#14579132	1	1	yes	characteristic
166	C₂×C₄	#14579157	1	1	yes	characteristic
167	C₂×C₄	#14579192	1	4	yes	—
168	C₈	—	2	4	no	—
169	C₈	—	316	2	no	—
170	C₈	—	316	2	no	—
171	C₈	—	632	2	no	—
172	C₂²	—	2	1	no	—
173	C₂²	—	2	1	no	—
174	C₂²	#19098466	1	1	yes	characteristic
175	C₂²	#19098931	1	2	yes	—
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	—	8	4	no	—
179	C₄	#19098572	1	2	yes	—
180	C₄	#19098830	1	2	yes	—
181	C₄	#19098835	1	2	yes	—
182	C₄	#19098955	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	#23764221	1	1	yes	characteristic
185	C₂	#23764311	1	1	yes	characteristic
186	C₂	#23764316	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23763538

C₂

yes

maximal, characteristic

#23762085

C₂

yes

maximal

#23762081

C₂

yes

maximal

#23761692

C₂

yes

maximal, characteristic

C₂×#500×C₇₉

C₂

yes

maximal, characteristic

#19098616

C₂²

yes

characteristic

#19098616

C₂²

yes

characteristic

#19098587

C₂²

yes

—

—

—

—

—

C₂×C₄⋊Q₈×C₇₉

C₂²

yes

—

#500×C₇₉

—

#500×C₇₉

C₄

yes

—

C₂×#107×C₇₉

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×#11929608

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

D₄

yes

—

C₄⋊(C₇₉⋊C₈)

D₄

yes

—

C₂²⋊(C₇₉⋊C₈)

—

C₂²⋊(C₇₉⋊C₈)

—

C₄×C₇₉⋊C₈

—

C₄×C₇₉⋊C₈

—

C₂²×Q₈×C₇₉

—

C₂²×C₈×C₇₉

D₄

yes

—

C₄⋊Q₈×C₇₉

—

C₄⋊Q₈×C₇₉

—

C₄⋊Q₈×C₇₉

—

C₄⋊Q₈×C₇₉

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

#107×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

D₄

yes

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

C₂²×C₄×C₇₉

yes

C₂²×C₄×C₇₉

yes

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₂³×C₇₉

#124

yes

characteristic

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄×D₄

yes

characteristic

C₂×C₄×C₇₉

C₄×D₄

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#390

yes

characteristic

C₂²×C₇₉

#491

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#419

yes

—

C₄×C₇₉

#472

yes

—

C₄×C₇₉

#473

yes

—

C₄×C₇₉

#497

yes

—

#9778

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

D₇₉

yes

—

—

—

—

C₇₉

#9778

yes

characteristic, complemented, Sylow

100

C₂×C₄⋊Q₈

C₂×D₇₉

yes

—

101

#500

—

102

#500

Dic₇₉

yes

—

103

#431

—

104

#431

—

105

C₂×#107

—

106

#423

—

107

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

108

#341

—

158

—

109

#333

—

158

—

110

#332

—

158

—

111

#322

—

158

—

112

C₂²×Q₈

—

113

C₂×M₁₆

—

158

—

114

C₂²×C₈

C₂²⋊D₇₉

yes

—

115

C₄⋊Q₈

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

—

118

C₄⋊Q₈

C₂×Dic₇₉

yes

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄²

C₂²×D₇₉

yes

characteristic

122

#107

—

123

C₄⋊C₈

—

124

C₄⋊C₈

—

125

C₄⋊C₈

—

158

—

126

C₄⋊C₈

—

158

—

127

C₂²⋊C₈

—

158

—

128

C₂²⋊C₈

—

158

—

129

C₄×C₈

—

158

—

130

C₄×C₈

—

158

—

131

C₄×C₈

C₂×Dic₇₉

yes

—

132

C₄×C₈

C₂²⋊D₇₉

yes

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂²×C₄

—

137

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

138

C₂²×C₄

#11929623

yes

characteristic

139

C₂²×C₄

#11929623

yes

characteristic

140

M₁₆

—

316

—

141

C₂×C₈

—

142

C₂×C₈

—

316

—

143

C₂×C₈

—

158

—

144

C₂×C₈

—

158

—

145

C₂×C₈

C₂²⋊Dic₇₉

yes

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄²

C₂²×Dic₇₉

yes

characteristic

151

C₄²

D₄×D₇₉

yes

characteristic

152

C₄²

D₄×D₇₉

yes

characteristic

153

C₄²

C₂×C₂²⋊D₇₉

yes

characteristic

154

C₂³

#14579148

yes

characteristic

155

Q₈

—

156

Q₈

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

C₂×C₂²⋊Dic₇₉

yes

characteristic

162

C₂×C₄

D₄×Dic₇₉

yes

characteristic

163

C₂×C₄

D₄×Dic₇₉

yes

characteristic

164

C₂×C₄

#14579132

yes

characteristic

165

C₂×C₄

#14579132

yes

characteristic

166

C₂×C₄

#14579157

yes

characteristic

167

C₂×C₄

#14579192

yes

—

168

C₈

—

169

C₈

—

316

—

170

C₈

—

316

—

171

C₈

—

632

—

172

C₂²

—

173

C₂²

—

174

C₂²

#19098466

yes

characteristic

175

C₂²

#19098931

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

#19098572

yes

—

180

C₄

#19098830

yes

—

181

C₄

#19098835

yes

—

182

C₄

#19098955

yes

—

183

C₂

—

184

C₂

#23764221

yes

characteristic

185

C₂

#23764311

yes

characteristic

186

C₂

#23764316

yes

characteristic

	subgroup	G/H
center	C₂²	#19098466
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×#500×C₇₉	C₂
socle	C₂²×C₇₉	#390
prime radical	C₂³×C₇₉	#124
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×#500	D₇₉
hypocenter	C₇₉	#9778
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

subgroup

G/H

center

C₂²

#19098466

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₂×#500×C₇₉

C₂

C₂²×C₇₉

C₂³×C₇₉

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

N	H ≅ G/N	type
99. C₇₉	#9778	semidirect

H ≅ G/N

type

99. C₇₉

#9778

semidirect

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

subgroup	index
#23763538	2
#23762085	2
#23762081	2
#23761692	2
C₂×#500×C₇₉	2
#9778	79

subgroup

index

C₂×#500×C₇₉

Subgroups — #34187766

Conjugacy classes of subgroups

Complements

Maximal subgroups