Group #34182034 — subgroups

Subgroups — #34182034

Order 20224.

382 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 163 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	#23763552	C₂	1	1	yes	maximal, characteristic
2	#23763146	C₂	1	1	yes	maximal, characteristic
3	C₂×#19098315	C₂	1	2	yes	maximal
4	C₂×#19098313	C₂	1	2	yes	maximal
5	C₇₉×#1831	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
7	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
9	#19098318	C₄	1	2	yes	—
10	#19098318	C₄	1	2	yes	—
11	#19098315	—	2	4	no	—
12	#19098313	—	2	4	no	—
13	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
15	C₄³×C₇₉	C₂²	1	1	yes	characteristic
16	#336×C₇₉	—	2	2	no	—
17	C₂²×C₇₉⋊C₈	—	2	2	no	—
18	C₂²×C₇₉⋊C₈	C₂×C₄	1	4	yes	—
19	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
20	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
21	C₄⋊(C₇₉⋊C₈)	—	4	4	no	—
22	C₄×C₇₉⋊C₈	D₄	1	4	yes	—
23	C₄×C₇₉⋊C₈	Q₈	1	4	yes	—
24	C₂²×C₈×C₇₉	—	2	1	no	—
25	C₂²×C₈×C₇₉	—	2	1	no	—
26	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₇₉	D₄	1	2	yes	—
29	C₂×C₄²×C₇₉	Q₈	1	2	yes	—
30	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₇₉	—	2	2	no	—
32	C₄⋊C₈×C₇₉	—	2	2	no	—
33	C₂×C₇₉⋊C₈	—	2	4	no	—
34	C₂×C₇₉⋊C₈	—	2	4	no	—
35	C₂×C₇₉⋊C₈	—	4	2	no	—
36	C₂×C₇₉⋊C₈	—	4	4	no	—
37	C₂×C₇₉⋊C₈	C₄⋊C₄	1	8	yes	—
38	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
39	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₇₉	—	4	1	no	—
44	C₂×C₈×C₇₉	—	4	1	no	—
45	C₂×C₈×C₇₉	—	4	2	no	—
46	C₂×C₈×C₇₉	—	4	2	no	—
47	C₄²×C₇₉	—	2	4	no	—
48	C₄²×C₇₉	—	2	2	no	—
49	C₄²×C₇₉	—	2	2	no	—
50	C₄²×C₇₉	—	2	2	no	—
51	C₄²×C₇₉	—	2	2	no	—
52	C₄²×C₇₉	C₂×D₄	1	2	yes	—
53	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
54	C₂³×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₇₉	—	2	2	no	—
56	C₂×C₄×C₇₉	—	2	2	no	—
57	C₂×C₄×C₇₉	—	2	2	no	—
58	C₂×C₄×C₇₉	—	2	2	no	—
59	C₂×C₄×C₇₉	—	2	2	no	—
60	C₂×C₄×C₇₉	—	2	2	no	—
61	C₂×C₄×C₇₉	#108	1	4	yes	—
62	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
63	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
64	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
66	C₂×C₄×C₇₉	#131	1	2	yes	—
67	C₂×C₄×C₇₉	#131	1	2	yes	—
68	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
69	C₂×C₄×C₇₉	#137	1	2	yes	—
70	C₈×C₇₉	—	8	1	no	—
71	C₈×C₇₉	—	8	1	no	—
72	C₇₉⋊C₈	—	2	8	no	—
73	C₇₉⋊C₈	—	8	2	no	—
74	C₂²×C₇₉	#384	1	1	yes	characteristic
75	C₂²×C₇₉	#424	1	1	yes	characteristic
76	C₂²×C₇₉	#424	1	1	yes	characteristic
77	C₂²×C₇₉	#428	1	2	yes	—
78	C₂²×C₇₉	C₂×#108	1	2	yes	—
79	C₄×C₇₉	—	2	4	no	—
80	C₄×C₇₉	—	2	2	no	—
81	C₄×C₇₉	—	4	2	no	—
82	C₄×C₇₉	#424	1	4	yes	—
83	C₄×C₇₉	#496	1	1	yes	characteristic
84	C₄×C₇₉	#497	1	1	yes	characteristic
85	C₄×C₇₉	#501	1	2	yes	—
86	#8812	—	79	1	no	maximal, Sylow
87	C₂×C₇₉	#1573	1	2	yes	—
88	C₂×C₇₉	#1574	1	2	yes	—
89	C₂×C₇₉	#1843	1	1	yes	characteristic
90	C₂×C₇₉	#1847	1	1	yes	characteristic
91	C₂×C₇₉	#1848	1	1	yes	characteristic
92	#1842	—	79	1	no	—
93	#1831	D₇₉	1	1	yes	characteristic
94	#1744	—	79	1	no	—
95	C₂×#335	—	79	2	no	—
96	C₂×#334	—	79	2	no	—
97	C₇₉	#8812	1	1	yes	characteristic, complemented, Sylow
98	C₂×C₄⋊C₈	—	79	2	no	—
99	C₂×C₄⋊C₈	—	158	2	no	—
100	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
101	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
102	C₂×C₄×C₈	—	79	2	no	—
103	C₄³	C₂×D₇₉	1	1	yes	characteristic
104	#336	—	2	2	no	—
105	#336	—	79	2	no	—
106	#336	—	79	2	no	—
107	#335	—	158	4	no	—
108	#334	—	158	4	no	—
109	C₂²×C₈	—	2	1	no	—
110	C₂²×C₈	—	2	1	no	—
111	C₂²×C₈	—	79	4	no	—
112	C₂²×C₈	—	158	2	no	—
113	C₂×C₄²	Dic₁₅₈	1	2	yes	—
114	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
115	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
116	C₂×C₄²	C₂²⋊D₇₉	1	2	yes	—
117	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
118	C₄⋊C₈	—	2	2	no	—
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	316	4	no	—
121	C₄⋊C₈	—	158	2	no	—
122	C₄⋊C₈	—	158	2	no	—
123	C₄×C₈	—	79	4	no	—
124	C₄×C₈	—	79	4	no	—
125	C₂²×C₄	Dic₇₉⋊C₄	1	2	yes	—
126	C₂²×C₄	Dic₇₉⋊C₄	1	2	yes	—
127	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
128	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
129	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
130	C₂×C₈	—	4	1	no	—
131	C₂×C₈	—	4	1	no	—
132	C₂×C₈	—	4	2	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	—	316	2	no	—
135	C₂×C₈	—	316	4	no	—
136	C₂×C₈	—	79	8	no	—
137	C₂×C₈	—	158	4	no	—
138	C₂×C₈	—	158	4	no	—
139	C₄²	—	2	4	no	—
140	C₄²	—	2	2	no	—
141	C₄²	—	2	2	no	—
142	C₄²	—	2	2	no	—
143	C₄²	—	2	2	no	—
144	C₄²	D₄×D₇₉	1	2	yes	—
145	C₄²	Q₈×D₇₉	1	2	yes	—
146	C₂³	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	#14579096	1	4	yes	—
154	C₂×C₄	C₄⋊C₄×D₇₉	1	2	yes	—
155	C₂×C₄	#14579157	1	1	yes	characteristic
156	C₂×C₄	#14579159	1	1	yes	characteristic
157	C₂×C₄	#14579160	1	2	yes	—
158	C₂×C₄	#14579167	1	2	yes	—
159	C₂×C₄	#14579169	1	2	yes	—
160	C₂×C₄	#14579186	1	2	yes	—
161	C₂×C₄	#14579191	1	2	yes	—
162	C₈	—	8	1	no	—
163	C₈	—	8	1	no	—
164	C₈	—	632	2	no	—
165	C₈	—	158	8	no	—
166	C₂²	#19098432	1	1	yes	characteristic
167	C₂²	#19098589	1	1	yes	characteristic
168	C₂²	#19098591	1	1	yes	characteristic
169	C₂²	#19098606	1	2	yes	—
170	C₂²	C₂×#14579096	1	2	yes	—
171	C₄	—	2	4	no	—
172	C₄	—	2	2	no	—
173	C₄	—	4	2	no	—
174	C₄	#19098588	1	4	yes	—
175	C₄	#19098953	1	1	yes	characteristic
176	C₄	#19098958	1	1	yes	characteristic
177	C₄	#19098973	1	2	yes	—
178	C₂	#23762186	1	2	yes	—
179	C₂	#23762188	1	2	yes	—
180	C₂	#23763553	1	1	yes	characteristic
181	C₂	#23763569	1	1	yes	characteristic
182	C₂	#23763571	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23763552

C₂

yes

maximal, characteristic

#23763146

C₂

yes

maximal, characteristic

C₂×#19098315

C₂

yes

maximal

C₂×#19098313

C₂

yes

maximal

C₇₉×#1831

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₇₉⋊C₈)

—

C₂×C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

C₂×C₄×C₇₉⋊C₈

C₂²

yes

—

#19098318

C₄

yes

—

#19098318

C₄

yes

—

#19098315

—

#19098313

—

C₂×C₄⋊C₈×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₇₉

C₂²

yes

characteristic

C₄³×C₇₉

C₂²

yes

characteristic

#336×C₇₉

—

C₂²×C₇₉⋊C₈

—

C₂²×C₇₉⋊C₈

C₂×C₄

yes

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₄×C₇₉⋊C₈

D₄

yes

—

C₄×C₇₉⋊C₈

Q₈

yes

—

C₂²×C₈×C₇₉

—

C₂²×C₈×C₇₉

—

C₂×C₄²×C₇₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

Q₈

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₄⋊C₄

yes

—

C₂²×C₄×C₇₉

C₄⋊C₄

yes

—

C₂²×C₄×C₇₉

C₄⋊C₄

yes

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂×D₄

yes

—

C₄²×C₇₉

C₂×Q₈

yes

—

C₂³×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#108

yes

—

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

C₄⋊Q₈

yes

—

C₂×C₄×C₇₉

#131

yes

—

C₂×C₄×C₇₉

#131

yes

—

C₂×C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇₉

#137

yes

—

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

—

C₂²×C₇₉

C₂×#108

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#424

yes

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

#8812

—

maximal, Sylow

C₂×C₇₉

#1573

yes

—

C₂×C₇₉

#1574

yes

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

D₇₉

yes

—

—

—

C₇₉

#8812

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

158

—

100

C₂×C₄⋊C₈

C₂×D₇₉

yes

characteristic

101

C₂×C₄⋊C₈

C₂×D₇₉

yes

characteristic

102

C₂×C₄×C₈

—

103

C₄³

C₂×D₇₉

yes

characteristic

104

#336

—

105

#336

—

106

#336

—

107

#335

—

158

—

108

#334

—

158

—

109

C₂²×C₈

—

110

C₂²×C₈

—

111

C₂²×C₈

—

112

C₂²×C₈

—

158

—

113

C₂×C₄²

Dic₁₅₈

yes

—

114

C₂×C₄²

C₄×D₇₉

yes

characteristic

115

C₂×C₄²

C₄×D₇₉

yes

characteristic

116

C₂×C₄²

C₂²⋊D₇₉

yes

—

117

C₂×C₄²

C₂²×D₇₉

yes

characteristic

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₄⋊C₈

—

316

—

121

C₄⋊C₈

—

158

—

122

C₄⋊C₈

—

158

—

123

C₄×C₈

—

124

C₄×C₈

—

125

C₂²×C₄

Dic₇₉⋊C₄

yes

—

126

C₂²×C₄

Dic₇₉⋊C₄

yes

—

127

C₂²×C₄

C₂×C₄×D₇₉

yes

characteristic

128

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

129

C₂²×C₄

C₂×Dic₁₅₈

yes

characteristic

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

316

—

135

C₂×C₈

—

316

—

136

C₂×C₈

—

137

C₂×C₈

—

158

—

138

C₂×C₈

—

158

—

139

C₄²

—

140

C₄²

—

141

C₄²

—

142

C₄²

—

143

C₄²

—

144

C₄²

D₄×D₇₉

yes

—

145

C₄²

Q₈×D₇₉

yes

—

146

C₂³

C₂×Dic₇₉⋊C₄

yes

characteristic

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

#14579096

yes

—

154

C₂×C₄

C₄⋊C₄×D₇₉

yes

—

155

C₂×C₄

#14579157

yes

characteristic

156

C₂×C₄

#14579159

yes

characteristic

157

C₂×C₄

#14579160

yes

—

158

C₂×C₄

#14579167

yes

—

159

C₂×C₄

#14579169

yes

—

160

C₂×C₄

#14579186

yes

—

161

C₂×C₄

#14579191

yes

—

162

C₈

—

163

C₈

—

164

C₈

—

632

—

165

C₈

—

158

—

166

C₂²

#19098432

yes

characteristic

167

C₂²

#19098589

yes

characteristic

168

C₂²

#19098591

yes

characteristic

169

C₂²

#19098606

yes

—

170

C₂²

C₂×#14579096

yes

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

#19098588

yes

—

175

C₄

#19098953

yes

characteristic

176

C₄

#19098958

yes

characteristic

177

C₄

#19098973

yes

—

178

C₂

#23762186

yes

—

179

C₂

#23762188

yes

—

180

C₂

#23763553

yes

characteristic

181

C₂

#23763569

yes

characteristic

182

C₂

#23763571

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₉
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1831	C₂
socle, prime radical	C₂³×C₇₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1831	D₇₉
hypocenter	C₇₉	#8812
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

subgroup

G/H

center

C₂²×C₄

C₂×C₂²⋊D₇₉

commutator

C₂²×C₄×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1831

C₂

socle, prime radical

C₂³×C₇₉

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

N	H ≅ G/N	type
97. C₇₉	#8812	semidirect

H ≅ G/N

type

97. C₇₉

#8812

semidirect

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

subgroup	index
#23763552	2
#23763146	2
C₂×#19098315	2
C₂×#19098313	2
C₇₉×#1831	2
#8812	79

subgroup

index

C₇₉×#1831

Subgroups — #34182034

Conjugacy classes of subgroups

Complements

Maximal subgroups