Group #34182074 — subgroups

Subgroups — #34182074

Order 20224.

322 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 139 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	#23762158	C₂	1	2	yes	maximal
3	#23762155	C₂	1	2	yes	maximal
4	C₂×C₈×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₇₉×#1837	C₂	1	1	yes	maximal, characteristic
6	#19098587	C₂²	1	2	yes	—
7	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
9	#19098328	—	2	2	no	—
10	#19098315	—	2	4	no	—
11	#19098313	—	2	4	no	—
12	C₈×C₇₉⋊C₈	C₄	1	1	yes	characteristic
13	C₈×C₇₉⋊C₈	C₄	1	1	yes	characteristic
14	C₈×C₇₉⋊C₈	C₄	1	1	yes	characteristic
15	C₈×C₇₉⋊C₈	C₄	1	1	yes	characteristic
16	#423×C₇₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
18	#341×C₇₉	—	2	2	no	—
19	C₂×#11929608	—	2	2	no	—
20	C₂²×C₇₉⋊C₈	Q₈	1	2	yes	—
21	C₂²×C₇₉⋊C₈	Q₈	1	2	yes	—
22	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
23	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
24	C₄×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
29	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
30	C₂×M₁₆×C₇₉	—	2	2	no	—
31	C₂²×C₈×C₇₉	D₄	1	2	yes	—
32	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₇₉	—	2	2	no	—
34	C₄⋊C₈×C₇₉	—	2	2	no	—
35	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₇₉	Q₈	1	2	yes	—
38	#11929608	—	4	4	no	—
39	C₂×C₇₉⋊C₈	—	2	4	no	—
40	C₂×C₇₉⋊C₈	—	2	4	no	—
41	C₂×C₇₉⋊C₈	—	4	2	no	—
42	C₂×C₇₉⋊C₈	C₄⋊C₄	1	2	yes	—
43	C₂×C₇₉⋊C₈	C₄⋊C₄	1	2	yes	—
44	C₂×C₇₉⋊C₈	C₄⋊C₄	1	2	yes	—
45	C₂×C₇₉⋊C₈	C₄⋊C₄	1	2	yes	—
46	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
49	M₁₆×C₇₉	—	4	4	no	—
50	C₂×C₈×C₇₉	—	2	4	no	—
51	C₂×C₈×C₇₉	—	4	2	no	—
52	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
53	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
54	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
55	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
58	C₂³×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₄×C₇₉	—	2	2	no	—
60	C₂×C₄×C₇₉	—	2	2	no	—
61	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
65	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₇₉	#135	1	4	yes	—
68	C₈×C₇₉	—	2	2	no	—
69	C₈×C₇₉	—	2	2	no	—
70	C₈×C₇₉	—	8	2	no	—
71	C₇₉⋊C₈	—	2	2	no	—
72	C₇₉⋊C₈	—	2	2	no	—
73	C₇₉⋊C₈	—	8	2	no	—
74	C₇₉⋊C₈	#108	1	4	yes	—
75	C₇₉⋊C₈	#108	1	4	yes	—
76	C₂²×C₇₉	—	2	1	no	—
77	C₂²×C₇₉	—	2	1	no	—
78	C₂²×C₇₉	#384	1	1	yes	characteristic
79	C₂²×C₇₉	#499	1	2	yes	—
80	C₄×C₇₉	—	2	1	no	—
81	C₄×C₇₉	—	2	1	no	—
82	C₄×C₇₉	#427	1	2	yes	—
83	C₄×C₇₉	C₂×#108	1	2	yes	—
84	C₄×C₇₉	C₂×#108	1	2	yes	—
85	C₄×C₇₉	#495	1	2	yes	—
86	#8824	—	79	1	no	maximal, Sylow
87	C₂×C₇₉	—	2	2	no	—
88	C₂×C₇₉	#1846	1	1	yes	characteristic
89	C₂×C₇₉	#1848	1	1	yes	characteristic
90	C₂×C₇₉	#1848	1	1	yes	characteristic
91	#1837	—	79	1	no	—
92	#1837	D₇₉	1	1	yes	characteristic
93	#1567	—	79	2	no	—
94	#1566	—	79	2	no	—
95	C₂×C₈²	—	79	1	no	—
96	C₇₉	#8824	1	1	yes	characteristic, complemented, Sylow
97	#423	—	79	2	no	—
98	#423	C₂×D₇₉	1	2	yes	—
99	C₂×C₄×C₈	—	79	1	no	—
100	C₂×C₄×C₈	—	79	1	no	—
101	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
102	#341	—	2	2	no	—
103	#341	—	158	2	no	—
104	#335	—	158	4	no	—
105	#334	—	158	4	no	—
106	C₈²	—	79	1	no	—
107	C₈²	—	79	1	no	—
108	C₈²	—	79	1	no	—
109	C₈²	—	79	1	no	—
110	C₂×M₁₆	—	2	2	no	—
111	C₂×M₁₆	—	158	2	no	—
112	C₂²×C₈	—	79	2	no	—
113	C₂²×C₈	—	79	2	no	—
114	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
115	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	158	2	no	—
119	C₄⋊C₈	—	158	2	no	—
120	C₄×C₈	—	79	1	no	—
121	C₄×C₈	—	79	1	no	—
122	C₄×C₈	—	79	1	no	—
123	C₄×C₈	—	79	1	no	—
124	C₄×C₈	—	79	2	no	—
125	C₄×C₈	—	79	2	no	—
126	C₄×C₈	Dic₁₅₈	1	2	yes	—
127	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
128	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
129	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
130	C₂²×C₄	Q₈×D₇₉	1	1	yes	characteristic
131	C₂²×C₄	Q₈×D₇₉	1	1	yes	characteristic
132	M₁₆	—	4	4	no	—
133	M₁₆	—	316	4	no	—
134	C₂×C₈	—	2	4	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	316	2	no	—
137	C₂×C₈	—	79	2	no	—
138	C₂×C₈	—	79	2	no	—
139	C₂×C₈	—	79	2	no	—
140	C₂×C₈	—	79	2	no	—
141	C₂×C₈	—	158	4	no	—
142	C₂×C₈	—	158	4	no	—
143	C₂×C₈	Dic₇₉⋊C₄	1	2	yes	—
144	C₂×C₈	Dic₇₉⋊C₄	1	2	yes	—
145	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
146	C₄²	D₄×D₇₉	1	1	yes	characteristic
147	C₄²	D₄×D₇₉	1	1	yes	characteristic
148	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
149	C₂³	#14579159	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
153	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
154	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
155	C₂×C₄	#14579157	1	1	yes	characteristic
156	C₂×C₄	#14579160	1	1	yes	characteristic
157	C₂×C₄	#14579160	1	1	yes	characteristic
158	C₂×C₄	#14579181	1	4	yes	—
159	C₈	—	2	2	no	—
160	C₈	—	2	2	no	—
161	C₈	—	8	2	no	—
162	C₈	—	79	4	no	—
163	C₈	—	79	4	no	—
164	C₈	—	632	2	no	—
165	C₈	—	158	2	no	—
166	C₈	—	158	2	no	—
167	C₂²	—	2	1	no	—
168	C₂²	—	2	1	no	—
169	C₂²	#19098432	1	1	yes	characteristic
170	C₂²	#19098962	1	2	yes	—
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	#19098599	1	2	yes	—
174	C₄	#108×D₇₉	1	2	yes	—
175	C₄	#108×D₇₉	1	2	yes	—
176	C₄	#19098945	1	2	yes	—
177	C₂	—	2	2	no	—
178	C₂	#23763562	1	1	yes	characteristic
179	C₂	#23763570	1	1	yes	characteristic
180	C₂	#23763570	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23763538

C₂

yes

maximal, characteristic

#23762158

C₂

yes

maximal

#23762155

C₂

yes

maximal

C₂×C₈×C₇₉⋊C₈

C₂

yes

maximal, characteristic

C₇₉×#1837

C₂

yes

maximal, characteristic

#19098587

C₂²

yes

—

C₂×C₄×C₇₉⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₇₉⋊C₈

C₂²

yes

—

—

—

C₈×C₇₉⋊C₈

C₄

yes

characteristic

C₈×C₇₉⋊C₈

C₄

yes

characteristic

C₈×C₇₉⋊C₈

C₄

yes

characteristic

C₈×C₇₉⋊C₈

C₄

yes

characteristic

#423×C₇₉

C₂²

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#341×C₇₉

—

C₂×#11929608

—

C₂²×C₇₉⋊C₈

Q₈

yes

—

C₂²×C₇₉⋊C₈

Q₈

yes

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₄×C₇₉⋊C₈

C₂×C₄

yes

characteristic

C₄×C₇₉⋊C₈

C₂×C₄

yes

characteristic

C₄×C₇₉⋊C₈

C₂×C₄

yes

characteristic

C₄×C₇₉⋊C₈

C₂×C₄

yes

characteristic

C₄×C₇₉⋊C₈

D₄

yes

—

C₄×C₇₉⋊C₈

D₄

yes

—

C₂×M₁₆×C₇₉

—

C₂²×C₈×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

Q₈

yes

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₄⋊C₄

yes

—

C₂×C₇₉⋊C₈

C₄⋊C₄

yes

—

C₂×C₇₉⋊C₈

C₄⋊C₄

yes

—

C₂×C₇₉⋊C₈

C₄⋊C₄

yes

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×Q₈

yes

characteristic

C₂²×C₄×C₇₉

C₂×Q₈

yes

characteristic

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

C₂×Q₈

yes

characteristic

C₂³×C₇₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

#135

yes

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

#108

yes

—

C₇₉⋊C₈

#108

yes

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#384

yes

characteristic

C₂²×C₇₉

#499

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#427

yes

—

C₄×C₇₉

C₂×#108

yes

—

C₄×C₇₉

C₂×#108

yes

—

C₄×C₇₉

#495

yes

—

#8824

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

D₇₉

yes

characteristic

#1567

—

#1566

—

C₂×C₈²

—

C₇₉

#8824

yes

characteristic, complemented, Sylow

#423

—

#423

C₂×D₇₉

yes

—

C₂×C₄×C₈

—

100

C₂×C₄×C₈

—

101

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

102

#341

—

103

#341

—

158

—

104

#335

—

158

—

105

#334

—

158

—

106

C₈²

—

107

C₈²

—

108

C₈²

—

109

C₈²

—

110

C₂×M₁₆

—

111

C₂×M₁₆

—

158

—

112

C₂²×C₈

—

113

C₂²×C₈

—

114

C₂²×C₈

C₂²⋊D₇₉

yes

—

115

C₂×C₄²

C₂²×D₇₉

yes

characteristic

116

C₄⋊C₈

—

117

C₄⋊C₈

—

118

C₄⋊C₈

—

158

—

119

C₄⋊C₈

—

158

—

120

C₄×C₈

—

121

C₄×C₈

—

122

C₄×C₈

—

123

C₄×C₈

—

124

C₄×C₈

—

125

C₄×C₈

—

126

C₄×C₈

Dic₁₅₈

yes

—

127

C₄×C₈

C₄×D₇₉

yes

characteristic

128

C₄×C₈

C₄×D₇₉

yes

characteristic

129

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

130

C₂²×C₄

Q₈×D₇₉

yes

characteristic

131

C₂²×C₄

Q₈×D₇₉

yes

characteristic

132

M₁₆

—

133

M₁₆

—

316

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

316

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

158

—

142

C₂×C₈

—

158

—

143

C₂×C₈

Dic₇₉⋊C₄

yes

—

144

C₂×C₈

Dic₇₉⋊C₄

yes

—

145

C₄²

C₂×C₄×D₇₉

yes

characteristic

146

C₄²

D₄×D₇₉

yes

characteristic

147

C₄²

D₄×D₇₉

yes

characteristic

148

C₄²

C₂×Dic₁₅₈

yes

characteristic

149

C₂³

#14579159

yes

characteristic

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

C₄⋊C₄×D₇₉

yes

characteristic

153

C₂×C₄

C₄⋊C₄×D₇₉

yes

characteristic

154

C₂×C₄

C₂×Dic₇₉⋊C₄

yes

characteristic

155

C₂×C₄

#14579157

yes

characteristic

156

C₂×C₄

#14579160

yes

characteristic

157

C₂×C₄

#14579160

yes

characteristic

158

C₂×C₄

#14579181

yes

—

159

C₈

—

160

C₈

—

161

C₈

—

162

C₈

—

163

C₈

—

164

C₈

—

632

—

165

C₈

—

158

—

166

C₈

—

158

—

167

C₂²

—

168

C₂²

—

169

C₂²

#19098432

yes

characteristic

170

C₂²

#19098962

yes

—

171

C₄

—

172

C₄

—

173

C₄

#19098599

yes

—

174

C₄

#108×D₇₉

yes

—

175

C₄

#108×D₇₉

yes

—

176

C₄

#19098945

yes

—

177

C₂

—

178

C₂

#23763562

yes

characteristic

179

C₂

#23763570

yes

characteristic

180

C₂

#23763570

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14579157
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1837	C₂
socle	C₂²×C₇₉	#384
prime radical	C₂³×C₇₉	C₄⋊Q₈
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1837	D₇₉
hypocenter	C₇₉	#8824
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1837

subgroup

G/H

center

C₂×C₄

#14579157

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1837

C₂

socle

C₂²×C₇₉

#384

prime radical

C₂³×C₇₉

C₄⋊Q₈

cosocle

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1837

N	H ≅ G/N	type
96. C₇₉	#8824	semidirect

H ≅ G/N

type

96. C₇₉

#8824

semidirect

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

subgroup	index
#23763538	2
#23762158	2
#23762155	2
C₂×C₈×C₇₉⋊C₈	2
C₇₉×#1837	2
#8824	79

subgroup

index

#23763538

#23762158

#23762155

C₂×C₈×C₇₉⋊C₈

C₇₉×#1837

#8824

Subgroups — #34182074

Conjugacy classes of subgroups

Complements

Maximal subgroups