Group #34188413 — subgroups

Subgroups — #34188413

Order 20224.

438 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 167 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	#23763253	C₂	1	1	yes	maximal, characteristic
3	#23761283	C₂	1	4	yes	maximal
4	C₇₉×#2859	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
6	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
8	C₂×#14579072	—	2	1	no	—
9	#19098318	—	2	1	no	—
10	C₂×#125×C₇₉	—	2	1	no	—
11	#389×C₇₉	—	2	2	no	—
12	#389×C₇₉	C₄	1	4	yes	—
13	#384×C₇₉	C₂²	1	2	yes	—
14	#382×C₇₉	—	2	1	no	—
15	C₄³×C₇₉	C₂²	1	1	yes	characteristic
16	C₂²×C₇₉⋊C₈	—	2	2	no	—
17	C₂²×C₇₉⋊C₈	—	2	2	no	—
18	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
19	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
20	C₄⋊(C₇₉⋊C₈)	—	4	4	no	—
21	C₄⋊(C₇₉⋊C₈)	Q₈	1	4	yes	—
22	#14579072	—	4	2	no	—
23	#125×C₇₉	—	4	4	no	—
24	C₂×C₄⋊C₄×C₇₉	—	2	4	no	—
25	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
26	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
27	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₇₉	D₄	1	2	yes	—
31	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
33	#95×C₇₉	—	2	4	no	—
34	C₂×C₇₉⋊C₈	—	2	4	no	—
35	C₂×C₇₉⋊C₈	—	4	2	no	—
36	C₂×C₇₉⋊C₈	—	4	2	no	—
37	C₂×C₇₉⋊C₈	—	4	2	no	—
38	C₂×C₇₉⋊C₈	—	4	4	no	—
39	C₂²×C₄×C₇₉	—	2	4	no	—
40	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
45	C₄⋊C₄×C₇₉	—	4	8	no	—
46	C₄⋊C₄×C₇₉	—	4	4	no	—
47	C₄⋊C₄×C₇₉	—	4	2	no	—
48	C₄⋊C₄×C₇₉	—	4	4	no	—
49	C₄⋊C₄×C₇₉	—	4	2	no	—
50	C₄²×C₇₉	—	2	2	no	—
51	C₄²×C₇₉	—	2	2	no	—
52	C₄²×C₇₉	—	2	2	no	—
53	C₄²×C₇₉	—	4	2	no	—
54	C₄²×C₇₉	SD₁₆	1	2	yes	—
55	C₄²×C₇₉	Q₁₆	1	2	yes	—
56	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
57	C₄²×C₇₉	#41	1	2	yes	—
58	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₇₉	—	2	2	no	—
60	C₂×C₄×C₇₉	—	2	2	no	—
61	C₂×C₄×C₇₉	—	2	2	no	—
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	—	4	4	no	—
66	C₂×C₄×C₇₉	—	4	4	no	—
67	C₂×C₄×C₇₉	—	4	4	no	—
68	C₂×C₄×C₇₉	#103	1	4	yes	—
69	C₂×C₄×C₇₉	C₄×Q₈	1	2	yes	—
70	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
71	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
72	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
75	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
76	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
77	C₂×C₄×C₇₉	#137	1	2	yes	—
78	C₇₉⋊C₈	—	4	4	no	—
79	C₇₉⋊C₈	—	8	2	no	—
80	C₂²×C₇₉	#391	1	1	yes	characteristic
81	C₂²×C₇₉	#413	1	2	yes	—
82	C₂²×C₇₉	C₂×#103	1	1	yes	characteristic
83	C₂²×C₇₉	#417	1	2	yes	—
84	C₂²×C₇₉	#418	1	1	yes	characteristic
85	C₄×C₇₉	—	2	4	no	—
86	C₄×C₇₉	—	2	2	no	—
87	C₄×C₇₉	—	4	2	no	—
88	C₄×C₇₉	—	8	4	no	—
89	C₄×C₇₉	#476	1	2	yes	—
90	C₄×C₇₉	#477	1	2	yes	—
91	C₄×C₇₉	#490	1	1	yes	characteristic
92	C₄×C₇₉	#491	1	1	yes	characteristic
93	C₄×C₇₉	#501	1	2	yes	—
94	#9902	—	79	1	no	maximal, Sylow
95	C₂×C₇₉	#1550	1	2	yes	—
96	C₂×C₇₉	#1557	1	2	yes	—
97	C₂×C₇₉	#1987	1	1	yes	characteristic
98	C₂×C₇₉	#1992	1	1	yes	characteristic
99	C₂×C₇₉	#1998	1	1	yes	characteristic
100	#2859	D₇₉	1	1	yes	characteristic
101	#1842	—	79	1	no	—
102	#1767	—	79	1	no	—
103	#1296	—	79	4	no	—
104	C₇₉	#9902	1	1	yes	characteristic, complemented, Sylow
105	C₂×#125	—	2	1	no	—
106	C₂×C₄⋊C₈	—	79	2	no	—
107	C₂×C₄⋊C₈	—	79	2	no	—
108	C₂×C₄⋊C₈	—	158	2	no	—
109	C₂×#97	—	158	1	no	—
110	#389	—	2	2	no	—
111	#389	Dic₇₉	1	4	yes	—
112	#384	C₂×D₇₉	1	2	yes	—
113	#382	—	2	1	no	—
114	C₄³	C₂×D₇₉	1	1	yes	characteristic
115	#336	—	158	1	no	—
116	C₂²×C₈	—	158	2	no	—
117	C₂²×C₈	—	158	2	no	—
118	#125	—	4	4	no	—
119	C₂×C₄⋊C₄	—	2	4	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄⋊C₄	C₂×Dic₇₉	1	4	yes	—
123	C₂×C₄²	C₂×Dic₇₉	1	2	yes	—
124	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
125	C₂×C₄²	C₂²⋊D₇₉	1	2	yes	—
126	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
127	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
128	C₄⋊C₈	—	316	4	no	—
129	C₄⋊C₈	—	79	4	no	—
130	C₄⋊C₈	—	158	4	no	—
131	C₄⋊C₈	—	158	2	no	—
132	#97	—	316	2	no	—
133	#95	—	2	4	no	—
134	C₂²×C₄	—	2	4	no	—
135	C₂²×C₄	C₂²⋊Dic₇₉	1	2	yes	—
136	C₂²×C₄	C₂²⋊Dic₇₉	1	2	yes	—
137	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
138	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
139	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
140	C₂×C₈	—	316	2	no	—
141	C₂×C₈	—	316	2	no	—
142	C₂×C₈	—	316	2	no	—
143	C₂×C₈	—	316	4	no	—
144	C₂×C₈	—	158	4	no	—
145	C₄⋊C₄	—	4	4	no	—
146	C₄⋊C₄	—	4	8	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	4	4	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	2	2	no	—
152	C₄²	—	2	2	no	—
153	C₄²	—	4	2	no	—
154	C₄²	#11929611	1	2	yes	—
155	C₄²	#11929614	1	2	yes	—
156	C₄²	Q₈×D₇₉	1	2	yes	—
157	C₄²	#11929625	1	2	yes	—
158	C₂³	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	#14579085	1	4	yes	—
169	C₂×C₄	Q₈×Dic₇₉	1	2	yes	—
170	C₂×C₄	#14579137	1	2	yes	—
171	C₂×C₄	#14579150	1	1	yes	characteristic
172	C₂×C₄	#14579159	1	1	yes	characteristic
173	C₂×C₄	C₂×#11929611	1	1	yes	characteristic
174	C₂×C₄	C₂×#11929614	1	1	yes	characteristic
175	C₂×C₄	#14579186	1	2	yes	—
176	C₂×C₄	#14579187	1	2	yes	—
177	C₂×C₄	#14579191	1	2	yes	—
178	C₈	—	316	4	no	—
179	C₈	—	632	2	no	—
180	C₂²	#19098470	1	1	yes	characteristic
181	C₂²	#19098547	1	2	yes	—
182	C₂²	C₂×#14579085	1	1	yes	characteristic
183	C₂²	#19098564	1	2	yes	—
184	C₂²	#19098568	1	1	yes	characteristic
185	C₄	—	2	4	no	—
186	C₄	—	2	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	8	4	no	—
189	C₄	#19098850	1	2	yes	—
190	C₄	#19098855	1	2	yes	—
191	C₄	#19098929	1	1	yes	characteristic
192	C₄	#19098933	1	1	yes	characteristic
193	C₄	#19098973	1	2	yes	—
194	C₂	#23762090	1	2	yes	—
195	C₂	#23762122	1	2	yes	—
196	C₂	#23764349	1	1	yes	characteristic
197	C₂	#23764370	1	1	yes	characteristic
198	C₂	#23764394	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23763552

C₂

yes

maximal, characteristic

#23763253

C₂

yes

maximal, characteristic

#23761283

C₂

yes

maximal

C₇₉×#2859

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₇₉⋊C₈)

—

C₂×C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

C₂×C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

C₂×#14579072

—

#19098318

—

C₂×#125×C₇₉

—

#389×C₇₉

—

#389×C₇₉

C₄

yes

—

#384×C₇₉

C₂²

yes

—

#382×C₇₉

—

C₄³×C₇₉

C₂²

yes

characteristic

C₂²×C₇₉⋊C₈

—

C₂²×C₇₉⋊C₈

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

Q₈

yes

—

#14579072

—

#125×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

C₂×C₄

yes

—

C₂×C₄²×C₇₉

C₂×C₄

yes

—

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₂×C₄²×C₇₉

C₂³

yes

characteristic

#95×C₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

SD₁₆

yes

—

C₄²×C₇₉

Q₁₆

yes

—

C₄²×C₇₉

C₂×Q₈

yes

—

C₄²×C₇₉

#41

yes

—

C₂³×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#103

yes

—

C₂×C₄×C₇₉

C₄×Q₈

yes

—

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₉

#124

yes

characteristic

C₂×C₄×C₇₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇₉

#137

yes

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

#391

yes

characteristic

C₂²×C₇₉

#413

yes

—

C₂²×C₇₉

C₂×#103

yes

characteristic

C₂²×C₇₉

#417

yes

—

C₂²×C₇₉

#418

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#476

yes

—

C₄×C₇₉

#477

yes

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

#9902

—

maximal, Sylow

C₂×C₇₉

#1550

yes

—

C₂×C₇₉

#1557

yes

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

100

D₇₉

yes

characteristic

101

#1842

—

102

#1767

—

103

#1296

—

104

C₇₉

#9902

yes

characteristic, complemented, Sylow

105

C₂×#125

—

106

C₂×C₄⋊C₈

—

107

C₂×C₄⋊C₈

—

108

C₂×C₄⋊C₈

—

158

—

109

C₂×#97

—

158

—

110

#389

—

111

#389

Dic₇₉

yes

—

112

#384

C₂×D₇₉

yes

—

113

#382

—

114

C₄³

C₂×D₇₉

yes

characteristic

115

#336

—

158

—

116

C₂²×C₈

—

158

—

117

C₂²×C₈

—

158

—

118

#125

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

C₂×Dic₇₉

yes

—

123

C₂×C₄²

C₂×Dic₇₉

yes

—

124

C₂×C₄²

C₂²⋊D₇₉

yes

characteristic

125

C₂×C₄²

C₂²⋊D₇₉

yes

—

126

C₂×C₄²

C₂²⋊D₇₉

yes

characteristic

127

C₂×C₄²

C₂²×D₇₉

yes

characteristic

128

C₄⋊C₈

—

316

—

129

C₄⋊C₈

—

130

C₄⋊C₈

—

158

—

131

C₄⋊C₈

—

158

—

132

#97

—

316

—

133

#95

—

134

C₂²×C₄

—

135

C₂²×C₄

C₂²⋊Dic₇₉

yes

—

136

C₂²×C₄

C₂²⋊Dic₇₉

yes

—

137

C₂²×C₄

C₂²×Dic₇₉

yes

characteristic

138

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

139

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

140

C₂×C₈

—

316

—

141

C₂×C₈

—

316

—

142

C₂×C₈

—

316

—

143

C₂×C₈

—

316

—

144

C₂×C₈

—

158

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄²

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

154

C₄²

#11929611

yes

—

155

C₄²

#11929614

yes

—

156

C₄²

Q₈×D₇₉

yes

—

157

C₄²

#11929625

yes

—

158

C₂³

C₂×C₂²⋊Dic₇₉

yes

characteristic

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

#14579085

yes

—

169

C₂×C₄

Q₈×Dic₇₉

yes

—

170

C₂×C₄

#14579137

yes

—

171

C₂×C₄

#14579150

yes

characteristic

172

C₂×C₄

#14579159

yes

characteristic

173

C₂×C₄

C₂×#11929611

yes

characteristic

174

C₂×C₄

C₂×#11929614

yes

characteristic

175

C₂×C₄

#14579186

yes

—

176

C₂×C₄

#14579187

yes

—

177

C₂×C₄

#14579191

yes

—

178

C₈

—

316

—

179

C₈

—

632

—

180

C₂²

#19098470

yes

characteristic

181

C₂²

#19098547

yes

—

182

C₂²

C₂×#14579085

yes

characteristic

183

C₂²

#19098564

yes

—

184

C₂²

#19098568

yes

characteristic

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

#19098850

yes

—

190

C₄

#19098855

yes

—

191

C₄

#19098929

yes

characteristic

192

C₄

#19098933

yes

characteristic

193

C₄

#19098973

yes

—

194

C₂

#23762090

yes

—

195

C₂

#23762122

yes

—

196

C₂

#23764349

yes

characteristic

197

C₂

#23764370

yes

characteristic

198

C₂

#23764394

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₇₉
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#2859	C₂
socle, prime radical	C₂³×C₇₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#2859	D₇₉
hypocenter	C₇₉	#9902
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₇₉

commutator

C₂²×C₄×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#2859

C₂

socle, prime radical

C₂³×C₇₉

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

N	H ≅ G/N	type
104. C₇₉	#9902	semidirect

H ≅ G/N

type

104. C₇₉

#9902

semidirect

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

subgroup	index
#23763552	2
#23763253	2
#23761283	2
C₇₉×#2859	2
#9902	79

subgroup

index

C₇₉×#2859

Subgroups — #34188413

Conjugacy classes of subgroups

Complements

Maximal subgroups