Group #34187363 — subgroups

Subgroups — #34187363

Order 20224.

538 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 171 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	#23771798	C₂	1	1	yes	maximal, characteristic
2	#23763517	C₂	1	1	yes	maximal, characteristic
3	C₂×#19098310	C₂	1	4	yes	maximal
4	C₇₉×#1767	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579161	—	2	1	no	—
6	C₂×#14579161	—	2	1	no	—
7	C₂×#14579161	C₂²	1	2	yes	—
8	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
9	#19098433	—	2	2	no	—
10	#19098433	C₄	1	4	yes	—
11	#19098318	—	2	2	no	—
12	#19098310	—	2	8	no	—
13	C₂×C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
14	C₂×#97×C₇₉	—	2	1	no	—
15	C₄³×C₇₉	C₂²	1	1	yes	characteristic
16	#336×C₇₉	—	2	1	no	—
17	C₂²×Dic₁₅₈	—	2	2	no	—
18	C₂²×C₇₉⋊C₈	—	2	2	no	—
19	#14579161	—	2	4	no	—
20	#14579161	—	2	4	no	—
21	#14579161	—	4	4	no	—
22	#14579161	—	4	2	no	—
23	#14579161	—	4	2	no	—
24	C₂×C₄⋊Dic₇₉	—	2	4	no	—
25	C₂×C₄⋊Dic₇₉	—	2	4	no	—
26	C₂×C₄⋊Dic₇₉	—	2	2	no	—
27	C₂×C₄⋊Dic₇₉	C₂×C₄	1	4	yes	—
28	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
29	C₂²×C₈×C₇₉	—	2	2	no	—
30	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₇₉	D₄	1	2	yes	—
34	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₇₉	—	2	2	no	—
36	C₄⋊C₈×C₇₉	D₄	1	4	yes	—
37	#97×C₇₉	—	4	2	no	—
38	C₂×Dic₁₅₈	—	4	4	no	—
39	C₂×Dic₁₅₈	—	4	2	no	—
40	C₂×Dic₁₅₈	—	4	2	no	—
41	C₂×Dic₁₅₈	—	4	4	no	—
42	C₂²×Dic₇₉	—	2	4	no	—
43	C₂×C₇₉⋊C₈	—	4	2	no	—
44	C₂×C₇₉⋊C₈	—	4	4	no	—
45	C₄⋊Dic₇₉	—	2	8	no	—
46	C₄⋊Dic₇₉	—	4	8	no	—
47	C₄⋊Dic₇₉	—	4	8	no	—
48	C₄⋊Dic₇₉	—	4	2	no	—
49	C₄⋊Dic₇₉	—	4	2	no	—
50	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₇₉	—	2	4	no	—
56	C₂×C₈×C₇₉	—	4	2	no	—
57	C₂×C₈×C₇₉	—	4	2	no	—
58	C₄²×C₇₉	—	2	4	no	—
59	C₄²×C₇₉	—	2	2	no	—
60	C₄²×C₇₉	—	2	4	no	—
61	C₄²×C₇₉	D₈	1	2	yes	—
62	C₄²×C₇₉	SD₁₆	1	2	yes	—
63	C₄²×C₇₉	C₂×D₄	1	2	yes	—
64	C₄²×C₇₉	#41	1	2	yes	—
65	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₇₉	—	2	2	no	—
67	C₂×C₄×C₇₉	—	2	2	no	—
68	C₂×C₄×C₇₉	—	2	2	no	—
69	C₂×C₄×C₇₉	—	2	2	no	—
70	C₂×C₄×C₇₉	—	2	2	no	—
71	C₂×C₄×C₇₉	#101	1	4	yes	—
72	C₂×C₄×C₇₉	#102	1	4	yes	—
73	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
74	C₂×C₄×C₇₉	#121	1	2	yes	—
75	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
76	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
77	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
80	C₂×C₄×C₇₉	#137	1	4	yes	—
81	C₂×Dic₇₉	—	4	4	no	—
82	C₂×Dic₇₉	—	4	4	no	—
83	C₂×Dic₇₉	—	4	4	no	—
84	Dic₁₅₈	—	8	4	no	—
85	Dic₁₅₈	—	8	4	no	—
86	C₈×C₇₉	—	4	4	no	—
87	C₇₉⋊C₈	—	8	2	no	—
88	C₂²×C₇₉	#390	1	1	yes	characteristic
89	C₂²×C₇₉	C₂×#101	1	2	yes	—
90	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
91	C₂²×C₇₉	#418	1	1	yes	characteristic
92	C₂²×C₇₉	#419	1	2	yes	—
93	C₄×C₇₉	—	2	4	no	—
94	C₄×C₇₉	—	2	4	no	—
95	C₄×C₇₉	—	2	2	no	—
96	C₄×C₇₉	#459	1	2	yes	—
97	C₄×C₇₉	#460	1	2	yes	—
98	C₄×C₇₉	#491	1	2	yes	—
99	C₄×C₇₉	#497	1	2	yes	—
100	Dic₇₉	—	8	4	no	—
101	#9705	—	79	1	no	maximal, Sylow
102	C₂×C₇₉	#1546	1	2	yes	—
103	C₂×C₇₉	#1556	1	2	yes	—
104	C₂×C₇₉	#1960	1	1	yes	characteristic
105	C₂×C₇₉	#1966	1	1	yes	characteristic
106	C₂×C₇₉	#1982	1	1	yes	characteristic
107	#2868	—	79	1	no	—
108	#1831	—	79	1	no	—
109	#1767	D₇₉	1	1	yes	characteristic
110	C₂×#332	—	79	4	no	—
111	C₇₉	#9705	1	1	yes	characteristic, complemented, Sylow
112	C₂×C₄⋊Q₈	—	79	2	no	—
113	C₂×C₄⋊Q₈	—	158	1	no	—
114	C₂×C₄⋊Q₈	—	158	1	no	—
115	C₂×C₄⋊C₈	—	79	2	no	—
116	C₂×C₄⋊C₈	C₂×D₇₉	1	2	yes	—
117	C₂×#97	—	2	1	no	—
118	#384	—	79	4	no	—
119	#384	—	158	2	no	—
120	C₄³	C₂×D₇₉	1	1	yes	characteristic
121	#336	—	2	1	no	—
122	#336	—	158	2	no	—
123	#332	—	158	8	no	—
124	C₂²×Q₈	—	158	2	no	—
125	C₂²×C₈	—	2	2	no	—
126	C₂²×C₈	—	158	2	no	—
127	C₄⋊Q₈	—	316	4	no	—
128	C₄⋊Q₈	—	316	2	no	—
129	C₄⋊Q₈	—	316	2	no	—
130	C₄⋊Q₈	—	158	4	no	—
131	C₄⋊Q₈	—	158	4	no	—
132	C₂×C₄⋊C₄	—	79	4	no	—
133	C₂×C₄⋊C₄	—	158	4	no	—
134	C₂×C₄⋊C₄	—	158	4	no	—
135	C₂×C₄⋊C₄	—	158	2	no	—
136	C₂×C₄²	C₄×D₇₉	1	2	yes	—
137	C₂×C₄²	D₃₁₆	1	2	yes	—
138	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
139	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
140	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
141	C₄⋊C₈	—	2	2	no	—
142	C₄⋊C₈	—	158	4	no	—
143	C₄⋊C₈	D₃₁₆	1	4	yes	—
144	#97	—	4	2	no	—
145	C₂×Q₈	—	316	4	no	—
146	C₂×Q₈	—	316	2	no	—
147	C₂×Q₈	—	316	2	no	—
148	C₂×Q₈	—	316	4	no	—
149	C₂²×C₄	—	158	4	no	—
150	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
151	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
152	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
154	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
155	C₂×C₈	—	2	4	no	—
156	C₂×C₈	—	4	2	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	316	2	no	—
159	C₂×C₈	—	316	4	no	—
160	C₄⋊C₄	—	316	8	no	—
161	C₄⋊C₄	—	316	8	no	—
162	C₄⋊C₄	—	316	2	no	—
163	C₄⋊C₄	—	316	2	no	—
164	C₄⋊C₄	—	158	8	no	—
165	C₄²	—	2	4	no	—
166	C₄²	—	2	2	no	—
167	C₄²	—	2	4	no	—
168	C₄²	#11929609	1	2	yes	—
169	C₄²	#11929611	1	2	yes	—
170	C₄²	C₂×D₃₁₆	1	2	yes	—
171	C₄²	#11929624	1	2	yes	—
172	C₂³	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
173	Q₈	—	632	4	no	—
174	Q₈	—	632	4	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	316	4	no	—
181	C₂×C₄	—	316	4	no	—
182	C₂×C₄	—	316	4	no	—
183	C₂×C₄	#14579079	1	4	yes	—
184	C₂×C₄	#14579081	1	4	yes	—
185	C₂×C₄	C₄×D₃₁₆	1	2	yes	—
186	C₂×C₄	#14579135	1	2	yes	—
187	C₂×C₄	#14579149	1	1	yes	characteristic
188	C₂×C₄	#14579158	1	1	yes	characteristic
189	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
190	C₂×C₄	C₂×#11929611	1	1	yes	characteristic
191	C₂×C₄	#14579186	1	2	yes	—
192	C₂×C₄	#14579194	1	4	yes	—
193	C₈	—	4	4	no	—
194	C₈	—	632	2	no	—
195	C₂²	#19098465	1	1	yes	characteristic
196	C₂²	C₂×#14579079	1	2	yes	—
197	C₂²	C₂×#14579081	1	1	yes	characteristic
198	C₂²	#19098567	1	1	yes	characteristic
199	C₂²	#19098573	1	2	yes	—
200	C₄	—	2	2	no	—
201	C₄	—	2	4	no	—
202	C₄	—	2	4	no	—
203	C₄	—	632	4	no	—
204	C₄	#19098758	1	2	yes	—
205	C₄	#19098763	1	2	yes	—
206	C₄	#19098932	1	2	yes	—
207	C₄	#19098957	1	2	yes	—
208	C₂	#23762073	1	2	yes	—
209	C₂	#23762118	1	2	yes	—
210	C₂	#23764220	1	1	yes	characteristic
211	C₂	#23764244	1	1	yes	characteristic
212	C₂	#23764327	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23771798

C₂

yes

maximal, characteristic

#23763517

C₂

yes

maximal, characteristic

C₂×#19098310

C₂

yes

maximal

C₇₉×#1767

C₂

yes

maximal, characteristic

C₂×#14579161

—

C₂×#14579161

—

C₂×#14579161

C₂²

yes

—

C₂×C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

#19098433

—

#19098433

C₄

yes

—

#19098318

—

#19098310

—

C₂×C₄⋊C₈×C₇₉

C₂²

yes

—

C₂×#97×C₇₉

—

C₄³×C₇₉

C₂²

yes

characteristic

#336×C₇₉

—

C₂²×Dic₁₅₈

—

C₂²×C₇₉⋊C₈

—

—

—

—

—

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

C₂×C₄

yes

—

C₄⋊(C₇₉⋊C₈)

—

C₂²×C₈×C₇₉

—

C₂×C₄²×C₇₉

C₂×C₄

yes

—

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

D₄

yes

—

#97×C₇₉

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

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

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

D₈

yes

—

C₄²×C₇₉

SD₁₆

yes

—

C₄²×C₇₉

C₂×D₄

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

#101

yes

—

C₂×C₄×C₇₉

#102

yes

—

C₂×C₄×C₇₉

C₄×D₄

yes

—

C₂×C₄×C₇₉

#121

yes

—

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇₉

#137

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

#390

yes

characteristic

C₂²×C₇₉

C₂×#101

yes

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#459

yes

—

C₄×C₇₉

#460

yes

—

C₄×C₇₉

#491

yes

—

C₄×C₇₉

#497

yes

—

100

Dic₇₉

—

101

#9705

—

maximal, Sylow

102

C₂×C₇₉

#1546

yes

—

103

C₂×C₇₉

#1556

yes

—

104

C₂×C₇₉

#1960

yes

characteristic

105

C₂×C₇₉

#1966

yes

characteristic

106

C₂×C₇₉

#1982

yes

characteristic

107

#2868

—

108

#1831

—

109

#1767

D₇₉

yes

characteristic

110

C₂×#332

—

111

C₇₉

#9705

yes

characteristic, complemented, Sylow

112

C₂×C₄⋊Q₈

—

113

C₂×C₄⋊Q₈

—

158

—

114

C₂×C₄⋊Q₈

—

158

—

115

C₂×C₄⋊C₈

—

116

C₂×C₄⋊C₈

C₂×D₇₉

yes

—

117

C₂×#97

—

118

#384

—

119

#384

—

158

—

120

C₄³

C₂×D₇₉

yes

characteristic

121

#336

—

122

#336

—

158

—

123

#332

—

158

—

124

C₂²×Q₈

—

158

—

125

C₂²×C₈

—

126

C₂²×C₈

—

158

—

127

C₄⋊Q₈

—

316

—

128

C₄⋊Q₈

—

316

—

129

C₄⋊Q₈

—

316

—

130

C₄⋊Q₈

—

158

—

131

C₄⋊Q₈

—

158

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

158

—

134

C₂×C₄⋊C₄

—

158

—

135

C₂×C₄⋊C₄

—

158

—

136

C₂×C₄²

C₄×D₇₉

yes

—

137

C₂×C₄²

D₃₁₆

yes

—

138

C₂×C₄²

C₂²⋊D₇₉

yes

characteristic

139

C₂×C₄²

C₂²⋊D₇₉

yes

characteristic

140

C₂×C₄²

C₂²×D₇₉

yes

characteristic

141

C₄⋊C₈

—

142

C₄⋊C₈

—

158

—

143

C₄⋊C₈

D₃₁₆

yes

—

144

#97

—

145

C₂×Q₈

—

316

—

146

C₂×Q₈

—

316

—

147

C₂×Q₈

—

316

—

148

C₂×Q₈

—

316

—

149

C₂²×C₄

—

158

—

150

C₂²×C₄

(C₂×D₇₉)⋊C₄

yes

—

151

C₂²×C₄

(C₂×D₇₉)⋊C₄

yes

—

152

C₂²×C₄

C₂×C₄×D₇₉

yes

characteristic

153

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

154

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

316

—

159

C₂×C₈

—

316

—

160

C₄⋊C₄

—

316

—

161

C₄⋊C₄

—

316

—

162

C₄⋊C₄

—

316

—

163

C₄⋊C₄

—

316

—

164

C₄⋊C₄

—

158

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

168

C₄²

#11929609

yes

—

169

C₄²

#11929611

yes

—

170

C₄²

C₂×D₃₁₆

yes

—

171

C₄²

#11929624

yes

—

172

C₂³

C₂×(C₂×D₇₉)⋊C₄

yes

characteristic

173

Q₈

—

632

—

174

Q₈

—

632

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

316

—

181

C₂×C₄

—

316

—

182

C₂×C₄

—

316

—

183

C₂×C₄

#14579079

yes

—

184

C₂×C₄

#14579081

yes

—

185

C₂×C₄

C₄×D₃₁₆

yes

—

186

C₂×C₄

#14579135

yes

—

187

C₂×C₄

#14579149

yes

characteristic

188

C₂×C₄

#14579158

yes

characteristic

189

C₂×C₄

C₂×#11929609

yes

characteristic

190

C₂×C₄

C₂×#11929611

yes

characteristic

191

C₂×C₄

#14579186

yes

—

192

C₂×C₄

#14579194

yes

—

193

C₈

—

194

C₈

—

632

—

195

C₂²

#19098465

yes

characteristic

196

C₂²

C₂×#14579079

yes

—

197

C₂²

C₂×#14579081

yes

characteristic

198

C₂²

#19098567

yes

characteristic

199

C₂²

#19098573

yes

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

—

632

—

204

C₄

#19098758

yes

—

205

C₄

#19098763

yes

—

206

C₄

#19098932

yes

—

207

C₄

#19098957

yes

—

208

C₂

#23762073

yes

—

209

C₂

#23762118

yes

—

210

C₂

#23764220

yes

characteristic

211

C₂

#23764244

yes

characteristic

212

C₂

#23764327

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₇₉)⋊C₄
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1767	C₂
socle, prime radical	C₂³×C₇₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1767	D₇₉
hypocenter	C₇₉	#9705
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

subgroup

G/H

center

C₂³

C₂×(C₂×D₇₉)⋊C₄

commutator

C₂²×C₄×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1767

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 ◁ #1767

N	H ≅ G/N	type
111. C₇₉	#9705	semidirect

H ≅ G/N

type

111. C₇₉

#9705

semidirect

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

subgroup	index
#23771798	2
#23763517	2
C₂×#19098310	2
C₇₉×#1767	2
#9705	79

subgroup

index

C₇₉×#1767

Subgroups — #34187363

Conjugacy classes of subgroups

Complements

Maximal subgroups