Group #34175484 — subgroups

Subgroups — #34175484

Order 20224.

298 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 75 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	#23764341	C₂	1	1	yes	maximal, characteristic
2	C₄×#14579080	C₂	1	1	yes	maximal, characteristic
3	C₄×#101×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₇₉⋊C₈)	C₄	1	1	yes	characteristic
5	C₂×#14579080	—	2	1	no	—
6	C₄×#11929608	—	2	1	no	—
7	C₂×#14579072	C₄	1	1	yes	characteristic
8	#19098332	C₄	1	2	yes	—
9	#19098318	—	2	1	no	—
10	#19098318	—	2	1	no	—
11	C₂×C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
12	C₂×#101×C₇₉	—	2	1	no	—
13	C₄×M₁₆×C₇₉	—	2	1	no	—
14	#343×C₇₉	C₄	1	2	yes	—
15	C₂×#11929608	—	2	2	no	—
16	C₂²×C₇₉⋊C₈	—	2	1	no	—
17	C₂²×C₇₉⋊C₈	—	2	1	no	—
18	C₄⋊(C₇₉⋊C₈)	—	4	1	no	—
19	#14579080	—	2	4	no	—
20	#14579072	—	4	1	no	—
21	#14579072	—	4	1	no	—
22	C₄×C₇₉⋊C₈	—	4	1	no	—
23	C₂²×Q₈×C₇₉	D₄	1	1	yes	characteristic
24	C₂×M₁₆×C₇₉	—	2	2	no	—
25	C₄×Q₈×C₇₉	—	4	1	no	—
26	C₄×Q₈×C₇₉	—	4	1	no	—
27	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₇₉	Q₈	1	1	yes	characteristic
30	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
33	#101×C₇₉	—	2	4	no	—
34	#97×C₇₉	—	4	1	no	—
35	C₄×C₈×C₇₉	—	4	1	no	—
36	#11929608	—	2	4	no	—
37	#11929608	—	4	2	no	—
38	C₂×C₇₉⋊C₈	—	2	2	no	—
39	C₂×C₇₉⋊C₈	—	4	2	no	—
40	C₂×C₇₉⋊C₈	—	4	1	no	—
41	C₂×C₇₉⋊C₈	—	4	2	no	—
42	C₂×C₇₉⋊C₈	—	4	1	no	—
43	C₂×C₇₉⋊C₈	—	4	1	no	—
44	C₂×Q₈×C₇₉	—	2	2	no	—
45	C₂×Q₈×C₇₉	—	2	2	no	—
46	C₂×Q₈×C₇₉	—	4	1	no	—
47	C₂²×C₄×C₇₉	C₄²	1	1	yes	characteristic
48	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
54	M₁₆×C₇₉	—	2	4	no	—
55	M₁₆×C₇₉	—	4	2	no	—
56	C₂×C₈×C₇₉	—	4	2	no	—
57	C₄⋊C₄×C₇₉	—	4	1	no	—
58	C₄⋊C₄×C₇₉	—	4	1	no	—
59	C₄⋊C₄×C₇₉	—	4	1	no	—
60	C₄²×C₇₉	—	2	1	no	—
61	C₄²×C₇₉	—	2	1	no	—
62	C₄²×C₇₉	—	2	1	no	—
63	C₄²×C₇₉	—	2	1	no	—
64	C₄²×C₇₉	—	2	1	no	—
65	C₄²×C₇₉	—	2	1	no	—
66	C₂³×C₇₉	#95	1	1	yes	characteristic
67	Q₈×C₇₉	—	4	4	no	—
68	C₂×C₄×C₇₉	—	2	1	no	—
69	C₂×C₄×C₇₉	—	2	1	no	—
70	C₂×C₄×C₇₉	—	2	1	no	—
71	C₂×C₄×C₇₉	—	2	1	no	—
72	C₂×C₄×C₇₉	—	2	1	no	—
73	C₂×C₄×C₇₉	—	2	1	no	—
74	C₂×C₄×C₇₉	—	2	1	no	—
75	C₂×C₄×C₇₉	—	2	1	no	—
76	C₂×C₄×C₇₉	—	2	1	no	—
77	C₂×C₄×C₇₉	—	2	1	no	—
78	C₂×C₄×C₇₉	—	2	1	no	—
79	C₂×C₄×C₇₉	—	2	1	no	—
80	C₂×C₄×C₇₉	—	4	1	no	—
81	C₂×C₄×C₇₉	—	4	1	no	—
82	C₂×C₄×C₇₉	—	4	1	no	—
83	C₂×C₄×C₇₉	#99	1	1	yes	characteristic
84	C₂×C₄×C₇₉	#99	1	1	yes	characteristic
85	C₂×C₄×C₇₉	#104	1	2	yes	—
86	C₂×C₄×C₇₉	#104	1	2	yes	—
87	C₈×C₇₉	—	4	4	no	—
88	C₇₉⋊C₈	—	4	2	no	—
89	C₇₉⋊C₈	—	4	4	no	—
90	C₇₉⋊C₈	—	8	1	no	—
91	C₂²×C₇₉	—	2	1	no	—
92	C₂²×C₇₉	—	2	1	no	—
93	C₂²×C₇₉	#339	1	1	yes	characteristic
94	C₂²×C₇₉	#339	1	1	yes	characteristic
95	C₂²×C₇₉	#342	1	1	yes	characteristic
96	C₄×C₇₉	—	2	2	no	—
97	C₄×C₇₉	—	2	2	no	—
98	C₄×C₇₉	—	2	1	no	—
99	C₄×C₇₉	—	4	1	no	—
100	C₄×C₇₉	—	4	1	no	—
101	C₄×C₇₉	—	4	1	no	—
102	C₄×C₇₉	—	4	1	no	—
103	C₄×C₇₉	#355	1	1	yes	characteristic
104	C₄×C₇₉	#356	1	1	yes	characteristic
105	#7284	—	79	1	no	maximal, Sylow
106	C₂×C₇₉	—	2	1	no	—
107	C₂×C₇₉	—	2	1	no	—
108	C₂×C₇₉	#1297	1	1	yes	characteristic
109	C₂×C₇₉	#1392	1	1	yes	characteristic
110	C₂×C₇₉	#1394	1	1	yes	characteristic
111	#1985	—	79	1	no	—
112	C₄×#101	—	79	1	no	—
113	C₄×#101	D₇₉	1	1	yes	characteristic
114	C₇₉	#7284	1	1	yes	characteristic, complemented, Sylow
115	C₂×C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
116	C₂×C₄⋊C₈	—	79	1	no	—
117	C₂×#101	—	2	1	no	—
118	C₂×#101	—	158	1	no	—
119	C₄×M₁₆	—	2	1	no	—
120	C₄×M₁₆	—	158	1	no	—
121	C₂×#97	—	79	1	no	—
122	#343	—	79	2	no	—
123	#343	Dic₇₉	1	2	yes	—
124	#336	—	158	1	no	—
125	#336	—	158	1	no	—
126	C₂²×Q₈	C₂²⋊D₇₉	1	1	yes	characteristic
127	C₂×M₁₆	—	2	2	no	—
128	C₂×M₁₆	—	158	2	no	—
129	C₂²×C₈	—	158	1	no	—
130	C₂²×C₈	—	158	1	no	—
131	C₄×Q₈	—	4	1	no	—
132	C₄×Q₈	—	4	1	no	—
133	C₂×C₄⋊C₄	Dic₁₅₈	1	1	yes	characteristic
134	C₂×C₄⋊C₄	C₄×D₇₉	1	1	yes	characteristic
135	C₂×C₄⋊C₄	C₂×Dic₇₉	1	1	yes	characteristic
136	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
137	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
138	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
139	C₄⋊C₈	—	316	1	no	—
140	#101	—	2	4	no	—
141	#101	—	158	4	no	—
142	#97	—	4	1	no	—
143	#97	—	316	1	no	—
144	#97	—	316	1	no	—
145	C₄×C₈	—	4	1	no	—
146	C₄×C₈	—	316	1	no	—
147	C₂×Q₈	—	2	2	no	—
148	C₂×Q₈	—	2	2	no	—
149	C₂×Q₈	—	4	1	no	—
150	C₂²×C₄	C₄×Dic₇₉	1	1	yes	characteristic
151	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
152	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
153	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
154	C₂²×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
155	C₂²×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
156	C₂²×C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
157	M₁₆	—	2	4	no	—
158	M₁₆	—	4	2	no	—
159	M₁₆	—	316	2	no	—
160	M₁₆	—	158	4	no	—
161	C₂×C₈	—	4	2	no	—
162	C₂×C₈	—	316	2	no	—
163	C₂×C₈	—	316	1	no	—
164	C₂×C₈	—	316	1	no	—
165	C₂×C₈	—	316	2	no	—
166	C₂×C₈	—	316	1	no	—
167	C₂×C₈	—	158	2	no	—
168	C₄⋊C₄	—	4	1	no	—
169	C₄⋊C₄	—	4	1	no	—
170	C₄⋊C₄	—	4	1	no	—
171	C₄²	—	2	1	no	—
172	C₄²	—	2	1	no	—
173	C₄²	—	2	1	no	—
174	C₄²	—	2	1	no	—
175	C₄²	—	2	1	no	—
176	C₄²	—	2	1	no	—
177	C₂³	C₇₉⋊#95	1	1	yes	characteristic
178	Q₈	—	4	4	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	—	4	1	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	#14579075	1	1	yes	characteristic
195	C₂×C₄	#14579075	1	1	yes	characteristic
196	C₂×C₄	#14579088	1	2	yes	—
197	C₂×C₄	#14579089	1	2	yes	—
198	C₈	—	4	4	no	—
199	C₈	—	316	2	no	—
200	C₈	—	316	4	no	—
201	C₈	—	632	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	—	2	1	no	—
204	C₂²	#19098322	1	1	yes	characteristic
205	C₂²	#19098323	1	1	yes	characteristic
206	C₂²	#19098329	1	1	yes	characteristic
207	C₄	—	2	1	no	—
208	C₄	—	2	2	no	—
209	C₄	—	2	2	no	—
210	C₄	—	4	1	no	—
211	C₄	—	4	1	no	—
212	C₄	—	4	1	no	—
213	C₄	—	4	1	no	—
214	C₄	#19098359	1	1	yes	characteristic
215	C₄	#19098361	1	1	yes	characteristic
216	C₂	—	2	1	no	—
217	C₂	—	2	1	no	—
218	C₂	#23761286	1	1	yes	characteristic
219	C₂	#23761522	1	1	yes	characteristic
220	C₂	#23761526	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23764341

C₂

yes

maximal, characteristic

C₄×#14579080

C₂

yes

maximal, characteristic

C₄×#101×C₇₉

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₇₉⋊C₈)

C₄

yes

—

—

C₄

yes

characteristic

#19098332

C₄

yes

—

#19098318

—

#19098318

—

C₂×C₄×Q₈×C₇₉

C₂²

yes

characteristic

C₂×#101×C₇₉

—

C₄×M₁₆×C₇₉

—

#343×C₇₉

C₄

yes

—

C₂×#11929608

—

C₂²×C₇₉⋊C₈

—

C₂²×C₇₉⋊C₈

—

C₄⋊(C₇₉⋊C₈)

—

#14579080

—

#14579072

—

#14579072

—

C₄×C₇₉⋊C₈

—

C₂²×Q₈×C₇₉

D₄

yes

characteristic

C₂×M₁₆×C₇₉

—

C₄×Q₈×C₇₉

—

C₄×Q₈×C₇₉

—

C₂×C₄⋊C₄×C₇₉

C₂×C₄

yes

characteristic

C₂×C₄⋊C₄×C₇₉

C₂×C₄

yes

characteristic

C₂×C₄⋊C₄×C₇₉

Q₈

yes

characteristic

C₂×C₄²×C₇₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

characteristic

#101×C₇₉

—

#97×C₇₉

—

C₄×C₈×C₇₉

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂²×C₄×C₇₉

C₄²

yes

characteristic

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₉

C₄⋊C₄

yes

characteristic

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₂³×C₇₉

#95

yes

characteristic

Q₈×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

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

—

C₂×C₄×C₇₉

#104

yes

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

100

C₄×C₇₉

—

101

C₄×C₇₉

—

102

C₄×C₇₉

—

103

C₄×C₇₉

#355

yes

characteristic

104

C₄×C₇₉

yes

105

—

106

C₂×C₇₉

—

107

C₂×C₇₉

—

108

C₂×C₇₉

#1297

yes

characteristic

109

C₂×C₇₉

#1392

yes

characteristic

110

C₂×C₇₉

#1394

yes

characteristic

111

#1985

—

112

C₄×#101

—

113

C₄×#101

D₇₉

yes

characteristic

114

C₇₉

#7284

yes

characteristic, complemented, Sylow

115

C₂×C₄×Q₈

C₂×D₇₉

yes

characteristic

116

C₂×C₄⋊C₈

—

117

C₂×#101

—

118

C₂×#101

—

158

—

119

C₄×M₁₆

—

120

C₄×M₁₆

—

158

—

121

C₂×#97

—

122

#343

—

123

#343

Dic₇₉

yes

—

124

#336

—

158

—

125

#336

—

158

—

126

C₂²×Q₈

C₂²⋊D₇₉

yes

characteristic

127

C₂×M₁₆

—

128

C₂×M₁₆

—

158

—

129

C₂²×C₈

—

158

—

130

C₂²×C₈

—

158

—

131

C₄×Q₈

—

132

C₄×Q₈

—

133

C₂×C₄⋊C₄

Dic₁₅₈

yes

characteristic

134

C₂×C₄⋊C₄

C₄×D₇₉

yes

characteristic

135

C₂×C₄⋊C₄

C₂×Dic₇₉

yes

characteristic

136

C₂×C₄²

C₄×D₇₉

yes

characteristic

137

C₂×C₄²

D₃₁₆

yes

characteristic

138

C₂×C₄²

C₂²⋊D₇₉

yes

characteristic

139

C₄⋊C₈

—

316

—

140

#101

—

141

#101

—

158

—

142

#97

—

143

#97

—

316

—

144

#97

—

316

—

145

C₄×C₈

—

146

C₄×C₈

—

316

—

147

C₂×Q₈

—

148

C₂×Q₈

—

149

C₂×Q₈

—

150

C₂²×C₄

C₄×Dic₇₉

yes

characteristic

151

C₂²×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

152

C₂²×C₄

(C₂×D₇₉)⋊C₄

yes

characteristic

153

C₂²×C₄

C₂²⋊Dic₇₉

yes

characteristic

154

C₂²×C₄

Dic₇₉⋊C₄

yes

characteristic

155

C₂²×C₄

Dic₇₉⋊C₄

yes

characteristic

156

C₂²×C₄

C₄⋊Dic₇₉

yes

characteristic

157

M₁₆

—

158

M₁₆

—

159

M₁₆

—

316

—

160

M₁₆

—

158

—

161

C₂×C₈

—

162

C₂×C₈

—

316

—

163

C₂×C₈

—

316

—

164

C₂×C₈

—

316

—

165

C₂×C₈

—

316

—

166

C₂×C₈

—

316

—

167

C₂×C₈

—

158

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₄²

—

172

C₄²

—

173

C₄²

—

174

C₄²

—

175

C₄²

—

176

C₄²

—

177

C₂³

C₇₉⋊#95

yes

characteristic

178

Q₈

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

#14579075

yes

characteristic

195

C₂×C₄

#14579075

yes

characteristic

196

C₂×C₄

#14579088

yes

—

197

C₂×C₄

#14579089

yes

—

198

C₈

—

199

C₈

—

316

—

200

C₈

—

316

—

201

C₈

—

632

—

202

C₂²

—

203

C₂²

—

204

C₂²

#19098322

yes

characteristic

205

C₂²

#19098323

yes

characteristic

206

C₂²

#19098329

yes

characteristic

207

C₄

—

208

C₄

—

209

C₄

—

210

C₄

—

211

C₄

—

212

C₄

—

213

C₄

—

214

C₄

#19098359

yes

characteristic

215

C₄

#19098361

yes

characteristic

216

C₂

—

217

C₂

—

218

C₂

#23761286

yes

characteristic

219

C₂

#23761522

yes

characteristic

220

C₂

#23761526

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14579075
commutator	C₂²×C₄×C₇₉	C₄²
Frattini	C₂×C₄×Q₈	C₂×D₇₉
Fitting	C₄×#101×C₇₉	C₂
socle	C₂²×C₇₉	#342
prime radical	C₂³×C₇₉	#95
cosocle	C₂×C₄×Q₈×C₇₉	C₂²
hypercenter	C₄×#101	D₇₉
hypocenter	C₇₉	#7284
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂³×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ C₄×#101

subgroup

G/H

center

C₂×C₄

#14579075

commutator

C₂²×C₄×C₇₉

C₄²

Frattini

C₂×C₄×Q₈

C₂×D₇₉

Fitting

C₄×#101×C₇₉

C₂

C₂²×C₇₉

C₂³×C₇₉

C₂×C₄×Q₈×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂³×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ C₄×#101

N	H ≅ G/N	type
114. C₇₉	#7284	semidirect

H ≅ G/N

type

114. C₇₉

#7284

semidirect

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

subgroup	index
#23764341	2
C₄×#14579080	2
C₄×#101×C₇₉	2
#7284	79

subgroup

index

#23764341

C₄×#14579080

C₄×#101×C₇₉

#7284

Subgroups — #34175484

Conjugacy classes of subgroups

Complements

Maximal subgroups