Group #34188300 — subgroups

Subgroups — #34188300

Order 20224.

486 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 159 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	C₄×#14579153	C₂	1	1	yes	maximal, characteristic
2	#23763552	C₂	1	1	yes	maximal, characteristic
3	#23761240	C₂	1	4	yes	maximal
4	C₄×#97×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579153	C₄	1	1	yes	characteristic
6	C₂×C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
7	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
8	#19098461	C₄	1	2	yes	—
9	#19098447	—	2	2	no	—
10	#19098429	C₄	1	1	yes	characteristic
11	C₄×Dic₇₉⋊C₄	—	2	2	no	—
12	C₄×C₄⋊Dic₇₉	C₂²	1	2	yes	—
13	C₂×#97×C₇₉	—	2	2	no	—
14	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
15	C₄³×C₇₉	C₂²	1	1	yes	characteristic
16	C₂²×C₇₉⋊C₈	—	2	2	no	—
17	#14579153	—	2	4	no	—
18	C₂×C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
19	C₂×C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
20	C₂×Dic₇₉⋊C₄	—	2	2	no	—
21	C₂×Dic₇₉⋊C₄	—	2	2	no	—
22	C₂×C₄×Dic₇₉	—	2	2	no	—
23	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
24	C₄⋊(C₇₉⋊C₈)	—	4	4	no	—
25	C₇₉⋊#95	—	2	2	no	—
26	C₇₉⋊#95	—	2	2	no	—
27	C₇₉⋊#95	—	2	2	no	—
28	C₂²×C₈×C₇₉	—	2	2	no	—
29	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₇₉	D₄	1	2	yes	—
32	C₂×C₄²×C₇₉	D₄	1	2	yes	—
33	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
34	#97×C₇₉	—	2	8	no	—
35	C₄×C₈×C₇₉	—	2	2	no	—
36	C₄×C₈×C₇₉	Q₈	1	4	yes	—
37	C₂²×Dic₇₉	—	2	2	no	—
38	C₂²×Dic₇₉	—	2	2	no	—
39	C₂×C₇₉⋊C₈	—	4	2	no	—
40	C₂×C₇₉⋊C₈	—	4	4	no	—
41	C₄⋊Dic₇₉	—	2	2	no	—
42	C₄⋊Dic₇₉	—	2	4	no	—
43	C₄⋊Dic₇₉	M₁₆	1	4	yes	—
44	Dic₇₉⋊C₄	—	2	8	no	—
45	Dic₇₉⋊C₄	—	4	4	no	—
46	C₄×Dic₇₉	—	4	2	no	—
47	C₄×Dic₇₉	—	4	2	no	—
48	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₇₉	—	2	8	no	—
54	C₂×C₈×C₇₉	—	2	4	no	—
55	C₄²×C₇₉	—	2	4	no	—
56	C₄²×C₇₉	—	2	2	no	—
57	C₄²×C₇₉	—	2	2	no	—
58	C₄²×C₇₉	—	2	2	no	—
59	C₄²×C₇₉	—	2	2	no	—
60	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
61	C₄²×C₇₉	#41	1	2	yes	—
62	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₇₉	—	2	2	no	—
64	C₂×C₄×C₇₉	—	2	2	no	—
65	C₂×C₄×C₇₉	—	2	2	no	—
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	#104	1	4	yes	—
70	C₂×C₄×C₇₉	C₄×Q₈	1	2	yes	—
71	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
72	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
73	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
75	C₂×C₄×C₇₉	#131	1	2	yes	—
76	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
77	C₂×C₄×C₇₉	#137	1	2	yes	—
78	C₂×Dic₇₉	—	2	4	no	—
79	C₂×Dic₇₉	—	4	2	no	—
80	C₂×Dic₇₉	—	4	2	no	—
81	C₂×Dic₇₉	—	4	2	no	—
82	C₂×Dic₇₉	—	4	2	no	—
83	C₂×Dic₇₉	—	4	2	no	—
84	C₈×C₇₉	—	2	8	no	—
85	C₇₉⋊C₈	—	8	2	no	—
86	C₂²×C₇₉	#391	1	1	yes	characteristic
87	C₂²×C₇₉	#407	1	1	yes	characteristic
88	C₂²×C₇₉	#408	1	1	yes	characteristic
89	C₂²×C₇₉	#417	1	2	yes	—
90	C₂²×C₇₉	C₂×#104	1	2	yes	—
91	C₄×C₇₉	—	2	2	no	—
92	C₄×C₇₉	—	2	4	no	—
93	C₄×C₇₉	—	4	2	no	—
94	C₄×C₇₉	#446	1	4	yes	—
95	C₄×C₇₉	#490	1	1	yes	characteristic
96	C₄×C₇₉	#491	1	1	yes	characteristic
97	C₄×C₇₉	#501	1	2	yes	—
98	Dic₇₉	—	4	4	no	—
99	Dic₇₉	—	8	2	no	—
100	#9876	—	79	1	no	maximal, Sylow
101	C₂×C₇₉	#1483	1	2	yes	—
102	C₂×C₇₉	#1489	1	2	yes	—
103	C₂×C₇₉	#1985	1	1	yes	characteristic
104	C₂×C₇₉	#1992	1	1	yes	characteristic
105	C₂×C₇₉	#1994	1	1	yes	characteristic
106	C₄×#125	—	79	1	no	—
107	#1842	—	79	1	no	—
108	C₄×#97	D₇₉	1	1	yes	characteristic
109	#1279	—	79	4	no	—
110	C₇₉	#9876	1	1	yes	characteristic, complemented, Sylow
111	C₂×#125	—	79	1	no	—
112	C₂×C₄⋊C₈	—	79	2	no	—
113	C₂×C₄⋊C₈	—	158	2	no	—
114	C₂×#97	—	2	2	no	—
115	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
116	#389	—	79	2	no	—
117	#387	—	158	2	no	—
118	#382	—	79	1	no	—
119	C₄×C₄⋊C₄	—	79	2	no	—
120	C₄×C₄⋊C₄	—	158	2	no	—
121	C₄³	C₂×D₇₉	1	1	yes	characteristic
122	C₂²×C₈	—	2	2	no	—
123	C₂²×C₈	—	158	2	no	—
124	#125	—	158	4	no	—
125	C₂×C₄⋊C₄	—	79	2	no	—
126	C₂×C₄⋊C₄	—	79	2	no	—
127	C₂×C₄⋊C₄	—	158	2	no	—
128	C₂×C₄⋊C₄	—	158	2	no	—
129	C₂×C₄²	—	158	2	no	—
130	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
131	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
132	C₂×C₄²	D₃₁₆	1	2	yes	—
133	C₂×C₄²	C₂²⋊D₇₉	1	2	yes	—
134	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
135	C₄⋊C₈	—	316	4	no	—
136	C₄⋊C₈	—	158	4	no	—
137	#97	—	2	8	no	—
138	C₄×C₈	—	2	2	no	—
139	C₄×C₈	Dic₁₅₈	1	4	yes	—
140	#95	—	158	2	no	—
141	#95	—	158	2	no	—
142	#95	—	158	2	no	—
143	C₂²×C₄	—	158	2	no	—
144	C₂²×C₄	—	158	2	no	—
145	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
146	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
147	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
149	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
150	C₂×C₈	—	2	8	no	—
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	316	2	no	—
153	C₂×C₈	—	316	4	no	—
154	C₄⋊C₄	—	316	4	no	—
155	C₄⋊C₄	—	79	4	no	—
156	C₄⋊C₄	—	158	8	no	—
157	C₄⋊C₄	—	158	4	no	—
158	C₄⋊C₄	—	158	2	no	—
159	C₄²	—	2	4	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	2	no	—
164	C₄²	—	316	2	no	—
165	C₄²	—	316	2	no	—
166	C₄²	C₂×Dic₁₅₈	1	2	yes	—
167	C₄²	#11929624	1	2	yes	—
168	C₂³	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	316	2	no	—
176	C₂×C₄	—	316	2	no	—
177	C₂×C₄	—	316	2	no	—
178	C₂×C₄	—	316	2	no	—
179	C₂×C₄	—	316	2	no	—
180	C₂×C₄	—	158	4	no	—
181	C₂×C₄	#14579089	1	4	yes	—
182	C₂×C₄	C₄×Dic₁₅₈	1	2	yes	—
183	C₂×C₄	#14579140	1	2	yes	—
184	C₂×C₄	#14579149	1	1	yes	characteristic
185	C₂×C₄	#14579161	1	1	yes	characteristic
186	C₂×C₄	M₁₆×D₇₉	1	2	yes	—
187	C₂×C₄	#14579169	1	2	yes	—
188	C₂×C₄	#14579189	1	2	yes	—
189	C₂×C₄	#14579194	1	2	yes	—
190	C₈	—	2	8	no	—
191	C₈	—	632	2	no	—
192	C₂²	#19098469	1	1	yes	characteristic
193	C₂²	#19098526	1	1	yes	characteristic
194	C₂²	#19098531	1	1	yes	characteristic
195	C₂²	#19098565	1	2	yes	—
196	C₂²	C₂×#14579089	1	2	yes	—
197	C₄	—	2	4	no	—
198	C₄	—	2	2	no	—
199	C₄	—	4	2	no	—
200	C₄	—	316	4	no	—
201	C₄	—	632	2	no	—
202	C₄	#19098691	1	4	yes	—
203	C₄	#19098928	1	1	yes	characteristic
204	C₄	#19098932	1	1	yes	characteristic
205	C₄	#19098972	1	2	yes	—
206	C₂	#23761814	1	2	yes	—
207	C₂	#23761840	1	2	yes	—
208	C₂	#23764340	1	1	yes	characteristic
209	C₂	#23764369	1	1	yes	characteristic
210	C₂	#23764377	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14579153

C₂

yes

maximal, characteristic

#23763552

C₂

yes

maximal, characteristic

#23761240

C₂

yes

maximal

C₄×#97×C₇₉

C₂

yes

maximal, characteristic

C₂×#14579153

C₄

yes

characteristic

C₂×C₄⋊(C₇₉⋊C₈)

—

C₂×C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

#19098461

C₄

yes

—

#19098447

—

#19098429

C₄

yes

characteristic

C₄×Dic₇₉⋊C₄

—

C₄×C₄⋊Dic₇₉

C₂²

yes

—

C₂×#97×C₇₉

—

C₂×C₄×C₈×C₇₉

C₂²

yes

—

C₄³×C₇₉

C₂²

yes

characteristic

C₂²×C₇₉⋊C₈

—

#14579153

—

C₂×C₄⋊Dic₇₉

C₂×C₄

yes

—

C₂×C₄⋊Dic₇₉

C₂×C₄

yes

—

C₂×Dic₇₉⋊C₄

—

C₂×Dic₇₉⋊C₄

—

C₂×C₄×Dic₇₉

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₇₉⋊#95

—

C₇₉⋊#95

—

C₇₉⋊#95

—

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

D₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

#97×C₇₉

—

C₄×C₈×C₇₉

—

C₄×C₈×C₇₉

Q₈

yes

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

M₁₆

yes

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

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

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

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

#104

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×M₁₆

yes

—

C₂×C₄×C₇₉

#131

yes

—

C₂×C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇₉

#137

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

—

C₂²×C₇₉

C₂×#104

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#446

yes

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

Dic₇₉

—

Dic₇₉

—

100

#9876

—

maximal, Sylow

101

C₂×C₇₉

#1483

yes

—

102

C₂×C₇₉

#1489

yes

—

103

C₂×C₇₉

#1985

yes

characteristic

104

C₂×C₇₉

#1992

yes

characteristic

105

C₂×C₇₉

#1994

yes

characteristic

106

C₄×#125

—

107

#1842

—

108

C₄×#97

D₇₉

yes

characteristic

109

#1279

—

110

C₇₉

#9876

yes

characteristic, complemented, Sylow

111

C₂×#125

—

112

C₂×C₄⋊C₈

—

113

C₂×C₄⋊C₈

—

158

—

114

C₂×#97

—

115

C₂×C₄×C₈

C₂×D₇₉

yes

—

116

#389

—

117

#387

—

158

—

118

#382

—

119

C₄×C₄⋊C₄

—

120

C₄×C₄⋊C₄

—

158

—

121

C₄³

C₂×D₇₉

yes

characteristic

122

C₂²×C₈

—

123

C₂²×C₈

—

158

—

124

#125

—

158

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄⋊C₄

—

158

—

128

C₂×C₄⋊C₄

—

158

—

129

C₂×C₄²

—

158

—

130

C₂×C₄²

C₄×D₇₉

yes

characteristic

131

C₂×C₄²

C₄×D₇₉

yes

characteristic

132

C₂×C₄²

D₃₁₆

yes

—

133

C₂×C₄²

C₂²⋊D₇₉

yes

—

134

C₂×C₄²

C₂²×D₇₉

yes

characteristic

135

C₄⋊C₈

—

316

—

136

C₄⋊C₈

—

158

—

137

#97

—

138

C₄×C₈

—

139

C₄×C₈

Dic₁₅₈

yes

—

140

#95

—

158

—

141

#95

—

158

—

142

#95

—

158

—

143

C₂²×C₄

—

158

—

144

C₂²×C₄

—

158

—

145

C₂²×C₄

(C₂×D₇₉)⋊C₄

yes

—

146

C₂²×C₄

(C₂×D₇₉)⋊C₄

yes

—

147

C₂²×C₄

C₂×C₄×D₇₉

yes

characteristic

148

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

149

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

316

—

153

C₂×C₈

—

316

—

154

C₄⋊C₄

—

316

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

158

—

157

C₄⋊C₄

—

158

—

158

C₄⋊C₄

—

158

—

159

C₄²

—

160

C₄²

—

161

C₄²

—

162

C₄²

—

163

C₄²

—

164

C₄²

—

316

—

165

C₄²

—

316

—

166

C₄²

C₂×Dic₁₅₈

yes

—

167

C₄²

#11929624

yes

—

168

C₂³

C₂×(C₂×D₇₉)⋊C₄

yes

characteristic

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

316

—

176

C₂×C₄

—

316

—

177

C₂×C₄

—

316

—

178

C₂×C₄

—

316

—

179

C₂×C₄

—

316

—

180

C₂×C₄

—

158

—

181

C₂×C₄

#14579089

yes

—

182

C₂×C₄

C₄×Dic₁₅₈

yes

—

183

C₂×C₄

#14579140

yes

—

184

C₂×C₄

#14579149

yes

characteristic

185

C₂×C₄

#14579161

yes

characteristic

186

C₂×C₄

M₁₆×D₇₉

yes

—

187

C₂×C₄

#14579169

yes

—

188

C₂×C₄

#14579189

yes

—

189

C₂×C₄

#14579194

yes

—

190

C₈

—

191

C₈

—

632

—

192

C₂²

#19098469

yes

characteristic

193

C₂²

#19098526

yes

characteristic

194

C₂²

#19098531

yes

characteristic

195

C₂²

#19098565

yes

—

196

C₂²

C₂×#14579089

yes

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

316

—

201

C₄

—

632

—

202

C₄

#19098691

yes

—

203

C₄

#19098928

yes

characteristic

204

C₄

#19098932

yes

characteristic

205

C₄

#19098972

yes

—

206

C₂

#23761814

yes

—

207

C₂

#23761840

yes

—

208

C₂

#23764340

yes

characteristic

209

C₂

#23764369

yes

characteristic

210

C₂

#23764377

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₁₆
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₄×#97×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₄×#97	D₇₉
hypocenter	C₇₉	#9876
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#97

subgroup

G/H

center

C₂²×C₄

C₂×D₃₁₆

commutator

C₂²×C₄×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₄×#97×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₇₉

C₂³

hypercenter

C₄×#97

D₇₉

hypocenter

C₇₉

#9876

LCS

G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#97

N	H ≅ G/N	type
110. C₇₉	#9876	semidirect

H ≅ G/N

type

110. C₇₉

#9876

semidirect

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

subgroup	index
C₄×#14579153	2
#23763552	2
#23761240	2
C₄×#97×C₇₉	2
#9876	79

subgroup

index

C₄×#97×C₇₉

Subgroups — #34188300

Conjugacy classes of subgroups

Complements

Maximal subgroups