Group #34187664 — subgroups

Subgroups — #34187664

Order 20224.

478 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 111 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	#23773828	C₂	1	1	yes	maximal, characteristic
2	#23761959	C₂	1	4	yes	maximal, complemented
3	#23761692	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₉×#1835	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14579183	—	2	2	no	—
6	#19099113	C₂²	1	2	yes	—
7	#19098947	—	2	2	no	—
8	#19098943	C₄	1	1	yes	characteristic, complemented
9	#19098940	C₄	1	1	yes	characteristic, complemented
10	#19098936	C₄	1	2	yes	complemented
11	#19098616	C₂²	1	2	yes	—
12	#19098296	—	2	8	no	—
13	#19098285	—	2	2	no	—
14	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
15	#382×C₇₉	C₂²	1	2	yes	—
16	#340×C₇₉	—	2	2	no	—
17	C₂×#11929624	—	2	2	no	—
18	#14579183	—	4	8	no	—
19	C₂×Dic₃₁₆	—	2	2	no	—
20	C₂×#11929613	—	2	4	no	—
21	C₂×D₆₃₂	—	2	2	no	—
22	#14579161	C₂×C₄	1	2	yes	—
23	#14579158	C₂×C₄	1	2	yes	—
24	#14579149	—	2	2	no	—
25	#14579135	—	2	4	no	—
26	C₄×D₃₁₆	—	2	4	no	—
27	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
28	C₂²⋊(C₇₉⋊C₈)	D₄	1	4	yes	—
29	C₄×C₇₉⋊C₈	—	2	2	no	—
30	C₂²×C₈×C₇₉	D₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
32	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
33	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₇₉	D₄	1	2	yes	—
36	#95×C₇₉	—	2	4	no	—
37	#11929624	—	4	8	no	—
38	C₂×Dic₁₅₈	—	2	2	no	—
39	C₂×D₃₁₆	—	2	2	no	—
40	C₂×D₃₁₆	—	2	4	no	—
41	C₂×C₂²⋊D₇₉	—	4	2	no	—
42	C₂×C₄×D₇₉	—	4	2	no	—
43	Dic₃₁₆	—	4	4	no	—
44	#11929613	—	4	8	no	—
45	D₆₃₂	—	4	4	no	—
46	C₂×C₇₉⋊C₈	—	2	4	no	—
47	C₄⋊Dic₇₉	—	2	4	no	—
48	(C₂×D₇₉)⋊C₄	—	4	4	no	—
49	C₂²×C₄×C₇₉	—	4	2	no	—
50	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
51	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₇₉	—	2	4	no	—
53	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
55	C₄⋊C₄×C₇₉	—	4	4	no	—
56	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₇₉	#41	1	2	yes	—
59	C₂³×C₇₉	#127	1	1	yes	characteristic
60	C₂²×D₇₉	—	4	2	no	—
61	C₂×C₄×C₇₉	—	4	2	no	—
62	C₂×C₄×C₇₉	—	8	4	no	—
63	C₂×C₄×C₇₉	—	8	2	no	—
64	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
66	C₂×C₄×C₇₉	#121	1	2	yes	—
67	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
68	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	4	yes	—
69	C₂²⋊D₇₉	—	8	4	no	—
70	C₂×Dic₇₉	—	4	2	no	—
71	D₃₁₆	—	4	4	no	—
72	D₃₁₆	—	8	4	no	—
73	C₄×D₇₉	—	8	4	no	—
74	Dic₁₅₈	—	4	4	no	—
75	C₈×C₇₉	—	2	2	no	—
76	C₈×C₇₉	—	2	2	no	—
77	C₇₉⋊C₈	—	4	4	no	—
78	C₂²×C₇₉	—	2	2	no	—
79	C₂²×C₇₉	#390	1	1	yes	characteristic
80	C₂²×C₇₉	#496	1	2	yes	—
81	C₂×D₇₉	—	8	2	no	—
82	C₂×D₇₉	—	8	4	no	—
83	C₄×C₇₉	—	2	2	no	—
84	C₄×C₇₉	—	2	2	no	—
85	C₄×C₇₉	—	16	2	no	—
86	C₄×C₇₉	#418	1	2	yes	—
87	C₄×C₇₉	#490	1	2	yes	—
88	Dic₇₉	—	8	2	no	—
89	#9760	—	79	1	no	maximal, Sylow
90	C₂×C₇₉	—	2	2	no	—
91	C₂×C₇₉	#1961	1	1	yes	characteristic
92	C₂×C₇₉	#1976	1	2	yes	—
93	D₇₉	—	16	2	no	—
94	#3147	—	79	1	no	—
95	#1835	D₇₉	1	1	yes	characteristic, complemented
96	#1517	—	79	4	no	—
97	#1452	—	79	1	no	—
98	C₇₉	#9760	1	1	yes	characteristic, complemented, Sylow
99	C₂×#135	—	158	2	no	—
100	#532	—	79	2	no	—
101	#495	—	158	2	no	—
102	#494	—	79	1	no	—
103	#493	—	79	1	no	—
104	#492	—	79	2	no	—
105	#431	—	79	2	no	—
106	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
107	#382	C₂×D₇₉	1	2	yes	—
108	#340	—	2	2	no	—
109	#327	—	158	8	no	—
110	#322	—	158	2	no	—
111	C₂×#41	—	158	2	no	—
112	#135	—	316	8	no	—
113	C₂×Q₁₆	—	158	2	no	—
114	C₂×SD₁₆	—	158	4	no	—
115	C₂×D₈	—	158	2	no	—
116	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
117	C₄⋊Q₈	—	79	2	no	—
118	#127	—	79	2	no	—
119	#124	—	158	2	no	—
120	#121	—	158	4	no	—
121	C₄×D₄	—	158	4	no	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
124	C₄⋊C₈	—	158	2	no	—
125	C₂²⋊C₈	—	79	4	no	—
126	C₄×C₈	—	158	2	no	—
127	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
128	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
129	C₄×C₈	D₃₁₆	1	2	yes	—
130	#95	—	2	4	no	—
131	#41	—	316	8	no	—
132	C₂×Q₈	—	158	2	no	—
133	C₂×D₄	—	316	2	no	—
134	C₂×D₄	—	158	2	no	—
135	C₂×D₄	—	158	4	no	—
136	C₂²×C₄	—	4	2	no	—
137	C₂²×C₄	—	316	2	no	—
138	C₂²×C₄	D₄×D₇₉	1	2	yes	—
139	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
140	Q₁₆	—	316	4	no	—
141	SD₁₆	—	316	8	no	—
142	D₈	—	316	4	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	—	158	4	no	—
145	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
146	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
147	C₄⋊C₄	—	4	4	no	—
148	C₄⋊C₄	—	158	4	no	—
149	C₂²⋊C₄	—	316	4	no	—
150	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
151	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
152	C₄²	#11929625	1	2	yes	—
153	C₂³	—	316	2	no	—
154	C₂³	#14579157	1	1	yes	characteristic
155	Q₈	—	316	4	no	—
156	D₄	—	316	4	no	—
157	D₄	—	632	4	no	—
158	D₄	—	632	4	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	—	8	4	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	—	316	2	no	—
163	C₂×C₄	—	632	4	no	—
164	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
165	C₂×C₄	#14579123	1	2	yes	—
166	C₂×C₄	#14579136	1	2	yes	—
167	C₂×C₄	#14579150	1	1	yes	characteristic
168	C₂×C₄	#14579186	1	4	yes	—
169	C₈	—	2	2	no	—
170	C₈	—	2	2	no	—
171	C₈	—	316	4	no	—
172	C₂²	—	2	2	no	—
173	C₂²	—	632	4	no	—
174	C₂²	—	632	2	no	—
175	C₂²	#19098464	1	1	yes	characteristic
176	C₂²	#19098953	1	2	yes	—
177	C₄	—	2	2	no	—
178	C₄	—	2	2	no	—
179	C₄	—	16	2	no	—
180	C₄	—	632	2	no	—
181	C₄	#19098567	1	2	yes	—
182	C₄	#19098929	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	—	1264	2	no	—
185	C₂	#23764223	1	1	yes	characteristic
186	C₂	#23764296	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23773828

C₂

yes

maximal, characteristic

#23761959

C₂

yes

maximal, complemented

#23761692

C₂

yes

maximal, characteristic, complemented

C₇₉×#1835

C₂

yes

maximal, characteristic, complemented

C₂×#14579183

—

#19099113

C₂²

yes

—

#19098947

—

#19098943

C₄

yes

characteristic, complemented

#19098940

C₄

yes

characteristic, complemented

#19098936

C₄

yes

complemented

#19098616

C₂²

yes

—

#19098296

—

#19098285

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#382×C₇₉

C₂²

yes

—

#340×C₇₉

—

C₂×#11929624

—

#14579183

—

C₂×Dic₃₁₆

—

C₂×#11929613

—

C₂×D₆₃₂

—

#14579161

C₂×C₄

yes

—

#14579158

C₂×C₄

yes

—

#14579149

—

#14579135

—

C₄×D₃₁₆

—

C₄⋊(C₇₉⋊C₈)

—

C₂²⋊(C₇₉⋊C₈)

D₄

yes

—

C₄×C₇₉⋊C₈

—

C₂²×C₈×C₇₉

D₄

yes

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

—

#95×C₇₉

—

#11929624

—

C₂×Dic₁₅₈

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×C₂²⋊D₇₉

—

C₂×C₄×D₇₉

—

Dic₃₁₆

—

#11929613

—

D₆₃₂

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

(C₂×D₇₉)⋊C₄

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

#41

yes

—

C₂³×C₇₉

#127

yes

characteristic

C₂²×D₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄×D₄

yes

—

C₂×C₄×C₇₉

#121

yes

—

C₂×C₄×C₇₉

#124

yes

characteristic

C₂×C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₂²⋊D₇₉

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

C₄×D₇₉

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

#390

yes

characteristic

C₂²×C₇₉

#496

yes

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#418

yes

—

C₄×C₇₉

#490

yes

—

Dic₇₉

—

#9760

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

#1961

yes

characteristic

C₂×C₇₉

#1976

yes

—

D₇₉

—

#3147

—

#1835

D₇₉

yes

characteristic, complemented

#1517

—

#1452

—

C₇₉

#9760

yes

characteristic, complemented, Sylow

C₂×#135

—

158

—

100

#532

—

101

#495

—

158

—

102

#494

—

103

#493

—

104

#492

—

105

#431

—

106

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

107

#382

C₂×D₇₉

yes

—

108

#340

—

109

#327

—

158

—

110

#322

—

158

—

111

C₂×#41

—

158

—

112

#135

—

316

—

113

C₂×Q₁₆

—

158

—

114

C₂×SD₁₆

—

158

—

115

C₂×D₈

—

158

—

116

C₂²×C₈

C₂²⋊D₇₉

yes

—

117

C₄⋊Q₈

—

118

#127

—

119

#124

—

158

—

120

#121

—

158

—

121

C₄×D₄

—

158

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄²

C₂²×D₇₉

yes

characteristic

124

C₄⋊C₈

—

158

—

125

C₂²⋊C₈

—

126

C₄×C₈

—

158

—

127

C₄×C₈

C₄×D₇₉

yes

characteristic

128

C₄×C₈

C₄×D₇₉

yes

characteristic

129

C₄×C₈

D₃₁₆

yes

—

130

#95

—

131

#41

—

316

—

132

C₂×Q₈

—

158

—

133

C₂×D₄

—

316

—

134

C₂×D₄

—

158

—

135

C₂×D₄

—

158

—

136

C₂²×C₄

—

137

C₂²×C₄

—

316

—

138

C₂²×C₄

D₄×D₇₉

yes

—

139

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

140

Q₁₆

—

316

—

141

SD₁₆

—

316

—

142

D₈

—

316

—

143

C₂×C₈

—

144

C₂×C₈

—

158

—

145

C₂×C₈

(C₂×D₇₉)⋊C₄

yes

—

146

C₂×C₈

(C₂×D₇₉)⋊C₄

yes

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

158

—

149

C₂²⋊C₄

—

316

—

150

C₄²

C₂×C₄×D₇₉

yes

characteristic

151

C₄²

C₂×D₃₁₆

yes

characteristic

152

C₄²

#11929625

yes

—

153

C₂³

—

316

—

154

C₂³

#14579157

yes

characteristic

155

Q₈

—

316

—

156

D₄

—

316

—

157

D₄

—

632

—

158

D₄

—

632

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

316

—

163

C₂×C₄

—

632

—

164

C₂×C₄

C₂×(C₂×D₇₉)⋊C₄

yes

characteristic

165

C₂×C₄

#14579123

yes

—

166

C₂×C₄

#14579136

yes

—

167

C₂×C₄

#14579150

yes

characteristic

168

C₂×C₄

#14579186

yes

—

169

C₈

—

170

C₈

—

171

C₈

—

316

—

172

C₂²

—

173

C₂²

—

632

—

174

C₂²

—

632

—

175

C₂²

#19098464

yes

characteristic

176

C₂²

#19098953

yes

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

632

—

181

C₄

#19098567

yes

—

182

C₄

#19098929

yes

—

183

C₂

—

184

C₂

—

1264

—

185

C₂

#23764223

yes

characteristic

186

C₂

#23764296

yes

—

	subgroup	G/H
center	C₂²	#19098464
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1835	C₂
socle	C₂²×C₇₉	#390
prime radical	#23773828	C₂
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1835	D₇₉
hypocenter	C₇₉	#9760
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1835

subgroup

G/H

center

C₂²

#19098464

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1835

C₂

C₂²×C₇₉

C₂

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1835

N	H ≅ G/N	type
2. #23761959	C₂	semidirect
3. #23761692	C₂	semidirect
4. C₇₉×#1835	C₂	semidirect
8. #19098943	C₄	semidirect
9. #19098940	C₄	semidirect
10. #19098936	C₄	semidirect
95. #1835	D₇₉	semidirect
98. C₇₉	#9760	semidirect

H ≅ G/N

type

2. #23761959

C₂

semidirect

3. #23761692

C₂

semidirect

4. C₇₉×#1835

C₂

semidirect

8. #19098943

C₄

semidirect

9. #19098940

C₄

semidirect

10. #19098936

C₄

semidirect

95. #1835

D₇₉

semidirect

98. C₇₉

#9760

semidirect

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

subgroup	index
#23773828	2
#23761959	2
#23761692	2
C₇₉×#1835	2
#9760	79

subgroup

index

C₇₉×#1835

Subgroups — #34187664

Conjugacy classes of subgroups

Complements

Maximal subgroups