Group #34193281 — subgroups

Subgroups — #34193281

Order 20224.

354 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 97 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	#23764232	C₂	1	1	yes	maximal, characteristic
2	#23762170	C₂	1	2	yes	maximal, complemented
3	#23761798	C₂	1	2	yes	maximal
4	#23761229	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₉×#1843	C₂	1	1	yes	maximal, characteristic, complemented
6	#19099113	C₂²	1	1	yes	characteristic
7	#19098616	C₂²	1	2	yes	—
8	#19098587	C₂²	1	1	yes	characteristic
9	C₂×#14579089	—	2	2	no	—
10	#19098313	—	2	4	no	—
11	#19098290	—	2	4	no	—
12	C₂×C₄⋊C₈×C₇₉	—	2	2	no	—
13	C₂×C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
14	#376×C₇₉	C₂²	1	1	yes	characteristic
15	C₂×#11929624	—	2	1	no	—
16	C₂×#11929608	—	2	1	no	—
17	#14579161	Q₈	1	1	yes	characteristic
18	#14579158	Q₈	1	1	yes	characteristic
19	#14579149	—	2	1	no	—
20	#14579135	—	2	2	no	—
21	C₄×D₃₁₆	—	2	2	no	—
22	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
23	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
24	#14579089	—	4	4	no	—
25	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
26	C₄×C₇₉⋊C₈	D₄	1	4	yes	—
27	C₂²×C₈×C₇₉	—	2	2	no	—
28	C₂×C₄²×C₇₉	D₄	1	2	yes	—
29	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₇₉	—	2	2	no	—
31	C₄⋊C₈×C₇₉	—	2	2	no	—
32	C₄⋊C₈×C₇₉	—	4	4	no	—
33	#95×C₇₉	—	2	2	no	—
34	#11929624	—	4	2	no	—
35	#11929624	—	4	2	no	—
36	C₂×Dic₁₅₈	—	2	1	no	—
37	C₂×D₃₁₆	—	2	1	no	—
38	C₂×D₃₁₆	—	2	2	no	—
39	C₂×C₂²⋊D₇₉	—	4	1	no	—
40	C₂×C₄×D₇₉	—	4	1	no	—
41	#11929608	—	4	2	no	—
42	C₂×C₇₉⋊C₈	—	2	4	no	—
43	C₂×C₇₉⋊C₈	—	4	1	no	—
44	C₄⋊Dic₇₉	—	2	2	no	—
45	(C₂×D₇₉)⋊C₄	—	4	2	no	—
46	C₂²×C₄×C₇₉	—	2	2	no	—
47	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₇₉	#41	1	2	yes	—
49	C₂×C₈×C₇₉	—	4	2	no	—
50	C₂×C₈×C₇₉	—	4	4	no	—
51	C₄²×C₇₉	—	2	2	no	—
52	C₄²×C₇₉	SD₁₆	1	4	yes	—
53	C₄²×C₇₉	C₂×D₄	1	2	yes	—
54	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
55	C₄²×C₇₉	#41	1	1	yes	characteristic
56	C₂³×C₇₉	#123	1	1	yes	characteristic
57	C₂²×D₇₉	—	4	1	no	—
58	C₂×C₄×C₇₉	—	2	4	no	—
59	C₂×C₄×C₇₉	—	4	2	no	—
60	C₂×C₄×C₇₉	—	4	4	no	—
61	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
63	C₂×C₄×C₇₉	#123	1	2	yes	—
64	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
65	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
66	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
67	C₂²⋊D₇₉	—	8	2	no	—
68	C₂×Dic₇₉	—	4	1	no	—
69	D₃₁₆	—	4	4	no	—
70	D₃₁₆	—	4	2	no	—
71	C₄×D₇₉	—	8	1	no	—
72	C₄×D₇₉	—	8	1	no	—
73	Dic₁₅₈	—	4	2	no	—
74	C₈×C₇₉	—	8	2	no	—
75	C₇₉⋊C₈	—	4	4	no	—
76	C₇₉⋊C₈	—	8	1	no	—
77	C₂²×C₇₉	—	2	2	no	—
78	C₂²×C₇₉	#399	1	1	yes	characteristic
79	C₂²×C₇₉	#481	1	2	yes	—
80	C₂×D₇₉	—	8	1	no	—
81	C₂×D₇₉	—	8	2	no	—
82	C₄×C₇₉	—	2	2	no	—
83	C₄×C₇₉	—	2	2	no	—
84	C₄×C₇₉	—	4	4	no	—
85	C₄×C₇₉	#476	1	2	yes	—
86	C₄×C₇₉	#492	1	1	yes	characteristic
87	C₄×C₇₉	#496	1	1	yes	characteristic
88	Dic₇₉	—	8	1	no	—
89	#10666	—	79	1	no	maximal, Sylow
90	C₂×C₇₉	—	2	2	no	—
91	C₂×C₇₉	#2094	1	1	yes	characteristic
92	C₂×C₇₉	#2098	1	2	yes	—
93	D₇₉	—	16	1	no	—
94	#1963	—	79	1	no	—
95	#1843	D₇₉	1	1	yes	characteristic, complemented
96	#1570	—	79	2	no	—
97	#1480	—	79	2	no	—
98	#1274	—	79	1	no	—
99	C₇₉	#10666	1	1	yes	characteristic, complemented, Sylow
100	#532	—	79	1	no	—
101	#431	—	79	2	no	—
102	#423	—	79	1	no	—
103	C₂×C₄⋊C₈	—	2	2	no	—
104	C₂×C₄⋊C₈	C₂×D₇₉	1	2	yes	—
105	C₂×#104	—	158	2	no	—
106	#376	C₂×D₇₉	1	1	yes	characteristic
107	#334	—	158	4	no	—
108	D₄⋊C₈	—	158	4	no	—
109	C₂×#41	—	158	1	no	—
110	C₂×M₁₆	—	158	1	no	—
111	C₂²×C₈	—	2	2	no	—
112	C₄⋊Q₈	—	79	1	no	—
113	#127	—	79	1	no	—
114	#124	—	158	1	no	—
115	#121	—	158	2	no	—
116	C₄×D₄	—	158	2	no	—
117	C₂×C₄²	C₂²⋊D₇₉	1	2	yes	—
118	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	4	4	no	—
122	C₄⋊C₈	—	158	2	no	—
123	C₄⋊C₈	—	158	2	no	—
124	#104	—	316	4	no	—
125	C₂²⋊C₈	—	158	2	no	—
126	C₄×C₈	—	79	4	no	—
127	#95	—	2	2	no	—
128	#41	—	316	2	no	—
129	#41	—	316	2	no	—
130	C₂×Q₈	—	158	1	no	—
131	C₂×D₄	—	316	1	no	—
132	C₂×D₄	—	158	1	no	—
133	C₂×D₄	—	158	2	no	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	—	316	1	no	—
136	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
137	C₂²×C₄	#11929624	1	2	yes	—
138	M₁₆	—	316	2	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	4	4	no	—
141	C₂×C₈	—	316	1	no	—
142	C₂×C₈	—	158	4	no	—
143	C₄⋊C₄	—	158	2	no	—
144	C₂²⋊C₄	—	316	2	no	—
145	C₄²	—	2	2	no	—
146	C₄²	#11929611	1	4	yes	—
147	C₄²	D₄×D₇₉	1	2	yes	—
148	C₄²	Q₈×D₇₉	1	1	yes	characteristic
149	C₄²	#11929625	1	1	yes	characteristic
150	C₂³	—	316	1	no	—
151	C₂³	#14579144	1	1	yes	characteristic
152	Q₈	—	316	2	no	—
153	D₄	—	316	4	no	—
154	D₄	—	316	2	no	—
155	D₄	—	632	2	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	—	316	1	no	—
160	C₂×C₄	—	632	1	no	—
161	C₂×C₄	—	632	1	no	—
162	C₂×C₄	#14579137	1	1	yes	characteristic
163	C₂×C₄	#14579139	1	2	yes	—
164	C₂×C₄	#14579146	1	2	yes	—
165	C₂×C₄	#14579157	1	1	yes	characteristic
166	C₂×C₄	C₂×#11929611	1	2	yes	—
167	C₂×C₄	#14579187	1	2	yes	—
168	C₈	—	8	2	no	—
169	C₈	—	316	4	no	—
170	C₈	—	632	1	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	632	1	no	—
173	C₂²	—	632	2	no	—
174	C₂²	#19098498	1	1	yes	characteristic
175	C₂²	#19098881	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	4	4	no	—
179	C₄	—	632	1	no	—
180	C₄	#19098850	1	2	yes	—
181	C₄	#19098937	1	1	yes	characteristic
182	C₄	#19098950	1	1	yes	characteristic
183	C₂	—	2	2	no	—
184	C₂	—	1264	1	no	—
185	C₂	#23764934	1	1	yes	characteristic
186	C₂	#23764956	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23764232

C₂

yes

maximal, characteristic

#23762170

C₂

yes

maximal, complemented

#23761798

C₂

yes

maximal

#23761229

C₂

yes

maximal, characteristic, complemented

C₇₉×#1843

C₂

yes

maximal, characteristic, complemented

#19099113

C₂²

yes

characteristic

#19098616

C₂²

yes

—

#19098587

C₂²

yes

—

—

—

C₂×C₄⋊C₈×C₇₉

—

C₂×C₄⋊C₈×C₇₉

C₂²

yes

—

#376×C₇₉

C₂²

yes

—

—

Q₈

yes

characteristic

#14579158

Q₈

yes

characteristic

#14579149

—

#14579135

—

C₄×D₃₁₆

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

#14579089

—

C₂²⋊(C₇₉⋊C₈)

—

C₄×C₇₉⋊C₈

D₄

yes

—

C₂²×C₈×C₇₉

—

C₂×C₄²×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

—

#95×C₇₉

—

#11929624

—

#11929624

—

C₂×Dic₁₅₈

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×C₂²⋊D₇₉

—

C₂×C₄×D₇₉

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

(C₂×D₇₉)⋊C₄

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

#41

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

SD₁₆

yes

—

C₄²×C₇₉

C₂×D₄

yes

—

C₄²×C₇₉

C₂×Q₈

yes

C₄²×C₇₉

yes

C₂³×C₇₉

yes

C₂²×D₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₉

#123

yes

—

C₂×C₄×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

C₂×SD₁₆

yes

—

C₂×C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₂²⋊D₇₉

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

C₄×D₇₉

—

C₄×D₇₉

—

Dic₁₅₈

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

#399

yes

characteristic

C₂²×C₇₉

#481

yes

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#476

yes

—

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

#10666

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

#2094

yes

characteristic

C₂×C₇₉

#2098

yes

—

D₇₉

—

#1963

—

#1843

D₇₉

yes

characteristic, complemented

#1570

—

#1480

—

#1274

—

C₇₉

#10666

yes

characteristic, complemented, Sylow

100

#532

—

101

#431

—

102

#423

—

103

C₂×C₄⋊C₈

—

104

C₂×C₄⋊C₈

C₂×D₇₉

yes

—

105

C₂×#104

—

158

—

106

#376

C₂×D₇₉

yes

characteristic

107

#334

—

158

—

108

D₄⋊C₈

—

158

—

109

C₂×#41

—

158

—

110

C₂×M₁₆

—

158

—

111

C₂²×C₈

—

112

C₄⋊Q₈

—

113

#127

—

114

#124

—

158

—

115

#121

—

158

—

116

C₄×D₄

—

158

—

117

C₂×C₄²

C₂²⋊D₇₉

yes

—

118

C₂×C₄²

C₂²×D₇₉

yes

characteristic

119

C₄⋊C₈

—

120

C₄⋊C₈

—

121

C₄⋊C₈

—

122

C₄⋊C₈

—

158

—

123

C₄⋊C₈

—

158

—

124

#104

—

316

—

125

C₂²⋊C₈

—

158

—

126

C₄×C₈

—

127

#95

—

128

#41

—

316

—

129

#41

—

316

—

130

C₂×Q₈

—

158

—

131

C₂×D₄

—

316

—

132

C₂×D₄

—

158

—

133

C₂×D₄

—

158

—

134

C₂²×C₄

—

135

C₂²×C₄

—

316

—

136

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

137

C₂²×C₄

#11929624

yes

—

138

M₁₆

—

316

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

316

—

142

C₂×C₈

—

158

—

143

C₄⋊C₄

—

158

—

144

C₂²⋊C₄

—

316

—

145

C₄²

—

146

C₄²

#11929611

yes

—

147

C₄²

D₄×D₇₉

yes

—

148

C₄²

Q₈×D₇₉

yes

characteristic

149

C₄²

#11929625

yes

characteristic

150

C₂³

—

316

—

151

C₂³

#14579144

yes

characteristic

152

Q₈

—

316

—

153

D₄

—

316

—

154

D₄

—

316

—

155

D₄

—

632

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

316

—

160

C₂×C₄

—

632

—

161

C₂×C₄

—

632

—

162

C₂×C₄

#14579137

yes

characteristic

163

C₂×C₄

#14579139

yes

—

164

C₂×C₄

#14579146

yes

—

165

C₂×C₄

#14579157

yes

characteristic

166

C₂×C₄

C₂×#11929611

yes

—

167

C₂×C₄

#14579187

yes

—

168

C₈

—

169

C₈

—

316

—

170

C₈

—

632

—

171

C₂²

—

172

C₂²

—

632

—

173

C₂²

—

632

—

174

C₂²

#19098498

yes

characteristic

175

C₂²

#19098881

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

632

—

180

C₄

#19098850

yes

—

181

C₄

#19098937

yes

characteristic

182

C₄

#19098950

yes

characteristic

183

C₂

—

184

C₂

—

1264

—

185

C₂

#23764934

yes

characteristic

186

C₂

#23764956

yes

—

	subgroup	G/H
center	C₂×C₄	#14579157
commutator, cosocle	C₂×C₄²×C₇₉	C₂³
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1843	C₂
socle	C₂²×C₇₉	#399
prime radical	#19099113	C₂²
hypercenter	#1843	D₇₉
hypocenter	C₇₉	#10666
LCS	G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1843

subgroup

G/H

center

C₂×C₄

#14579157

commutator, cosocle

C₂×C₄²×C₇₉

C₂³

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1843

C₂

C₂²×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1843

N	H ≅ G/N	type
2. #23762170	C₂	semidirect
4. #23761229	C₂	semidirect
5. C₇₉×#1843	C₂	semidirect
95. #1843	D₇₉	semidirect
99. C₇₉	#10666	semidirect

H ≅ G/N

type

2. #23762170

C₂

semidirect

4. #23761229

C₂

semidirect

5. C₇₉×#1843

C₂

semidirect

95. #1843

D₇₉

semidirect

99. C₇₉

#10666

semidirect

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

subgroup	index
#23764232	2
#23762170	2
#23761798	2
#23761229	2
C₇₉×#1843	2
#10666	79

subgroup

index

C₇₉×#1843

Subgroups — #34193281

Conjugacy classes of subgroups

Complements

Maximal subgroups