Group #35303158 — subgroups

Subgroups — #35303158

Order 21248.

268 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 115 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	#24906252	C₂	1	2	yes	maximal
2	#24906219	C₂	1	1	yes	maximal, characteristic
3	C₈×#13050574	C₂	1	1	yes	maximal, characteristic
4	#24905354	C₂	1	2	yes	maximal
5	C₈₃×#1564	C₂	1	1	yes	maximal, characteristic
6	#19127754	C₂²	1	1	yes	characteristic
7	#19127725	C₂²	1	2	yes	—
8	C₄×#13050574	C₂²	1	1	yes	characteristic
9	#19127454	—	2	2	no	—
10	#19127452	—	2	2	no	—
11	#19127452	—	2	1	no	—
12	#19127452	—	2	1	no	—
13	#19127423	—	2	1	no	—
14	C₈×C₈₃⋊C₈	—	2	1	no	—
15	#431×C₈₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
17	#335×C₈₃	C₄	1	2	yes	—
18	#334×C₈₃	C₄	1	2	yes	—
19	C₂×#13050574	—	2	1	no	—
20	C₂×#13050574	—	2	2	no	—
21	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
22	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
23	C₄⋊(C₈₃⋊C₈)	D₄	1	2	yes	—
24	C₂²⋊(C₈₃⋊C₈)	—	2	1	no	—
25	#14581703	Q₈	1	2	yes	—
26	C₄×C₈₃⋊C₈	—	2	1	no	—
27	C₄×C₈₃⋊C₈	—	2	1	no	—
28	C₂²×C₈×C₈₃	Q₈	1	1	yes	characteristic
29	C₂²×C₈×C₈₃	Q₈	1	1	yes	characteristic
30	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₈₃	C₂×C₄	1	4	yes	—
32	C₂²⋊C₈×C₈₃	—	2	2	no	—
33	C₄×C₈×C₈₃	—	2	2	no	—
34	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
37	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
38	#13050574	—	2	4	no	—
39	#13050574	—	8	2	no	—
40	C₂×C₈₃⋊C₈	—	2	2	no	—
41	C₂×C₈₃⋊C₈	—	2	2	no	—
42	C₂×C₈₃⋊C₈	—	4	2	no	—
43	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
44	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
45	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
46	C₂×C₈×C₈₃	—	2	2	no	—
47	C₂×C₈×C₈₃	—	2	4	no	—
48	C₂×C₈×C₈₃	—	4	1	no	—
49	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
53	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
54	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
57	C₂³×C₈₃	#125	1	1	yes	characteristic
58	C₂×C₄×C₈₃	—	2	1	no	—
59	C₂×C₄×C₈₃	—	2	1	no	—
60	C₂×C₄×C₈₃	—	2	2	no	—
61	C₂×C₄×C₈₃	—	2	2	no	—
62	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
64	C₂×C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
65	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
66	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
68	C₈×C₈₃	—	2	1	no	—
69	C₈×C₈₃	—	2	1	no	—
70	C₈×C₈₃	—	2	1	no	—
71	C₈×C₈₃	—	4	4	no	—
72	C₈×C₈₃	#108	1	2	yes	—
73	C₈₃⋊C₈	—	2	4	no	—
74	C₈₃⋊C₈	—	4	2	no	—
75	C₈₃⋊C₈	—	8	2	no	—
76	C₂²×C₈₃	—	2	1	no	—
77	C₂²×C₈₃	—	2	1	no	—
78	C₂²×C₈₃	—	2	1	no	—
79	C₂²×C₈₃	#389	1	1	yes	characteristic
80	C₄×C₈₃	—	2	1	no	—
81	C₄×C₈₃	—	2	1	no	—
82	C₄×C₈₃	—	2	1	no	—
83	C₄×C₈₃	C₂×#108	1	1	yes	characteristic
84	C₄×C₈₃	#430	1	1	yes	characteristic
85	C₄×C₈₃	#443	1	2	yes	—
86	C₄×C₈₃	#471	1	2	yes	—
87	#9633	—	83	1	no	maximal, Sylow
88	C₂×C₈₃	—	4	1	no	—
89	C₂×C₈₃	#1945	1	1	yes	characteristic
90	C₂×C₈₃	#1950	1	1	yes	characteristic
91	C₂×C₈₃	#1951	1	1	yes	characteristic
92	#1574	—	83	2	no	—
93	#1566	—	83	1	no	—
94	#1564	D₈₃	1	1	yes	characteristic
95	C₈×M₁₆	—	83	1	no	—
96	#1299	—	83	2	no	—
97	C₈₃	#9633	1	1	yes	characteristic, complemented, Sylow
98	#431	—	83	1	no	—
99	#431	C₂×D₈₃	1	2	yes	—
100	#423	—	83	2	no	—
101	C₄×M₁₆	—	83	1	no	—
102	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
103	#335	—	166	2	no	—
104	#335	Dic₈₃	1	2	yes	—
105	#334	—	166	2	no	—
106	#334	—	166	1	no	—
107	#334	—	166	1	no	—
108	#334	Dic₈₃	1	2	yes	—
109	#322	—	166	1	no	—
110	C₈²	—	166	1	no	—
111	C₂×M₁₆	—	166	2	no	—
112	C₂×M₁₆	—	166	1	no	—
113	C₂²×C₈	Dic₁₆₆	1	1	yes	characteristic
114	C₂²×C₈	Dic₁₆₆	1	1	yes	characteristic
115	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
116	C₄⋊C₈	—	83	2	no	—
117	C₄⋊C₈	—	166	2	no	—
118	C₄⋊C₈	—	166	2	no	—
119	C₄⋊C₈	C₂×Dic₈₃	1	4	yes	—
120	C₂²⋊C₈	—	2	2	no	—
121	C₂²⋊C₈	—	166	1	no	—
122	#97	—	83	2	no	—
123	C₄×C₈	—	2	2	no	—
124	C₄×C₈	—	166	1	no	—
125	C₄×C₈	—	166	1	no	—
126	C₄×C₈	D₃₃₂	1	1	yes	characteristic
127	C₄×C₈	D₃₃₂	1	1	yes	characteristic
128	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
129	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
130	C₂²×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
131	C₂²×C₄	#13050589	1	1	yes	characteristic
132	C₂²×C₄	#13050591	1	1	yes	characteristic
133	M₁₆	—	664	2	no	—
134	M₁₆	—	166	4	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	2	4	no	—
137	C₂×C₈	—	4	1	no	—
138	C₂×C₈	—	332	2	no	—
139	C₂×C₈	—	166	2	no	—
140	C₂×C₈	—	166	2	no	—
141	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
142	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
143	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
144	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
145	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
146	C₄²	D₄×D₈₃	1	1	yes	characteristic
147	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
148	C₄²	Q₈×D₈₃	1	1	yes	characteristic
149	C₂³	#14581783	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
155	C₂×C₄	D₄×Dic₈₃	1	1	yes	characteristic
156	C₂×C₄	Q₈×Dic₈₃	1	1	yes	characteristic
157	C₂×C₄	#14581767	1	1	yes	characteristic
158	C₂×C₄	#14581772	1	1	yes	characteristic
159	C₂×C₄	#14581791	1	1	yes	characteristic
160	C₈	—	2	1	no	—
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	—	4	4	no	—
164	C₈	—	332	2	no	—
165	C₈	—	664	2	no	—
166	C₈	—	166	4	no	—
167	C₈	#14581728	1	2	yes	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	#19127600	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	C₂×#14581728	1	1	yes	characteristic
176	C₄	#19127750	1	1	yes	characteristic
177	C₄	#19127818	1	2	yes	—
178	C₄	#19127965	1	2	yes	—
179	C₂	—	4	1	no	—
180	C₂	#24908212	1	1	yes	characteristic
181	C₂	#24908239	1	1	yes	characteristic
182	C₂	#24908244	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24906252

C₂

yes

maximal

#24906219

C₂

yes

maximal, characteristic

C₈×#13050574

C₂

yes

maximal, characteristic

#24905354

C₂

yes

maximal

C₈₃×#1564

C₂

yes

maximal, characteristic

#19127754

C₂²

yes

characteristic

#19127725

C₂²

yes

—

C₄×#13050574

C₂²

yes

—

—

—

—

—

C₈×C₈₃⋊C₈

—

#431×C₈₃

C₂²

yes

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

#335×C₈₃

C₄

yes

—

#334×C₈₃

C₄

yes

—

C₂×#13050574

—

C₂×#13050574

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

D₄

yes

—

C₂²⋊(C₈₃⋊C₈)

—

#14581703

Q₈

yes

—

C₄×C₈₃⋊C₈

—

C₄×C₈₃⋊C₈

—

C₂²×C₈×C₈₃

Q₈

yes

characteristic

C₂²×C₈×C₈₃

Q₈

yes

characteristic

C₂×C₄²×C₈₃

C₂³

yes

characteristic

C₄⋊C₈×C₈₃

C₂×C₄

yes

—

C₂²⋊C₈×C₈₃

—

C₄×C₈×C₈₃

—

C₄×C₈×C₈₃

C₂×C₄

yes

characteristic

C₄×C₈×C₈₃

C₂×C₄

yes

characteristic

C₄×C₈×C₈₃

D₄

yes

characteristic

C₄×C₈×C₈₃

D₄

yes

characteristic

#13050574

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂²×C₄×C₈₃

C₂×Q₈

yes

C₂²×C₄×C₈₃

yes

C₂²×C₄×C₈₃

yes

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₈₃

C₄⋊C₄

yes

characteristic

C₄²×C₈₃

C₂²×C₄

yes

characteristic

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₄²×C₈₃

C₂×Q₈

yes

characteristic

C₂³×C₈₃

#125

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄×D₄

yes

characteristic

C₂×C₄×C₈₃

C₄×Q₈

yes

characteristic

C₂×C₄×C₈₃

#121

yes

characteristic

C₂×C₄×C₈₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₈₃

C₄⋊Q₈

yes

characteristic

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

#108

yes

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#389

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

—

C₄×C₈₃

#471

yes

—

#9633

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

—

D₈₃

yes

characteristic

C₈×M₁₆

—

#1299

—

C₈₃

#9633

yes

characteristic, complemented, Sylow

#431

—

#431

C₂×D₈₃

yes

—

100

#423

—

101

C₄×M₁₆

—

102

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

103

#335

—

166

—

104

#335

Dic₈₃

yes

—

105

#334

—

166

—

106

#334

—

166

—

107

#334

—

166

—

108

#334

Dic₈₃

yes

—

109

#322

—

166

—

110

C₈²

—

166

—

111

C₂×M₁₆

—

166

—

112

C₂×M₁₆

—

166

—

113

C₂²×C₈

Dic₁₆₆

yes

characteristic

114

C₂²×C₈

Dic₁₆₆

yes

characteristic

115

C₂×C₄²

C₂²×D₈₃

yes

characteristic

116

C₄⋊C₈

—

117

C₄⋊C₈

—

166

—

118

C₄⋊C₈

—

166

—

119

C₄⋊C₈

C₂×Dic₈₃

yes

—

120

C₂²⋊C₈

—

121

C₂²⋊C₈

—

166

—

122

#97

—

123

C₄×C₈

—

124

C₄×C₈

—

166

—

125

C₄×C₈

—

166

—

126

C₄×C₈

D₃₃₂

yes

characteristic

127

C₄×C₈

D₃₃₂

yes

characteristic

128

C₄×C₈

C₂×Dic₈₃

yes

characteristic

129

C₄×C₈

C₂×Dic₈₃

yes

characteristic

130

C₂²×C₄

C₂×Dic₁₆₆

yes

characteristic

131

C₂²×C₄

#13050589

yes

characteristic

132

C₂²×C₄

#13050591

yes

characteristic

133

M₁₆

—

664

—

134

M₁₆

—

166

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

332

—

139

C₂×C₈

—

166

—

140

C₂×C₈

—

166

—

141

C₂×C₈

C₄⋊Dic₈₃

yes

characteristic

142

C₂×C₈

C₄⋊Dic₈₃

yes

characteristic

143

C₂×C₈

C₄⋊Dic₈₃

yes

characteristic

144

C₂×C₈

C₄⋊Dic₈₃

yes

characteristic

145

C₄²

C₂²×Dic₈₃

yes

characteristic

146

C₄²

D₄×D₈₃

yes

characteristic

147

C₄²

C₂×D₃₃₂

yes

characteristic

148

C₄²

Q₈×D₈₃

yes

characteristic

149

C₂³

#14581783

yes

characteristic

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

C₂×C₄⋊Dic₈₃

yes

characteristic

155

C₂×C₄

D₄×Dic₈₃

yes

characteristic

156

C₂×C₄

Q₈×Dic₈₃

yes

characteristic

157

C₂×C₄

#14581767

yes

characteristic

158

C₂×C₄

#14581772

yes

characteristic

159

C₂×C₄

#14581791

yes

characteristic

160

C₈

—

161

C₈

—

162

C₈

—

163

C₈

—

164

C₈

—

332

—

165

C₈

—

664

—

166

C₈

—

166

—

167

C₈

#14581728

yes

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

#19127600

yes

characteristic

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

C₂×#14581728

yes

characteristic

176

C₄

#19127750

yes

characteristic

177

C₄

#19127818

yes

—

178

C₄

#19127965

yes

—

179

C₂

—

180

C₂

#24908212

yes

characteristic

181

C₂

#24908239

yes

characteristic

182

C₂

#24908244

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14581767
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1564	C₂
socle	C₂²×C₈₃	#389
prime radical	C₂³×C₈₃	#125
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1564	D₈₃
hypocenter	C₈₃	#9633
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

subgroup

G/H

center

C₂×C₄

#14581767

commutator

C₄²×C₈₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₈₃×#1564

C₂

C₂²×C₈₃

C₂³×C₈₃

C₂×C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

N	H ≅ G/N	type
97. C₈₃	#9633	semidirect

H ≅ G/N

type

97. C₈₃

#9633

semidirect

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

subgroup	index
#24906252	2
#24906219	2
C₈×#13050574	2
#24905354	2
C₈₃×#1564	2
#9633	83

subgroup

index

C₈₃×#1564

Subgroups — #35303158

Conjugacy classes of subgroups

Complements

Maximal subgroups