Group #35320588 — subgroups

Subgroups — #35320588

Order 21248.

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

normal subgroups only

This group has 374 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	C₂²×#14581731	C₂	1	1	yes	maximal, characteristic
2	C₂×#19127530	C₂	1	2	yes	maximal, complemented
3	#24907498	C₂	1	2	yes	maximal
4	#24905557	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₃×#1381	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₈₃	C₂²	1	1	yes	characteristic
7	C₂×#14581731	—	2	4	no	—
8	C₂×#14581731	C₄	1	4	yes	—
9	C₂²×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×#14581728	C₂²	1	2	yes	—
11	#19127664	—	2	2	no	—
12	#19127530	—	2	4	no	—
13	#19127465	—	2	2	no	—
14	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
15	C₂×#108×C₈₃	C₂²	1	2	yes	—
16	C₂²×C₄×D₈₃	D₄	1	1	yes	characteristic
17	C₂×#13050574	—	2	2	no	—
18	C₂²×C₈₃⋊C₈	D₄	1	1	yes	characteristic
19	C₂×C₈×D₈₃	—	2	2	no	—
20	C₂×C₈×D₈₃	—	2	2	no	—
21	C₂×C₈×D₈₃	C₂×C₄	1	2	yes	—
22	C₂×C₈×D₈₃	Q₈	1	2	yes	—
23	#14581731	—	2	8	no	—
24	C₂×C₈₃⋊C₁₆	C₂×C₄	1	2	yes	—
25	C₂×C₈₃⋊C₁₆	D₄	1	4	yes	—
26	#14581728	—	2	2	no	—
27	#14581728	—	2	2	no	—
28	(C₂×D₈₃)⋊C₈	—	4	4	no	—
29	C₂×M₁₆×C₈₃	—	2	2	no	—
30	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
31	C₂×C₁₆×C₈₃	—	2	2	no	—
32	C₂×C₁₆×C₈₃	C₂×C₄	1	2	yes	—
33	#108×C₈₃	—	2	2	no	—
34	#108×C₈₃	—	2	2	no	—
35	C₂³×D₈₃	Q₁₆	1	1	yes	characteristic
36	C₂²×Dic₈₃	Q₁₆	1	1	yes	characteristic
37	C₂×C₄×D₈₃	—	2	2	no	—
38	C₂×C₄×D₈₃	—	2	2	no	—
39	C₂×C₄×D₈₃	C₄⋊C₄	1	2	yes	—
40	C₂×C₄×D₈₃	D₈	1	2	yes	—
41	#13050574	—	4	2	no	—
42	#13050574	—	4	2	no	—
43	C₂×C₈₃⋊C₈	—	2	2	no	—
44	C₂×C₈₃⋊C₈	—	4	2	no	—
45	C₂×C₈₃⋊C₈	C₄⋊C₄	1	2	yes	—
46	C₈×D₈₃	—	2	2	no	—
47	C₈×D₈₃	—	2	2	no	—
48	C₈×D₈₃	—	2	4	no	—
49	C₈₃⋊C₁₆	C₂²⋊C₄	1	4	yes	—
50	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
51	M₁₆×C₈₃	—	4	2	no	—
52	M₁₆×C₈₃	—	4	2	no	—
53	C₂×C₈×C₈₃	—	4	2	no	—
54	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₈₃	#41	1	2	yes	—
58	C₁₆×C₈₃	—	2	2	no	—
59	C₂³×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
60	C₂²×D₈₃	—	2	2	no	—
61	C₂²×D₈₃	—	4	2	no	—
62	C₂²×D₈₃	#107	1	2	yes	—
63	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₈₃	#121	1	2	yes	—
65	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₈₃	#135	1	2	yes	—
67	C₂×Dic₈₃	—	2	2	no	—
68	C₂×Dic₈₃	#107	1	2	yes	—
69	C₄×D₈₃	—	2	2	no	—
70	C₄×D₈₃	—	2	2	no	—
71	C₄×D₈₃	—	2	4	no	—
72	C₈×C₈₃	—	8	2	no	—
73	C₈×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₈×C₈₃	#117	1	1	yes	characteristic
75	C₈×C₈₃	C₂²⋊Q₈	1	2	yes	—
76	C₈₃⋊C₈	—	2	2	no	—
77	C₈₃⋊C₈	—	8	2	no	—
78	C₂²×C₈₃	—	2	2	no	—
79	C₂²×C₈₃	C₂×#107	1	1	yes	characteristic
80	C₂²×C₈₃	#467	1	2	yes	—
81	C₂×D₈₃	—	2	2	no	—
82	C₂×D₈₃	—	2	4	no	—
83	C₂×D₈₃	—	4	4	no	—
84	C₄×C₈₃	#380	1	1	yes	characteristic
85	C₄×C₈₃	#427	1	1	yes	characteristic
86	C₄×C₈₃	#466	1	2	yes	—
87	Dic₈₃	—	2	2	no	—
88	#12838	—	83	1	no	maximal, Sylow
89	C₂×C₈₃	—	2	2	no	—
90	C₂×C₈₃	#1808	1	1	yes	characteristic
91	C₂×C₈₃	#2161	1	2	yes	—
92	D₈₃	—	4	2	no	—
93	C₂²×M₃₂	—	83	1	no	—
94	C₂×#368	—	83	2	no	—
95	#1810	—	83	2	no	—
96	#1381	—	83	1	no	—
97	#1381	D₈₃	1	1	yes	characteristic, complemented
98	C₈₃	#12838	1	1	yes	characteristic, complemented, Sylow
99	C₂³×C₈	—	83	1	no	—
100	C₂×M₃₂	—	83	4	no	—
101	C₂×M₃₂	—	166	4	no	—
102	C₂²×C₁₆	—	83	1	no	—
103	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
104	C₂×#108	—	83	2	no	—
105	C₂×#108	C₂×D₈₃	1	2	yes	—
106	#407	—	166	2	no	—
107	#368	—	166	4	no	—
108	#341	—	166	2	no	—
109	C₂³×C₄	—	83	1	no	—
110	C₂×M₁₆	—	2	2	no	—
111	C₂×M₁₆	—	166	2	no	—
112	C₂²×C₈	—	83	1	no	—
113	C₂²×C₈	—	83	2	no	—
114	C₂²×C₈	—	83	2	no	—
115	C₂²×C₈	—	166	2	no	—
116	C₂²×C₈	—	166	2	no	—
117	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
118	M₃₂	—	166	8	no	—
119	C₂×C₁₆	—	2	2	no	—
120	C₂×C₁₆	—	83	2	no	—
121	C₂×C₁₆	—	83	4	no	—
122	C₂×C₁₆	C₄×D₈₃	1	2	yes	—
123	#108	—	2	2	no	—
124	#108	—	2	2	no	—
125	#108	—	166	2	no	—
126	#108	—	166	2	no	—
127	C₂²⋊C₈	—	332	4	no	—
128	C₂⁴	—	83	1	no	—
129	C₂²×C₄	—	83	1	no	—
130	C₂²×C₄	—	83	2	no	—
131	C₂²×C₄	—	83	2	no	—
132	C₂²×C₄	—	166	2	no	—
133	C₂²×C₄	—	166	2	no	—
134	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
135	M₁₆	—	4	2	no	—
136	M₁₆	—	4	2	no	—
137	M₁₆	—	332	2	no	—
138	M₁₆	—	332	2	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	332	2	no	—
141	C₂×C₈	—	83	2	no	—
142	C₂×C₈	—	166	2	no	—
143	C₂×C₈	—	166	4	no	—
144	C₂×C₈	—	166	2	no	—
145	C₂×C₈	—	166	2	no	—
146	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
147	C₂×C₈	D₄×D₈₃	1	2	yes	—
148	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
149	C₂×C₈	#13050590	1	2	yes	—
150	C₁₆	—	2	2	no	—
151	C₁₆	—	83	4	no	—
152	C₂³	—	332	2	no	—
153	C₂³	—	83	2	no	—
154	C₂³	—	166	2	no	—
155	C₂³	Q₁₆×D₈₃	1	1	yes	characteristic
156	C₂×C₄	—	83	2	no	—
157	C₂×C₄	—	166	2	no	—
158	C₂×C₄	—	166	2	no	—
159	C₂×C₄	—	166	4	no	—
160	C₂×C₄	—	166	2	no	—
161	C₂×C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
162	C₂×C₄	#14581765	1	2	yes	—
163	C₂×C₄	D₈×D₈₃	1	1	yes	characteristic
164	C₂×C₄	#14581813	1	2	yes	—
165	C₈	—	8	2	no	—
166	C₈	—	664	2	no	—
167	C₈	—	166	2	no	—
168	C₈	C₂²⋊C₄×D₈₃	1	1	yes	characteristic
169	C₈	#14581749	1	1	yes	characteristic
170	C₈	#14581770	1	2	yes	—
171	C₂²	—	2	2	no	—
172	C₂²	—	332	4	no	—
173	C₂²	—	166	4	no	—
174	C₂²	—	166	2	no	—
175	C₂²	#107×D₈₃	1	1	yes	characteristic
176	C₂²	#19127941	1	2	yes	—
177	C₄	—	166	2	no	—
178	C₄	#19127560	1	1	yes	characteristic
179	C₄	#19127736	1	1	yes	characteristic
180	C₄	#19127936	1	2	yes	—
181	C₂	—	2	2	no	—
182	C₂	—	332	2	no	—
183	C₂	#24907491	1	1	yes	characteristic
184	C₂	#24909256	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#14581731

C₂

yes

maximal, characteristic

C₂×#19127530

C₂

yes

maximal, complemented

#24907498

C₂

yes

maximal

#24905557

C₂

yes

maximal, characteristic, complemented

C₈₃×#1381

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×D₈₃

C₂²

yes

characteristic

C₂×#14581731

—

C₂×#14581731

C₄

yes

—

C₂²×C₈₃⋊C₁₆

C₂²

yes

characteristic

C₂×#14581728

C₂²

yes

—

#19127664

—

#19127530

—

#19127465

—

C₂²×C₁₆×C₈₃

C₂²

yes

characteristic

C₂×#108×C₈₃

C₂²

yes

—

C₂²×C₄×D₈₃

D₄

yes

characteristic

C₂×#13050574

—

C₂²×C₈₃⋊C₈

D₄

yes

characteristic

C₂×C₈×D₈₃

—

C₂×C₈×D₈₃

—

C₂×C₈×D₈₃

C₂×C₄

yes

—

C₂×C₈×D₈₃

Q₈

yes

—

#14581731

—

C₂×C₈₃⋊C₁₆

C₂×C₄

yes

—

C₂×C₈₃⋊C₁₆

D₄

yes

—

#14581728

—

#14581728

—

(C₂×D₈₃)⋊C₈

—

C₂×M₁₆×C₈₃

—

C₂²×C₈×C₈₃

C₂³

yes

characteristic

C₂×C₁₆×C₈₃

—

C₂×C₁₆×C₈₃

C₂×C₄

yes

—

#108×C₈₃

—

#108×C₈₃

—

C₂³×D₈₃

Q₁₆

yes

characteristic

C₂²×Dic₈₃

Q₁₆

yes

characteristic

C₂×C₄×D₈₃

—

C₂×C₄×D₈₃

—

C₂×C₄×D₈₃

C₄⋊C₄

yes

—

C₂×C₄×D₈₃

D₈

yes

—

#13050574

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

C₄⋊C₄

yes

—

C₈×D₈₃

—

C₈×D₈₃

—

C₈×D₈₃

—

C₈₃⋊C₁₆

C₂²⋊C₄

yes

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

M₁₆×C₈₃

—

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₃

C₂×D₄

yes

—

C₂×C₈×C₈₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₃

#41

yes

—

C₁₆×C₈₃

—

C₂³×C₈₃

C₂×Q₁₆

yes

characteristic

C₂²×D₈₃

—

C₂²×D₈₃

—

C₂²×D₈₃

#107

yes

—

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

#121

yes

—

C₂×C₄×C₈₃

C₂×D₈

yes

characteristic

C₂×C₄×C₈₃

#135

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

#107

yes

—

C₄×D₈₃

—

C₄×D₈₃

—

C₄×D₈₃

—

C₈×C₈₃

—

C₈×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₈₃

#117

yes

characteristic

C₈×C₈₃

C₂²⋊Q₈

yes

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂×#107

yes

characteristic

C₂²×C₈₃

#467

yes

—

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

—

Dic₈₃

—

#12838

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

#1808

yes

characteristic

C₂×C₈₃

#2161

yes

—

D₈₃

—

C₂²×M₃₂

—

—

—

—

D₈₃

yes

characteristic, complemented

C₈₃

#12838

yes

characteristic, complemented, Sylow

C₂³×C₈

—

100

C₂×M₃₂

—

101

C₂×M₃₂

—

166

—

102

C₂²×C₁₆

—

103

C₂²×C₁₆

C₂×D₈₃

yes

characteristic

104

C₂×#108

—

105

C₂×#108

C₂×D₈₃

yes

—

106

#407

—

166

—

107

#368

—

166

—

108

#341

—

166

—

109

C₂³×C₄

—

110

C₂×M₁₆

—

111

C₂×M₁₆

—

166

—

112

C₂²×C₈

—

113

C₂²×C₈

—

114

C₂²×C₈

—

115

C₂²×C₈

—

166

—

116

C₂²×C₈

—

166

—

117

C₂²×C₈

C₂²×D₈₃

yes

characteristic

118

M₃₂

—

166

—

119

C₂×C₁₆

—

120

C₂×C₁₆

—

121

C₂×C₁₆

—

122

C₂×C₁₆

C₄×D₈₃

yes

—

123

#108

—

124

#108

—

125

#108

—

166

—

126

#108

—

166

—

127

C₂²⋊C₈

—

332

—

128

C₂⁴

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

166

—

133

C₂²×C₄

—

166

—

134

C₂²×C₄

D₄×D₈₃

yes

characteristic

135

M₁₆

—

136

M₁₆

—

137

M₁₆

—

332

—

138

M₁₆

—

332

—

139

C₂×C₈

—

140

C₂×C₈

—

332

—

141

C₂×C₈

—

142

C₂×C₈

—

166

—

143

C₂×C₈

—

166

—

144

C₂×C₈

—

166

—

145

C₂×C₈

—

166

—

146

C₂×C₈

C₂×C₄×D₈₃

yes

characteristic

147

C₂×C₈

D₄×D₈₃

yes

—

148

C₂×C₈

Q₈×D₈₃

yes

characteristic

149

C₂×C₈

#13050590

yes

—

150

C₁₆

—

151

C₁₆

—

152

C₂³

—

332

—

153

C₂³

—

154

C₂³

—

166

—

155

C₂³

Q₁₆×D₈₃

yes

characteristic

156

C₂×C₄

—

157

C₂×C₄

—

166

—

158

C₂×C₄

—

166

—

159

C₂×C₄

—

166

—

160

C₂×C₄

—

166

—

161

C₂×C₄

C₄⋊C₄×D₈₃

yes

characteristic

162

C₂×C₄

#14581765

yes

—

163

C₂×C₄

D₈×D₈₃

yes

characteristic

164

C₂×C₄

#14581813

yes

—

165

C₈

—

166

C₈

—

664

—

167

C₈

—

166

—

168

C₈

C₂²⋊C₄×D₈₃

yes

characteristic

169

C₈

#14581749

yes

characteristic

170

C₈

#14581770

yes

—

171

C₂²

—

172

C₂²

—

332

—

173

C₂²

—

166

—

174

C₂²

—

166

—

175

C₂²

#107×D₈₃

yes

characteristic

176

C₂²

#19127941

yes

—

177

C₄

—

166

—

178

C₄

#19127560

yes

characteristic

179

C₄

#19127736

yes

characteristic

180

C₄

#19127936

yes

—

181

C₂

—

182

C₂

—

332

—

183

C₂

#24907491

yes

characteristic

184

C₂

#24909256

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₈₃
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#1381	C₂
socle	C₂²×C₈₃	C₂×#107
prime radical	C₂³×D₈₃	Q₁₆
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	#1381	D₈₃
hypocenter	C₈₃	#12838
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

subgroup

G/H

center

C₂×C₄

D₈×D₈₃

commutator

C₂×C₈×C₈₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₈₃

Fitting

C₈₃×#1381

C₂

socle

C₂²×C₈₃

C₂×#107

prime radical

C₂³×D₈₃

Q₁₆

cosocle

C₂²×C₈×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
2. C₂×#19127530	C₂	semidirect
4. #24905557	C₂	semidirect
5. C₈₃×#1381	C₂	semidirect
97. #1381	D₈₃	semidirect
98. C₈₃	#12838	semidirect

H ≅ G/N

type

2. C₂×#19127530

C₂

semidirect

4. #24905557

C₂

semidirect

5. C₈₃×#1381

C₂

semidirect

97. #1381

D₈₃

semidirect

98. C₈₃

#12838

semidirect

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

subgroup	index
C₂²×#14581731	2
C₂×#19127530	2
#24907498	2
#24905557	2
C₈₃×#1381	2
#12838	83

subgroup

index

C₈₃×#1381

Subgroups — #35320588

Conjugacy classes of subgroups

Complements

Maximal subgroups