Group #35320618 — subgroups

Subgroups — #35320618

Order 21248.

454 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 171 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	C₂×C₂²⋊(C₈₃⋊C₁₆)	C₂	1	2	yes	maximal
2	#24905541	C₂	1	4	yes	maximal, complemented
3	C₂×#405×C₈₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₈₃⋊C₁₆	C₂²	1	4	yes	—
5	C₂²⋊(C₈₃⋊C₁₆)	—	2	4	no	—
6	C₂²×M₁₆×C₈₃	C₄	1	1	yes	characteristic
7	C₂³×C₈×C₈₃	C₂²	1	1	yes	characteristic
8	C₂×#117×C₈₃	C₄	1	1	yes	characteristic
9	#405×C₈₃	—	2	4	no	—
10	C₂×#97×C₈₃	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₈₃	C₂²	1	2	yes	—
12	C₂×C₈₃⋊C₁₆	—	2	4	no	—
13	C₂×C₈₃⋊C₁₆	—	2	8	no	—
14	C₂³×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
15	C₂×M₁₆×C₈₃	—	2	2	no	—
16	C₂×M₁₆×C₈₃	—	2	4	no	—
17	C₂²×C₈×C₈₃	—	2	4	no	—
18	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₈₃	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₈₃	D₄	1	4	yes	—
21	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
22	#117×C₈₃	—	2	2	no	—
23	#117×C₈₃	—	2	2	no	—
24	C₂×C₄⋊C₄×C₈₃	C₈	1	2	yes	—
25	C₂×C₂²⋊C₄×C₈₃	C₈	1	2	yes	—
26	C₂×C₄²×C₈₃	C₂×C₄	1	2	yes	—
27	#97×C₈₃	—	2	4	no	—
28	C₄×C₈×C₈₃	—	2	4	no	—
29	C₈₃⋊C₁₆	—	4	4	no	—
30	C₂⁴×C₈₃	C₂×C₈	1	1	yes	characteristic
31	C₂²×C₄×C₈₃	—	2	2	no	—
32	C₂²×C₄×C₈₃	—	2	2	no	—
33	C₂²×C₄×C₈₃	C₂²⋊C₄	1	2	yes	—
34	C₂²×C₄×C₈₃	C₂×C₈	1	4	yes	—
35	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₄×C₈₃	M₁₆	1	2	yes	—
37	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
38	M₁₆×C₈₃	—	2	8	no	—
39	C₂×C₈×C₈₃	—	2	2	no	—
40	C₂×C₈×C₈₃	—	2	2	no	—
41	C₂×C₈×C₈₃	—	2	2	no	—
42	C₂×C₈×C₈₃	—	2	8	no	—
43	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
44	C₂×C₈×C₈₃	M₁₆	1	4	yes	—
45	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
46	C₂×C₈×C₈₃	#41	1	4	yes	—
47	C₄⋊C₄×C₈₃	—	2	4	no	—
48	C₂²⋊C₄×C₈₃	—	2	4	no	—
49	C₄²×C₈₃	—	2	2	no	—
50	C₄²×C₈₃	—	2	2	no	—
51	C₂³×C₈₃	—	2	2	no	—
52	C₂³×C₈₃	—	2	4	no	—
53	C₂³×C₈₃	C₂²⋊C₈	1	2	yes	—
54	C₂³×C₈₃	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄×C₈₃	—	2	2	no	—
56	C₂×C₄×C₈₃	—	2	2	no	—
57	C₂×C₄×C₈₃	—	2	2	no	—
58	C₂×C₄×C₈₃	—	2	4	no	—
59	C₂×C₄×C₈₃	—	2	4	no	—
60	C₂×C₄×C₈₃	—	2	8	no	—
61	C₂×C₄×C₈₃	C₂²⋊C₈	1	2	yes	—
62	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₈₃	#117	1	2	yes	—
64	C₂×C₄×C₈₃	C₂×M₁₆	1	2	yes	—
65	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
66	C₈×C₈₃	—	2	2	no	—
67	C₈×C₈₃	—	2	4	no	—
68	C₈×C₈₃	#123	1	4	yes	—
69	C₂²×C₈₃	—	2	2	no	—
70	C₂²×C₈₃	—	2	4	no	—
71	C₂²×C₈₃	—	4	4	no	—
72	C₂²×C₈₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
73	C₂²×C₈₃	#431	1	2	yes	—
74	C₂²×C₈₃	#504	1	4	yes	—
75	C₄×C₈₃	—	2	2	no	—
76	C₄×C₈₃	—	4	4	no	—
77	C₄×C₈₃	#381	1	1	yes	characteristic
78	C₄×C₈₃	#407	1	1	yes	characteristic
79	C₄×C₈₃	#432	1	2	yes	—
80	#12845	—	83	1	no	maximal, Sylow
81	C₂×C₈₃	—	4	2	no	—
82	C₂×C₈₃	#1817	1	1	yes	characteristic
83	C₂×C₈₃	#2113	1	2	yes	—
84	C₂×C₈₃	#2164	1	4	yes	—
85	C₂×#405	D₈₃	1	1	yes	characteristic
86	C₂×C₂²⋊C₁₆	—	83	2	no	—
87	#1374	—	83	4	no	—
88	C₈₃	#12845	1	1	yes	characteristic, complemented, Sylow
89	C₂²×M₁₆	Dic₈₃	1	1	yes	characteristic
90	C₂³×C₈	C₂×D₈₃	1	1	yes	characteristic
91	C₂×#117	Dic₈₃	1	1	yes	characteristic
92	C₂²×C₁₆	—	83	4	no	—
93	#405	—	2	4	no	—
94	C₂×#97	Dic₈₃	1	2	yes	—
95	C₂×C₄×C₈	C₂×D₈₃	1	2	yes	—
96	C₂²⋊C₁₆	—	166	4	no	—
97	C₂³×C₄	C₂×Dic₈₃	1	1	yes	characteristic
98	C₂×M₁₆	—	2	2	no	—
99	C₂×M₁₆	—	2	4	no	—
100	C₂²×C₈	—	2	4	no	—
101	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
102	C₂²×C₈	C₂×Dic₈₃	1	2	yes	—
103	C₂²×C₈	C₂²⋊D₈₃	1	4	yes	—
104	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
105	#117	—	2	2	no	—
106	#117	—	2	2	no	—
107	C₂×C₄⋊C₄	C₈₃⋊C₈	1	2	yes	—
108	C₂×C₂²⋊C₄	C₈₃⋊C₈	1	2	yes	—
109	C₂×C₄²	C₂×Dic₈₃	1	2	yes	—
110	C₂×C₁₆	—	166	4	no	—
111	C₂×C₁₆	—	166	8	no	—
112	#97	—	2	4	no	—
113	C₄×C₈	—	2	4	no	—
114	C₂⁴	C₂×C₈₃⋊C₈	1	1	yes	characteristic
115	C₂²×C₄	—	2	2	no	—
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	C₂²⋊Dic₈₃	1	2	yes	—
118	C₂²×C₄	C₂×C₈₃⋊C₈	1	4	yes	—
119	C₂²×C₄	C₂×C₈₃⋊C₈	1	1	yes	characteristic
120	C₂²×C₄	#13050574	1	2	yes	—
121	C₂²×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
122	M₁₆	—	2	8	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	8	no	—
127	C₂×C₈	C₂²⋊Dic₈₃	1	2	yes	—
128	C₂×C₈	#13050574	1	4	yes	—
129	C₂×C₈	C₂×C₂²⋊D₈₃	1	2	yes	—
130	C₂×C₈	#13050590	1	4	yes	—
131	C₄⋊C₄	—	2	4	no	—
132	C₂²⋊C₄	—	2	4	no	—
133	C₄²	—	2	2	no	—
134	C₄²	—	2	2	no	—
135	C₁₆	—	332	4	no	—
136	C₂³	—	2	4	no	—
137	C₂³	—	2	2	no	—
138	C₂³	C₂²⋊(C₈₃⋊C₈)	1	2	yes	—
139	C₂³	C₂²×C₈₃⋊C₈	1	1	yes	characteristic
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	8	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	C₂²⋊(C₈₃⋊C₈)	1	2	yes	—
147	C₂×C₄	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
148	C₂×C₄	#14581751	1	2	yes	—
149	C₂×C₄	C₂×#13050574	1	2	yes	—
150	C₂×C₄	C₂×#13050574	1	1	yes	characteristic
151	C₈	—	2	2	no	—
152	C₈	—	2	4	no	—
153	C₈	#14581775	1	4	yes	—
154	C₂²	—	2	2	no	—
155	C₂²	—	2	4	no	—
156	C₂²	—	4	4	no	—
157	C₂²	C₂×C₂²⋊(C₈₃⋊C₈)	1	1	yes	characteristic
158	C₂²	#19127754	1	2	yes	—
159	C₂²	#19128118	1	4	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	4	4	no	—
162	C₄	#19127564	1	1	yes	characteristic
163	C₄	#19127665	1	1	yes	characteristic
164	C₄	#19127758	1	2	yes	—
165	C₂	—	4	2	no	—
166	C₂	#24907531	1	1	yes	characteristic
167	C₂	#24909077	1	2	yes	—
168	C₂	#24909270	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂²⋊(C₈₃⋊C₁₆)

C₂

yes

maximal

#24905541

C₂

yes

maximal, complemented

C₂×#405×C₈₃

C₂

yes

maximal, characteristic

C₂²×C₈₃⋊C₁₆

C₂²

yes

—

C₂²⋊(C₈₃⋊C₁₆)

—

C₂²×M₁₆×C₈₃

C₄

yes

characteristic

C₂³×C₈×C₈₃

C₂²

yes

characteristic

C₂×#117×C₈₃

C₄

yes

characteristic

#405×C₈₃

—

C₂×#97×C₈₃

C₄

yes

—

C₂×C₄×C₈×C₈₃

C₂²

yes

—

C₂×C₈₃⋊C₁₆

—

C₂×C₈₃⋊C₁₆

—

C₂³×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂×M₁₆×C₈₃

—

C₂×M₁₆×C₈₃

—

C₂²×C₈×C₈₃

—

C₂²×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₈₃

C₂×C₄

yes

—

C₂²×C₈×C₈₃

D₄

yes

—

C₂²×C₈×C₈₃

C₂³

yes

characteristic

#117×C₈₃

—

#117×C₈₃

—

C₂×C₄⋊C₄×C₈₃

C₈

yes

—

C₂×C₂²⋊C₄×C₈₃

C₈

yes

—

C₂×C₄²×C₈₃

C₂×C₄

yes

—

#97×C₈₃

—

C₄×C₈×C₈₃

—

C₈₃⋊C₁₆

—

C₂⁴×C₈₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₈₃

C₂×C₈

yes

—

C₂²×C₄×C₈₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₈₃

M₁₆

yes

—

C₂²×C₄×C₈₃

C₂²×C₄

yes

characteristic

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂²⋊C₄

yes

—

C₂×C₈×C₈₃

M₁₆

yes

—

C₂×C₈×C₈₃

C₂×D₄

yes

—

C₂×C₈×C₈₃

#41

yes

—

C₄⋊C₄×C₈₃

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

C₂²⋊C₈

yes

—

C₂³×C₈₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²⋊C₈

yes

—

C₂×C₄×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

#117

yes

—

C₂×C₄×C₈₃

C₂×M₁₆

yes

—

C₂×C₄×C₈₃

C₂×M₁₆

yes

characteristic

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

#123

yes

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₈₃

#431

yes

—

C₂²×C₈₃

#504

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

—

#12845

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

#1817

yes

characteristic

C₂×C₈₃

#2113

yes

—

C₂×C₈₃

#2164

yes

—

C₂×#405

D₈₃

yes

characteristic

C₂×C₂²⋊C₁₆

—

#1374

—

C₈₃

#12845

yes

characteristic, complemented, Sylow

C₂²×M₁₆

Dic₈₃

yes

characteristic

C₂³×C₈

C₂×D₈₃

yes

characteristic

C₂×#117

Dic₈₃

yes

characteristic

C₂²×C₁₆

—

#405

—

C₂×#97

Dic₈₃

yes

—

C₂×C₄×C₈

C₂×D₈₃

yes

—

C₂²⋊C₁₆

—

166

—

C₂³×C₄

C₂×Dic₈₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

100

C₂²×C₈

—

101

C₂²×C₈

C₂×Dic₈₃

yes

characteristic

102

C₂²×C₈

C₂×Dic₈₃

yes

—

103

C₂²×C₈

C₂²⋊D₈₃

yes

—

104

C₂²×C₈

C₂²×D₈₃

yes

characteristic

105

#117

—

106

#117

—

107

C₂×C₄⋊C₄

C₈₃⋊C₈

yes

—

108

C₂×C₂²⋊C₄

C₈₃⋊C₈

yes

—

109

C₂×C₄²

C₂×Dic₈₃

yes

—

110

C₂×C₁₆

—

166

—

111

C₂×C₁₆

—

166

—

112

#97

—

113

C₄×C₈

—

114

C₂⁴

C₂×C₈₃⋊C₈

yes

characteristic

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

C₂²⋊Dic₈₃

yes

—

118

C₂²×C₄

C₂×C₈₃⋊C₈

yes

—

119

C₂²×C₄

C₂×C₈₃⋊C₈

yes

characteristic

120

C₂²×C₄

#13050574

yes

—

121

C₂²×C₄

C₂²×Dic₈₃

yes

characteristic

122

M₁₆

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

C₂²⋊Dic₈₃

yes

—

128

C₂×C₈

#13050574

yes

—

129

C₂×C₈

C₂×C₂²⋊D₈₃

yes

—

130

C₂×C₈

#13050590

yes

—

131

C₄⋊C₄

—

132

C₂²⋊C₄

—

133

C₄²

—

134

C₄²

—

135

C₁₆

—

332

—

136

C₂³

—

137

C₂³

—

138

C₂³

C₂²⋊(C₈₃⋊C₈)

yes

—

139

C₂³

C₂²×C₈₃⋊C₈

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

C₂²⋊(C₈₃⋊C₈)

yes

—

147

C₂×C₄

C₂×C₂²⋊Dic₈₃

yes

characteristic

148

C₂×C₄

#14581751

yes

—

149

C₂×C₄

C₂×#13050574

yes

—

150

C₂×C₄

C₂×#13050574

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

#14581775

yes

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

C₂×C₂²⋊(C₈₃⋊C₈)

yes

characteristic

158

C₂²

#19127754

yes

—

159

C₂²

#19128118

yes

—

160

C₄

—

161

C₄

—

162

C₄

#19127564

yes

characteristic

163

C₄

#19127665

yes

characteristic

164

C₄

#19127758

yes

—

165

C₂

—

166

C₂

#24907531

yes

characteristic

167

C₂

#24909077

yes

—

168

C₂

#24909270

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₈₃
commutator, socle	C₂³×C₈₃	C₂²×C₈
Fitting	C₂×#405×C₈₃	C₂
prime radical	C₂⁴×C₈₃	C₂×C₈
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₂×#405	D₈₃
hypocenter	C₈₃	#12845
LCS	G ▷ C₂³×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#405

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₈₃

commutator, socle

C₂³×C₈₃

C₂²×C₈

Fitting

C₂×#405×C₈₃

C₂

prime radical

C₂⁴×C₈₃

C₂×C₈

cosocle

C₂²×C₈×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂³×C₈₃ ▷ C₈₃

C₁ ◁ C₂²×C₈ ◁ C₂×#405

N	H ≅ G/N	type
2. #24905541	C₂	semidirect
88. C₈₃	#12845	semidirect

H ≅ G/N

type

2. #24905541

C₂

semidirect

88. C₈₃

#12845

semidirect

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

subgroup	index
C₂×C₂²⋊(C₈₃⋊C₁₆)	2
#24905541	2
C₂×#405×C₈₃	2
#12845	83

subgroup

index

C₂×C₂²⋊(C₈₃⋊C₁₆)

#24905541

C₂×#405×C₈₃

#12845

Subgroups — #35320618

Conjugacy classes of subgroups

Complements

Maximal subgroups