Group #35324608 — subgroups

Subgroups — #35324608

Order 21248.

290 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 143 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×#19127535	C₂	1	2	yes	maximal, complemented
2	#24905603	C₂	1	4	yes	maximal, complemented
3	C₂×#504×C₈₃	C₂	1	1	yes	maximal, characteristic
4	#19127535	—	2	2	no	—
5	#19127535	C₂²	1	4	yes	—
6	C₂×C₈₃⋊C₃₂	—	2	2	no	—
7	C₂×#131×C₈₃	C₄	1	1	yes	characteristic
8	#504×C₈₃	—	2	2	no	—
9	#504×C₈₃	—	2	2	no	—
10	C₂×M₃₂×C₈₃	C₄	1	1	yes	characteristic
11	C₂×M₃₂×C₈₃	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₈₃	C₄	1	2	yes	—
13	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
14	C₈₃⋊C₃₂	—	2	4	no	—
15	C₂×#41×C₈₃	C₈	1	1	yes	characteristic
16	#131×C₈₃	—	2	2	no	—
17	#131×C₈₃	—	2	2	no	—
18	C₂×M₁₆×C₈₃	C₈	1	1	yes	characteristic
19	C₂×M₁₆×C₈₃	C₈	1	2	yes	—
20	C₂²×C₈×C₈₃	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
22	M₃₂×C₈₃	—	2	2	no	—
23	M₃₂×C₈₃	—	2	4	no	—
24	C₂×C₁₆×C₈₃	—	2	2	no	—
25	C₂×C₁₆×C₈₃	—	2	2	no	—
26	C₂×C₁₆×C₈₃	C₂×C₄	1	2	yes	—
27	C₂×C₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₈₃	D₄	1	2	yes	—
29	C₂×C₁₆×C₈₃	D₄	1	2	yes	—
30	C₂×C₁₆×C₈₃	C₂³	1	1	yes	characteristic
31	#41×C₈₃	—	2	2	no	—
32	#41×C₈₃	—	2	2	no	—
33	C₂×Q₈×C₈₃	C₁₆	1	1	yes	characteristic
34	C₂×D₄×C₈₃	C₁₆	1	2	yes	—
35	C₂×D₄×C₈₃	C₁₆	1	1	yes	characteristic
36	C₂²×C₄×C₈₃	C₂×C₈	1	2	yes	—
37	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
38	M₁₆×C₈₃	—	2	2	no	—
39	M₁₆×C₈₃	—	2	4	no	—
40	C₂×C₈×C₈₃	—	2	2	no	—
41	C₂×C₈×C₈₃	—	2	2	no	—
42	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
43	C₂×C₈×C₈₃	C₂×C₈	1	2	yes	—
44	C₂×C₈×C₈₃	C₂×C₈	1	1	yes	characteristic
45	C₂×C₈×C₈₃	M₁₆	1	2	yes	—
46	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
47	C₁₆×C₈₃	—	2	2	no	—
48	C₁₆×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₁₆×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₁₆×C₈₃	C₂×D₄	1	2	yes	—
51	C₂³×C₈₃	C₂×C₁₆	1	2	yes	—
52	C₂³×C₈₃	C₂×C₁₆	1	1	yes	characteristic
53	Q₈×C₈₃	—	2	2	no	—
54	D₄×C₈₃	—	2	4	no	—
55	D₄×C₈₃	—	2	2	no	—
56	C₂×C₄×C₈₃	—	2	2	no	—
57	C₂×C₄×C₈₃	—	2	2	no	—
58	C₂×C₄×C₈₃	C₂²⋊C₈	1	2	yes	—
59	C₂×C₄×C₈₃	C₂×C₁₆	1	2	yes	—
60	C₂×C₄×C₈₃	C₂×C₁₆	1	1	yes	characteristic
61	C₂×C₄×C₈₃	M₃₂	1	2	yes	—
62	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
63	C₈×C₈₃	—	2	2	no	—
64	C₈×C₈₃	C₂²⋊C₈	1	2	yes	—
65	C₈×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₈×C₈₃	C₂×M₁₆	1	1	yes	characteristic
67	C₂²×C₈₃	—	2	2	no	—
68	C₂²×C₈₃	—	4	1	no	—
69	C₂²×C₈₃	—	4	2	no	—
70	C₂²×C₈₃	C₂²⋊C₁₆	1	2	yes	—
71	C₂²×C₈₃	C₂²×C₁₆	1	1	yes	characteristic
72	C₄×C₈₃	—	2	2	no	—
73	C₄×C₈₃	C₂²⋊C₁₆	1	2	yes	—
74	C₄×C₈₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
75	C₄×C₈₃	C₂×M₃₂	1	1	yes	characteristic
76	#13556	—	83	1	no	maximal, Sylow
77	C₂×C₈₃	—	2	1	no	—
78	C₂×C₈₃	—	2	2	no	—
79	C₂×C₈₃	—	4	2	no	—
80	C₂×C₈₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
81	C₂×#504	D₈₃	1	1	yes	characteristic
82	C₂×M₆₄	—	83	2	no	—
83	#1400	—	83	4	no	—
84	C₈₃	#13556	1	1	yes	characteristic, complemented, Sylow
85	C₂×#131	Dic₈₃	1	1	yes	characteristic
86	#504	—	2	2	no	—
87	#504	—	2	2	no	—
88	C₂×M₃₂	Dic₈₃	1	1	yes	characteristic
89	C₂×M₃₂	C₂×D₈₃	1	2	yes	—
90	C₂²×C₁₆	Dic₈₃	1	2	yes	—
91	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
92	M₆₄	—	83	4	no	—
93	M₆₄	—	166	2	no	—
94	C₂×C₃₂	—	166	2	no	—
95	C₂×#41	C₈₃⋊C₈	1	1	yes	characteristic
96	#131	—	2	2	no	—
97	#131	—	2	2	no	—
98	C₂×M₁₆	C₈₃⋊C₈	1	1	yes	characteristic
99	C₂×M₁₆	C₈₃⋊C₈	1	2	yes	—
100	C₂²×C₈	C₂×Dic₈₃	1	2	yes	—
101	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
102	M₃₂	—	2	2	no	—
103	M₃₂	—	2	4	no	—
104	C₂×C₁₆	—	2	2	no	—
105	C₂×C₁₆	—	2	2	no	—
106	C₂×C₁₆	C₂×Dic₈₃	1	2	yes	—
107	C₂×C₁₆	C₂×Dic₈₃	1	1	yes	characteristic
108	C₂×C₁₆	C₂²⋊D₈₃	1	2	yes	—
109	C₂×C₁₆	C₂²⋊D₈₃	1	2	yes	—
110	C₂×C₁₆	C₂²×D₈₃	1	1	yes	characteristic
111	C₃₂	—	166	4	no	—
112	#41	—	2	2	no	—
113	#41	—	2	2	no	—
114	C₂×Q₈	C₈₃⋊C₁₆	1	1	yes	characteristic
115	C₂×D₄	C₈₃⋊C₁₆	1	2	yes	—
116	C₂×D₄	C₈₃⋊C₁₆	1	1	yes	characteristic
117	C₂²×C₄	C₂×C₈₃⋊C₈	1	2	yes	—
118	C₂²×C₄	C₂×C₈₃⋊C₈	1	1	yes	characteristic
119	M₁₆	—	2	2	no	—
120	M₁₆	—	2	4	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	C₂²⋊Dic₈₃	1	2	yes	—
124	C₂×C₈	C₂×C₈₃⋊C₈	1	2	yes	—
125	C₂×C₈	C₂×C₈₃⋊C₈	1	1	yes	characteristic
126	C₂×C₈	#13050574	1	2	yes	—
127	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
128	C₁₆	—	2	2	no	—
129	C₁₆	C₂²⋊Dic₈₃	1	1	yes	characteristic
130	C₁₆	C₂²⋊Dic₈₃	1	1	yes	characteristic
131	C₁₆	C₂×C₂²⋊D₈₃	1	2	yes	—
132	C₂³	C₂×C₈₃⋊C₁₆	1	2	yes	—
133	C₂³	C₂×C₈₃⋊C₁₆	1	1	yes	characteristic
134	Q₈	—	2	2	no	—
135	D₄	—	2	4	no	—
136	D₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	C₂²⋊(C₈₃⋊C₈)	1	2	yes	—
140	C₂×C₄	C₂×C₈₃⋊C₁₆	1	2	yes	—
141	C₂×C₄	C₂×C₈₃⋊C₁₆	1	1	yes	characteristic
142	C₂×C₄	#14581732	1	2	yes	—
143	C₂×C₄	C₂²×C₈₃⋊C₈	1	1	yes	characteristic
144	C₈	—	2	2	no	—
145	C₈	C₂²⋊(C₈₃⋊C₈)	1	2	yes	—
146	C₈	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
147	C₈	C₂×#13050574	1	1	yes	characteristic
148	C₂²	—	2	2	no	—
149	C₂²	—	4	1	no	—
150	C₂²	—	4	2	no	—
151	C₂²	C₂²⋊(C₈₃⋊C₁₆)	1	2	yes	—
152	C₂²	C₂²×C₈₃⋊C₁₆	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	C₂²⋊(C₈₃⋊C₁₆)	1	2	yes	—
155	C₄	C₂×C₂²⋊(C₈₃⋊C₈)	1	1	yes	characteristic
156	C₄	C₂×#14581732	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	2	no	—
159	C₂	—	4	2	no	—
160	C₂	C₂×C₂²⋊(C₈₃⋊C₁₆)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19127535

C₂

yes

maximal, complemented

#24905603

C₂

yes

maximal, complemented

C₂×#504×C₈₃

C₂

yes

maximal, characteristic

#19127535

—

#19127535

C₂²

yes

—

C₂×C₈₃⋊C₃₂

—

C₂×#131×C₈₃

C₄

yes

characteristic

#504×C₈₃

—

#504×C₈₃

—

C₂×M₃₂×C₈₃

C₄

yes

characteristic

C₂×M₃₂×C₈₃

C₂²

yes

—

C₂²×C₁₆×C₈₃

C₄

yes

—

C₂²×C₁₆×C₈₃

C₂²

yes

characteristic

C₈₃⋊C₃₂

—

C₂×#41×C₈₃

C₈

yes

characteristic

#131×C₈₃

—

#131×C₈₃

—

C₂×M₁₆×C₈₃

C₈

yes

characteristic

C₂×M₁₆×C₈₃

C₈

yes

—

C₂²×C₈×C₈₃

C₂×C₄

yes

—

C₂²×C₈×C₈₃

C₂×C₄

yes

characteristic

M₃₂×C₈₃

—

M₃₂×C₈₃

—

C₂×C₁₆×C₈₃

—

C₂×C₁₆×C₈₃

—

C₂×C₁₆×C₈₃

C₂×C₄

yes

—

C₂×C₁₆×C₈₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₃

D₄

yes

—

C₂×C₁₆×C₈₃

D₄

yes

—

C₂×C₁₆×C₈₃

C₂³

yes

characteristic

#41×C₈₃

—

#41×C₈₃

—

C₂×Q₈×C₈₃

C₁₆

yes

characteristic

C₂×D₄×C₈₃

C₁₆

yes

—

C₂×D₄×C₈₃

C₁₆

yes

characteristic

C₂²×C₄×C₈₃

C₂×C₈

yes

—

C₂²×C₄×C₈₃

C₂×C₈

yes

characteristic

M₁₆×C₈₃

—

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂²⋊C₄

yes

—

C₂×C₈×C₈₃

C₂×C₈

yes

—

C₂×C₈×C₈₃

C₂×C₈

yes

characteristic

C₂×C₈×C₈₃

M₁₆

yes

—

C₂×C₈×C₈₃

C₂²×C₄

yes

characteristic

C₁₆×C₈₃

—

C₁₆×C₈₃

C₂²⋊C₄

yes

characteristic

C₁₆×C₈₃

C₂²⋊C₄

yes

characteristic

C₁₆×C₈₃

C₂×D₄

yes

—

C₂³×C₈₃

C₂×C₁₆

yes

—

C₂³×C₈₃

C₂×C₁₆

yes

characteristic

Q₈×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²⋊C₈

yes

—

C₂×C₄×C₈₃

C₂×C₁₆

yes

—

C₂×C₄×C₈₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₈₃

M₃₂

yes

—

C₂×C₄×C₈₃

C₂²×C₈

yes

characteristic

C₈×C₈₃

—

C₈×C₈₃

C₂²⋊C₈

yes

—

C₈×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₈₃

C₂×M₁₆

yes

characteristic

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂²⋊C₁₆

yes

—

C₂²×C₈₃

C₂²×C₁₆

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

C₂²⋊C₁₆

yes

—

C₄×C₈₃

C₂×C₂²⋊C₈

yes

characteristic

C₄×C₈₃

C₂×M₃₂

yes

characteristic

#13556

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

C₂×C₂²⋊C₁₆

yes

characteristic

C₂×#504

D₈₃

yes

characteristic

C₂×M₆₄

—

#1400

—

C₈₃

#13556

yes

characteristic, complemented, Sylow

C₂×#131

Dic₈₃

yes

characteristic

#504

—

#504

—

C₂×M₃₂

Dic₈₃

yes

characteristic

C₂×M₃₂

C₂×D₈₃

yes

—

C₂²×C₁₆

Dic₈₃

yes

—

C₂²×C₁₆

C₂×D₈₃

yes

characteristic

M₆₄

—

M₆₄

—

166

—

C₂×C₃₂

—

166

—

C₂×#41

C₈₃⋊C₈

yes

characteristic

#131

—

#131

—

C₂×M₁₆

C₈₃⋊C₈

yes

characteristic

C₂×M₁₆

C₈₃⋊C₈

yes

—

100

C₂²×C₈

C₂×Dic₈₃

yes

—

101

C₂²×C₈

C₂×Dic₈₃

yes

characteristic

102

M₃₂

—

103

M₃₂

—

104

C₂×C₁₆

—

105

C₂×C₁₆

—

106

C₂×C₁₆

C₂×Dic₈₃

yes

—

107

C₂×C₁₆

C₂×Dic₈₃

yes

characteristic

108

C₂×C₁₆

C₂²⋊D₈₃

yes

—

109

C₂×C₁₆

C₂²⋊D₈₃

yes

—

110

C₂×C₁₆

C₂²×D₈₃

yes

characteristic

111

C₃₂

—

166

—

112

#41

—

113

#41

—

114

C₂×Q₈

C₈₃⋊C₁₆

yes

characteristic

115

C₂×D₄

C₈₃⋊C₁₆

yes

—

116

C₂×D₄

C₈₃⋊C₁₆

yes

characteristic

117

C₂²×C₄

C₂×C₈₃⋊C₈

yes

—

118

C₂²×C₄

C₂×C₈₃⋊C₈

yes

characteristic

119

M₁₆

—

120

M₁₆

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

C₂²⋊Dic₈₃

yes

—

124

C₂×C₈

C₂×C₈₃⋊C₈

yes

—

125

C₂×C₈

C₂×C₈₃⋊C₈

yes

characteristic

126

C₂×C₈

#13050574

yes

—

127

C₂×C₈

C₂²×Dic₈₃

yes

characteristic

128

C₁₆

—

129

C₁₆

C₂²⋊Dic₈₃

yes

characteristic

130

C₁₆

C₂²⋊Dic₈₃

yes

characteristic

131

C₁₆

C₂×C₂²⋊D₈₃

yes

—

132

C₂³

C₂×C₈₃⋊C₁₆

yes

—

133

C₂³

C₂×C₈₃⋊C₁₆

yes

characteristic

134

Q₈

—

135

D₄

—

136

D₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

C₂²⋊(C₈₃⋊C₈)

yes

—

140

C₂×C₄

C₂×C₈₃⋊C₁₆

yes

—

141

C₂×C₄

C₂×C₈₃⋊C₁₆

yes

characteristic

142

C₂×C₄

#14581732

yes

—

143

C₂×C₄

C₂²×C₈₃⋊C₈

yes

characteristic

144

C₈

—

145

C₈

C₂²⋊(C₈₃⋊C₈)

yes

—

146

C₈

C₂×C₂²⋊Dic₈₃

yes

characteristic

147

C₈

C₂×#13050574

yes

characteristic

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

C₂²⋊(C₈₃⋊C₁₆)

yes

—

152

C₂²

C₂²×C₈₃⋊C₁₆

yes

characteristic

153

C₄

—

154

C₄

C₂²⋊(C₈₃⋊C₁₆)

yes

—

155

C₄

C₂×C₂²⋊(C₈₃⋊C₈)

yes

characteristic

156

C₄

C₂×#14581732

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

—

160

C₂

C₂×C₂²⋊(C₈₃⋊C₁₆)

yes

characteristic

	subgroup	G/H
center	C₁₆	C₂²⋊Dic₈₃
commutator	C₂²×C₈₃	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₈₃
Fitting	C₂×#504×C₈₃	C₂
socle	C₂×C₈₃	C₂×C₂²⋊C₁₆
prime radical	C₂×#41×C₈₃	C₈
cosocle	C₂×C₁₆×C₈₃	C₂³
hypercenter	C₂×#504	D₈₃
hypocenter	C₈₃	#13556
LCS	G ▷ C₂²×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ C₂×#504

subgroup

G/H

center

C₁₆

C₂²⋊Dic₈₃

commutator

C₂²×C₈₃

C₂²×C₁₆

Frattini

C₂×C₁₆

C₂²×D₈₃

Fitting

C₂×#504×C₈₃

C₂

socle

C₂×C₈₃

C₂×C₂²⋊C₁₆

prime radical

C₂×#41×C₈₃

C₈

cosocle

C₂×C₁₆×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂²×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ C₂×#504

N	H ≅ G/N	type
1. C₂×#19127535	C₂	semidirect
2. #24905603	C₂	semidirect
84. C₈₃	#13556	semidirect

H ≅ G/N

type

1. C₂×#19127535

C₂

semidirect

2. #24905603

C₂

semidirect

84. C₈₃

#13556

semidirect

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

subgroup	index
C₂×#19127535	2
#24905603	2
C₂×#504×C₈₃	2
#13556	83

subgroup

index

C₂×#19127535

#24905603

C₂×#504×C₈₃

#13556

Subgroups — #35324608

Conjugacy classes of subgroups

Complements

Maximal subgroups