Group #35324931 — subgroups

Subgroups — #35324931

Order 21248.

194 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 93 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×C₁₆×Dic₈₃	C₂	1	1	yes	maximal, characteristic
2	Dic₈₃⋊C₃₂	C₂	1	2	yes	maximal, complemented
3	C₂²⋊(C₈₃⋊C₃₂)	C₂	1	1	yes	maximal, characteristic
4	#24905596	C₂	1	2	yes	maximal, complemented
5	C₈₃×C₂²⋊C₃₂	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈₃⋊C₁₆	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₈₃	C₄	1	1	yes	characteristic
8	C₂×C₈₃⋊C₃₂	C₂²	1	2	yes	—
9	C₁₆×Dic₈₃	C₄	1	1	yes	characteristic
10	C₁₆×Dic₈₃	C₄	1	1	yes	characteristic
11	C₁₆×Dic₈₃	C₂²	1	1	yes	characteristic
12	C₁₆×Dic₈₃	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
14	C₂×C₃₂×C₈₃	C₂²	1	2	yes	—
15	C₂²×C₈₃⋊C₈	C₈	1	1	yes	characteristic
16	C₂×C₄×Dic₈₃	C₈	1	1	yes	characteristic
17	C₂×C₈₃⋊C₁₆	—	2	2	no	—
18	C₂×C₈₃⋊C₁₆	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈₃⋊C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₈×Dic₈₃	C₈	1	1	yes	characteristic
21	C₈×Dic₈₃	C₈	1	1	yes	characteristic
22	C₈×Dic₈₃	C₂×C₄	1	1	yes	characteristic
23	C₈×Dic₈₃	C₂×C₄	1	1	yes	characteristic
24	C₈₃⋊C₃₂	—	2	2	no	—
25	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₈₃	—	2	2	no	—
27	C₂×C₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₈₃	C₂³	1	1	yes	characteristic
29	C₃₂×C₈₃	—	2	2	no	—
30	C₂²×Dic₈₃	C₁₆	1	1	yes	characteristic
31	C₂²×Dic₈₃	C₁₆	1	1	yes	characteristic
32	C₂×C₈₃⋊C₈	—	2	2	no	—
33	C₂×C₈₃⋊C₈	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈₃⋊C₈	C₂×C₈	1	1	yes	characteristic
35	C₄×Dic₈₃	C₁₆	1	1	yes	characteristic
36	C₄×Dic₈₃	C₁₆	1	1	yes	characteristic
37	C₄×Dic₈₃	C₂×C₈	1	1	yes	characteristic
38	C₄×Dic₈₃	C₂×C₈	1	1	yes	characteristic
39	C₈₃⋊C₁₆	—	2	1	no	—
40	C₈₃⋊C₁₆	M₁₆	1	2	yes	—
41	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
42	C₂×C₈×C₈₃	—	2	1	no	—
43	C₂×C₈×C₈₃	—	2	1	no	—
44	C₂×C₈×C₈₃	C₂×C₈	1	1	yes	characteristic
45	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
46	C₁₆×C₈₃	—	2	1	no	—
47	C₁₆×C₈₃	#41	1	2	yes	—
48	C₂³×C₈₃	C₂×C₁₆	1	1	yes	characteristic
49	C₂×C₄×C₈₃	—	2	1	no	—
50	C₂×C₄×C₈₃	—	2	1	no	—
51	C₂×C₄×C₈₃	C₂×C₁₆	1	1	yes	characteristic
52	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
53	C₂×Dic₈₃	—	2	2	no	—
54	C₂×Dic₈₃	—	2	2	no	—
55	C₂×Dic₈₃	C₂×C₁₆	1	1	yes	characteristic
56	C₂×Dic₈₃	C₂×C₁₆	1	1	yes	characteristic
57	C₂×Dic₈₃	C₂×C₁₆	1	1	yes	characteristic
58	C₂×Dic₈₃	C₂×C₁₆	1	1	yes	characteristic
59	C₈×C₈₃	—	2	1	no	—
60	C₈×C₈₃	#117	1	1	yes	characteristic
61	C₈×C₈₃	C₂×M₁₆	1	1	yes	characteristic
62	C₈₃⋊C₈	—	2	1	no	—
63	C₈₃⋊C₈	M₃₂	1	2	yes	—
64	C₂²×C₈₃	—	2	1	no	—
65	C₂²×C₈₃	—	2	1	no	—
66	C₂²×C₈₃	M₆₄	1	2	yes	—
67	C₂²×C₈₃	C₂²×C₁₆	1	1	yes	characteristic
68	C₄×C₈₃	—	2	1	no	—
69	C₄×C₈₃	#431	1	1	yes	characteristic
70	C₄×C₈₃	C₂×M₃₂	1	1	yes	characteristic
71	Dic₈₃	—	2	1	no	—
72	Dic₈₃	—	2	1	no	—
73	Dic₈₃	—	2	1	no	—
74	Dic₈₃	M₆₄	1	2	yes	—
75	#13640	—	83	1	no	maximal, Sylow
76	C₂×C₈₃	—	2	2	no	—
77	C₂×C₈₃	#2162	1	1	yes	characteristic
78	C₂×C₈₃	C₂×M₆₄	1	1	yes	characteristic
79	C₂×C₈₃	C₂×M₆₄	1	1	yes	characteristic
80	C₂×C₄×C₁₆	—	83	1	no	—
81	C₄⋊C₃₂	—	83	2	no	—
82	C₂²⋊C₃₂	—	83	1	no	—
83	C₂²⋊C₃₂	D₈₃	1	1	yes	characteristic
84	#1397	—	83	2	no	—
85	C₈₃	#13640	1	1	yes	characteristic, complemented, Sylow
86	C₂²×C₁₆	—	83	1	no	—
87	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
88	C₂×C₄×C₈	—	83	1	no	—
89	C₂×C₃₂	—	83	2	no	—
90	C₂×C₃₂	C₂×D₈₃	1	2	yes	—
91	C₄×C₁₆	—	83	1	no	—
92	C₄×C₁₆	—	83	1	no	—
93	C₄×C₁₆	—	83	1	no	—
94	C₄×C₁₆	—	83	1	no	—
95	C₂²×C₈	—	83	1	no	—
96	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
97	C₂×C₄²	—	83	1	no	—
98	C₂×C₁₆	—	2	2	no	—
99	C₂×C₁₆	—	83	1	no	—
100	C₂×C₁₆	—	83	1	no	—
101	C₂×C₁₆	—	166	2	no	—
102	C₂×C₁₆	C₄×D₈₃	1	1	yes	characteristic
103	C₂×C₁₆	C₂²×D₈₃	1	1	yes	characteristic
104	C₄×C₈	—	83	1	no	—
105	C₄×C₈	—	83	1	no	—
106	C₄×C₈	—	83	1	no	—
107	C₄×C₈	—	83	1	no	—
108	C₃₂	—	2	2	no	—
109	C₃₂	—	166	2	no	—
110	C₂²×C₄	—	83	1	no	—
111	C₂²×C₄	—	83	1	no	—
112	C₂²×C₄	C₈×D₈₃	1	1	yes	characteristic
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	83	1	no	—
116	C₂×C₈	—	83	1	no	—
117	C₂×C₈	—	166	2	no	—
118	C₂×C₈	C₈×D₈₃	1	1	yes	characteristic
119	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
120	C₄²	—	83	1	no	—
121	C₄²	—	83	1	no	—
122	C₄²	—	83	1	no	—
123	C₄²	—	83	1	no	—
124	C₁₆	—	2	1	no	—
125	C₁₆	—	83	2	no	—
126	C₁₆	—	166	1	no	—
127	C₁₆	#13050589	1	2	yes	—
128	C₂³	C₁₆×D₈₃	1	1	yes	characteristic
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	83	1	no	—
132	C₂×C₄	—	83	1	no	—
133	C₂×C₄	—	83	1	no	—
134	C₂×C₄	—	83	1	no	—
135	C₂×C₄	—	166	2	no	—
136	C₂×C₄	—	166	2	no	—
137	C₂×C₄	C₁₆×D₈₃	1	1	yes	characteristic
138	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
139	C₈	—	2	1	no	—
140	C₈	—	83	2	no	—
141	C₈	—	166	1	no	—
142	C₈	#14581748	1	1	yes	characteristic
143	C₈	M₁₆×D₈₃	1	1	yes	characteristic
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	#19127534	1	2	yes	—
147	C₂²	C₂×C₁₆×D₈₃	1	1	yes	characteristic
148	C₄	—	2	1	no	—
149	C₄	—	83	2	no	—
150	C₄	—	166	1	no	—
151	C₄	—	166	1	no	—
152	C₄	—	166	1	no	—
153	C₄	#19127751	1	1	yes	characteristic
154	C₄	M₃₂×D₈₃	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	#24909259	1	1	yes	characteristic
157	C₂	M₆₄×D₈₃	1	1	yes	characteristic
158	C₂	C₂×#19127534	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₆×Dic₈₃

C₂

yes

maximal, characteristic

Dic₈₃⋊C₃₂

C₂

yes

maximal, complemented

C₂²⋊(C₈₃⋊C₃₂)

C₂

yes

maximal, characteristic

#24905596

C₂

yes

maximal, complemented

C₈₃×C₂²⋊C₃₂

C₂

yes

maximal, characteristic

C₂²×C₈₃⋊C₁₆

C₄

yes

characteristic

C₂×C₈×Dic₈₃

C₄

yes

characteristic

C₂×C₈₃⋊C₃₂

C₂²

yes

—

C₁₆×Dic₈₃

C₄

yes

characteristic

C₁₆×Dic₈₃

C₄

yes

characteristic

C₁₆×Dic₈₃

C₂²

yes

characteristic

C₁₆×Dic₈₃

C₂²

yes

characteristic

C₂²×C₁₆×C₈₃

C₂²

yes

characteristic

C₂×C₃₂×C₈₃

C₂²

yes

—

C₂²×C₈₃⋊C₈

C₈

yes

characteristic

C₂×C₄×Dic₈₃

C₈

yes

characteristic

C₂×C₈₃⋊C₁₆

—

C₂×C₈₃⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₈₃⋊C₁₆

C₂×C₄

yes

characteristic

C₈×Dic₈₃

C₈

yes

characteristic

C₈×Dic₈₃

C₈

yes

characteristic

C₈×Dic₈₃

C₂×C₄

yes

characteristic

C₈×Dic₈₃

C₂×C₄

yes

characteristic

C₈₃⋊C₃₂

—

C₂²×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₃

—

C₂×C₁₆×C₈₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₃

C₂³

yes

characteristic

C₃₂×C₈₃

—

C₂²×Dic₈₃

C₁₆

yes

characteristic

C₂²×Dic₈₃

C₁₆

yes

characteristic

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

C₂×C₈

yes

characteristic

C₂×C₈₃⋊C₈

C₂×C₈

yes

characteristic

C₄×Dic₈₃

C₁₆

yes

characteristic

C₄×Dic₈₃

C₁₆

yes

characteristic

C₄×Dic₈₃

C₂×C₈

yes

characteristic

C₄×Dic₈₃

C₂×C₈

yes

characteristic

C₈₃⋊C₁₆

—

C₈₃⋊C₁₆

M₁₆

yes

—

C₂²×C₄×C₈₃

C₂×C₈

yes

characteristic

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂×C₈

yes

characteristic

C₂×C₈×C₈₃

C₂²×C₄

yes

characteristic

C₁₆×C₈₃

—

C₁₆×C₈₃

#41

yes

—

C₂³×C₈₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₈₃

C₂²×C₈

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

C₂×C₁₆

yes

characteristic

C₂×Dic₈₃

C₂×C₁₆

yes

characteristic

C₂×Dic₈₃

C₂×C₁₆

yes

characteristic

C₂×Dic₈₃

C₂×C₁₆

yes

characteristic

C₈×C₈₃

—

C₈×C₈₃

#117

yes

characteristic

C₈×C₈₃

C₂×M₁₆

yes

characteristic

C₈₃⋊C₈

—

C₈₃⋊C₈

M₃₂

yes

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

M₆₄

yes

—

C₂²×C₈₃

C₂²×C₁₆

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

#431

yes

characteristic

C₄×C₈₃

C₂×M₃₂

yes

characteristic

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

Dic₈₃

M₆₄

yes

—

#13640

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

#2162

yes

characteristic

C₂×C₈₃

C₂×M₆₄

yes

characteristic

C₂×C₈₃

C₂×M₆₄

yes

characteristic

C₂×C₄×C₁₆

—

C₄⋊C₃₂

—

C₂²⋊C₃₂

—

C₂²⋊C₃₂

D₈₃

yes

characteristic

#1397

—

C₈₃

#13640

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₈₃

yes

characteristic

C₂×C₄×C₈

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₈₃

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₂²×C₈

—

C₂²×C₈

C₄×D₈₃

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

100

C₂×C₁₆

—

101

C₂×C₁₆

—

166

—

102

C₂×C₁₆

C₄×D₈₃

yes

characteristic

103

C₂×C₁₆

C₂²×D₈₃

yes

characteristic

104

C₄×C₈

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₃₂

—

109

C₃₂

—

166

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

C₈×D₈₃

yes

characteristic

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

166

—

118

C₂×C₈

C₈×D₈₃

yes

characteristic

119

C₂×C₈

C₂×C₄×D₈₃

yes

characteristic

120

C₄²

—

121

C₄²

—

122

C₄²

—

123

C₄²

—

124

C₁₆

—

125

C₁₆

—

126

C₁₆

—

166

—

127

C₁₆

#13050589

yes

—

128

C₂³

C₁₆×D₈₃

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

166

—

136

C₂×C₄

—

166

—

137

C₂×C₄

C₁₆×D₈₃

yes

characteristic

138

C₂×C₄

C₂×C₈×D₈₃

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

—

166

—

142

C₈

#14581748

yes

characteristic

143

C₈

M₁₆×D₈₃

yes

characteristic

144

C₂²

—

145

C₂²

—

146

C₂²

#19127534

yes

—

147

C₂²

C₂×C₁₆×D₈₃

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

—

166

—

151

C₄

—

166

—

152

C₄

—

166

—

153

C₄

#19127751

yes

characteristic

154

C₄

M₃₂×D₈₃

yes

characteristic

155

C₂

—

156

C₂

#24909259

yes

characteristic

157

C₂

M₆₄×D₈₃

yes

characteristic

158

C₂

C₂×#19127534

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₈₃
commutator, socle	C₂²×C₈₃	C₂²×C₁₆
Fitting	C₈₃×C₂²⋊C₃₂	C₂
prime radical	C₂³×C₈₃	C₂×C₁₆
cosocle	C₂×C₁₆×C₈₃	C₂³
hypercenter	C₂²⋊C₃₂	D₈₃
hypocenter	C₈₃	#13640
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²⋊C₃₂

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₈₃

commutator, socle

C₂²×C₈₃

C₂²×C₁₆

Fitting

C₈₃×C₂²⋊C₃₂

C₂

prime radical

C₂³×C₈₃

C₂×C₁₆

cosocle

C₂×C₁₆×C₈₃

C₂³

hypercenter

C₂²⋊C₃₂

D₈₃

C₈₃

G ▷ C₂²×C₈₃ ▷ C₈₃

C₁ ◁ C₂×C₁₆ ◁ C₂²⋊C₃₂

N	H ≅ G/N	type
2. Dic₈₃⋊C₃₂	C₂	semidirect
4. #24905596	C₂	semidirect
85. C₈₃	#13640	semidirect

H ≅ G/N

type

2. Dic₈₃⋊C₃₂

C₂

semidirect

4. #24905596

C₂

semidirect

85. C₈₃

#13640

semidirect

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

subgroup	index
C₂×C₁₆×Dic₈₃	2
Dic₈₃⋊C₃₂	2
C₂²⋊(C₈₃⋊C₃₂)	2
#24905596	2
C₈₃×C₂²⋊C₃₂	2
#13640	83

subgroup

index

C₂×C₁₆×Dic₈₃

Dic₈₃⋊C₃₂

C₂²⋊(C₈₃⋊C₃₂)

#24905596

C₈₃×C₂²⋊C₃₂

#13640

Subgroups — #35324931

Conjugacy classes of subgroups

Complements

Maximal subgroups