Group #35298113 — subgroups

Subgroups — #35298113

Order 21248.

228 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 99 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	#24906226	C₂	1	1	yes	maximal, characteristic
2	#24905746	C₂	1	1	yes	maximal, characteristic
3	#24905326	C₂	1	2	yes	maximal
4	#24905324	C₂	1	2	yes	maximal
5	C₈₃×#1568	C₂	1	1	yes	maximal, characteristic
6	#19127758	C₂²	1	2	yes	—
7	#19127754	C₂²	1	1	yes	characteristic
8	#19127754	C₂²	1	1	yes	characteristic
9	#19127454	—	2	1	no	—
10	#19127452	—	2	1	no	—
11	#19127422	C₄	1	2	yes	—
12	C₈×C₈₃⋊C₈	C₄	1	2	yes	—
13	#432×C₈₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
15	#335×C₈₃	—	2	1	no	—
16	#334×C₈₃	—	2	1	no	—
17	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
18	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
19	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
20	C₂²⋊(C₈₃⋊C₈)	—	2	1	no	—
21	C₂²⋊(C₈₃⋊C₈)	—	2	1	no	—
22	C₂²⋊(C₈₃⋊C₈)	—	2	2	no	—
23	#14581703	—	2	2	no	—
24	C₄×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
25	C₄×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₈₃	Q₈	1	1	yes	characteristic
27	C₂²×C₈×C₈₃	Q₈	1	1	yes	characteristic
28	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₈₃	—	2	2	no	—
30	C₂²⋊C₈×C₈₃	—	2	2	no	—
31	#97×C₈₃	—	2	2	no	—
32	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
35	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
36	C₂×C₈₃⋊C₈	—	2	2	no	—
37	C₂×C₈₃⋊C₈	—	2	2	no	—
38	C₂×C₈₃⋊C₈	—	4	2	no	—
39	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
41	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
42	C₂×C₈×C₈₃	—	2	2	no	—
43	C₂×C₈×C₈₃	—	2	2	no	—
44	C₂×C₈×C₈₃	—	4	2	no	—
45	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
50	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
51	C₄²×C₈₃	#41	1	1	yes	characteristic
52	C₄²×C₈₃	#41	1	1	yes	characteristic
53	C₂³×C₈₃	#125	1	1	yes	characteristic
54	C₂×C₄×C₈₃	—	2	1	no	—
55	C₂×C₄×C₈₃	—	2	1	no	—
56	C₂×C₄×C₈₃	—	4	1	no	—
57	C₂×C₄×C₈₃	—	4	1	no	—
58	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
60	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
61	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
62	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
63	C₂×C₄×C₈₃	#125	1	1	yes	characteristic
64	C₈×C₈₃	—	2	1	no	—
65	C₈×C₈₃	—	2	1	no	—
66	C₈×C₈₃	—	8	2	no	—
67	C₈×C₈₃	#108	1	2	yes	—
68	C₈×C₈₃	#108	1	2	yes	—
69	C₈₃⋊C₈	—	4	2	no	—
70	C₈₃⋊C₈	—	4	2	no	—
71	C₈₃⋊C₈	—	8	2	no	—
72	C₂²×C₈₃	—	2	1	no	—
73	C₂²×C₈₃	—	2	1	no	—
74	C₂²×C₈₃	—	2	1	no	—
75	C₂²×C₈₃	#382	1	1	yes	characteristic
76	C₄×C₈₃	—	2	1	no	—
77	C₄×C₈₃	—	2	1	no	—
78	C₄×C₈₃	—	2	1	no	—
79	C₄×C₈₃	—	2	1	no	—
80	C₄×C₈₃	—	2	1	no	—
81	C₄×C₈₃	C₂×#108	1	1	yes	characteristic
82	C₄×C₈₃	C₂×#108	1	1	yes	characteristic
83	#8797	—	83	1	no	maximal, Sylow
84	C₂×C₈₃	—	4	1	no	—
85	C₂×C₈₃	#1833	1	1	yes	characteristic
86	C₂×C₈₃	#1833	1	1	yes	characteristic
87	C₂×C₈₃	#1837	1	1	yes	characteristic
88	#1568	—	83	1	no	—
89	#1568	D₈₃	1	1	yes	characteristic
90	#1450	—	83	1	no	—
91	#1287	—	83	2	no	—
92	#1287	—	83	2	no	—
93	C₈₃	#8797	1	1	yes	characteristic, complemented, Sylow
94	#432	—	83	2	no	—
95	#432	C₂×D₈₃	1	2	yes	—
96	#431	—	83	1	no	—
97	#431	—	83	1	no	—
98	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
99	#335	—	2	1	no	—
100	#335	—	166	1	no	—
101	#334	—	2	1	no	—
102	#334	—	166	1	no	—
103	#322	—	83	2	no	—
104	C₈²	—	83	2	no	—
105	C₂²×C₈	Dic₁₆₆	1	1	yes	characteristic
106	C₂²×C₈	Dic₁₆₆	1	1	yes	characteristic
107	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
108	C₄⋊C₈	—	2	2	no	—
109	C₄⋊C₈	—	166	2	no	—
110	C₄⋊C₈	—	166	1	no	—
111	C₄⋊C₈	—	166	1	no	—
112	C₂²⋊C₈	—	2	2	no	—
113	C₂²⋊C₈	—	166	2	no	—
114	C₂²⋊C₈	—	166	1	no	—
115	C₂²⋊C₈	—	166	1	no	—
116	#97	—	2	2	no	—
117	#97	—	166	2	no	—
118	C₄×C₈	—	83	2	no	—
119	C₄×C₈	—	83	2	no	—
120	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
121	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
122	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
123	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
124	C₂²×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
125	C₂²×C₄	#13050589	1	1	yes	characteristic
126	C₂²×C₄	#13050591	1	1	yes	characteristic
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	—	332	2	no	—
131	C₂×C₈	—	166	2	no	—
132	C₂×C₈	—	166	2	no	—
133	C₂×C₈	Dic₈₃⋊C₄	1	1	yes	characteristic
134	C₂×C₈	Dic₈₃⋊C₄	1	1	yes	characteristic
135	C₂×C₈	Dic₈₃⋊C₄	1	1	yes	characteristic
136	C₂×C₈	Dic₈₃⋊C₄	1	1	yes	characteristic
137	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
138	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
139	C₄²	#13050589	1	1	yes	characteristic
140	C₄²	#13050591	1	1	yes	characteristic
141	C₂³	#14581783	1	1	yes	characteristic
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
147	C₂×C₄	#14581750	1	1	yes	characteristic
148	C₂×C₄	#14581750	1	1	yes	characteristic
149	C₂×C₄	#14581779	1	1	yes	characteristic
150	C₂×C₄	#14581781	1	1	yes	characteristic
151	C₂×C₄	#14581783	1	1	yes	characteristic
152	C₈	—	2	1	no	—
153	C₈	—	2	1	no	—
154	C₈	—	8	2	no	—
155	C₈	—	332	2	no	—
156	C₈	—	332	2	no	—
157	C₈	—	664	2	no	—
158	C₈	#14581727	1	2	yes	—
159	C₈	#14581727	1	2	yes	—
160	C₂²	—	2	1	no	—
161	C₂²	—	2	1	no	—
162	C₂²	—	2	1	no	—
163	C₂²	#19127566	1	1	yes	characteristic
164	C₄	—	2	1	no	—
165	C₄	—	2	1	no	—
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	C₂×#14581727	1	1	yes	characteristic
170	C₄	C₂×#14581727	1	1	yes	characteristic
171	C₂	—	4	1	no	—
172	C₂	#24907586	1	1	yes	characteristic
173	C₂	#24907588	1	1	yes	characteristic
174	C₂	#24907600	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24906226

C₂

yes

maximal, characteristic

#24905746

C₂

yes

maximal, characteristic

#24905326

C₂

yes

maximal

#24905324

C₂

yes

maximal

C₈₃×#1568

C₂

yes

maximal, characteristic

#19127758

C₂²

yes

—

#19127754

C₂²

yes

characteristic

#19127754

C₂²

yes

—

—

C₄

yes

—

C₈×C₈₃⋊C₈

C₄

yes

—

#432×C₈₃

C₂²

yes

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

#335×C₈₃

—

#334×C₈₃

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

—

C₂²⋊(C₈₃⋊C₈)

—

C₂²⋊(C₈₃⋊C₈)

—

C₂²⋊(C₈₃⋊C₈)

—

#14581703

—

C₄×C₈₃⋊C₈

C₂×C₄

yes

—

C₄×C₈₃⋊C₈

C₂×C₄

yes

—

C₂²×C₈×C₈₃

Q₈

yes

characteristic

C₂²×C₈×C₈₃

Q₈

yes

characteristic

C₂×C₄²×C₈₃

C₂³

yes

characteristic

C₄⋊C₈×C₈₃

—

C₂²⋊C₈×C₈₃

—

#97×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

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

C₄²×C₈₃

yes

C₄²×C₈₃

yes

C₂³×C₈₃

yes

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

#108

yes

—

C₈×C₈₃

#108

yes

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#382

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

—

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

D₈₃

yes

—

—

—

C₈₃

#8797

yes

characteristic, complemented, Sylow

#432

—

#432

C₂×D₈₃

yes

—

#431

—

#431

—

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

#335

—

100

#335

—

166

—

101

#334

—

102

#334

—

166

—

103

#322

—

104

C₈²

—

105

C₂²×C₈

Dic₁₆₆

yes

characteristic

106

C₂²×C₈

Dic₁₆₆

yes

characteristic

107

C₂×C₄²

C₂²×D₈₃

yes

characteristic

108

C₄⋊C₈

—

109

C₄⋊C₈

—

166

—

110

C₄⋊C₈

—

166

—

111

C₄⋊C₈

—

166

—

112

C₂²⋊C₈

—

113

C₂²⋊C₈

—

166

—

114

C₂²⋊C₈

—

166

—

115

C₂²⋊C₈

—

166

—

116

#97

—

117

#97

—

166

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₄×C₈

C₄×D₈₃

yes

characteristic

121

C₄×C₈

C₄×D₈₃

yes

characteristic

122

C₄×C₈

C₂²⋊D₈₃

yes

characteristic

123

C₄×C₈

C₂²⋊D₈₃

yes

characteristic

124

C₂²×C₄

C₂×Dic₁₆₆

yes

characteristic

125

C₂²×C₄

#13050589

yes

characteristic

126

C₂²×C₄

#13050591

yes

characteristic

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

332

—

131

C₂×C₈

—

166

—

132

C₂×C₈

—

166

—

133

C₂×C₈

Dic₈₃⋊C₄

yes

characteristic

134

C₂×C₈

Dic₈₃⋊C₄

yes

characteristic

135

C₂×C₈

Dic₈₃⋊C₄

yes

characteristic

136

C₂×C₈

Dic₈₃⋊C₄

yes

characteristic

137

C₄²

C₂×C₄×D₈₃

yes

characteristic

138

C₄²

C₂×C₂²⋊D₈₃

yes

characteristic

139

C₄²

#13050589

yes

characteristic

140

C₄²

#13050591

yes

characteristic

141

C₂³

#14581783

yes

characteristic

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

C₂×Dic₈₃⋊C₄

yes

characteristic

147

C₂×C₄

#14581750

yes

characteristic

148

C₂×C₄

#14581750

yes

characteristic

149

C₂×C₄

#14581779

yes

characteristic

150

C₂×C₄

#14581781

yes

characteristic

151

C₂×C₄

#14581783

yes

characteristic

152

C₈

—

153

C₈

—

154

C₈

—

155

C₈

—

332

—

156

C₈

—

332

—

157

C₈

—

664

—

158

C₈

#14581727

yes

—

159

C₈

#14581727

yes

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

#19127566

yes

characteristic

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

C₂×#14581727

yes

characteristic

170

C₄

C₂×#14581727

yes

characteristic

171

C₂

—

172

C₂

#24907586

yes

characteristic

173

C₂

#24907588

yes

characteristic

174

C₂

#24907600

yes

characteristic

	subgroup	G/H
center	C₂²	#19127566
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1568	C₂
socle	C₂²×C₈₃	#382
prime radical	C₂³×C₈₃	#125
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1568	D₈₃
hypocenter	C₈₃	#8797
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1568

subgroup

G/H

center

C₂²

#19127566

commutator

C₄²×C₈₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₈₃×#1568

C₂

C₂²×C₈₃

C₂³×C₈₃

C₂×C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1568

N	H ≅ G/N	type
93. C₈₃	#8797	semidirect

H ≅ G/N

type

93. C₈₃

#8797

semidirect

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

subgroup	index
#24906226	2
#24905746	2
#24905326	2
#24905324	2
C₈₃×#1568	2
#8797	83

subgroup

index

C₈₃×#1568

Subgroups — #35298113

Conjugacy classes of subgroups

Complements

Maximal subgroups