Group #35297399 — subgroups

Subgroups — #35297399

Order 21248.

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

normal subgroups only

This group has 320 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	#24916393	C₂	1	1	yes	maximal, characteristic
2	#24906224	C₂	1	2	yes	maximal
3	#24906100	C₂	1	2	yes	maximal
4	#24905374	C₂	1	1	yes	maximal, characteristic
5	C₈₃×#1307	C₂	1	1	yes	maximal, characteristic
6	#19128258	C₂²	1	1	yes	characteristic
7	#19127828	—	2	2	no	—
8	#19127825	—	2	2	no	—
9	#19127758	C₂²	1	2	yes	—
10	#19127729	C₄	1	2	yes	—
11	#19127659	C₄	1	2	yes	—
12	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
13	#19127453	—	2	2	no	—
14	#19127451	—	2	2	no	—
15	#19127442	—	2	4	no	—
16	#432×C₈₃	C₂²	1	2	yes	—
17	#423×C₈₃	C₂²	1	1	yes	characteristic
18	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
20	#14581792	Q₈	1	2	yes	—
21	#14581784	D₄	1	2	yes	—
22	#14581771	—	2	2	no	—
23	C₄×Dic₁₆₆	—	2	2	no	—
24	#14581751	C₂×C₄	1	2	yes	—
25	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
26	Dic₈₃⋊C₈	—	2	4	no	—
27	C₂²⋊(C₈₃⋊C₈)	—	2	2	no	—
28	#14581703	—	2	2	no	—
29	#14581702	—	2	2	no	—
30	C₄×C₈₃⋊C₈	D₄	1	2	yes	—
31	C₄×C₈₃⋊C₈	D₄	1	2	yes	—
32	C₈×Dic₈₃	—	2	2	no	—
33	C₂×M₁₆×C₈₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₈₃	—	2	2	no	—
36	C₄⋊C₈×C₈₃	—	2	2	no	—
37	C₂²⋊C₈×C₈₃	—	2	2	no	—
38	#97×C₈₃	—	2	2	no	—
39	C₂×Dic₁₆₆	—	2	2	no	—
40	C₂×C₈₃⋊C₈	—	2	2	no	—
41	C₂×C₈₃⋊C₈	—	4	1	no	—
42	C₂×C₈₃⋊C₈	—	4	2	no	—
43	C₂×C₈₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₈₃⋊C₈	C₂²⋊C₄	1	2	yes	—
45	C₂×C₈₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
46	C₄⋊Dic₈₃	—	2	2	no	—
47	C₄⋊Dic₈₃	C₄⋊C₄	1	4	yes	—
48	Dic₈₃⋊C₄	—	2	4	no	—
49	C₂²⋊Dic₈₃	—	2	2	no	—
50	C₄×Dic₈₃	—	2	2	no	—
51	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₈₃	#41	1	2	yes	—
53	M₁₆×C₈₃	—	4	2	no	—
54	C₂×C₈×C₈₃	—	2	2	no	—
55	C₂×C₈×C₈₃	—	4	2	no	—
56	C₄²×C₈₃	C₂×D₄	1	2	yes	—
57	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
59	C₂³×C₈₃	#117	1	1	yes	characteristic
60	C₂×C₄×C₈₃	—	2	1	no	—
61	C₂×C₄×C₈₃	—	2	1	no	—
62	C₂×C₄×C₈₃	—	4	2	no	—
63	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₈₃	#121	1	2	yes	—
66	C₂×C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
67	C₂×Dic₈₃	—	2	4	no	—
68	Dic₁₆₆	—	4	4	no	—
69	C₈×C₈₃	—	4	2	no	—
70	C₈×C₈₃	—	8	2	no	—
71	C₈₃⋊C₈	—	2	1	no	—
72	C₈₃⋊C₈	—	2	2	no	—
73	C₈₃⋊C₈	—	8	2	no	—
74	C₈₃⋊C₈	#101	1	2	yes	—
75	C₂²×C₈₃	—	2	1	no	—
76	C₂²×C₈₃	—	2	2	no	—
77	C₂²×C₈₃	#380	1	1	yes	characteristic
78	C₄×C₈₃	—	2	1	no	—
79	C₄×C₈₃	—	2	2	no	—
80	C₄×C₈₃	—	2	2	no	—
81	C₄×C₈₃	C₂×#101	1	1	yes	characteristic
82	C₄×C₈₃	#413	1	1	yes	characteristic
83	Dic₈₃	—	4	4	no	—
84	#8631	—	83	1	no	maximal, Sylow
85	C₂×C₈₃	—	4	1	no	—
86	C₂×C₈₃	#1801	1	1	yes	characteristic
87	C₂×C₈₃	#1814	1	2	yes	—
88	#2960	—	83	1	no	—
89	#1568	—	83	2	no	—
90	#1537	—	83	2	no	—
91	#1307	—	83	1	no	—
92	#1307	D₈₃	1	1	yes	characteristic
93	C₈₃	#8631	1	1	yes	characteristic, complemented, Sylow
94	#533	—	83	1	no	—
95	#446	—	166	2	no	—
96	C₈×Q₈	—	166	2	no	—
97	#432	—	83	2	no	—
98	#432	C₂×D₈₃	1	2	yes	—
99	#424	—	83	2	no	—
100	#423	C₂×D₈₃	1	1	yes	characteristic
101	#405	—	83	2	no	—
102	C₂×C₄×C₈	—	83	1	no	—
103	#335	—	166	2	no	—
104	#334	—	166	2	no	—
105	#330	—	166	4	no	—
106	C₂×M₁₆	C₄×D₈₃	1	2	yes	—
107	C₂²×C₈	—	83	1	no	—
108	C₂²×C₈	—	83	1	no	—
109	C₄⋊Q₈	—	83	2	no	—
110	#125	—	83	2	no	—
111	C₂²⋊Q₈	—	166	2	no	—
112	C₄×Q₈	—	166	2	no	—
113	#117	—	83	2	no	—
114	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
115	C₄⋊C₈	—	2	2	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	166	4	no	—
118	C₄⋊C₈	—	166	2	no	—
119	C₂²⋊C₈	—	2	2	no	—
120	C₂²⋊C₈	—	166	2	no	—
121	#97	—	2	2	no	—
122	#97	—	166	2	no	—
123	#97	—	166	2	no	—
124	C₄×C₈	—	83	2	no	—
125	C₄×C₈	—	83	2	no	—
126	C₄×C₈	—	166	2	no	—
127	C₂×Q₈	—	166	2	no	—
128	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
129	C₂²×C₄	#13050590	1	2	yes	—
130	M₁₆	—	4	2	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	4	2	no	—
133	C₂×C₈	—	332	2	no	—
134	C₂×C₈	—	332	1	no	—
135	C₂×C₈	—	83	1	no	—
136	C₂×C₈	—	83	1	no	—
137	C₂×C₈	—	83	2	no	—
138	C₂×C₈	—	166	2	no	—
139	C₄⋊C₄	—	83	4	no	—
140	C₄⋊C₄	—	166	4	no	—
141	C₄⋊C₄	—	166	2	no	—
142	C₂²⋊C₄	—	166	2	no	—
143	C₄²	—	166	2	no	—
144	C₄²	D₄×D₈₃	1	1	yes	characteristic
145	C₄²	D₄×D₈₃	1	2	yes	—
146	C₄²	Q₈×D₈₃	1	1	yes	characteristic
147	C₂³	#14581749	1	1	yes	characteristic
148	Q₈	—	332	4	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	166	4	no	—
153	C₂×C₄	C₂²⋊C₄×D₈₃	1	1	yes	characteristic
154	C₂×C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
155	C₂×C₄	#14581765	1	2	yes	—
156	C₂×C₄	#14581770	1	2	yes	—
157	C₈	—	4	2	no	—
158	C₈	—	8	2	no	—
159	C₈	—	83	2	no	—
160	C₈	—	664	2	no	—
161	C₈	—	166	1	no	—
162	C₈	—	166	2	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	2	no	—
165	C₂²	#19127560	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	2	2	no	—
168	C₄	—	2	2	no	—
169	C₄	—	332	4	no	—
170	C₄	#101×D₈₃	1	1	yes	characteristic
171	C₄	#19127682	1	1	yes	characteristic
172	C₂	—	4	1	no	—
173	C₂	#24907458	1	1	yes	characteristic
174	C₂	#24907516	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24916393

C₂

yes

maximal, characteristic

#24906224

C₂

yes

maximal

#24906100

C₂

yes

maximal

#24905374

C₂

yes

maximal, characteristic

C₈₃×#1307

C₂

yes

maximal, characteristic

#19128258

C₂²

yes

—

—

C₂²

yes

—

#19127729

C₄

yes

—

#19127659

C₄

yes

—

C₂×C₄×C₈₃⋊C₈

C₂²

yes

—

—

—

#432×C₈₃

C₂²

yes

—

#423×C₈₃

C₂²

yes

characteristic

C₂²×C₈₃⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₈₃⋊C₈

C₂×C₄

yes

characteristic

#14581792

Q₈

yes

—

#14581784

D₄

yes

—

#14581771

—

C₄×Dic₁₆₆

—

#14581751

C₂×C₄

yes

—

C₄⋊(C₈₃⋊C₈)

—

Dic₈₃⋊C₈

—

C₂²⋊(C₈₃⋊C₈)

—

#14581703

—

#14581702

—

C₄×C₈₃⋊C₈

D₄

yes

—

C₄×C₈₃⋊C₈

D₄

yes

—

C₈×Dic₈₃

—

C₂×M₁₆×C₈₃

C₂×C₄

yes

—

C₂×C₄²×C₈₃

C₂³

yes

characteristic

C₄⋊C₈×C₈₃

—

C₄⋊C₈×C₈₃

—

C₂²⋊C₈×C₈₃

—

#97×C₈₃

—

C₂×Dic₁₆₆

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈₃⋊C₈

C₂²⋊C₄

yes

—

C₂×C₈₃⋊C₈

C₂²⋊C₄

yes

characteristic

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

C₄⋊C₄

yes

—

Dic₈₃⋊C₄

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

—

C₂²×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₃

#41

yes

—

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₄²×C₈₃

C₂×D₄

yes

—

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₄²×C₈₃

C₂×Q₈

yes

characteristic

C₂³×C₈₃

#117

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

#121

yes

—

C₂×C₄×C₈₃

C₂²⋊Q₈

yes

—

C₂×Dic₈₃

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

#101

yes

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#380

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

Dic₈₃

—

#8631

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

#1801

yes

characteristic

C₂×C₈₃

#1814

yes

—

—

—

—

—

D₈₃

yes

characteristic

C₈₃

#8631

yes

characteristic, complemented, Sylow

#533

—

#446

—

166

—

C₈×Q₈

—

166

—

#432

—

#432

C₂×D₈₃

yes

—

#424

—

100

#423

C₂×D₈₃

yes

characteristic

101

#405

—

102

C₂×C₄×C₈

—

103

#335

—

166

—

104

#334

—

166

—

105

#330

—

166

—

106

C₂×M₁₆

C₄×D₈₃

yes

—

107

C₂²×C₈

—

108

C₂²×C₈

—

109

C₄⋊Q₈

—

110

#125

—

111

C₂²⋊Q₈

—

166

—

112

C₄×Q₈

—

166

—

113

#117

—

114

C₂×C₄²

C₂²×D₈₃

yes

characteristic

115

C₄⋊C₈

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

166

—

118

C₄⋊C₈

—

166

—

119

C₂²⋊C₈

—

120

C₂²⋊C₈

—

166

—

121

#97

—

122

#97

—

166

—

123

#97

—

166

—

124

C₄×C₈

—

125

C₄×C₈

—

126

C₄×C₈

—

166

—

127

C₂×Q₈

—

166

—

128

C₂²×C₄

C₂×C₄×D₈₃

yes

characteristic

129

C₂²×C₄

#13050590

yes

—

130

M₁₆

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

332

—

134

C₂×C₈

—

332

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

166

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

166

—

141

C₄⋊C₄

—

166

—

142

C₂²⋊C₄

—

166

—

143

C₄²

—

166

—

144

C₄²

D₄×D₈₃

yes

characteristic

145

C₄²

D₄×D₈₃

yes

—

146

C₄²

Q₈×D₈₃

yes

characteristic

147

C₂³

#14581749

yes

characteristic

148

Q₈

—

332

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

166

—

153

C₂×C₄

C₂²⋊C₄×D₈₃

yes

characteristic

154

C₂×C₄

C₄⋊C₄×D₈₃

yes

characteristic

155

C₂×C₄

#14581765

yes

—

156

C₂×C₄

#14581770

yes

—

157

C₈

—

158

C₈

—

159

C₈

—

160

C₈

—

664

—

161

C₈

—

166

—

162

C₈

—

166

—

163

C₂²

—

164

C₂²

—

165

C₂²

#19127560

yes

characteristic

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

332

—

170

C₄

#101×D₈₃

yes

characteristic

171

C₄

#19127682

yes

characteristic

172

C₂

—

173

C₂

#24907458

yes

characteristic

174

C₂

#24907516

yes

—

	subgroup	G/H
center	C₂²	#19127560
commutator	C₂²×C₄×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1307	C₂
socle	C₂²×C₈₃	#380
prime radical	C₂³×C₈₃	#117
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1307	D₈₃
hypocenter	C₈₃	#8631
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1307

subgroup

G/H

center

C₂²

#19127560

commutator

C₂²×C₄×C₈₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₈₃×#1307

C₂

C₂²×C₈₃

C₂³×C₈₃

C₂×C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂²×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1307

N	H ≅ G/N	type
93. C₈₃	#8631	semidirect

H ≅ G/N

type

93. C₈₃

#8631

semidirect

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

subgroup	index
#24916393	2
#24906224	2
#24906100	2
#24905374	2
C₈₃×#1307	2
#8631	83

subgroup

index

C₈₃×#1307

Subgroups — #35297399

Conjugacy classes of subgroups

Complements

Maximal subgroups