Group #35297398 — subgroups

Subgroups — #35297398

Order 21248.

320 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 101 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	#24916398	C₂	1	1	yes	maximal, characteristic
2	#24906224	C₂	1	2	yes	maximal
3	#24906101	C₂	1	1	yes	maximal, characteristic
4	#24905373	C₂	1	2	yes	maximal
5	C₈₃×#1537	C₂	1	1	yes	maximal, characteristic
6	#19127832	—	2	2	no	—
7	Q₈×C₈₃⋊C₈	—	2	2	no	—
8	#19127758	C₂²	1	2	yes	—
9	#19127728	C₄	1	2	yes	—
10	#19127725	C₂²	1	1	yes	characteristic
11	#19127657	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	#19127443	—	2	2	no	—
16	#533×C₈₃	C₂²	1	1	yes	characteristic
17	#432×C₈₃	C₂²	1	2	yes	—
18	#330×C₈₃	—	2	2	no	—
19	C₂×#13050574	C₂×C₄	1	2	yes	—
20	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
22	C₄⋊(C₈₃⋊C₈)	—	2	4	no	—
23	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
24	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
25	C₂²⋊(C₈₃⋊C₈)	—	2	2	no	—
26	#14581703	—	2	2	no	—
27	#14581703	—	2	2	no	—
28	C₄×C₈₃⋊C₈	—	2	2	no	—
29	C₄×C₈₃⋊C₈	D₄	1	2	yes	—
30	C₄×C₈₃⋊C₈	D₄	1	2	yes	—
31	C₄⋊Q₈×C₈₃	Q₈	1	2	yes	—
32	#125×C₈₃	D₄	1	2	yes	—
33	C₂²⋊Q₈×C₈₃	—	2	2	no	—
34	C₄×Q₈×C₈₃	—	2	2	no	—
35	#117×C₈₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₈₃	—	2	2	no	—
38	C₂²⋊C₈×C₈₃	—	2	2	no	—
39	#97×C₈₃	—	2	2	no	—
40	#13050574	—	4	2	no	—
41	C₂×C₈₃⋊C₈	—	2	2	no	—
42	C₂×C₈₃⋊C₈	—	2	2	no	—
43	C₂×C₈₃⋊C₈	—	4	1	no	—
44	C₂×C₈₃⋊C₈	—	4	2	no	—
45	C₂×C₈₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₈₃⋊C₈	C₂²⋊C₄	1	2	yes	—
47	C₂×C₈₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×Q₈×C₈₃	—	2	2	no	—
49	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₈₃	#41	1	2	yes	—
51	C₂×C₈×C₈₃	—	4	2	no	—
52	C₄⋊C₄×C₈₃	—	2	4	no	—
53	C₄⋊C₄×C₈₃	—	2	2	no	—
54	C₄⋊C₄×C₈₃	C₄⋊C₄	1	4	yes	—
55	C₂²⋊C₄×C₈₃	—	2	2	no	—
56	C₄²×C₈₃	—	2	2	no	—
57	C₄²×C₈₃	C₂×D₄	1	2	yes	—
58	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
60	C₂³×C₈₃	#117	1	1	yes	characteristic
61	Q₈×C₈₃	—	4	4	no	—
62	C₂×C₄×C₈₃	—	2	1	no	—
63	C₂×C₄×C₈₃	—	2	1	no	—
64	C₂×C₄×C₈₃	—	2	4	no	—
65	C₂×C₄×C₈₃	—	4	2	no	—
66	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₈₃	#121	1	2	yes	—
69	C₂×C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
70	C₈×C₈₃	—	8	2	no	—
71	C₈₃⋊C₈	—	2	1	no	—
72	C₈₃⋊C₈	—	2	2	no	—
73	C₈₃⋊C₈	—	4	2	no	—
74	C₈₃⋊C₈	—	8	2	no	—
75	C₈₃⋊C₈	#101	1	2	yes	—
76	C₂²×C₈₃	—	2	1	no	—
77	C₂²×C₈₃	—	2	2	no	—
78	C₂²×C₈₃	#380	1	1	yes	characteristic
79	C₄×C₈₃	—	2	1	no	—
80	C₄×C₈₃	—	2	2	no	—
81	C₄×C₈₃	—	2	2	no	—
82	C₄×C₈₃	—	4	4	no	—
83	C₄×C₈₃	C₂×#101	1	1	yes	characteristic
84	C₄×C₈₃	#413	1	1	yes	characteristic
85	#8631	—	83	1	no	maximal, Sylow
86	C₂×C₈₃	—	4	1	no	—
87	C₂×C₈₃	#1801	1	1	yes	characteristic
88	C₂×C₈₃	#1814	1	2	yes	—
89	#2960	—	83	1	no	—
90	#1568	—	83	2	no	—
91	#1537	—	83	1	no	—
92	#1537	D₈₃	1	1	yes	characteristic
93	#1307	—	83	2	no	—
94	C₈₃	#8631	1	1	yes	characteristic, complemented, Sylow
95	#533	C₂×D₈₃	1	1	yes	characteristic
96	#446	—	166	2	no	—
97	C₈×Q₈	—	166	2	no	—
98	#432	—	83	2	no	—
99	#432	C₂×D₈₃	1	2	yes	—
100	#424	—	83	2	no	—
101	#423	—	83	1	no	—
102	#405	—	83	2	no	—
103	C₂×C₄×C₈	—	83	1	no	—
104	#335	—	166	2	no	—
105	#334	—	166	2	no	—
106	#330	—	2	2	no	—
107	#330	—	166	2	no	—
108	C₂×M₁₆	—	83	2	no	—
109	C₂²×C₈	—	83	1	no	—
110	C₂²×C₈	—	83	1	no	—
111	C₄⋊Q₈	Dic₁₆₆	1	2	yes	—
112	#125	C₂²⋊D₈₃	1	2	yes	—
113	C₂²⋊Q₈	—	2	2	no	—
114	C₄×Q₈	—	2	2	no	—
115	#117	C₄×D₈₃	1	2	yes	—
116	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	166	4	no	—
119	C₄⋊C₈	—	166	2	no	—
120	C₄⋊C₈	—	166	2	no	—
121	C₂²⋊C₈	—	2	2	no	—
122	C₂²⋊C₈	—	166	2	no	—
123	#97	—	2	2	no	—
124	#97	—	166	2	no	—
125	#97	—	166	2	no	—
126	C₄×C₈	—	83	2	no	—
127	C₄×C₈	—	83	2	no	—
128	C₄×C₈	—	166	2	no	—
129	C₂×Q₈	—	2	2	no	—
130	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
131	C₂²×C₄	#13050589	1	2	yes	—
132	M₁₆	—	332	2	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	—	332	2	no	—
135	C₂×C₈	—	332	1	no	—
136	C₂×C₈	—	83	1	no	—
137	C₂×C₈	—	83	1	no	—
138	C₂×C₈	—	83	2	no	—
139	C₂×C₈	—	166	2	no	—
140	C₂×C₈	—	166	2	no	—
141	C₄⋊C₄	—	2	4	no	—
142	C₄⋊C₄	—	2	2	no	—
143	C₄⋊C₄	Dic₈₃⋊C₄	1	4	yes	—
144	C₂²⋊C₄	—	2	2	no	—
145	C₄²	—	2	2	no	—
146	C₄²	D₄×D₈₃	1	2	yes	—
147	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
148	C₄²	C₂×Dic₁₆₆	1	1	yes	characteristic
149	C₂³	#14581748	1	1	yes	characteristic
150	Q₈	—	4	4	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	C₂²⋊C₄×D₈₃	1	1	yes	characteristic
156	C₂×C₄	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
157	C₂×C₄	#14581764	1	2	yes	—
158	C₂×C₄	#14581769	1	2	yes	—
159	C₈	—	8	2	no	—
160	C₈	—	332	2	no	—
161	C₈	—	83	2	no	—
162	C₈	—	664	2	no	—
163	C₈	—	166	1	no	—
164	C₈	—	166	2	no	—
165	C₂²	—	2	1	no	—
166	C₂²	—	2	2	no	—
167	C₂²	#19127559	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	2	no	—
170	C₄	—	2	2	no	—
171	C₄	—	4	4	no	—
172	C₄	#101×D₈₃	1	1	yes	characteristic
173	C₄	#19127684	1	1	yes	characteristic
174	C₂	—	4	1	no	—
175	C₂	#24907457	1	1	yes	characteristic
176	C₂	#24907515	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24916398

C₂

yes

maximal, characteristic

#24906224

C₂

yes

maximal

#24906101

C₂

yes

maximal, characteristic

#24905373

C₂

yes

maximal

C₈₃×#1537

C₂

yes

maximal, characteristic

#19127832

—

Q₈×C₈₃⋊C₈

—

#19127758

C₂²

yes

—

#19127728

C₄

yes

—

#19127725

C₂²

yes

characteristic

#19127657

C₄

yes

—

C₂×C₄×C₈₃⋊C₈

C₂²

yes

—

—

—

#533×C₈₃

C₂²

yes

characteristic

#432×C₈₃

C₂²

yes

—

#330×C₈₃

—

C₂×#13050574

C₂×C₄

yes

—

C₂²×C₈₃⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₈₃⋊C₈

C₂×C₄

yes

characteristic

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

—

C₂²⋊(C₈₃⋊C₈)

—

#14581703

—

#14581703

—

C₄×C₈₃⋊C₈

—

C₄×C₈₃⋊C₈

D₄

yes

—

C₄×C₈₃⋊C₈

D₄

yes

—

C₄⋊Q₈×C₈₃

Q₈

yes

—

#125×C₈₃

D₄

yes

—

C₂²⋊Q₈×C₈₃

—

C₄×Q₈×C₈₃

—

#117×C₈₃

C₂×C₄

yes

—

C₂×C₄²×C₈₃

C₂³

yes

characteristic

C₄⋊C₈×C₈₃

—

C₂²⋊C₈×C₈₃

—

#97×C₈₃

—

#13050574

—

C₂×C₈₃⋊C₈

—

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

C₂×Q₈×C₈₃

—

C₂²×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₃

#41

yes

—

C₂×C₈×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

C₄⋊C₄

yes

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

C₂×D₄

yes

—

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₄²×C₈₃

C₂×Q₈

yes

characteristic

C₂³×C₈₃

#117

yes

characteristic

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

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

—

C₈×C₈₃

—

C₈₃⋊C₈

—

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

—

C₄×C₈₃

yes

C₄×C₈₃

yes

—

C₂×C₈₃

—

C₂×C₈₃

#1801

yes

characteristic

C₂×C₈₃

#1814

yes

—

—

—

—

D₈₃

yes

characteristic

#1307

—

C₈₃

#8631

yes

characteristic, complemented, Sylow

#533

C₂×D₈₃

yes

characteristic

#446

—

166

—

C₈×Q₈

—

166

—

#432

—

#432

C₂×D₈₃

yes

—

100

#424

—

101

#423

—

102

#405

—

103

C₂×C₄×C₈

—

104

#335

—

166

—

105

#334

—

166

—

106

#330

—

107

#330

—

166

—

108

C₂×M₁₆

—

109

C₂²×C₈

—

110

C₂²×C₈

—

111

C₄⋊Q₈

Dic₁₆₆

yes

—

112

#125

C₂²⋊D₈₃

yes

—

113

C₂²⋊Q₈

—

114

C₄×Q₈

—

115

#117

C₄×D₈₃

yes

—

116

C₂×C₄²

C₂²×D₈₃

yes

characteristic

117

C₄⋊C₈

—

118

C₄⋊C₈

—

166

—

119

C₄⋊C₈

—

166

—

120

C₄⋊C₈

—

166

—

121

C₂²⋊C₈

—

122

C₂²⋊C₈

—

166

—

123

#97

—

124

#97

—

166

—

125

#97

—

166

—

126

C₄×C₈

—

127

C₄×C₈

—

128

C₄×C₈

—

166

—

129

C₂×Q₈

—

130

C₂²×C₄

C₂×C₄×D₈₃

yes

characteristic

131

C₂²×C₄

#13050589

yes

—

132

M₁₆

—

332

—

133

C₂×C₈

—

134

C₂×C₈

—

332

—

135

C₂×C₈

—

332

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

166

—

140

C₂×C₈

—

166

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

Dic₈₃⋊C₄

yes

—

144

C₂²⋊C₄

—

145

C₄²

—

146

C₄²

D₄×D₈₃

yes

—

147

C₄²

C₂×C₂²⋊D₈₃

yes

characteristic

148

C₄²

C₂×Dic₁₆₆

yes

characteristic

149

C₂³

#14581748

yes

characteristic

150

Q₈

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

C₂²⋊C₄×D₈₃

yes

characteristic

156

C₂×C₄

C₂×Dic₈₃⋊C₄

yes

characteristic

157

C₂×C₄

#14581764

yes

—

158

C₂×C₄

#14581769

yes

—

159

C₈

—

160

C₈

—

332

—

161

C₈

—

162

C₈

—

664

—

163

C₈

—

166

—

164

C₈

—

166

—

165

C₂²

—

166

C₂²

—

167

C₂²

#19127559

yes

characteristic

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

#101×D₈₃

yes

characteristic

173

C₄

#19127684

yes

characteristic

174

C₂

—

175

C₂

#24907457

yes

characteristic

176

C₂

#24907515

yes

—

	subgroup	G/H
center	C₂²	#19127559
commutator	C₂²×C₄×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1537	C₂
socle	C₂²×C₈₃	#380
prime radical	C₂³×C₈₃	#117
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1537	D₈₃
hypocenter	C₈₃	#8631
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1537

subgroup

G/H

center

C₂²

#19127559

commutator

C₂²×C₄×C₈₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₈₃×#1537

C₂

C₂²×C₈₃

C₂³×C₈₃

C₂×C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂²×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1537

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

H ≅ G/N

type

94. C₈₃

#8631

semidirect

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

subgroup	index
#24916398	2
#24906224	2
#24906101	2
#24905373	2
C₈₃×#1537	2
#8631	83

subgroup

index

C₈₃×#1537

Subgroups — #35297398

Conjugacy classes of subgroups

Complements

Maximal subgroups