Group #35309416 — subgroups

Subgroups — #35309416

Order 21248.

354 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 97 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	#24908295	C₂	1	1	yes	maximal, characteristic
2	#24906233	C₂	1	2	yes	maximal, complemented
3	#24905861	C₂	1	2	yes	maximal
4	#24905292	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₃×#1843	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128251	C₂²	1	1	yes	characteristic
7	#19127754	C₂²	1	2	yes	—
8	#19127725	C₂²	1	1	yes	characteristic
9	C₂×#14581720	—	2	2	no	—
10	#19127451	—	2	4	no	—
11	#19127428	—	2	4	no	—
12	C₂×C₄⋊C₈×C₈₃	—	2	2	no	—
13	C₂×C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
14	#376×C₈₃	C₂²	1	1	yes	characteristic
15	C₂×#13050590	—	2	1	no	—
16	C₂×#13050574	—	2	1	no	—
17	#14581792	Q₈	1	1	yes	characteristic
18	#14581789	Q₈	1	1	yes	characteristic
19	#14581780	—	2	1	no	—
20	#14581766	—	2	2	no	—
21	C₄×D₃₃₂	—	2	2	no	—
22	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
23	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
24	#14581720	—	4	4	no	—
25	C₂²⋊(C₈₃⋊C₈)	—	2	2	no	—
26	C₄×C₈₃⋊C₈	D₄	1	4	yes	—
27	C₂²×C₈×C₈₃	—	2	2	no	—
28	C₂×C₄²×C₈₃	D₄	1	2	yes	—
29	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₈₃	—	2	2	no	—
31	C₄⋊C₈×C₈₃	—	2	2	no	—
32	C₄⋊C₈×C₈₃	—	4	4	no	—
33	#95×C₈₃	—	2	2	no	—
34	#13050590	—	4	2	no	—
35	#13050590	—	4	2	no	—
36	C₂×Dic₁₆₆	—	2	1	no	—
37	C₂×D₃₃₂	—	2	2	no	—
38	C₂×D₃₃₂	—	2	1	no	—
39	C₂×C₂²⋊D₈₃	—	4	1	no	—
40	C₂×C₄×D₈₃	—	4	1	no	—
41	#13050574	—	4	2	no	—
42	C₂×C₈₃⋊C₈	—	2	4	no	—
43	C₂×C₈₃⋊C₈	—	4	1	no	—
44	C₄⋊Dic₈₃	—	2	2	no	—
45	(C₂×D₈₃)⋊C₄	—	4	2	no	—
46	C₂²×C₄×C₈₃	—	2	2	no	—
47	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₈₃	#41	1	2	yes	—
49	C₂×C₈×C₈₃	—	4	2	no	—
50	C₂×C₈×C₈₃	—	4	4	no	—
51	C₄²×C₈₃	—	2	2	no	—
52	C₄²×C₈₃	SD₁₆	1	4	yes	—
53	C₄²×C₈₃	C₂×D₄	1	2	yes	—
54	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
55	C₄²×C₈₃	#41	1	1	yes	characteristic
56	C₂³×C₈₃	#123	1	1	yes	characteristic
57	C₂²×D₈₃	—	4	1	no	—
58	C₂×C₄×C₈₃	—	2	4	no	—
59	C₂×C₄×C₈₃	—	4	2	no	—
60	C₂×C₄×C₈₃	—	4	4	no	—
61	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
63	C₂×C₄×C₈₃	#123	1	2	yes	—
64	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
65	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
66	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
67	C₂²⋊D₈₃	—	8	2	no	—
68	C₂×Dic₈₃	—	4	1	no	—
69	D₃₃₂	—	4	4	no	—
70	D₃₃₂	—	4	2	no	—
71	C₄×D₈₃	—	8	1	no	—
72	C₄×D₈₃	—	8	1	no	—
73	Dic₁₆₆	—	4	2	no	—
74	C₈×C₈₃	—	8	2	no	—
75	C₈₃⋊C₈	—	4	4	no	—
76	C₈₃⋊C₈	—	8	1	no	—
77	C₂²×C₈₃	—	2	2	no	—
78	C₂²×C₈₃	#399	1	1	yes	characteristic
79	C₂²×C₈₃	#481	1	2	yes	—
80	C₂×D₈₃	—	8	1	no	—
81	C₂×D₈₃	—	8	2	no	—
82	C₄×C₈₃	—	2	2	no	—
83	C₄×C₈₃	—	2	2	no	—
84	C₄×C₈₃	—	4	4	no	—
85	C₄×C₈₃	#476	1	2	yes	—
86	C₄×C₈₃	#492	1	1	yes	characteristic
87	C₄×C₈₃	#496	1	1	yes	characteristic
88	Dic₈₃	—	8	1	no	—
89	#10666	—	83	1	no	maximal, Sylow
90	C₂×C₈₃	—	2	2	no	—
91	C₂×C₈₃	#2094	1	1	yes	characteristic
92	C₂×C₈₃	#2098	1	2	yes	—
93	D₈₃	—	16	1	no	—
94	#1963	—	83	1	no	—
95	#1843	D₈₃	1	1	yes	characteristic, complemented
96	#1570	—	83	2	no	—
97	#1480	—	83	2	no	—
98	#1274	—	83	1	no	—
99	C₈₃	#10666	1	1	yes	characteristic, complemented, Sylow
100	#532	—	83	1	no	—
101	#431	—	83	2	no	—
102	#423	—	83	1	no	—
103	C₂×C₄⋊C₈	—	2	2	no	—
104	C₂×C₄⋊C₈	C₂×D₈₃	1	2	yes	—
105	C₂×#104	—	166	2	no	—
106	#376	C₂×D₈₃	1	1	yes	characteristic
107	#334	—	166	4	no	—
108	D₄⋊C₈	—	166	4	no	—
109	C₂×#41	—	166	1	no	—
110	C₂×M₁₆	—	166	1	no	—
111	C₂²×C₈	—	2	2	no	—
112	C₄⋊Q₈	—	83	1	no	—
113	#127	—	83	1	no	—
114	#124	—	166	1	no	—
115	#121	—	166	2	no	—
116	C₄×D₄	—	166	2	no	—
117	C₂×C₄²	C₂²⋊D₈₃	1	2	yes	—
118	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	4	4	no	—
122	C₄⋊C₈	—	166	2	no	—
123	C₄⋊C₈	—	166	2	no	—
124	#104	—	332	4	no	—
125	C₂²⋊C₈	—	166	2	no	—
126	C₄×C₈	—	83	4	no	—
127	#95	—	2	2	no	—
128	#41	—	332	2	no	—
129	#41	—	332	2	no	—
130	C₂×Q₈	—	166	1	no	—
131	C₂×D₄	—	332	1	no	—
132	C₂×D₄	—	166	1	no	—
133	C₂×D₄	—	166	2	no	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	—	332	1	no	—
136	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
137	C₂²×C₄	#13050590	1	2	yes	—
138	M₁₆	—	332	2	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	4	4	no	—
141	C₂×C₈	—	332	1	no	—
142	C₂×C₈	—	166	4	no	—
143	C₄⋊C₄	—	166	2	no	—
144	C₂²⋊C₄	—	332	2	no	—
145	C₄²	—	2	2	no	—
146	C₄²	#13050577	1	4	yes	—
147	C₄²	D₄×D₈₃	1	2	yes	—
148	C₄²	Q₈×D₈₃	1	1	yes	characteristic
149	C₄²	#13050591	1	1	yes	characteristic
150	C₂³	—	332	1	no	—
151	C₂³	#14581775	1	1	yes	characteristic
152	Q₈	—	332	2	no	—
153	D₄	—	332	4	no	—
154	D₄	—	332	2	no	—
155	D₄	—	664	2	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	—	332	1	no	—
160	C₂×C₄	—	664	1	no	—
161	C₂×C₄	—	664	1	no	—
162	C₂×C₄	#14581768	1	1	yes	characteristic
163	C₂×C₄	#14581770	1	2	yes	—
164	C₂×C₄	#14581777	1	2	yes	—
165	C₂×C₄	#14581788	1	1	yes	characteristic
166	C₂×C₄	C₂×#13050577	1	2	yes	—
167	C₂×C₄	#14581818	1	2	yes	—
168	C₈	—	8	2	no	—
169	C₈	—	332	4	no	—
170	C₈	—	664	1	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	664	1	no	—
173	C₂²	—	664	2	no	—
174	C₂²	#19127636	1	1	yes	characteristic
175	C₂²	#19128019	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	4	4	no	—
179	C₄	—	664	1	no	—
180	C₄	#19127988	1	2	yes	—
181	C₄	#19128075	1	1	yes	characteristic
182	C₄	#19128088	1	1	yes	characteristic
183	C₂	—	2	2	no	—
184	C₂	—	1328	1	no	—
185	C₂	#24908997	1	1	yes	characteristic
186	C₂	#24909019	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24908295

C₂

yes

maximal, characteristic

#24906233

C₂

yes

maximal, complemented

#24905861

C₂

yes

maximal

#24905292

C₂

yes

maximal, characteristic, complemented

C₈₃×#1843

C₂

yes

maximal, characteristic, complemented

#19128251

C₂²

yes

characteristic

#19127754

C₂²

yes

—

#19127725

C₂²

yes

—

—

—

C₂×C₄⋊C₈×C₈₃

—

C₂×C₄⋊C₈×C₈₃

C₂²

yes

—

#376×C₈₃

C₂²

yes

—

—

Q₈

yes

characteristic

#14581789

Q₈

yes

characteristic

#14581780

—

#14581766

—

C₄×D₃₃₂

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

—

#14581720

—

C₂²⋊(C₈₃⋊C₈)

—

C₄×C₈₃⋊C₈

D₄

yes

—

C₂²×C₈×C₈₃

—

C₂×C₄²×C₈₃

D₄

yes

—

C₂×C₄²×C₈₃

C₂³

yes

characteristic

C₄⋊C₈×C₈₃

—

C₄⋊C₈×C₈₃

—

C₄⋊C₈×C₈₃

—

#95×C₈₃

—

#13050590

—

#13050590

—

C₂×Dic₁₆₆

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

C₂×C₂²⋊D₈₃

—

C₂×C₄×D₈₃

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

(C₂×D₈₃)⋊C₄

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₃

#41

yes

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

SD₁₆

yes

—

C₄²×C₈₃

C₂×D₄

yes

—

C₄²×C₈₃

C₂×Q₈

yes

C₄²×C₈₃

yes

C₂³×C₈₃

yes

C₂²×D₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₈₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₃

#123

yes

—

C₂×C₄×C₈₃

#127

yes

characteristic

C₂×C₄×C₈₃

C₂×SD₁₆

yes

—

C₂×C₄×C₈₃

AGL(1,ℤ/8)

yes

—

C₂²⋊D₈₃

—

C₂×Dic₈₃

—

D₃₃₂

—

D₃₃₂

—

C₄×D₈₃

—

C₄×D₈₃

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

#399

yes

characteristic

C₂²×C₈₃

#481

yes

—

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#476

yes

—

C₄×C₈₃

yes

C₄×C₈₃

yes

Dic₈₃

—

#10666

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

#2094

yes

characteristic

C₂×C₈₃

#2098

yes

—

D₈₃

—

#1963

—

#1843

D₈₃

yes

characteristic, complemented

#1570

—

#1480

—

#1274

—

C₈₃

#10666

yes

characteristic, complemented, Sylow

100

#532

—

101

#431

—

102

#423

—

103

C₂×C₄⋊C₈

—

104

C₂×C₄⋊C₈

C₂×D₈₃

yes

—

105

C₂×#104

—

166

—

106

#376

C₂×D₈₃

yes

characteristic

107

#334

—

166

—

108

D₄⋊C₈

—

166

—

109

C₂×#41

—

166

—

110

C₂×M₁₆

—

166

—

111

C₂²×C₈

—

112

C₄⋊Q₈

—

113

#127

—

114

#124

—

166

—

115

#121

—

166

—

116

C₄×D₄

—

166

—

117

C₂×C₄²

C₂²⋊D₈₃

yes

—

118

C₂×C₄²

C₂²×D₈₃

yes

characteristic

119

C₄⋊C₈

—

120

C₄⋊C₈

—

121

C₄⋊C₈

—

122

C₄⋊C₈

—

166

—

123

C₄⋊C₈

—

166

—

124

#104

—

332

—

125

C₂²⋊C₈

—

166

—

126

C₄×C₈

—

127

#95

—

128

#41

—

332

—

129

#41

—

332

—

130

C₂×Q₈

—

166

—

131

C₂×D₄

—

332

—

132

C₂×D₄

—

166

—

133

C₂×D₄

—

166

—

134

C₂²×C₄

—

135

C₂²×C₄

—

332

—

136

C₂²×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

137

C₂²×C₄

#13050590

yes

—

138

M₁₆

—

332

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

332

—

142

C₂×C₈

—

166

—

143

C₄⋊C₄

—

166

—

144

C₂²⋊C₄

—

332

—

145

C₄²

—

146

C₄²

#13050577

yes

—

147

C₄²

D₄×D₈₃

yes

—

148

C₄²

Q₈×D₈₃

yes

characteristic

149

C₄²

#13050591

yes

characteristic

150

C₂³

—

332

—

151

C₂³

#14581775

yes

characteristic

152

Q₈

—

332

—

153

D₄

—

332

—

154

D₄

—

332

—

155

D₄

—

664

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

332

—

160

C₂×C₄

—

664

—

161

C₂×C₄

—

664

—

162

C₂×C₄

#14581768

yes

characteristic

163

C₂×C₄

#14581770

yes

—

164

C₂×C₄

#14581777

yes

—

165

C₂×C₄

#14581788

yes

characteristic

166

C₂×C₄

C₂×#13050577

yes

—

167

C₂×C₄

#14581818

yes

—

168

C₈

—

169

C₈

—

332

—

170

C₈

—

664

—

171

C₂²

—

172

C₂²

—

664

—

173

C₂²

—

664

—

174

C₂²

#19127636

yes

characteristic

175

C₂²

#19128019

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

664

—

180

C₄

#19127988

yes

—

181

C₄

#19128075

yes

characteristic

182

C₄

#19128088

yes

characteristic

183

C₂

—

184

C₂

—

1328

—

185

C₂

#24908997

yes

characteristic

186

C₂

#24909019

yes

—

	subgroup	G/H
center	C₂×C₄	#14581788
commutator, cosocle	C₂×C₄²×C₈₃	C₂³
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1843	C₂
socle	C₂²×C₈₃	#399
prime radical	#19128251	C₂²
hypercenter	#1843	D₈₃
hypocenter	C₈₃	#10666
LCS	G ▷ C₂×C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1843

subgroup

G/H

center

C₂×C₄

#14581788

commutator, cosocle

C₂×C₄²×C₈₃

C₂³

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₈₃×#1843

C₂

C₂²×C₈₃

C₂²

D₈₃

C₈₃

G ▷ C₂×C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1843

N	H ≅ G/N	type
2. #24906233	C₂	semidirect
4. #24905292	C₂	semidirect
5. C₈₃×#1843	C₂	semidirect
95. #1843	D₈₃	semidirect
99. C₈₃	#10666	semidirect

H ≅ G/N

type

2. #24906233

C₂

semidirect

4. #24905292

C₂

semidirect

5. C₈₃×#1843

C₂

semidirect

95. #1843

D₈₃

semidirect

99. C₈₃

#10666

semidirect

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

subgroup	index
#24908295	2
#24906233	2
#24905861	2
#24905292	2
C₈₃×#1843	2
#10666	83

subgroup

index

C₈₃×#1843

Subgroups — #35309416

Conjugacy classes of subgroups

Complements

Maximal subgroups