Group #35303801 — subgroups

Subgroups — #35303801

Order 21248.

478 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 119 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	#24907594	C₂	1	1	yes	maximal, characteristic, complemented
2	#24906023	C₂	1	4	yes	maximal, complemented
3	#24905755	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₃×#3147	C₂	1	1	yes	maximal, characteristic
5	#19127754	C₂²	1	2	yes	—
6	#19127567	C₂²	1	2	yes	—
7	#19127464	—	2	2	no	—
8	#19127435	—	2	8	no	—
9	#19127423	—	2	2	no	—
10	C₂×#135×C₈₃	—	2	2	no	—
11	#532×C₈₃	C₂²	1	2	yes	—
12	#495×C₈₃	—	2	2	no	—
13	#494×C₈₃	C₄	1	1	yes	characteristic, complemented
14	#493×C₈₃	C₄	1	1	yes	characteristic, complemented
15	#492×C₈₃	C₄	1	2	yes	complemented
16	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₈₃	—	2	2	no	—
18	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
19	C₂²⋊(C₈₃⋊C₈)	D₄	1	4	yes	—
20	C₄×C₈₃⋊C₈	—	2	2	no	—
21	C₈₃⋊#95	—	2	4	no	—
22	C₂×#41×C₈₃	—	2	2	no	—
23	#135×C₈₃	—	4	8	no	—
24	C₂×Q₁₆×C₈₃	—	2	2	no	—
25	C₂×SD₁₆×C₈₃	—	2	4	no	—
26	C₂×D₈×C₈₃	—	2	2	no	—
27	C₂²×C₈×C₈₃	D₄	1	2	yes	—
28	C₄⋊Q₈×C₈₃	C₂×C₄	1	2	yes	—
29	#127×C₈₃	C₂×C₄	1	2	yes	—
30	#124×C₈₃	—	2	2	no	—
31	#121×C₈₃	—	2	4	no	—
32	C₄×D₄×C₈₃	—	2	4	no	—
33	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
34	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₈₃	D₄	1	2	yes	—
37	C₂²×Dic₈₃	—	4	2	no	—
38	C₂×C₈₃⋊C₈	—	2	4	no	—
39	C₄⋊Dic₈₃	—	4	4	no	—
40	#41×C₈₃	—	4	8	no	—
41	C₂×Q₈×C₈₃	—	2	2	no	—
42	C₂×D₄×C₈₃	—	2	2	no	—
43	C₂×D₄×C₈₃	—	2	4	no	—
44	C₂×D₄×C₈₃	—	4	2	no	—
45	C₂²×C₄×C₈₃	—	4	2	no	—
46	C₂²×C₄×C₈₃	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
48	Q₁₆×C₈₃	—	4	4	no	—
49	SD₁₆×C₈₃	—	4	8	no	—
50	D₈×C₈₃	—	4	4	no	—
51	C₂×C₈×C₈₃	—	2	4	no	—
52	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
53	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
54	C₄⋊C₄×C₈₃	—	2	4	no	—
55	C₂²⋊C₄×C₈₃	—	4	4	no	—
56	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₈₃	#41	1	2	yes	—
59	C₂³×C₈₃	—	4	2	no	—
60	C₂³×C₈₃	#127	1	1	yes	characteristic
61	Q₈×C₈₃	—	4	4	no	—
62	D₄×C₈₃	—	4	4	no	—
63	D₄×C₈₃	—	8	4	no	—
64	D₄×C₈₃	—	8	4	no	—
65	C₂×C₄×C₈₃	—	4	2	no	—
66	C₂×C₄×C₈₃	—	4	2	no	—
67	C₂×C₄×C₈₃	—	8	4	no	—
68	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₈₃	C₄×D₄	1	2	yes	—
70	C₂×C₄×C₈₃	#121	1	2	yes	—
71	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
72	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	4	yes	—
73	C₂×Dic₈₃	—	8	4	no	—
74	C₂×Dic₈₃	—	8	2	no	—
75	C₈×C₈₃	—	2	2	no	—
76	C₈×C₈₃	—	2	2	no	—
77	C₈₃⋊C₈	—	4	4	no	—
78	C₂²×C₈₃	—	2	2	no	—
79	C₂²×C₈₃	—	8	2	no	—
80	C₂²×C₈₃	—	8	4	no	—
81	C₂²×C₈₃	#390	1	1	yes	characteristic
82	C₂²×C₈₃	#496	1	2	yes	—
83	C₄×C₈₃	—	2	2	no	—
84	C₄×C₈₃	—	2	2	no	—
85	C₄×C₈₃	—	8	2	no	—
86	C₄×C₈₃	#418	1	2	yes	—
87	C₄×C₈₃	#490	1	2	yes	—
88	Dic₈₃	—	16	2	no	—
89	#9760	—	83	1	no	maximal, Sylow
90	C₂×C₈₃	—	2	2	no	—
91	C₂×C₈₃	—	16	2	no	—
92	C₂×C₈₃	#1961	1	1	yes	characteristic
93	C₂×C₈₃	#1976	1	2	yes	—
94	#3147	D₈₃	1	1	yes	characteristic
95	#1835	—	83	1	no	—
96	#1517	—	83	4	no	—
97	#1452	—	83	1	no	—
98	C₈₃	#9760	1	1	yes	characteristic, complemented, Sylow
99	C₂×#135	—	2	2	no	—
100	#532	C₂×D₈₃	1	2	yes	—
101	#495	—	2	2	no	—
102	#494	Dic₈₃	1	1	yes	characteristic, complemented
103	#493	Dic₈₃	1	1	yes	characteristic, complemented
104	#492	Dic₈₃	1	2	yes	complemented
105	#431	—	83	2	no	—
106	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
107	#382	—	83	2	no	—
108	#340	—	166	2	no	—
109	#327	—	166	8	no	—
110	#322	—	166	2	no	—
111	C₂×#41	—	2	2	no	—
112	#135	—	4	8	no	—
113	C₂×Q₁₆	—	2	2	no	—
114	C₂×SD₁₆	—	2	4	no	—
115	C₂×D₈	—	2	2	no	—
116	C₂²×C₈	C₂²⋊D₈₃	1	2	yes	—
117	C₄⋊Q₈	C₂×Dic₈₃	1	2	yes	—
118	#127	C₂×Dic₈₃	1	2	yes	—
119	#124	—	2	2	no	—
120	#121	—	2	4	no	—
121	C₄×D₄	—	2	4	no	—
122	C₂×C₄⋊C₄	—	166	2	no	—
123	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
124	C₄⋊C₈	—	166	2	no	—
125	C₂²⋊C₈	—	83	4	no	—
126	C₄×C₈	—	166	2	no	—
127	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
128	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
129	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
130	#95	—	166	4	no	—
131	#41	—	4	8	no	—
132	C₂×Q₈	—	2	2	no	—
133	C₂×D₄	—	2	2	no	—
134	C₂×D₄	—	2	4	no	—
135	C₂×D₄	—	4	2	no	—
136	C₂²×C₄	—	4	2	no	—
137	C₂²×C₄	—	332	2	no	—
138	C₂²×C₄	D₄×D₈₃	1	2	yes	—
139	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
140	Q₁₆	—	4	4	no	—
141	SD₁₆	—	4	8	no	—
142	D₈	—	4	4	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	—	166	4	no	—
145	C₂×C₈	C₂²⋊Dic₈₃	1	2	yes	—
146	C₂×C₈	C₂²⋊Dic₈₃	1	2	yes	—
147	C₄⋊C₄	—	2	4	no	—
148	C₄⋊C₄	—	332	4	no	—
149	C₂²⋊C₄	—	4	4	no	—
150	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
151	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
152	C₄²	#13050589	1	2	yes	—
153	C₂³	—	4	2	no	—
154	C₂³	#14581788	1	1	yes	characteristic
155	Q₈	—	4	4	no	—
156	D₄	—	4	4	no	—
157	D₄	—	8	4	no	—
158	D₄	—	8	4	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	8	4	no	—
162	C₂×C₄	—	664	4	no	—
163	C₂×C₄	—	664	2	no	—
164	C₂×C₄	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
165	C₂×C₄	D₄×Dic₈₃	1	2	yes	—
166	C₂×C₄	#14581763	1	2	yes	—
167	C₂×C₄	#14581779	1	1	yes	characteristic
168	C₂×C₄	#14581818	1	4	yes	—
169	C₈	—	2	2	no	—
170	C₈	—	2	2	no	—
171	C₈	—	332	4	no	—
172	C₂²	—	2	2	no	—
173	C₂²	—	8	2	no	—
174	C₂²	—	8	4	no	—
175	C₂²	#19127604	1	1	yes	characteristic
176	C₂²	#19128088	1	2	yes	—
177	C₄	—	2	2	no	—
178	C₄	—	2	2	no	—
179	C₄	—	8	2	no	—
180	C₄	—	1328	2	no	—
181	C₄	#19127706	1	2	yes	—
182	C₄	#19128065	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	—	16	2	no	—
185	C₂	#24908288	1	1	yes	characteristic
186	C₂	#24908362	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24907594

C₂

yes

maximal, characteristic, complemented

#24906023

C₂

yes

maximal, complemented

#24905755

C₂

yes

maximal, characteristic, complemented

C₈₃×#3147

C₂

yes

maximal, characteristic

#19127754

C₂²

yes

—

#19127567

C₂²

yes

—

#19127464

—

#19127435

—

#19127423

—

C₂×#135×C₈₃

—

#532×C₈₃

C₂²

yes

—

#495×C₈₃

—

#494×C₈₃

C₄

yes

characteristic, complemented

#493×C₈₃

C₄

yes

characteristic, complemented

#492×C₈₃

C₄

yes

complemented

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₃

—

C₄⋊(C₈₃⋊C₈)

—

C₂²⋊(C₈₃⋊C₈)

D₄

yes

—

C₄×C₈₃⋊C₈

—

C₈₃⋊#95

—

C₂×#41×C₈₃

—

#135×C₈₃

—

C₂×Q₁₆×C₈₃

—

C₂×SD₁₆×C₈₃

—

C₂×D₈×C₈₃

—

C₂²×C₈×C₈₃

D₄

yes

—

C₄⋊Q₈×C₈₃

C₂×C₄

yes

—

#127×C₈₃

C₂×C₄

yes

—

#124×C₈₃

—

#121×C₈₃

—

C₄×D₄×C₈₃

—

C₂×C₄²×C₈₃

C₂³

yes

characteristic

C₄×C₈×C₈₃

C₂×C₄

yes

characteristic

C₄×C₈×C₈₃

C₂×C₄

yes

characteristic

C₄×C₈×C₈₃

D₄

yes

—

C₂²×Dic₈₃

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

#41×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂×D₄

yes

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

Q₁₆×C₈₃

—

SD₁₆×C₈₃

—

D₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂²⋊C₄

yes

—

C₂×C₈×C₈₃

C₂²⋊C₄

yes

—

C₄⋊C₄×C₈₃

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

C₂²×C₄

yes

characteristic

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₄²×C₈₃

#41

yes

—

C₂³×C₈₃

—

C₂³×C₈₃

#127

yes

characteristic

Q₈×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄×D₄

yes

—

C₂×C₄×C₈₃

#121

yes

—

C₂×C₄×C₈₃

#124

yes

characteristic

C₂×C₄×C₈₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#390

yes

characteristic

C₂²×C₈₃

#496

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#418

yes

—

C₄×C₈₃

#490

yes

—

Dic₈₃

—

#9760

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#1961

yes

characteristic

C₂×C₈₃

#1976

yes

—

#3147

D₈₃

yes

—

—

—

C₈₃

#9760

yes

characteristic, complemented, Sylow

C₂×#135

—

100

#532

C₂×D₈₃

yes

—

101

#495

—

102

#494

Dic₈₃

yes

characteristic, complemented

103

#493

Dic₈₃

yes

characteristic, complemented

104

#492

Dic₈₃

yes

complemented

105

#431

—

106

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

107

#382

—

108

#340

—

166

—

109

#327

—

166

—

110

#322

—

166

—

111

C₂×#41

—

112

#135

—

113

C₂×Q₁₆

—

114

C₂×SD₁₆

—

115

C₂×D₈

—

116

C₂²×C₈

C₂²⋊D₈₃

yes

—

117

C₄⋊Q₈

C₂×Dic₈₃

yes

—

118

#127

C₂×Dic₈₃

yes

—

119

#124

—

120

#121

—

121

C₄×D₄

—

122

C₂×C₄⋊C₄

—

166

—

123

C₂×C₄²

C₂²×D₈₃

yes

characteristic

124

C₄⋊C₈

—

166

—

125

C₂²⋊C₈

—

126

C₄×C₈

—

166

—

127

C₄×C₈

C₂×Dic₈₃

yes

characteristic

128

C₄×C₈

C₂×Dic₈₃

yes

characteristic

129

C₄×C₈

C₂²⋊D₈₃

yes

—

130

#95

—

166

—

131

#41

—

132

C₂×Q₈

—

133

C₂×D₄

—

134

C₂×D₄

—

135

C₂×D₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

332

—

138

C₂²×C₄

D₄×D₈₃

yes

—

139

C₂²×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

140

Q₁₆

—

141

SD₁₆

—

142

D₈

—

143

C₂×C₈

—

144

C₂×C₈

—

166

—

145

C₂×C₈

C₂²⋊Dic₈₃

yes

—

146

C₂×C₈

C₂²⋊Dic₈₃

yes

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

332

—

149

C₂²⋊C₄

—

150

C₄²

C₂²×Dic₈₃

yes

characteristic

151

C₄²

C₂×C₂²⋊D₈₃

yes

characteristic

152

C₄²

#13050589

yes

—

153

C₂³

—

154

C₂³

#14581788

yes

characteristic

155

Q₈

—

156

D₄

—

157

D₄

—

158

D₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

664

—

163

C₂×C₄

—

664

—

164

C₂×C₄

C₂×C₂²⋊Dic₈₃

yes

characteristic

165

C₂×C₄

D₄×Dic₈₃

yes

—

166

C₂×C₄

#14581763

yes

—

167

C₂×C₄

#14581779

yes

characteristic

168

C₂×C₄

#14581818

yes

—

169

C₈

—

170

C₈

—

171

C₈

—

332

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

#19127604

yes

characteristic

176

C₂²

#19128088

yes

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

1328

—

181

C₄

#19127706

yes

—

182

C₄

#19128065

yes

—

183

C₂

—

184

C₂

—

185

C₂

#24908288

yes

characteristic

186

C₂

#24908362

yes

—

	subgroup	G/H
center	C₂²	#19127604
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting, prime radical	C₈₃×#3147	C₂
socle	C₂²×C₈₃	#390
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#3147	D₈₃
hypocenter	C₈₃	#9760
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3147

subgroup

G/H

center

C₂²

#19127604

commutator

C₄²×C₈₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting, prime radical

C₈₃×#3147

C₂

socle

C₂²×C₈₃

#390

cosocle

C₂×C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3147

N	H ≅ G/N	type
1. #24907594	C₂	semidirect
2. #24906023	C₂	semidirect
3. #24905755	C₂	semidirect
13. #494×C₈₃	C₄	semidirect
14. #493×C₈₃	C₄	semidirect
15. #492×C₈₃	C₄	semidirect
98. C₈₃	#9760	semidirect
102. #494	Dic₈₃	semidirect
103. #493	Dic₈₃	semidirect
104. #492	Dic₈₃	semidirect

H ≅ G/N

type

1. #24907594

C₂

semidirect

2. #24906023

C₂

semidirect

3. #24905755

C₂

semidirect

13. #494×C₈₃

C₄

semidirect

14. #493×C₈₃

C₄

semidirect

15. #492×C₈₃

C₄

semidirect

98. C₈₃

#9760

semidirect

102. #494

Dic₈₃

semidirect

103. #493

Dic₈₃

semidirect

104. #492

Dic₈₃

semidirect

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

subgroup	index
#24907594	2
#24906023	2
#24905755	2
C₈₃×#3147	2
#9760	83

subgroup

index

C₈₃×#3147

Subgroups — #35303801

Conjugacy classes of subgroups

Complements

Maximal subgroups