Group #35303840 — subgroups

Subgroups — #35303840

Order 21248.

414 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 159 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	#24907635	C₂	1	1	yes	maximal, characteristic
2	#24906148	C₂	1	2	yes	maximal
3	#24906144	C₂	1	2	yes	maximal
4	#24905746	C₂	1	1	yes	maximal, characteristic
5	C₂×#498×C₈₃	C₂	1	1	yes	maximal, characteristic
6	#19127754	C₂²	1	2	yes	—
7	C₂×#14581728	—	2	2	no	—
8	#19127725	C₂²	1	1	yes	characteristic
9	#19127725	C₂²	1	1	yes	characteristic
10	#19127450	—	2	4	no	—
11	#19127447	—	2	4	no	—
12	#19127422	—	2	1	no	—
13	C₈×C₈₃⋊C₈	—	2	1	no	—
14	C₂×C₄⋊Q₈×C₈₃	C₂²	1	2	yes	—
15	#498×C₈₃	—	2	1	no	—
16	#498×C₈₃	—	2	1	no	—
17	#498×C₈₃	C₄	1	4	yes	—
18	C₂×#106×C₈₃	—	2	2	no	—
19	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
20	C₂×#13050574	—	2	1	no	—
21	C₂×#13050574	—	2	1	no	—
22	#14581728	—	4	4	no	—
23	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
24	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
25	C₄⋊(C₈₃⋊C₈)	D₄	1	4	yes	—
26	C₂²⋊(C₈₃⋊C₈)	—	2	2	no	—
27	C₄×C₈₃⋊C₈	—	2	2	no	—
28	C₂²×Q₈×C₈₃	—	2	2	no	—
29	C₂²×C₈×C₈₃	D₄	1	2	yes	—
30	C₄⋊Q₈×C₈₃	—	2	2	no	—
31	C₄⋊Q₈×C₈₃	—	2	2	no	—
32	C₄⋊Q₈×C₈₃	—	2	2	no	—
33	C₄⋊Q₈×C₈₃	C₂×C₄	1	4	yes	—
34	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
36	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
37	#106×C₈₃	—	2	4	no	—
38	#106×C₈₃	—	2	4	no	—
39	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
41	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
42	#13050574	—	4	2	no	—
43	#13050574	—	4	2	no	—
44	C₂×C₈₃⋊C₈	—	2	4	no	—
45	C₂×C₈₃⋊C₈	—	4	1	no	—
46	C₂×C₈₃⋊C₈	—	4	1	no	—
47	C₂×Q₈×C₈₃	—	2	4	no	—
48	C₂×Q₈×C₈₃	—	4	2	no	—
49	C₂×Q₈×C₈₃	—	8	4	no	—
50	C₂²×C₄×C₈₃	—	4	2	no	—
51	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₈₃	#41	1	2	yes	—
53	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
55	C₂×C₈×C₈₃	SD₁₆	1	4	yes	—
56	C₂×C₈×C₈₃	Q₁₆	1	4	yes	—
57	C₄⋊C₄×C₈₃	—	2	4	no	—
58	C₄⋊C₄×C₈₃	—	2	4	no	—
59	C₄⋊C₄×C₈₃	—	4	2	no	—
60	C₄⋊C₄×C₈₃	—	4	2	no	—
61	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₈₃	C₂×D₄	1	2	yes	—
63	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
64	C₂³×C₈₃	#124	1	1	yes	characteristic
65	Q₈×C₈₃	—	8	4	no	—
66	Q₈×C₈₃	—	8	4	no	—
67	C₂×C₄×C₈₃	—	2	4	no	—
68	C₂×C₄×C₈₃	—	4	4	no	—
69	C₂×C₄×C₈₃	—	8	4	no	—
70	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₈₃	C₄×D₄	1	2	yes	—
72	C₂×C₄×C₈₃	#121	1	2	yes	—
73	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
74	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
75	C₂×C₄×C₈₃	C₂×Q₁₆	1	2	yes	—
76	C₈×C₈₃	#103	1	4	yes	—
77	C₈×C₈₃	#103	1	4	yes	—
78	C₈₃⋊C₈	—	4	4	no	—
79	C₈₃⋊C₈	—	8	1	no	—
80	C₈₃⋊C₈	—	8	1	no	—
81	C₂²×C₈₃	—	2	2	no	—
82	C₂²×C₈₃	#390	1	1	yes	characteristic
83	C₂²×C₈₃	#487	1	2	yes	—
84	C₄×C₈₃	—	2	2	no	—
85	C₄×C₈₃	—	8	4	no	—
86	C₄×C₈₃	C₂×#103	1	2	yes	—
87	C₄×C₈₃	#471	1	4	yes	—
88	C₄×C₈₃	#492	1	1	yes	characteristic
89	C₄×C₈₃	#494	1	1	yes	characteristic
90	#9767	—	83	1	no	maximal, Sylow
91	C₂×C₈₃	—	2	2	no	—
92	C₂×C₈₃	#1956	1	1	yes	characteristic
93	C₂×C₈₃	#1977	1	2	yes	—
94	C₂×#498	D₈₃	1	1	yes	characteristic
95	#1848	—	83	1	no	—
96	#1549	—	83	2	no	—
97	#1548	—	83	2	no	—
98	#1450	—	83	1	no	—
99	C₈₃	#9767	1	1	yes	characteristic, complemented, Sylow
100	C₂×C₄⋊Q₈	C₂×D₈₃	1	2	yes	—
101	#498	—	2	1	no	—
102	#498	—	2	1	no	—
103	#498	Dic₈₃	1	4	yes	—
104	#431	—	83	2	no	—
105	C₂×#108	—	166	2	no	—
106	C₂×#106	—	2	2	no	—
107	#423	—	83	1	no	—
108	#423	—	83	1	no	—
109	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
110	#333	—	166	4	no	—
111	#332	—	166	4	no	—
112	#322	—	166	1	no	—
113	C₈²	—	166	1	no	—
114	C₂²×Q₈	—	2	2	no	—
115	C₂×M₁₆	—	166	1	no	—
116	C₂×M₁₆	—	166	1	no	—
117	C₂²×C₈	C₂²⋊D₈₃	1	2	yes	—
118	C₄⋊Q₈	—	2	2	no	—
119	C₄⋊Q₈	—	2	2	no	—
120	C₄⋊Q₈	—	2	2	no	—
121	C₄⋊Q₈	C₂×Dic₈₃	1	4	yes	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
125	#108	—	332	4	no	—
126	#106	—	2	4	no	—
127	#106	—	2	4	no	—
128	C₄⋊C₈	—	83	4	no	—
129	C₄⋊C₈	—	166	2	no	—
130	C₄⋊C₈	—	166	2	no	—
131	C₂²⋊C₈	—	166	2	no	—
132	C₄×C₈	—	166	2	no	—
133	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
134	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
135	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
136	C₂×Q₈	—	2	4	no	—
137	C₂×Q₈	—	4	2	no	—
138	C₂×Q₈	—	8	4	no	—
139	C₂²×C₄	—	4	2	no	—
140	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
141	C₂²×C₄	#13050589	1	2	yes	—
142	M₁₆	—	332	2	no	—
143	M₁₆	—	332	2	no	—
144	C₂×C₈	—	332	1	no	—
145	C₂×C₈	—	332	1	no	—
146	C₂×C₈	—	166	4	no	—
147	C₂×C₈	C₂²⋊Dic₈₃	1	2	yes	—
148	C₂×C₈	C₂²⋊Dic₈₃	1	2	yes	—
149	C₂×C₈	#13050577	1	4	yes	—
150	C₂×C₈	#13050580	1	4	yes	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄⋊C₄	—	4	2	no	—
155	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
156	C₄²	D₄×D₈₃	1	2	yes	—
157	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
158	C₂³	#14581779	1	1	yes	characteristic
159	Q₈	—	8	4	no	—
160	Q₈	—	8	4	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	—	8	4	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	#14581788	1	1	yes	characteristic
168	C₂×C₄	C₂×#13050577	1	2	yes	—
169	C₂×C₄	C₂×#13050580	1	2	yes	—
170	C₈	—	332	4	no	—
171	C₈	—	664	1	no	—
172	C₈	—	664	1	no	—
173	C₈	#14581716	1	4	yes	—
174	C₈	#14581716	1	4	yes	—
175	C₂²	—	2	2	no	—
176	C₂²	#19127604	1	1	yes	characteristic
177	C₂²	#19128051	1	2	yes	—
178	C₄	—	2	2	no	—
179	C₄	—	8	4	no	—
180	C₄	C₂×#14581716	1	2	yes	—
181	C₄	#19127961	1	4	yes	—
182	C₄	#19128075	1	1	yes	characteristic
183	C₄	#19128080	1	1	yes	characteristic
184	C₂	—	2	2	no	—
185	C₂	#24908266	1	1	yes	characteristic
186	C₂	#24908368	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24907635

C₂

yes

maximal, characteristic

#24906148

C₂

yes

maximal

#24906144

C₂

yes

maximal

#24905746

C₂

yes

maximal, characteristic

C₂×#498×C₈₃

C₂

yes

maximal, characteristic

#19127754

C₂²

yes

—

C₂×#14581728

—

#19127725

C₂²

yes

characteristic

#19127725

C₂²

yes

—

—

—

C₈×C₈₃⋊C₈

—

C₂×C₄⋊Q₈×C₈₃

C₂²

yes

—

#498×C₈₃

—

#498×C₈₃

—

#498×C₈₃

C₄

yes

—

C₂×#106×C₈₃

—

C₂×C₄×C₈×C₈₃

C₂²

yes

—

—

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

D₄

yes

—

C₂²⋊(C₈₃⋊C₈)

—

C₄×C₈₃⋊C₈

—

C₂²×Q₈×C₈₃

—

C₂²×C₈×C₈₃

D₄

yes

—

C₄⋊Q₈×C₈₃

—

C₄⋊Q₈×C₈₃

—

C₄⋊Q₈×C₈₃

—

C₄⋊Q₈×C₈₃

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄²×C₈₃

C₂³

yes

characteristic

#106×C₈₃

—

#106×C₈₃

—

C₄×C₈×C₈₃

C₂×C₄

yes

—

C₄×C₈×C₈₃

D₄

yes

characteristic

C₄×C₈×C₈₃

D₄

yes

characteristic

#13050574

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₃

#41

yes

—

C₂×C₈×C₈₃

C₂²⋊C₄

yes

—

C₂×C₈×C₈₃

C₂²⋊C₄

yes

—

C₂×C₈×C₈₃

SD₁₆

yes

—

C₂×C₈×C₈₃

Q₁₆

yes

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

C₂²×C₄

yes

characteristic

C₄²×C₈₃

C₂×D₄

yes

—

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₂³×C₈₃

#124

yes

characteristic

Q₈×C₈₃

—

Q₈×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

#127

yes

characteristic

C₂×C₄×C₈₃

C₂×SD₁₆

yes

—

C₂×C₄×C₈₃

C₂×Q₁₆

yes

—

C₈×C₈₃

#103

yes

—

C₈×C₈₃

#103

yes

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

#390

yes

characteristic

C₂²×C₈₃

#487

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂×#103

yes

—

C₄×C₈₃

#471

yes

—

C₄×C₈₃

yes

C₄×C₈₃

yes

—

C₂×C₈₃

—

C₂×C₈₃

#1956

yes

characteristic

C₂×C₈₃

#1977

yes

—

C₂×#498

D₈₃

yes

—

—

—

—

C₈₃

#9767

yes

characteristic, complemented, Sylow

100

C₂×C₄⋊Q₈

C₂×D₈₃

yes

—

101

#498

—

102

#498

—

103

#498

Dic₈₃

yes

—

104

#431

—

105

C₂×#108

—

166

—

106

C₂×#106

—

107

#423

—

108

#423

—

109

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

110

#333

—

166

—

111

#332

—

166

—

112

#322

—

166

—

113

C₈²

—

166

—

114

C₂²×Q₈

—

115

C₂×M₁₆

—

166

—

116

C₂×M₁₆

—

166

—

117

C₂²×C₈

C₂²⋊D₈₃

yes

—

118

C₄⋊Q₈

—

119

C₄⋊Q₈

—

120

C₄⋊Q₈

—

121

C₄⋊Q₈

C₂×Dic₈₃

yes

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄²

C₂²×D₈₃

yes

characteristic

125

#108

—

332

—

126

#106

—

127

#106

—

128

C₄⋊C₈

—

129

C₄⋊C₈

—

166

—

130

C₄⋊C₈

—

166

—

131

C₂²⋊C₈

—

166

—

132

C₄×C₈

—

166

—

133

C₄×C₈

C₂×Dic₈₃

yes

—

134

C₄×C₈

C₂²⋊D₈₃

yes

characteristic

135

C₄×C₈

C₂²⋊D₈₃

yes

characteristic

136

C₂×Q₈

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂²×C₄

—

140

C₂²×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

141

C₂²×C₄

#13050589

yes

—

142

M₁₆

—

332

—

143

M₁₆

—

332

—

144

C₂×C₈

—

332

—

145

C₂×C₈

—

332

—

146

C₂×C₈

—

166

—

147

C₂×C₈

C₂²⋊Dic₈₃

yes

—

148

C₂×C₈

C₂²⋊Dic₈₃

yes

—

149

C₂×C₈

#13050577

yes

—

150

C₂×C₈

#13050580

yes

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄²

C₂²×Dic₈₃

yes

characteristic

156

C₄²

D₄×D₈₃

yes

—

157

C₄²

C₂×C₂²⋊D₈₃

yes

characteristic

158

C₂³

#14581779

yes

characteristic

159

Q₈

—

160

Q₈

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

C₂×C₂²⋊Dic₈₃

yes

characteristic

165

C₂×C₄

D₄×Dic₈₃

yes

—

166

C₂×C₄

#14581763

yes

—

167

C₂×C₄

#14581788

yes

characteristic

168

C₂×C₄

C₂×#13050577

yes

—

169

C₂×C₄

C₂×#13050580

yes

—

170

C₈

—

332

—

171

C₈

—

664

—

172

C₈

—

664

—

173

C₈

#14581716

yes

—

174

C₈

#14581716

yes

—

175

C₂²

—

176

C₂²

#19127604

yes

characteristic

177

C₂²

#19128051

yes

—

178

C₄

—

179

C₄

—

180

C₄

C₂×#14581716

yes

—

181

C₄

#19127961

yes

—

182

C₄

#19128075

yes

characteristic

183

C₄

#19128080

yes

characteristic

184

C₂

—

185

C₂

#24908266

yes

characteristic

186

C₂

#24908368

yes

—

	subgroup	G/H
center	C₂²	#19127604
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₂×#498×C₈₃	C₂
socle	C₂²×C₈₃	#390
prime radical	C₂³×C₈₃	#124
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	C₂×#498	D₈₃
hypocenter	C₈₃	#9767
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#498

subgroup

G/H

center

C₂²

#19127604

commutator

C₄²×C₈₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₂×#498×C₈₃

C₂

C₂²×C₈₃

C₂³×C₈₃

C₂×C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#498

N	H ≅ G/N	type
99. C₈₃	#9767	semidirect

H ≅ G/N

type

99. C₈₃

#9767

semidirect

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

subgroup	index
#24907635	2
#24906148	2
#24906144	2
#24905746	2
C₂×#498×C₈₃	2
#9767	83

subgroup

index

C₂×#498×C₈₃

Subgroups — #35303840

Conjugacy classes of subgroups

Complements

Maximal subgroups