Group #35303868 — subgroups

Subgroups — #35303868

Order 21248.

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

normal subgroups only

This group has 478 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	#24906135	C₂	1	4	yes	maximal
3	#24905746	C₂	1	1	yes	maximal, characteristic
4	C₂×#494×C₈₃	C₂	1	1	yes	maximal, characteristic
5	#19127754	C₂²	1	2	yes	—
6	C₂×#14581728	—	2	2	no	—
7	#19127725	C₂²	1	2	yes	—
8	#19127447	—	2	8	no	—
9	#19127422	—	2	1	no	—
10	C₈×C₈₃⋊C₈	—	2	1	no	—
11	C₂²×Q₁₆×C₈₃	—	2	2	no	—
12	C₂×C₄⋊Q₈×C₈₃	C₂²	1	2	yes	—
13	#494×C₈₃	—	2	1	no	—
14	#494×C₈₃	—	2	1	no	—
15	#494×C₈₃	C₄	1	4	yes	—
16	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
17	C₂×#13050574	—	2	2	no	—
18	#14581728	—	4	4	no	—
19	C₄⋊(C₈₃⋊C₈)	—	2	4	no	—
20	C₄⋊(C₈₃⋊C₈)	D₄	1	4	yes	—
21	C₂²⋊(C₈₃⋊C₈)	—	2	2	no	—
22	C₄×C₈₃⋊C₈	—	2	2	no	—
23	C₂²×Q₈×C₈₃	—	2	2	no	—
24	C₂×Q₁₆×C₈₃	—	2	4	no	—
25	C₂×Q₁₆×C₈₃	—	2	4	no	—
26	C₂×Q₁₆×C₈₃	—	4	4	no	—
27	C₂×Q₁₆×C₈₃	—	4	4	no	—
28	C₂²×C₈×C₈₃	D₄	1	2	yes	—
29	C₄⋊Q₈×C₈₃	—	2	4	no	—
30	C₄⋊Q₈×C₈₃	—	2	2	no	—
31	C₄⋊Q₈×C₈₃	C₂×C₄	1	4	yes	—
32	C₂×C₄⋊C₄×C₈₃	—	2	4	no	—
33	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
34	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
36	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
37	#13050574	—	4	4	no	—
38	C₂×C₈₃⋊C₈	—	2	4	no	—
39	C₂×C₈₃⋊C₈	—	4	2	no	—
40	C₂×Q₈×C₈₃	—	2	4	no	—
41	C₂×Q₈×C₈₃	—	4	2	no	—
42	C₂×Q₈×C₈₃	—	4	4	no	—
43	C₂×Q₈×C₈₃	—	4	4	no	—
44	C₂²×C₄×C₈₃	—	4	2	no	—
45	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₈₃	#41	1	2	yes	—
47	Q₁₆×C₈₃	—	4	8	no	—
48	Q₁₆×C₈₃	—	4	8	no	—
49	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
50	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈×C₈₃	D₈	1	4	yes	—
52	C₂×C₈×C₈₃	SD₁₆	1	4	yes	—
53	C₄⋊C₄×C₈₃	—	2	8	no	—
54	C₄⋊C₄×C₈₃	—	4	4	no	—
55	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₈₃	C₂×D₄	1	2	yes	—
57	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
58	C₂³×C₈₃	#124	1	1	yes	characteristic
59	Q₈×C₈₃	—	4	8	no	—
60	Q₈×C₈₃	—	8	2	no	—
61	Q₈×C₈₃	—	8	2	no	—
62	C₂×C₄×C₈₃	—	2	4	no	—
63	C₂×C₄×C₈₃	—	4	4	no	—
64	C₂×C₄×C₈₃	—	8	4	no	—
65	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₈₃	C₄×D₄	1	2	yes	—
67	C₂×C₄×C₈₃	#121	1	2	yes	—
68	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
69	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
70	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
71	C₈×C₈₃	#102	1	4	yes	—
72	C₈×C₈₃	#102	1	4	yes	—
73	C₈₃⋊C₈	—	4	4	no	—
74	C₈₃⋊C₈	—	8	2	no	—
75	C₂²×C₈₃	—	2	2	no	—
76	C₂²×C₈₃	#390	1	1	yes	characteristic
77	C₂²×C₈₃	#486	1	2	yes	—
78	C₄×C₈₃	—	2	2	no	—
79	C₄×C₈₃	—	8	4	no	—
80	C₄×C₈₃	C₂×#102	1	2	yes	—
81	C₄×C₈₃	#473	1	4	yes	—
82	C₄×C₈₃	#492	1	1	yes	characteristic
83	C₄×C₈₃	#493	1	1	yes	characteristic
84	#9772	—	83	1	no	maximal, Sylow
85	C₂×C₈₃	—	2	2	no	—
86	C₂×C₈₃	#1957	1	1	yes	characteristic
87	C₂×C₈₃	#1979	1	2	yes	—
88	C₂×#494	D₈₃	1	1	yes	characteristic
89	#1848	—	83	1	no	—
90	#1546	—	83	4	no	—
91	#1450	—	83	1	no	—
92	C₈₃	#9772	1	1	yes	characteristic, complemented, Sylow
93	C₂²×Q₁₆	—	2	2	no	—
94	C₂×C₄⋊Q₈	C₂×D₈₃	1	2	yes	—
95	#494	—	2	1	no	—
96	#494	—	2	1	no	—
97	#494	Dic₈₃	1	4	yes	—
98	#431	—	83	2	no	—
99	C₂×#108	—	166	2	no	—
100	#423	—	83	2	no	—
101	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
102	#332	—	166	8	no	—
103	#322	—	166	1	no	—
104	C₈²	—	166	1	no	—
105	C₂²×Q₈	—	2	2	no	—
106	C₂×Q₁₆	—	2	4	no	—
107	C₂×Q₁₆	—	2	4	no	—
108	C₂×Q₁₆	—	4	4	no	—
109	C₂×Q₁₆	—	4	4	no	—
110	C₂×M₁₆	—	166	2	no	—
111	C₂²×C₈	C₂²⋊D₈₃	1	2	yes	—
112	C₄⋊Q₈	—	2	4	no	—
113	C₄⋊Q₈	—	2	2	no	—
114	C₄⋊Q₈	C₂×Dic₈₃	1	4	yes	—
115	C₂×C₄⋊C₄	—	2	4	no	—
116	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
117	#108	—	332	4	no	—
118	C₄⋊C₈	—	83	4	no	—
119	C₄⋊C₈	—	166	4	no	—
120	C₂²⋊C₈	—	166	2	no	—
121	C₄×C₈	—	166	2	no	—
122	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
123	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
124	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
125	C₂×Q₈	—	2	4	no	—
126	C₂×Q₈	—	4	2	no	—
127	C₂×Q₈	—	4	4	no	—
128	C₂×Q₈	—	4	4	no	—
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
131	C₂²×C₄	#13050589	1	2	yes	—
132	Q₁₆	—	4	8	no	—
133	Q₁₆	—	4	8	no	—
134	M₁₆	—	332	4	no	—
135	C₂×C₈	—	332	2	no	—
136	C₂×C₈	—	166	4	no	—
137	C₂×C₈	C₂²⋊Dic₈₃	1	2	yes	—
138	C₂×C₈	C₂²⋊Dic₈₃	1	2	yes	—
139	C₂×C₈	#13050575	1	4	yes	—
140	C₂×C₈	#13050578	1	4	yes	—
141	C₄⋊C₄	—	2	8	no	—
142	C₄⋊C₄	—	4	4	no	—
143	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
144	C₄²	D₄×D₈₃	1	2	yes	—
145	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
146	C₂³	#14581779	1	1	yes	characteristic
147	Q₈	—	4	8	no	—
148	Q₈	—	8	2	no	—
149	Q₈	—	8	2	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	4	4	no	—
152	C₂×C₄	—	8	4	no	—
153	C₂×C₄	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
154	C₂×C₄	D₄×Dic₈₃	1	2	yes	—
155	C₂×C₄	#14581763	1	2	yes	—
156	C₂×C₄	#14581788	1	1	yes	characteristic
157	C₂×C₄	C₂×#13050575	1	2	yes	—
158	C₂×C₄	C₂×#13050578	1	2	yes	—
159	C₈	—	332	4	no	—
160	C₈	—	664	2	no	—
161	C₈	#14581713	1	4	yes	—
162	C₈	#14581713	1	4	yes	—
163	C₂²	—	2	2	no	—
164	C₂²	#19127604	1	1	yes	characteristic
165	C₂²	#19128047	1	2	yes	—
166	C₄	—	2	2	no	—
167	C₄	—	8	4	no	—
168	C₄	C₂×#14581713	1	2	yes	—
169	C₄	#19127973	1	4	yes	—
170	C₄	#19128073	1	1	yes	characteristic
171	C₄	#19128077	1	1	yes	characteristic
172	C₂	—	2	2	no	—
173	C₂	#24908270	1	1	yes	characteristic
174	C₂	#24908379	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24907635

C₂

yes

maximal, characteristic

#24906135

C₂

yes

maximal

#24905746

C₂

yes

maximal, characteristic

C₂×#494×C₈₃

C₂

yes

maximal, characteristic

#19127754

C₂²

yes

—

C₂×#14581728

—

#19127725

C₂²

yes

—

#19127447

—

#19127422

—

C₈×C₈₃⋊C₈

—

C₂²×Q₁₆×C₈₃

—

C₂×C₄⋊Q₈×C₈₃

C₂²

yes

—

#494×C₈₃

—

#494×C₈₃

—

#494×C₈₃

C₄

yes

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂×#13050574

—

#14581728

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

D₄

yes

—

C₂²⋊(C₈₃⋊C₈)

—

C₄×C₈₃⋊C₈

—

C₂²×Q₈×C₈₃

—

C₂×Q₁₆×C₈₃

—

C₂×Q₁₆×C₈₃

—

C₂×Q₁₆×C₈₃

—

C₂×Q₁₆×C₈₃

—

C₂²×C₈×C₈₃

D₄

yes

—

C₄⋊Q₈×C₈₃

—

C₄⋊Q₈×C₈₃

—

C₄⋊Q₈×C₈₃

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄²×C₈₃

C₂³

yes

characteristic

C₄×C₈×C₈₃

C₂×C₄

yes

—

C₄×C₈×C₈₃

D₄

yes

characteristic

C₄×C₈×C₈₃

D₄

yes

characteristic

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₃

#41

yes

—

Q₁₆×C₈₃

—

Q₁₆×C₈₃

—

C₂×C₈×C₈₃

C₂²⋊C₄

yes

—

C₂×C₈×C₈₃

C₂²⋊C₄

yes

—

C₂×C₈×C₈₃

D₈

yes

—

C₂×C₈×C₈₃

SD₁₆

yes

—

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

—

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×D₈

yes

—

C₂×C₄×C₈₃

C₂×SD₁₆

yes

—

C₈×C₈₃

#102

yes

—

C₈×C₈₃

#102

yes

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

#390

yes

characteristic

C₂²×C₈₃

#486

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂×#102

yes

—

C₄×C₈₃

#473

yes

—

C₄×C₈₃

yes

C₄×C₈₃

yes

—

C₂×C₈₃

—

C₂×C₈₃

#1957

yes

characteristic

C₂×C₈₃

#1979

yes

—

C₂×#494

D₈₃

yes

—

—

—

C₈₃

#9772

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

C₂×D₈₃

yes

—

#494

—

#494

—

#494

Dic₈₃

yes

—

#431

—

C₂×#108

—

166

—

100

#423

—

101

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

102

#332

—

166

—

103

#322

—

166

—

104

C₈²

—

166

—

105

C₂²×Q₈

—

106

C₂×Q₁₆

—

107

C₂×Q₁₆

—

108

C₂×Q₁₆

—

109

C₂×Q₁₆

—

110

C₂×M₁₆

—

166

—

111

C₂²×C₈

C₂²⋊D₈₃

yes

—

112

C₄⋊Q₈

—

113

C₄⋊Q₈

—

114

C₄⋊Q₈

C₂×Dic₈₃

yes

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄²

C₂²×D₈₃

yes

characteristic

117

#108

—

332

—

118

C₄⋊C₈

—

119

C₄⋊C₈

—

166

—

120

C₂²⋊C₈

—

166

—

121

C₄×C₈

—

166

—

122

C₄×C₈

C₂×Dic₈₃

yes

—

123

C₄×C₈

C₂²⋊D₈₃

yes

characteristic

124

C₄×C₈

C₂²⋊D₈₃

yes

characteristic

125

C₂×Q₈

—

126

C₂×Q₈

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂²×C₄

—

130

C₂²×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

131

C₂²×C₄

#13050589

yes

—

132

Q₁₆

—

133

Q₁₆

—

134

M₁₆

—

332

—

135

C₂×C₈

—

332

—

136

C₂×C₈

—

166

—

137

C₂×C₈

C₂²⋊Dic₈₃

yes

—

138

C₂×C₈

C₂²⋊Dic₈₃

yes

—

139

C₂×C₈

#13050575

yes

—

140

C₂×C₈

#13050578

yes

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄²

C₂²×Dic₈₃

yes

characteristic

144

C₄²

D₄×D₈₃

yes

—

145

C₄²

C₂×C₂²⋊D₈₃

yes

characteristic

146

C₂³

#14581779

yes

characteristic

147

Q₈

—

148

Q₈

—

149

Q₈

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

C₂×C₂²⋊Dic₈₃

yes

characteristic

154

C₂×C₄

D₄×Dic₈₃

yes

—

155

C₂×C₄

#14581763

yes

—

156

C₂×C₄

#14581788

yes

characteristic

157

C₂×C₄

C₂×#13050575

yes

—

158

C₂×C₄

C₂×#13050578

yes

—

159

C₈

—

332

—

160

C₈

—

664

—

161

C₈

#14581713

yes

—

162

C₈

#14581713

yes

—

163

C₂²

—

164

C₂²

#19127604

yes

characteristic

165

C₂²

#19128047

yes

—

166

C₄

—

167

C₄

—

168

C₄

C₂×#14581713

yes

—

169

C₄

#19127973

yes

—

170

C₄

#19128073

yes

characteristic

171

C₄

#19128077

yes

characteristic

172

C₂

—

173

C₂

#24908270

yes

characteristic

174

C₂

#24908379

yes

—

	subgroup	G/H
center	C₂²	#19127604
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₂×#494×C₈₃	C₂
socle	C₂²×C₈₃	#390
prime radical	C₂³×C₈₃	#124
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	C₂×#494	D₈₃
hypocenter	C₈₃	#9772
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#494

subgroup

G/H

center

C₂²

#19127604

commutator

C₄²×C₈₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₂×#494×C₈₃

C₂

C₂²×C₈₃

C₂³×C₈₃

C₂×C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#494

N	H ≅ G/N	type
92. C₈₃	#9772	semidirect

H ≅ G/N

type

92. C₈₃

#9772

semidirect

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

subgroup	index
#24907635	2
#24906135	2
#24905746	2
C₂×#494×C₈₃	2
#9772	83

subgroup

index

C₂×#494×C₈₃

Subgroups — #35303868

Conjugacy classes of subgroups

Complements

Maximal subgroups