Group #35322939 — subgroups

Subgroups — #35322939

Order 21248.

334 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 139 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	#24916397	C₂	1	1	yes	maximal, characteristic
2	#24909219	C₂	1	1	yes	maximal, characteristic
3	#24905551	C₂	1	4	yes	maximal
4	C₂×#346×C₈₃	C₂	1	1	yes	maximal, characteristic
5	#19128258	C₄	1	1	yes	characteristic
6	C₂×#14581732	C₂²	1	2	yes	—
7	#19127831	—	2	2	no	—
8	C₈×Dic₁₆₆	—	2	2	no	—
9	#19127727	C₄	1	2	yes	—
10	#19127725	C₄	1	1	yes	characteristic
11	#19127656	C₂²	1	2	yes	—
12	C₄⋊(C₈₃⋊C₁₆)	—	2	2	no	—
13	C₂²×C₁₆×C₈₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
15	#346×C₈₃	—	2	2	no	—
16	C₂×#13050574	C₂×C₄	1	2	yes	—
17	#14581792	C₈	1	2	yes	—
18	#14581784	C₈	1	2	yes	—
19	#14581771	—	2	2	no	—
20	C₄×Dic₁₆₆	—	2	2	no	—
21	#14581751	C₂×C₄	1	2	yes	—
22	#14581732	—	2	4	no	—
23	C₂×C₈₃⋊C₁₆	—	2	2	no	—
24	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
25	Dic₈₃⋊C₈	—	2	4	no	—
26	#14581702	—	2	2	no	—
27	C₈×Dic₈₃	—	2	2	no	—
28	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₈₃	—	2	2	no	—
32	C₂×C₁₆×C₈₃	—	2	2	no	—
33	C₂×C₁₆×C₈₃	Q₈	1	4	yes	complemented
34	C₄×C₈×C₈₃	D₄	1	2	yes	—
35	C₄×C₈×C₈₃	D₄	1	2	yes	—
36	C₂×Dic₁₆₆	—	2	2	no	—
37	#13050574	—	2	4	no	—
38	C₂×C₈₃⋊C₈	—	2	2	no	—
39	C₄⋊Dic₈₃	—	2	2	no	—
40	C₄⋊Dic₈₃	C₂×C₈	1	4	yes	—
41	Dic₈₃⋊C₄	—	2	4	no	—
42	C₂²⋊Dic₈₃	—	2	2	no	—
43	C₄×Dic₈₃	—	2	2	no	—
44	C₈₃⋊C₁₆	—	4	2	no	—
45	C₂²×C₄×C₈₃	M₁₆	1	2	yes	—
46	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₈₃	—	2	2	no	—
48	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
49	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₈₃	C₂×Q₈	1	2	yes	—
52	C₂×C₈×C₈₃	#41	1	2	yes	—
53	C₄²×C₈₃	C₂²⋊C₄	1	2	yes	—
54	C₄²×C₈₃	C₂×C₈	1	1	yes	characteristic
55	C₄²×C₈₃	C₂×C₈	1	1	yes	characteristic
56	C₁₆×C₈₃	—	2	4	no	—
57	C₂³×C₈₃	C₂×M₁₆	1	1	yes	characteristic
58	C₂×C₄×C₈₃	—	4	2	no	—
59	C₂×C₄×C₈₃	C₂²⋊C₈	1	2	yes	—
60	C₂×C₄×C₈₃	C₂²⋊C₈	1	2	yes	—
61	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₈₃	C₄×Q₈	1	2	yes	—
63	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₈₃	#131	1	2	yes	—
65	C₂×Dic₈₃	—	2	4	no	—
66	Dic₁₆₆	—	4	4	no	—
67	C₈×C₈₃	—	2	2	no	—
68	C₈×C₈₃	C₂²⋊Q₈	1	2	yes	—
69	C₈×C₈₃	#124	1	1	yes	characteristic
70	C₈×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
71	C₈₃⋊C₈	—	4	2	no	—
72	C₂²×C₈₃	—	2	2	no	—
73	C₂²×C₈₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
74	C₂²×C₈₃	#446	1	2	yes	—
75	C₄×C₈₃	—	2	2	no	—
76	C₄×C₈₃	—	2	2	no	—
77	C₄×C₈₃	#391	1	1	yes	characteristic
78	C₄×C₈₃	#408	1	1	yes	characteristic
79	C₄×C₈₃	C₈×Q₈	1	2	yes	—
80	Dic₈₃	—	4	4	no	—
81	#13267	—	83	1	no	maximal, Sylow
82	C₂×C₈₃	—	2	2	no	—
83	C₂×C₈₃	#1984	1	1	yes	characteristic
84	C₂×C₈₃	#2117	1	2	yes	—
85	#2960	—	83	1	no	—
86	#2150	—	83	1	no	—
87	C₂×#346	D₈₃	1	1	yes	characteristic
88	#1379	—	83	4	no	—
89	C₈₃	#13267	1	1	yes	characteristic, complemented, Sylow
90	#533	—	83	1	no	—
91	C₂×M₃₂	—	83	2	no	—
92	C₂²×C₁₆	C₂×D₈₃	1	2	yes	—
93	#446	—	166	2	no	—
94	C₈×Q₈	—	166	2	no	—
95	#424	—	83	2	no	—
96	#423	—	83	1	no	—
97	#405	—	83	2	no	—
98	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
99	C₄⋊C₁₆	—	166	2	no	—
100	#346	—	2	2	no	—
101	C₂×M₁₆	—	83	2	no	—
102	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
104	C₄⋊Q₈	—	83	2	no	—
105	#125	—	83	2	no	—
106	C₂²⋊Q₈	—	166	2	no	—
107	C₄×Q₈	—	166	2	no	—
108	#117	—	83	2	no	—
109	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
110	M₃₂	—	166	4	no	—
111	C₂×C₁₆	—	2	2	no	—
112	C₂×C₁₆	—	2	2	no	—
113	C₂×C₁₆	—	166	2	no	—
114	C₂×C₁₆	Dic₁₆₆	1	4	yes	complemented
115	C₄⋊C₈	—	166	4	no	—
116	C₄⋊C₈	—	166	2	no	—
117	#97	—	166	2	no	—
118	C₄×C₈	—	166	2	no	—
119	C₄×C₈	D₃₃₂	1	2	yes	—
120	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
121	C₂×Q₈	—	166	2	no	—
122	C₂²×C₄	#13050573	1	2	yes	—
123	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
124	M₁₆	—	166	4	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	166	2	no	—
127	C₂×C₈	(C₂×D₈₃)⋊C₄	1	2	yes	—
128	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
129	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
130	C₂×C₈	C₂×Dic₁₆₆	1	2	yes	—
131	C₂×C₈	#13050590	1	2	yes	—
132	C₄⋊C₄	—	83	4	no	—
133	C₄⋊C₄	—	166	4	no	—
134	C₄⋊C₄	—	166	2	no	—
135	C₂²⋊C₄	—	166	2	no	—
136	C₄²	—	166	2	no	—
137	C₄²	(C₂×D₈₃)⋊C₄	1	2	yes	—
138	C₄²	C₈×D₈₃	1	1	yes	characteristic
139	C₄²	C₈×D₈₃	1	1	yes	characteristic
140	C₁₆	—	2	4	no	—
141	C₁₆	—	332	2	no	—
142	C₂³	C₂×#13050573	1	1	yes	characteristic
143	Q₈	—	332	4	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	166	4	no	—
146	C₂×C₄	(C₂×D₈₃)⋊C₈	1	2	yes	—
147	C₂×C₄	(C₂×D₈₃)⋊C₈	1	2	yes	—
148	C₂×C₄	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
149	C₂×C₄	C₄×Dic₁₆₆	1	2	yes	—
150	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
151	C₂×C₄	#14581798	1	2	yes	—
152	C₈	—	2	2	no	—
153	C₈	—	332	2	no	—
154	C₈	#14581771	1	2	yes	—
155	C₈	#14581780	1	1	yes	characteristic
156	C₈	#14581792	1	1	yes	characteristic
157	C₂²	—	2	2	no	—
158	C₂²	C₂×(C₂×D₈₃)⋊C₈	1	1	yes	characteristic
159	C₂²	#19127831	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	332	4	no	—
163	C₄	#19127607	1	1	yes	characteristic
164	C₄	#19127667	1	1	yes	characteristic
165	C₄	C₈×Dic₁₆₆	1	2	yes	—
166	C₂	—	2	2	no	—
167	C₂	#24908399	1	1	yes	characteristic
168	C₂	#24909090	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24916397

C₂

yes

maximal, characteristic

#24909219

C₂

yes

maximal, characteristic

#24905551

C₂

yes

maximal

C₂×#346×C₈₃

C₂

yes

maximal, characteristic

#19128258

C₄

yes

characteristic

C₂×#14581732

C₂²

yes

—

#19127831

—

C₈×Dic₁₆₆

—

#19127727

C₄

yes

—

#19127725

C₄

yes

characteristic

#19127656

C₂²

yes

—

C₄⋊(C₈₃⋊C₁₆)

—

C₂²×C₁₆×C₈₃

C₂²

yes

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

#346×C₈₃

—

C₂×#13050574

C₂×C₄

yes

—

#14581792

C₈

yes

—

#14581784

C₈

yes

—

#14581771

—

C₄×Dic₁₆₆

—

#14581751

C₂×C₄

yes

—

#14581732

—

C₂×C₈₃⋊C₁₆

—

C₄⋊(C₈₃⋊C₈)

—

Dic₈₃⋊C₈

—

#14581702

—

C₈×Dic₈₃

—

C₂²×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₈₃

C₂³

yes

characteristic

C₂×C₄²×C₈₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₃

—

C₂×C₁₆×C₈₃

—

C₂×C₁₆×C₈₃

Q₈

yes

complemented

C₄×C₈×C₈₃

D₄

yes

—

C₄×C₈×C₈₃

D₄

yes

—

C₂×Dic₁₆₆

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

C₂×C₈

yes

—

Dic₈₃⋊C₄

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

—

C₈₃⋊C₁₆

—

C₂²×C₄×C₈₃

M₁₆

yes

—

C₂²×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂²⋊C₄

yes

—

C₂×C₈×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

C₂×Q₈

yes

—

C₂×C₈×C₈₃

#41

yes

—

C₄²×C₈₃

C₂²⋊C₄

yes

—

C₄²×C₈₃

C₂×C₈

yes

characteristic

C₄²×C₈₃

C₂×C₈

yes

characteristic

C₁₆×C₈₃

—

C₂³×C₈₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²⋊C₈

yes

—

C₂×C₄×C₈₃

C₂²⋊C₈

yes

—

C₂×C₄×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄×Q₈

yes

—

C₂×C₄×C₈₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₈₃

#131

yes

—

C₂×Dic₈₃

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

C₂²⋊Q₈

yes

—

C₈×C₈₃

#124

yes

characteristic

C₈×C₈₃

C₄⋊Q₈

yes

characteristic

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₈₃

#446

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

C₈×Q₈

yes

—

Dic₈₃

—

#13267

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

#1984

yes

characteristic

C₂×C₈₃

#2117

yes

—

#2960

—

#2150

—

C₂×#346

D₈₃

yes

characteristic

#1379

—

C₈₃

#13267

yes

characteristic, complemented, Sylow

#533

—

C₂×M₃₂

—

C₂²×C₁₆

C₂×D₈₃

yes

—

#446

—

166

—

C₈×Q₈

—

166

—

#424

—

#423

—

#405

—

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

C₄⋊C₁₆

—

166

—

100

#346

—

101

C₂×M₁₆

—

102

C₂²×C₈

C₄×D₈₃

yes

characteristic

103

C₂²×C₈

C₂²×D₈₃

yes

characteristic

104

C₄⋊Q₈

—

105

#125

—

106

C₂²⋊Q₈

—

166

—

107

C₄×Q₈

—

166

—

108

#117

—

109

C₂×C₄²

C₄×D₈₃

yes

characteristic

110

M₃₂

—

166

—

111

C₂×C₁₆

—

112

C₂×C₁₆

—

113

C₂×C₁₆

—

166

—

114

C₂×C₁₆

Dic₁₆₆

yes

complemented

115

C₄⋊C₈

—

166

—

116

C₄⋊C₈

—

166

—

117

#97

—

166

—

118

C₄×C₈

—

166

—

119

C₄×C₈

D₃₃₂

yes

—

120

C₄×C₈

C₂²⋊D₈₃

yes

—

121

C₂×Q₈

—

166

—

122

C₂²×C₄

#13050573

yes

—

123

C₂²×C₄

C₂×C₄×D₈₃

yes

characteristic

124

M₁₆

—

166

—

125

C₂×C₈

—

126

C₂×C₈

—

166

—

127

C₂×C₈

(C₂×D₈₃)⋊C₄

yes

—

128

C₂×C₈

C₂×C₂²⋊D₈₃

yes

characteristic

129

C₂×C₈

C₂×D₃₃₂

yes

characteristic

130

C₂×C₈

C₂×Dic₁₆₆

yes

—

131

C₂×C₈

#13050590

yes

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

166

—

134

C₄⋊C₄

—

166

—

135

C₂²⋊C₄

—

166

—

136

C₄²

—

166

—

137

C₄²

(C₂×D₈₃)⋊C₄

yes

—

138

C₄²

C₈×D₈₃

yes

characteristic

139

C₄²

C₈×D₈₃

yes

characteristic

140

C₁₆

—

141

C₁₆

—

332

—

142

C₂³

C₂×#13050573

yes

characteristic

143

Q₈

—

332

—

144

C₂×C₄

—

145

C₂×C₄

—

166

—

146

C₂×C₄

(C₂×D₈₃)⋊C₈

yes

—

147

C₂×C₄

(C₂×D₈₃)⋊C₈

yes

—

148

C₂×C₄

C₂×(C₂×D₈₃)⋊C₄

yes

characteristic

149

C₂×C₄

C₄×Dic₁₆₆

yes

—

150

C₂×C₄

C₂×C₈×D₈₃

yes

characteristic

151

C₂×C₄

#14581798

yes

—

152

C₈

—

153

C₈

—

332

—

154

C₈

#14581771

yes

—

155

C₈

#14581780

yes

characteristic

156

C₈

#14581792

yes

characteristic

157

C₂²

—

158

C₂²

C₂×(C₂×D₈₃)⋊C₈

yes

characteristic

159

C₂²

#19127831

yes

—

160

C₄

—

161

C₄

—

162

C₄

—

332

—

163

C₄

#19127607

yes

characteristic

164

C₄

#19127667

yes

characteristic

165

C₄

C₈×Dic₁₆₆

yes

—

166

C₂

—

167

C₂

#24908399

yes

characteristic

168

C₂

#24909090

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×D₃₃₂
commutator	C₂×C₄×C₈₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₂×#346×C₈₃	C₂
socle	C₂²×C₈₃	C₂×C₂²⋊C₈
prime radical	C₂³×C₈₃	C₂×M₁₆
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₂×#346	D₈₃
hypocenter	C₈₃	#13267
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₈

C₂×D₃₃₂

commutator

C₂×C₄×C₈₃

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₈₃

Fitting

C₂×#346×C₈₃

C₂

socle

C₂²×C₈₃

C₂×C₂²⋊C₈

prime radical

C₂³×C₈₃

C₂×M₁₆

cosocle

C₂²×C₈×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
33. C₂×C₁₆×C₈₃	Q₈	semidirect
89. C₈₃	#13267	semidirect
114. C₂×C₁₆	Dic₁₆₆	semidirect

H ≅ G/N

type

33. C₂×C₁₆×C₈₃

Q₈

semidirect

89. C₈₃

#13267

semidirect

114. C₂×C₁₆

Dic₁₆₆

semidirect

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

subgroup	index
#24916397	2
#24909219	2
#24905551	2
C₂×#346×C₈₃	2
#13267	83

subgroup

index

C₂×#346×C₈₃

Subgroups — #35322939

Conjugacy classes of subgroups

Complements

Maximal subgroups