Group #35298221 — subgroups

Subgroups — #35298221

Order 21248.

306 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 131 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	#24907600	C₂	1	2	yes	maximal
2	#24906222	C₂	1	1	yes	maximal, characteristic
3	#24906218	C₂	1	2	yes	maximal
4	C₂×#19127423	C₂	1	1	yes	maximal, characteristic
5	C₈₃×#1567	C₂	1	1	yes	maximal, characteristic
6	#19127725	C₂²	1	2	yes	—
7	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
9	#19127465	—	2	4	no	—
10	#19127454	—	2	2	no	—
11	#19127451	—	2	4	no	—
12	#19127423	C₄	1	4	yes	—
13	#423×C₈₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
15	#335×C₈₃	—	2	2	no	—
16	C₂×#13050574	—	2	2	no	—
17	C₂²×C₈₃⋊C₈	D₄	1	2	yes	—
18	C₂²×C₈₃⋊C₈	Q₈	1	2	yes	—
19	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
20	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
21	C₄×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
22	C₄×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
23	C₄×C₈₃⋊C₈	D₄	1	2	yes	—
24	C₄×C₈₃⋊C₈	Q₈	1	2	yes	—
25	C₂×M₁₆×C₈₃	—	2	2	no	—
26	C₂²×C₈×C₈₃	Q₈	1	2	yes	—
27	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₈₃	—	2	2	no	—
29	C₄⋊C₈×C₈₃	—	2	2	no	—
30	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₈₃	D₄	1	2	yes	—
32	#13050574	—	4	4	no	—
33	C₂×C₈₃⋊C₈	—	2	2	no	—
34	C₂×C₈₃⋊C₈	—	2	2	no	—
35	C₂×C₈₃⋊C₈	—	4	2	no	—
36	C₂×C₈₃⋊C₈	—	4	2	no	—
37	C₂×C₈₃⋊C₈	C₄⋊C₄	1	4	yes	—
38	C₂×C₈₃⋊C₈	C₄⋊C₄	1	4	yes	—
39	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
42	M₁₆×C₈₃	—	4	4	no	—
43	C₂×C₈×C₈₃	—	4	2	no	—
44	C₂×C₈×C₈₃	—	4	2	no	—
45	C₂×C₈×C₈₃	C₄⋊C₄	1	4	yes	—
46	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
50	C₂³×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
51	C₂×C₄×C₈₃	—	2	2	no	—
52	C₂×C₄×C₈₃	—	2	2	no	—
53	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
57	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₄×C₈₃	#135	1	2	yes	—
60	C₂×C₄×C₈₃	#135	1	2	yes	—
61	C₈×C₈₃	—	2	4	no	—
62	C₈×C₈₃	—	8	2	no	—
63	C₈₃⋊C₈	—	2	4	no	—
64	C₈₃⋊C₈	—	2	4	no	—
65	C₈₃⋊C₈	—	8	2	no	—
66	C₂²×C₈₃	—	2	1	no	—
67	C₂²×C₈₃	—	2	1	no	—
68	C₂²×C₈₃	#384	1	1	yes	characteristic
69	C₂²×C₈₃	#499	1	2	yes	—
70	C₄×C₈₃	—	2	1	no	—
71	C₄×C₈₃	—	2	1	no	—
72	C₄×C₈₃	#427	1	2	yes	—
73	C₄×C₈₃	#430	1	2	yes	—
74	C₄×C₈₃	#430	1	2	yes	—
75	C₄×C₈₃	#495	1	1	yes	characteristic
76	C₄×C₈₃	#495	1	1	yes	characteristic
77	#8827	—	83	1	no	maximal, Sylow
78	C₂×C₈₃	—	2	2	no	—
79	C₂×C₈₃	#1846	1	1	yes	characteristic
80	C₂×C₈₃	#1849	1	1	yes	characteristic
81	C₂×C₈₃	#1849	1	1	yes	characteristic
82	#1837	—	83	2	no	—
83	#1567	—	83	1	no	—
84	#1567	D₈₃	1	1	yes	characteristic
85	#1566	—	83	2	no	—
86	C₂×#322	—	83	1	no	—
87	C₈₃	#8827	1	1	yes	characteristic, complemented, Sylow
88	#423	—	83	2	no	—
89	#423	C₂×D₈₃	1	2	yes	—
90	C₂×C₄×C₈	—	83	1	no	—
91	C₂×C₄×C₈	—	83	1	no	—
92	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
93	#341	—	166	4	no	—
94	#335	—	2	2	no	—
95	#335	—	166	2	no	—
96	#334	—	166	4	no	—
97	#322	—	83	4	no	—
98	C₂×M₁₆	—	2	2	no	—
99	C₂×M₁₆	—	166	2	no	—
100	C₂²×C₈	—	83	2	no	—
101	C₂²×C₈	—	83	2	no	—
102	C₂²×C₈	Dic₁₆₆	1	2	yes	—
103	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	2	2	no	—
106	C₄⋊C₈	—	166	2	no	—
107	C₄⋊C₈	—	166	2	no	—
108	C₄×C₈	—	83	2	no	—
109	C₄×C₈	—	83	2	no	—
110	C₄×C₈	—	83	2	no	—
111	C₄×C₈	—	83	2	no	—
112	C₄×C₈	C₄×D₈₃	1	2	yes	—
113	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
114	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
115	C₂²×C₄	Q₈×D₈₃	1	1	yes	characteristic
116	C₂²×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
117	M₁₆	—	4	4	no	—
118	M₁₆	—	332	4	no	—
119	C₂×C₈	—	4	2	no	—
120	C₂×C₈	—	4	2	no	—
121	C₂×C₈	—	332	2	no	—
122	C₂×C₈	—	332	2	no	—
123	C₂×C₈	—	83	4	no	—
124	C₂×C₈	—	83	4	no	—
125	C₂×C₈	—	166	2	no	—
126	C₂×C₈	—	166	2	no	—
127	C₂×C₈	Dic₈₃⋊C₄	1	4	yes	—
128	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
129	C₄²	D₄×D₈₃	1	1	yes	characteristic
130	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
131	C₄²	Q₈×D₈₃	1	1	yes	characteristic
132	C₂³	#14581791	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
136	C₂×C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
137	C₂×C₄	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
138	C₂×C₄	#14581788	1	1	yes	characteristic
139	C₂×C₄	#14581790	1	1	yes	characteristic
140	C₂×C₄	#14581791	1	1	yes	characteristic
141	C₂×C₄	#14581811	1	2	yes	—
142	C₂×C₄	#14581815	1	2	yes	—
143	C₈	—	2	4	no	—
144	C₈	—	8	2	no	—
145	C₈	—	664	2	no	—
146	C₈	—	166	4	no	—
147	C₈	—	166	4	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	#19127570	1	1	yes	characteristic
151	C₂²	#19128102	1	2	yes	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	#19127738	1	2	yes	—
155	C₄	#19127748	1	2	yes	—
156	C₄	#19127749	1	2	yes	—
157	C₄	#19128084	1	1	yes	characteristic
158	C₄	#19128086	1	1	yes	characteristic
159	C₂	—	2	2	no	—
160	C₂	#24907626	1	1	yes	characteristic
161	C₂	#24907636	1	1	yes	characteristic
162	C₂	#24907637	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24907600

C₂

yes

maximal

#24906222

C₂

yes

maximal, characteristic

#24906218

C₂

yes

maximal

C₂×#19127423

C₂

yes

maximal, characteristic

C₈₃×#1567

C₂

yes

maximal, characteristic

#19127725

C₂²

yes

—

C₂×C₄×C₈₃⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₈₃⋊C₈

C₂²

yes

—

—

—

C₄

yes

—

#423×C₈₃

C₂²

yes

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

#335×C₈₃

—

C₂×#13050574

—

C₂²×C₈₃⋊C₈

D₄

yes

—

C₂²×C₈₃⋊C₈

Q₈

yes

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

—

C₄×C₈₃⋊C₈

C₂×C₄

yes

—

C₄×C₈₃⋊C₈

C₂×C₄

yes

—

C₄×C₈₃⋊C₈

D₄

yes

—

C₄×C₈₃⋊C₈

Q₈

yes

—

C₂×M₁₆×C₈₃

—

C₂²×C₈×C₈₃

Q₈

yes

—

C₂×C₄²×C₈₃

C₂³

yes

characteristic

C₄⋊C₈×C₈₃

—

C₄⋊C₈×C₈₃

—

C₄×C₈×C₈₃

C₂×C₄

yes

—

C₄×C₈×C₈₃

D₄

yes

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

C₄⋊C₄

yes

—

C₂×C₈₃⋊C₈

C₄⋊C₄

yes

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₈₃

C₂×Q₈

yes

characteristic

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₄⋊C₄

yes

—

C₄²×C₈₃

C₂²×C₄

yes

characteristic

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₄²×C₈₃

C₂×Q₈

yes

characteristic

C₂³×C₈₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

#127

yes

characteristic

C₂×C₄×C₈₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₃

#135

yes

—

C₂×C₄×C₈₃

#135

yes

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#384

yes

characteristic

C₂²×C₈₃

#499

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#427

yes

—

C₄×C₈₃

#430

yes

—

C₄×C₈₃

#430

yes

—

C₄×C₈₃

yes

C₄×C₈₃

yes

—

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

—

D₈₃

yes

characteristic

#1566

—

C₂×#322

—

C₈₃

#8827

yes

characteristic, complemented, Sylow

#423

—

#423

C₂×D₈₃

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

#341

—

166

—

#335

—

#335

—

166

—

#334

—

166

—

#322

—

C₂×M₁₆

—

C₂×M₁₆

—

166

—

100

C₂²×C₈

—

101

C₂²×C₈

—

102

C₂²×C₈

Dic₁₆₆

yes

—

103

C₂×C₄²

C₂²×D₈₃

yes

characteristic

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

166

—

107

C₄⋊C₈

—

166

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

C₄×D₈₃

yes

—

113

C₄×C₈

C₂²⋊D₈₃

yes

—

114

C₂²×C₄

D₄×D₈₃

yes

characteristic

115

C₂²×C₄

Q₈×D₈₃

yes

characteristic

116

C₂²×C₄

C₂×Dic₁₆₆

yes

characteristic

117

M₁₆

—

118

M₁₆

—

332

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

332

—

122

C₂×C₈

—

332

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

166

—

126

C₂×C₈

—

166

—

127

C₂×C₈

Dic₈₃⋊C₄

yes

—

128

C₄²

C₂×C₄×D₈₃

yes

characteristic

129

C₄²

D₄×D₈₃

yes

characteristic

130

C₄²

C₂×C₂²⋊D₈₃

yes

characteristic

131

C₄²

Q₈×D₈₃

yes

characteristic

132

C₂³

#14581791

yes

characteristic

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₄⋊C₄×D₈₃

yes

characteristic

136

C₂×C₄

C₄⋊C₄×D₈₃

yes

characteristic

137

C₂×C₄

C₂×Dic₈₃⋊C₄

yes

characteristic

138

C₂×C₄

#14581788

yes

characteristic

139

C₂×C₄

#14581790

yes

characteristic

140

C₂×C₄

#14581791

yes

characteristic

141

C₂×C₄

#14581811

yes

—

142

C₂×C₄

#14581815

yes

—

143

C₈

—

144

C₈

—

145

C₈

—

664

—

146

C₈

—

166

—

147

C₈

—

166

—

148

C₂²

—

149

C₂²

—

150

C₂²

#19127570

yes

characteristic

151

C₂²

#19128102

yes

—

152

C₄

—

153

C₄

—

154

C₄

#19127738

yes

—

155

C₄

#19127748

yes

—

156

C₄

#19127749

yes

—

157

C₄

#19128084

yes

characteristic

158

C₄

#19128086

yes

characteristic

159

C₂

—

160

C₂

#24907626

yes

characteristic

161

C₂

#24907636

yes

characteristic

162

C₂

#24907637

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14581788
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1567	C₂
socle	C₂²×C₈₃	#384
prime radical	C₂³×C₈₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1567	D₈₃
hypocenter	C₈₃	#8827
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

subgroup

G/H

center

C₂×C₄

#14581788

commutator

C₄²×C₈₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₈₃×#1567

C₂

socle

C₂²×C₈₃

#384

prime radical

C₂³×C₈₃

C₄⋊Q₈

cosocle

C₂×C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

N	H ≅ G/N	type
87. C₈₃	#8827	semidirect

H ≅ G/N

type

87. C₈₃

#8827

semidirect

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

subgroup	index
#24907600	2
#24906222	2
#24906218	2
C₂×#19127423	2
C₈₃×#1567	2
#8827	83

subgroup

index

C₈₃×#1567

Subgroups — #35298221

Conjugacy classes of subgroups

Complements

Maximal subgroups