Group #46782742 — subgroups

Subgroups — #46782742

Order 33664.

272 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 87 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	C₂×#22554435	C₂	1	1	yes	maximal, characteristic
2	#30681961	C₂	1	2	yes	maximal
3	C₂×#22554388	C₂	1	2	yes	maximal
4	#30681707	C₂	1	1	yes	maximal, characteristic
5	#343×C₂₆₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₅₂₆	C₂²	1	1	yes	characteristic
7	C₂×#16862360	C₂²	1	2	yes	—
8	#22554435	—	2	2	no	—
9	#22554435	—	2	2	no	—
10	C₂×Dic₂₆₃⋊C₄	C₄	1	2	yes	—
11	C₂×C₄×Dic₂₆₃	C₄	1	2	yes	—
12	C₂×C₄×Dic₂₆₃	C₂²	1	1	yes	characteristic
13	Dic₂₆₃⋊C₈	—	2	2	no	—
14	Dic₂₆₃⋊C₈	—	2	2	no	—
15	#22554388	—	2	4	no	—
16	C₂×M₁₆×C₂₆₃	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₂₆₃	C₂²	1	1	yes	characteristic
18	C₂×Dic₅₂₆	—	2	2	no	—
19	C₂×Dic₅₂₆	—	2	2	no	—
20	C₂×Dic₅₂₆	D₄	1	2	yes	—
21	C₂×Dic₅₂₆	Q₈	1	2	yes	—
22	C₂²×Dic₂₆₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₂₆₃	C₂×C₄	1	2	yes	—
24	C₂²×Dic₂₆₃	C₂×C₄	1	1	yes	characteristic
25	#16862360	—	2	2	no	—
26	#16862360	—	2	2	no	—
27	C₂×C₂₆₃⋊C₈	—	2	2	no	—
28	Dic₂₆₃⋊C₄	—	2	4	no	—
29	C₄×Dic₂₆₃	—	2	2	no	—
30	C₄×Dic₂₆₃	—	2	2	no	—
31	C₄×Dic₂₆₃	D₄	1	2	yes	—
32	C₄×Dic₂₆₃	D₄	1	2	yes	—
33	C₂²×C₄×C₂₆₃	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₂₆₃	C₂³	1	1	yes	characteristic
35	M₁₆×C₂₆₃	—	2	2	no	—
36	M₁₆×C₂₆₃	—	2	2	no	—
37	C₂×C₈×C₂₆₃	—	2	2	no	—
38	C₄⋊C₄×C₂₆₃	—	2	2	no	—
39	Dic₅₂₆	—	2	4	no	—
40	Dic₅₂₆	—	2	4	no	—
41	C₂×Dic₂₆₃	—	2	2	no	—
42	C₂×Dic₂₆₃	—	2	2	no	—
43	C₂×Dic₂₆₃	—	2	2	no	—
44	C₂×Dic₂₆₃	—	4	1	no	—
45	C₂×Dic₂₆₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×Dic₂₆₃	C₂²⋊C₄	1	2	yes	—
47	C₂×Dic₂₆₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×Dic₂₆₃	C₄⋊C₄	1	4	yes	—
49	C₂₆₃⋊C₈	—	4	2	no	—
50	C₂³×C₂₆₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₂₆₃	—	2	2	no	—
52	C₂×C₄×C₂₆₃	—	4	2	no	—
53	C₂×C₄×C₂₆₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₂₆₃	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₂₆₃	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₄×C₂₆₃	#41	1	2	yes	—
57	C₈×C₂₆₃	—	4	2	no	—
58	C₂²×C₂₆₃	—	2	1	no	—
59	C₂²×C₂₆₃	—	2	1	no	—
60	C₂²×C₂₆₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₂₆₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₂₆₃	#117	1	1	yes	characteristic
63	C₄×C₂₆₃	—	4	2	no	—
64	C₄×C₂₆₃	#121	1	2	yes	—
65	C₄×C₂₆₃	C₂²⋊Q₈	1	2	yes	—
66	Dic₂₆₃	—	2	4	no	—
67	Dic₂₆₃	—	2	1	no	—
68	Dic₂₆₃	—	2	2	no	—
69	Dic₂₆₃	#101	1	2	yes	complemented
70	C₂×C₂₆₃	—	2	1	no	—
71	C₂×C₂₆₃	—	2	1	no	—
72	C₂×C₂₆₃	#380	1	1	yes	characteristic
73	C₂×C₂₆₃	C₂×#101	1	1	yes	characteristic
74	C₂×C₂₆₃	#413	1	1	yes	characteristic
75	C₂₆₃	#1801	1	1	yes	characteristic, complemented, Sylow
76	#1801	—	263	1	no	maximal, Sylow
77	C₂×C₄×Q₈	—	263	1	no	—
78	#423	—	263	2	no	—
79	C₂×#101	—	263	2	no	—
80	#343	—	263	1	no	—
81	#343	D₂₆₃	1	1	yes	characteristic
82	C₂²×Q₈	—	263	1	no	—
83	C₂×M₁₆	—	263	2	no	—
84	C₂×M₁₆	C₂×D₂₆₃	1	2	yes	—
85	C₄×Q₈	—	526	2	no	—
86	C₄×Q₈	—	526	2	no	—
87	C₂×C₄⋊C₄	—	263	2	no	—
88	C₂×C₄⋊C₄	C₂×D₂₆₃	1	1	yes	characteristic
89	C₂×C₄²	—	263	1	no	—
90	C₂×C₄²	—	263	2	no	—
91	C₄⋊C₈	—	526	2	no	—
92	C₄⋊C₈	—	526	2	no	—
93	#101	—	526	4	no	—
94	C₂×Q₈	—	263	2	no	—
95	C₂×Q₈	—	263	2	no	—
96	C₂×Q₈	—	526	2	no	—
97	C₂×Q₈	—	526	2	no	—
98	C₂²×C₄	—	263	1	no	—
99	C₂²×C₄	—	263	2	no	—
100	C₂²×C₄	—	263	1	no	—
101	C₂²×C₄	C₄×D₂₆₃	1	2	yes	—
102	C₂²×C₄	C₂²×D₂₆₃	1	1	yes	characteristic
103	M₁₆	—	2	2	no	—
104	M₁₆	—	2	2	no	—
105	M₁₆	—	526	2	no	—
106	M₁₆	—	526	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	526	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	526	4	no	—
111	C₄²	—	263	2	no	—
112	C₄²	—	263	2	no	—
113	C₄²	—	526	2	no	—
114	C₄²	—	526	2	no	—
115	C₂³	C₂×C₄×D₂₆₃	1	1	yes	characteristic
116	Q₈	—	526	4	no	—
117	Q₈	—	526	4	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	263	1	no	—
121	C₂×C₄	—	263	1	no	—
122	C₂×C₄	—	263	2	no	—
123	C₂×C₄	—	263	4	no	—
124	C₂×C₄	—	526	2	no	—
125	C₂×C₄	—	526	2	no	—
126	C₂×C₄	—	526	2	no	—
127	C₂×C₄	—	1052	1	no	—
128	C₂×C₄	D₄×D₂₆₃	1	1	yes	characteristic
129	C₂×C₄	D₄×D₂₆₃	1	2	yes	—
130	C₂×C₄	Q₈×D₂₆₃	1	1	yes	characteristic
131	C₂×C₄	#16862376	1	2	yes	—
132	C₈	—	4	2	no	—
133	C₈	—	1052	2	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	C₂²⋊C₄×D₂₆₃	1	1	yes	characteristic
137	C₂²	C₄⋊C₄×D₂₆₃	1	1	yes	characteristic
138	C₂²	#22554427	1	1	yes	characteristic
139	C₄	—	4	2	no	—
140	C₄	—	263	2	no	—
141	C₄	—	526	1	no	—
142	C₄	—	526	2	no	—
143	C₄	—	526	4	no	—
144	C₄	#22554443	1	2	yes	—
145	C₄	#22554448	1	2	yes	—
146	C₂	—	2	1	no	—
147	C₂	—	2	1	no	—
148	C₂	#30681797	1	1	yes	characteristic
149	C₂	#101×D₂₆₃	1	1	yes	characteristic
150	C₂	#30681919	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22554435

C₂

yes

maximal, characteristic

#30681961

C₂

yes

maximal

C₂×#22554388

C₂

yes

maximal

#30681707

C₂

yes

maximal, characteristic

#343×C₂₆₃

C₂

yes

maximal, characteristic

C₂²×Dic₅₂₆

C₂²

yes

characteristic

C₂×#16862360

C₂²

yes

—

#22554435

—

#22554435

—

C₂×Dic₂₆₃⋊C₄

C₄

yes

—

C₂×C₄×Dic₂₆₃

C₄

yes

—

C₂×C₄×Dic₂₆₃

C₂²

yes

characteristic

Dic₂₆₃⋊C₈

—

Dic₂₆₃⋊C₈

—

#22554388

—

C₂×M₁₆×C₂₆₃

C₂²

yes

—

C₂×C₄⋊C₄×C₂₆₃

C₂²

yes

characteristic

C₂×Dic₅₂₆

—

C₂×Dic₅₂₆

—

C₂×Dic₅₂₆

D₄

yes

—

C₂×Dic₅₂₆

Q₈

yes

—

C₂²×Dic₂₆₃

C₂×C₄

yes

characteristic

C₂²×Dic₂₆₃

C₂×C₄

yes

—

C₂²×Dic₂₆₃

C₂×C₄

yes

characteristic

#16862360

—

#16862360

—

C₂×C₂₆₃⋊C₈

—

Dic₂₆₃⋊C₄

—

C₄×Dic₂₆₃

—

C₄×Dic₂₆₃

—

C₄×Dic₂₆₃

D₄

yes

—

C₄×Dic₂₆₃

D₄

yes

—

C₂²×C₄×C₂₆₃

C₂×C₄

yes

—

C₂²×C₄×C₂₆₃

C₂³

yes

characteristic

M₁₆×C₂₆₃

—

M₁₆×C₂₆₃

—

C₂×C₈×C₂₆₃

—

C₄⋊C₄×C₂₆₃

—

Dic₅₂₆

—

Dic₅₂₆

—

C₂×Dic₂₆₃

—

C₂×Dic₂₆₃

—

C₂×Dic₂₆₃

—

C₂×Dic₂₆₃

—

C₂×Dic₂₆₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₆₃

C₂²⋊C₄

yes

—

C₂×Dic₂₆₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₆₃

C₄⋊C₄

yes

—

C₂₆₃⋊C₈

—

C₂³×C₂₆₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₆₃

—

C₂×C₄×C₂₆₃

—

C₂×C₄×C₂₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₃

C₂×D₄

yes

—

C₂×C₄×C₂₆₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₆₃

#41

yes

—

C₈×C₂₆₃

—

C₂²×C₂₆₃

—

C₂²×C₂₆₃

—

C₂²×C₂₆₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₆₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₆₃

#117

yes

characteristic

C₄×C₂₆₃

—

C₄×C₂₆₃

#121

yes

—

C₄×C₂₆₃

C₂²⋊Q₈

yes

—

Dic₂₆₃

—

Dic₂₆₃

—

Dic₂₆₃

—

Dic₂₆₃

#101

yes

complemented

C₂×C₂₆₃

—

C₂×C₂₆₃

—

C₂×C₂₆₃

yes

C₂×C₂₆₃

yes

C₂×C₂₆₃

yes

C₂₆₃

yes

characteristic, complemented, Sylow

#1801

—

263

maximal, Sylow

C₂×C₄×Q₈

—

263

—

#423

—

263

—

C₂×#101

—

263

—

#343

—

263

—

#343

D₂₆₃

yes

characteristic

C₂²×Q₈

—

263

—

C₂×M₁₆

—

263

—

C₂×M₁₆

C₂×D₂₆₃

yes

—

C₄×Q₈

—

526

—

C₄×Q₈

—

526

—

C₂×C₄⋊C₄

—

263

—

C₂×C₄⋊C₄

C₂×D₂₆₃

yes

characteristic

C₂×C₄²

—

263

—

C₂×C₄²

—

263

—

C₄⋊C₈

—

526

—

C₄⋊C₈

—

526

—

#101

—

526

—

C₂×Q₈

—

263

—

C₂×Q₈

—

263

—

C₂×Q₈

—

526

—

C₂×Q₈

—

526

—

C₂²×C₄

—

263

—

C₂²×C₄

—

263

—

100

C₂²×C₄

—

263

—

101

C₂²×C₄

C₄×D₂₆₃

yes

—

102

C₂²×C₄

C₂²×D₂₆₃

yes

characteristic

103

M₁₆

—

104

M₁₆

—

105

M₁₆

—

526

—

106

M₁₆

—

526

—

107

C₂×C₈

—

108

C₂×C₈

—

526

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

526

—

111

C₄²

—

263

—

112

C₄²

—

263

—

113

C₄²

—

526

—

114

C₄²

—

526

—

115

C₂³

C₂×C₄×D₂₆₃

yes

characteristic

116

Q₈

—

526

—

117

Q₈

—

526

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

263

—

121

C₂×C₄

—

263

—

122

C₂×C₄

—

263

—

123

C₂×C₄

—

263

—

124

C₂×C₄

—

526

—

125

C₂×C₄

—

526

—

126

C₂×C₄

—

526

—

127

C₂×C₄

—

1052

—

128

C₂×C₄

D₄×D₂₆₃

yes

characteristic

129

C₂×C₄

D₄×D₂₆₃

yes

—

130

C₂×C₄

Q₈×D₂₆₃

yes

characteristic

131

C₂×C₄

#16862376

yes

—

132

C₈

—

133

C₈

—

1052

—

134

C₂²

—

135

C₂²

—

136

C₂²

C₂²⋊C₄×D₂₆₃

yes

characteristic

137

C₂²

C₄⋊C₄×D₂₆₃

yes

characteristic

138

C₂²

#22554427

yes

characteristic

139

C₄

—

140

C₄

—

263

—

141

C₄

—

526

—

142

C₄

—

526

—

143

C₄

—

526

—

144

C₄

#22554443

yes

—

145

C₄

#22554448

yes

—

146

C₂

—

147

C₂

—

148

C₂

#30681797

yes

characteristic

149

C₂

#101×D₂₆₃

yes

characteristic

150

C₂

#30681919

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₂₆₃
commutator, prime radical	C₂³×C₂₆₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₆₃
Fitting	#343×C₂₆₃	C₂
socle	C₂²×C₂₆₃	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₂₆₃	C₂³
hypercenter	#343	D₂₆₃
hypocenter	C₂₆₃	#1801
LCS	G ▷ C₂³×C₂₆₃ ▷ C₂×C₂₆₃ ▷ C₂₆₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₂₆₃

commutator, prime radical

C₂³×C₂₆₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₆₃

Fitting

#343×C₂₆₃

C₂

socle

C₂²×C₂₆₃

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₂₆₃

C₂³

D₂₆₃

C₂₆₃

G ▷ C₂³×C₂₆₃ ▷ C₂×C₂₆₃ ▷ C₂₆₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

N	H ≅ G/N	type
69. Dic₂₆₃	#101	semidirect
75. C₂₆₃	#1801	semidirect

H ≅ G/N

type

69. Dic₂₆₃

#101

semidirect

75. C₂₆₃

#1801

semidirect

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

subgroup	index
C₂×#22554435	2
#30681961	2
C₂×#22554388	2
#30681707	2
#343×C₂₆₃	2
#1801	263

subgroup

index

#343×C₂₆₃

263

Subgroups — #46782742

Conjugacy classes of subgroups

Complements

Maximal subgroups