Group #46784072 — subgroups

Subgroups — #46784072

Order 33664.

282 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 99 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×#22554403	C₂	1	2	yes	maximal
2	#30681836	C₂	1	1	yes	maximal, characteristic
3	#30681700	C₂	1	2	yes	maximal
4	#30681693	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₂₆₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₂₆₃⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₂₆₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₂₆₃	C₂²	1	1	yes	characteristic
9	C₂×Dic₂₆₃⋊C₄	—	2	1	no	—
10	#22554403	—	2	4	no	—
11	C₂₆₃⋊#95	—	2	1	no	—
12	C₂²×Q₈×C₂₆₃	—	2	1	no	—
13	C₄⋊Q₈×C₂₆₃	—	2	4	no	—
14	C₂×C₄⋊C₄×C₂₆₃	—	2	1	no	—
15	C₂×C₄⋊C₄×C₂₆₃	C₂²	1	2	yes	—
16	C₂×C₄²×C₂₆₃	C₂²	1	1	yes	characteristic
17	C₂²×Dic₂₆₃	—	2	1	no	—
18	C₂²×Dic₂₆₃	—	2	1	no	—
19	C₂×C₂₆₃⋊C₈	—	2	2	no	—
20	C₂×C₂₆₃⋊C₈	—	2	2	no	—
21	C₂×C₂₆₃⋊C₈	Q₈	1	4	yes	complemented
22	C₄⋊Dic₂₆₃	—	2	2	no	—
23	C₄⋊Dic₂₆₃	—	2	1	no	—
24	C₄⋊Dic₂₆₃	Q₈	1	2	yes	complemented
25	Dic₂₆₃⋊C₄	—	4	2	no	—
26	C₂×Q₈×C₂₆₃	—	2	4	no	—
27	C₂×Q₈×C₂₆₃	—	4	2	no	—
28	C₂×Q₈×C₂₆₃	—	4	2	no	—
29	C₂²×C₄×C₂₆₃	—	2	2	no	—
30	C₂²×C₄×C₂₆₃	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₂₆₃	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₂₆₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₂₆₃	—	2	4	no	—
34	C₄⋊C₄×C₂₆₃	—	2	2	no	—
35	C₄⋊C₄×C₂₆₃	D₄	1	4	yes	—
36	C₄²×C₂₆₃	—	2	2	no	—
37	C₂×Dic₂₆₃	—	2	2	no	—
38	C₂×Dic₂₆₃	—	4	1	no	—
39	C₂×Dic₂₆₃	—	4	1	no	—
40	C₂×Dic₂₆₃	—	4	1	no	—
41	C₂×Dic₂₆₃	—	4	1	no	—
42	C₂×Dic₂₆₃	—	4	1	no	—
43	C₂₆₃⋊C₈	—	2	4	no	—
44	C₂³×C₂₆₃	C₂×D₄	1	1	yes	characteristic
45	Q₈×C₂₆₃	—	4	8	no	—
46	C₂×C₄×C₂₆₃	—	2	1	no	—
47	C₂×C₄×C₂₆₃	—	2	1	no	—
48	C₂×C₄×C₂₆₃	—	2	1	no	—
49	C₂×C₄×C₂₆₃	—	2	1	no	—
50	C₂×C₄×C₂₆₃	—	2	4	no	—
51	C₂×C₄×C₂₆₃	—	4	2	no	—
52	C₂×C₄×C₂₆₃	—	4	2	no	—
53	C₂×C₄×C₂₆₃	D₈	1	2	yes	—
54	C₂×C₄×C₂₆₃	Q₁₆	1	2	yes	—
55	C₂×C₄×C₂₆₃	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₂₆₃	C₂×Q₈	1	2	yes	—
57	C₂×C₄×C₂₆₃	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₄×C₂₆₃	#41	1	1	yes	characteristic
59	C₂²×C₂₆₃	#120	1	1	yes	characteristic
60	C₂²×C₂₆₃	C₂²⋊Q₈	1	1	yes	characteristic
61	C₂²×C₂₆₃	C₄⋊Q₈	1	1	yes	characteristic
62	C₂²×C₂₆₃	C₂×D₈	1	1	yes	characteristic
63	C₂²×C₂₆₃	C₂×Q₁₆	1	1	yes	characteristic
64	C₂²×C₂₆₃	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₂²×C₂₆₃	#137	1	1	yes	characteristic
66	C₄×C₂₆₃	—	2	2	no	—
67	C₄×C₂₆₃	—	2	2	no	—
68	C₄×C₂₆₃	—	4	4	no	—
69	C₄×C₂₆₃	C₂²⋊Q₈	1	2	yes	—
70	C₄×C₂₆₃	C₄⋊Q₈	1	2	yes	—
71	Dic₂₆₃	—	4	2	no	—
72	Dic₂₆₃	—	8	1	no	—
73	C₂×C₂₆₃	#395	1	1	yes	characteristic
74	C₂×C₂₆₃	#447	1	1	yes	characteristic
75	C₂×C₂₆₃	#451	1	1	yes	characteristic
76	C₂×C₂₆₃	#474	1	1	yes	characteristic
77	C₂×C₂₆₃	#477	1	1	yes	characteristic
78	C₂×C₂₆₃	#500	1	1	yes	characteristic
79	C₂×C₂₆₃	#501	1	1	yes	characteristic
80	C₂₆₃	#2057	1	1	yes	characteristic, complemented, Sylow
81	#2057	—	263	1	no	maximal, Sylow
82	C₂×C₄⋊Q₈	D₂₆₃	1	1	yes	characteristic
83	C₂×#107	—	263	2	no	—
84	#389	—	263	1	no	—
85	#340	—	263	2	no	—
86	#336	—	263	1	no	—
87	C₂²×Q₈	—	2	1	no	—
88	C₂²×C₈	—	263	2	no	—
89	C₄⋊Q₈	—	2	4	no	—
90	C₂×C₄⋊C₄	—	2	1	no	—
91	C₂×C₄⋊C₄	—	263	1	no	—
92	C₂×C₄⋊C₄	—	263	1	no	—
93	C₂×C₄⋊C₄	—	526	1	no	—
94	C₂×C₄⋊C₄	C₂×D₂₆₃	1	2	yes	—
95	C₂×C₄²	C₂×D₂₆₃	1	1	yes	characteristic
96	#107	—	526	4	no	—
97	#95	—	526	1	no	—
98	C₂×Q₈	—	2	4	no	—
99	C₂×Q₈	—	4	2	no	—
100	C₂×Q₈	—	4	2	no	—
101	C₂²×C₄	—	2	2	no	—
102	C₂²×C₄	—	526	1	no	—
103	C₂²×C₄	—	526	1	no	—
104	C₂²×C₄	C₂²⋊D₂₆₃	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊D₂₆₃	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₂₆₃	1	1	yes	characteristic
107	C₂×C₈	—	263	4	no	—
108	C₂×C₈	—	526	2	no	—
109	C₂×C₈	—	526	2	no	—
110	C₄⋊C₄	—	2	4	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	263	2	no	—
113	C₄⋊C₄	—	526	1	no	—
114	C₄⋊C₄	—	526	2	no	—
115	C₄⋊C₄	—	1052	2	no	—
116	C₄⋊C₄	C₂²⋊D₂₆₃	1	4	yes	—
117	C₄²	—	2	2	no	—
118	C₂³	C₂×C₂²⋊D₂₆₃	1	1	yes	characteristic
119	Q₈	—	4	8	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	526	2	no	—
128	C₂×C₄	—	1052	1	no	—
129	C₂×C₄	—	1052	1	no	—
130	C₂×C₄	—	1052	1	no	—
131	C₂×C₄	—	1052	1	no	—
132	C₂×C₄	—	1052	1	no	—
133	C₂×C₄	#16862361	1	2	yes	—
134	C₂×C₄	#16862366	1	2	yes	—
135	C₂×C₄	C₂×C₂²⋊D₂₆₃	1	2	yes	—
136	C₂×C₄	Q₈×D₂₆₃	1	2	yes	—
137	C₂×C₄	Q₈×D₂₆₃	1	1	yes	characteristic
138	C₂×C₄	#16862377	1	1	yes	characteristic
139	C₈	—	526	4	no	—
140	C₂²	#22554439	1	1	yes	characteristic
141	C₂²	#22554446	1	1	yes	characteristic
142	C₂²	#22554468	1	1	yes	characteristic
143	C₂²	C₂×#16862361	1	1	yes	characteristic
144	C₂²	C₂×#16862366	1	1	yes	characteristic
145	C₂²	#22554496	1	1	yes	characteristic
146	C₂²	#22554501	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	2	no	—
149	C₄	—	4	4	no	—
150	C₄	—	1052	2	no	—
151	C₄	—	2104	1	no	—
152	C₄	#22554446	1	2	yes	—
153	C₄	#22554468	1	2	yes	—
154	C₂	#30681857	1	1	yes	characteristic
155	C₂	#30682072	1	1	yes	characteristic
156	C₂	#30682094	1	1	yes	characteristic
157	C₂	#30682215	1	1	yes	characteristic
158	C₂	#30682230	1	1	yes	characteristic
159	C₂	#30682341	1	1	yes	characteristic
160	C₂	#30682344	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22554403

C₂

yes

maximal

#30681836

C₂

yes

maximal, characteristic

#30681700

C₂

yes

maximal

#30681693

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₂₆₃

C₂

yes

maximal, characteristic

C₂²×C₂₆₃⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₂₆₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₆₃

C₂²

yes

characteristic

C₂×Dic₂₆₃⋊C₄

—

#22554403

—

C₂₆₃⋊#95

—

C₂²×Q₈×C₂₆₃

—

C₄⋊Q₈×C₂₆₃

—

C₂×C₄⋊C₄×C₂₆₃

—

C₂×C₄⋊C₄×C₂₆₃

C₂²

yes

—

C₂×C₄²×C₂₆₃

C₂²

yes

characteristic

C₂²×Dic₂₆₃

—

C₂²×Dic₂₆₃

—

C₂×C₂₆₃⋊C₈

—

C₂×C₂₆₃⋊C₈

—

C₂×C₂₆₃⋊C₈

Q₈

yes

complemented

C₄⋊Dic₂₆₃

—

C₄⋊Dic₂₆₃

—

C₄⋊Dic₂₆₃

Q₈

yes

complemented

Dic₂₆₃⋊C₄

—

C₂×Q₈×C₂₆₃

—

C₂×Q₈×C₂₆₃

—

C₂×Q₈×C₂₆₃

—

C₂²×C₄×C₂₆₃

—

C₂²×C₄×C₂₆₃

D₄

yes

characteristic

C₂²×C₄×C₂₆₃

D₄

yes

characteristic

C₂²×C₄×C₂₆₃

C₂³

yes

characteristic

C₄⋊C₄×C₂₆₃

—

C₄⋊C₄×C₂₆₃

—

C₄⋊C₄×C₂₆₃

D₄

yes

—

C₄²×C₂₆₃

—

C₂×Dic₂₆₃

—

C₂×Dic₂₆₃

—

C₂×Dic₂₆₃

—

C₂×Dic₂₆₃

—

C₂×Dic₂₆₃

—

C₂×Dic₂₆₃

—

C₂₆₃⋊C₈

—

C₂³×C₂₆₃

C₂×D₄

yes

characteristic

Q₈×C₂₆₃

—

C₂×C₄×C₂₆₃

—

C₂×C₄×C₂₆₃

—

C₂×C₄×C₂₆₃

—

C₂×C₄×C₂₆₃

—

C₂×C₄×C₂₆₃

—

C₂×C₄×C₂₆₃

—

C₂×C₄×C₂₆₃

—

C₂×C₄×C₂₆₃

D₈

yes

—

C₂×C₄×C₂₆₃

Q₁₆

yes

—

C₂×C₄×C₂₆₃

C₂×D₄

yes

—

C₂×C₄×C₂₆₃

C₂×Q₈

yes

—

C₂×C₄×C₂₆₃

C₂×Q₈

yes

C₂×C₄×C₂₆₃

yes

C₂²×C₂₆₃

yes

C₂²×C₂₆₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₂₆₃

C₄⋊Q₈

yes

characteristic

C₂²×C₂₆₃

C₂×D₈

yes

characteristic

C₂²×C₂₆₃

C₂×Q₁₆

yes

C₂²×C₂₆₃

yes

C₂²×C₂₆₃

yes

C₄×C₂₆₃

—

C₄×C₂₆₃

—

C₄×C₂₆₃

—

C₄×C₂₆₃

C₂²⋊Q₈

yes

—

C₄×C₂₆₃

C₄⋊Q₈

yes

—

Dic₂₆₃

—

Dic₂₆₃

—

C₂×C₂₆₃

yes

C₂×C₂₆₃

yes

C₂×C₂₆₃

yes

C₂×C₂₆₃

yes

C₂×C₂₆₃

yes

C₂×C₂₆₃

yes

C₂×C₂₆₃

yes

C₂₆₃

yes

characteristic, complemented, Sylow

#2057

—

263

maximal, Sylow

C₂×C₄⋊Q₈

D₂₆₃

yes

characteristic

C₂×#107

—

263

—

#389

—

263

—

#340

—

263

—

#336

—

263

—

C₂²×Q₈

—

C₂²×C₈

—

263

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

263

—

C₂×C₄⋊C₄

—

263

—

C₂×C₄⋊C₄

—

526

—

C₂×C₄⋊C₄

C₂×D₂₆₃

yes

—

C₂×C₄²

C₂×D₂₆₃

yes

characteristic

#107

—

526

—

#95

—

526

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂²×C₄

—

102

C₂²×C₄

—

526

—

103

C₂²×C₄

—

526

—

104

C₂²×C₄

C₂²⋊D₂₆₃

yes

characteristic

105

C₂²×C₄

C₂²⋊D₂₆₃

yes

characteristic

106

C₂²×C₄

C₂²×D₂₆₃

yes

characteristic

107

C₂×C₈

—

263

—

108

C₂×C₈

—

526

—

109

C₂×C₈

—

526

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

263

—

113

C₄⋊C₄

—

526

—

114

C₄⋊C₄

—

526

—

115

C₄⋊C₄

—

1052

—

116

C₄⋊C₄

C₂²⋊D₂₆₃

yes

—

117

C₄²

—

118

C₂³

C₂×C₂²⋊D₂₆₃

yes

characteristic

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

526

—

128

C₂×C₄

—

1052

—

129

C₂×C₄

—

1052

—

130

C₂×C₄

—

1052

—

131

C₂×C₄

—

1052

—

132

C₂×C₄

—

1052

—

133

C₂×C₄

#16862361

yes

—

134

C₂×C₄

#16862366

yes

—

135

C₂×C₄

C₂×C₂²⋊D₂₆₃

yes

—

136

C₂×C₄

Q₈×D₂₆₃

yes

—

137

C₂×C₄

Q₈×D₂₆₃

yes

characteristic

138

C₂×C₄

#16862377

yes

characteristic

139

C₈

—

526

—

140

C₂²

#22554439

yes

characteristic

141

C₂²

#22554446

yes

characteristic

142

C₂²

#22554468

yes

characteristic

143

C₂²

C₂×#16862361

yes

characteristic

144

C₂²

C₂×#16862366

yes

characteristic

145

C₂²

#22554496

yes

characteristic

146

C₂²

#22554501

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

1052

—

151

C₄

—

2104

—

152

C₄

#22554446

yes

—

153

C₄

#22554468

yes

—

154

C₂

#30681857

yes

characteristic

155

C₂

#30682072

yes

characteristic

156

C₂

#30682094

yes

characteristic

157

C₂

#30682215

yes

characteristic

158

C₂

#30682230

yes

characteristic

159

C₂

#30682341

yes

characteristic

160

C₂

#30682344

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₆₃
commutator, cosocle	C₂²×C₄×C₂₆₃	C₂³
Frattini	C₂²×C₄	C₂²×D₂₆₃
Fitting	C₂×C₄⋊Q₈×C₂₆₃	C₂
socle, prime radical	C₂³×C₂₆₃	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₂₆₃
hypocenter	C₂₆₃	#2057
LCS	G ▷ C₂²×C₄×C₂₆₃ ▷ C₂×C₂₆₃ ▷ C₂₆₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₆₃

commutator, cosocle

C₂²×C₄×C₂₆₃

C₂³

Frattini

C₂²×C₄

C₂²×D₂₆₃

Fitting

C₂×C₄⋊Q₈×C₂₆₃

C₂

socle, prime radical

C₂³×C₂₆₃

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₂₆₃

hypocenter

C₂₆₃

#2057

LCS

G ▷ C₂²×C₄×C₂₆₃ ▷ C₂×C₂₆₃ ▷ C₂₆₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
21. C₂×C₂₆₃⋊C₈	Q₈	semidirect
24. C₄⋊Dic₂₆₃	Q₈	semidirect
80. C₂₆₃	#2057	semidirect

H ≅ G/N

type

21. C₂×C₂₆₃⋊C₈

Q₈

semidirect

24. C₄⋊Dic₂₆₃

Q₈

semidirect

80. C₂₆₃

#2057

semidirect

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

subgroup	index
C₂×#22554403	2
#30681836	2
#30681700	2
#30681693	2
C₂×C₄⋊Q₈×C₂₆₃	2
#2057	263

subgroup

index

C₂×C₄⋊Q₈×C₂₆₃

263

Subgroups — #46784072

Conjugacy classes of subgroups

Complements

Maximal subgroups