Group #46783297 — subgroups

Subgroups — #46783297

Order 33664.

308 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 85 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	#30682705	C₂	1	1	yes	maximal, characteristic
2	#30682509	C₂	1	1	yes	maximal, characteristic
3	C₂×#22554388	C₂	1	2	yes	maximal, complemented
4	#30681709	C₂	1	2	yes	maximal, complemented
5	#413×C₂₆₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₅₂₆	C₂²	1	1	yes	characteristic
7	#22554477	—	2	4	no	—
8	C₂×#16862360	—	2	2	no	—
9	C₂×#16862360	C₂²	1	2	yes	—
10	#22554468	—	2	1	no	—
11	#22554457	—	2	1	no	—
12	#22554451	C₄	1	4	yes	—
13	#22554449	—	2	2	no	—
14	#22554429	C₂²	1	1	yes	characteristic
15	#22554388	—	2	4	no	—
16	C₂×#41×C₂₆₃	C₂²	1	1	yes	characteristic
17	C₂×M₁₆×C₂₆₃	C₂²	1	2	yes	—
18	#101×C₂₆₃	—	2	1	no	—
19	#100×C₂₆₃	—	2	1	no	—
20	C₂×Dic₅₂₆	—	2	2	no	—
21	C₂×Dic₅₂₆	—	2	1	no	—
22	C₂×Dic₅₂₆	—	2	1	no	—
23	C₂×Dic₅₂₆	—	2	2	no	—
24	C₂²×Dic₂₆₃	C₂×C₄	1	2	yes	—
25	#16862360	—	2	4	no	—
26	#16862360	—	2	2	no	—
27	#16862360	D₄	1	4	yes	—
28	C₂×C₂₆₃⋊C₈	—	2	4	no	—
29	C₂×C₂₆₃⋊C₈	—	2	2	no	—
30	C₄⋊Dic₂₆₃	—	2	1	no	—
31	Dic₂₆₃⋊C₄	—	2	4	no	—
32	C₂²⋊Dic₂₆₃	—	2	4	no	—
33	C₂²⋊Dic₂₆₃	C₂×C₄	1	2	yes	—
34	C₄×Dic₂₆₃	—	2	1	no	—
35	#41×C₂₆₃	—	2	2	no	—
36	#41×C₂₆₃	—	2	2	no	—
37	C₂×Q₈×C₂₆₃	D₄	1	1	yes	characteristic
38	C₂×D₄×C₂₆₃	C₂×C₄	1	2	yes	—
39	C₂×D₄×C₂₆₃	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₂₆₃	D₄	1	2	yes	—
41	C₂²×C₄×C₂₆₃	C₂³	1	1	yes	characteristic
42	M₁₆×C₂₆₃	—	2	2	no	—
43	M₁₆×C₂₆₃	—	2	2	no	—
44	C₂×C₈×C₂₆₃	—	2	2	no	—
45	Dic₅₂₆	—	4	1	no	—
46	Dic₅₂₆	—	4	2	no	—
47	Dic₅₂₆	—	4	1	no	—
48	C₂×Dic₂₆₃	—	2	2	no	—
49	C₂×Dic₂₆₃	—	2	2	no	—
50	C₂×Dic₂₆₃	—	2	4	no	—
51	C₂₆₃⋊C₈	—	2	4	no	—
52	C₂³×C₂₆₃	C₂²⋊C₄	1	2	yes	—
53	C₂³×C₂₆₃	C₂²×C₄	1	1	yes	characteristic
54	Q₈×C₂₆₃	—	2	2	no	—
55	D₄×C₂₆₃	—	2	4	no	—
56	D₄×C₂₆₃	—	2	2	no	—
57	C₂×C₄×C₂₆₃	—	2	2	no	—
58	C₂×C₄×C₂₆₃	—	2	2	no	—
59	C₂×C₄×C₂₆₃	C₂²⋊C₄	1	2	yes	—
60	C₂×C₄×C₂₆₃	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₂₆₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₂₆₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₂₆₃	#41	1	2	yes	—
64	C₈×C₂₆₃	—	4	2	no	—
65	C₂²×C₂₆₃	—	2	2	no	—
66	C₂²×C₂₆₃	—	4	1	no	—
67	C₂²×C₂₆₃	—	4	2	no	—
68	C₂²×C₂₆₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₂₆₃	C₄×D₄	1	2	yes	—
70	C₄×C₂₆₃	—	2	2	no	—
71	C₄×C₂₆₃	#120	1	1	yes	characteristic
72	C₄×C₂₆₃	#121	1	2	yes	—
73	C₄×C₂₆₃	#123	1	1	yes	characteristic
74	Dic₂₆₃	—	4	2	no	—
75	Dic₂₆₃	—	4	2	no	—
76	C₂×C₂₆₃	—	2	1	no	—
77	C₂×C₂₆₃	—	2	2	no	—
78	C₂×C₂₆₃	—	4	2	no	—
79	C₂×C₂₆₃	#386	1	1	yes	characteristic
80	C₂₆₃	#1915	1	1	yes	characteristic, complemented, Sylow
81	#1915	—	263	1	no	maximal, Sylow
82	#568	—	263	1	no	—
83	#536	—	263	1	no	—
84	#413	D₂₆₃	1	1	yes	characteristic
85	C₂×#101	—	263	2	no	—
86	#344	—	263	2	no	—
87	C₂×#41	C₂×D₂₆₃	1	1	yes	characteristic
88	C₂²×Q₈	—	263	1	no	—
89	#131	—	526	4	no	—
90	C₂×M₁₆	—	263	2	no	—
91	C₂×M₁₆	—	526	2	no	—
92	C₂×M₁₆	C₂×D₂₆₃	1	2	yes	—
93	C₄⋊Q₈	—	526	1	no	—
94	#124	—	526	1	no	—
95	#123	—	263	4	no	—
96	C₂²⋊Q₈	—	526	2	no	—
97	#117	—	263	1	no	—
98	#101	—	2	1	no	—
99	#101	—	526	4	no	—
100	#100	—	2	1	no	—
101	#41	—	2	2	no	—
102	#41	—	2	2	no	—
103	C₂×Q₈	—	526	2	no	—
104	C₂×Q₈	—	526	1	no	—
105	C₂×Q₈	—	526	1	no	—
106	C₂×Q₈	—	526	2	no	—
107	C₂×Q₈	C₂²⋊D₂₆₃	1	1	yes	characteristic
108	C₂×D₄	C₄×D₂₆₃	1	2	yes	—
109	C₂×D₄	C₂²⋊D₂₆₃	1	1	yes	characteristic
110	C₂²×C₄	—	263	2	no	—
111	C₂²×C₄	D₁₀₅₂	1	2	yes	—
112	C₂²×C₄	C₂²×D₂₆₃	1	1	yes	characteristic
113	M₁₆	—	2	2	no	—
114	M₁₆	—	2	2	no	—
115	M₁₆	—	263	4	no	—
116	M₁₆	—	526	4	no	—
117	M₁₆	—	526	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	526	4	no	—
120	C₂×C₈	—	526	2	no	—
121	C₄⋊C₄	—	526	4	no	—
122	C₄⋊C₄	—	526	1	no	—
123	C₂²⋊C₄	—	263	2	no	—
124	C₂²⋊C₄	—	526	4	no	—
125	C₄²	—	526	1	no	—
126	C₂³	(C₂×D₂₆₃)⋊C₄	1	2	yes	—
127	C₂³	C₂×C₄×D₂₆₃	1	1	yes	characteristic
128	Q₈	—	2	2	no	—
129	Q₈	—	1052	1	no	—
130	Q₈	—	1052	2	no	—
131	Q₈	—	1052	1	no	—
132	D₄	—	2	4	no	—
133	D₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	526	2	no	—
137	C₂×C₄	—	526	2	no	—
138	C₂×C₄	—	526	4	no	—
139	C₂×C₄	(C₂×D₂₆₃)⋊C₄	1	2	yes	—
140	C₂×C₄	D₄×D₂₆₃	1	2	yes	—
141	C₂×C₄	C₂×C₂²⋊D₂₆₃	1	1	yes	characteristic
142	C₂×C₄	C₂×D₁₀₅₂	1	1	yes	characteristic
143	C₂×C₄	#16862375	1	2	yes	—
144	C₈	—	4	2	no	—
145	C₈	—	526	4	no	—
146	C₂²	—	2	2	no	—
147	C₂²	—	4	1	no	—
148	C₂²	—	4	2	no	—
149	C₂²	C₂×(C₂×D₂₆₃)⋊C₄	1	1	yes	characteristic
150	C₂²	(C₂²⋊D₂₆₃)⋊C₄	1	2	yes	—
151	C₄	—	2	2	no	—
152	C₄	—	1052	2	no	—
153	C₄	—	1052	2	no	—
154	C₄	#22554440	1	1	yes	characteristic
155	C₄	#22554442	1	2	yes	—
156	C₄	#22554454	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	2	no	—
159	C₂	—	4	2	no	—
160	C₂	#30681815	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30682705

C₂

yes

maximal, characteristic

#30682509

C₂

yes

maximal, characteristic

C₂×#22554388

C₂

yes

maximal, complemented

#30681709

C₂

yes

maximal, complemented

#413×C₂₆₃

C₂

yes

maximal, characteristic

C₂²×Dic₅₂₆

C₂²

yes

—

—

C₂²

yes

—

#22554468

—

#22554457

—

#22554451

C₄

yes

—

#22554449

—

#22554429

C₂²

yes

characteristic

#22554388

—

C₂×#41×C₂₆₃

C₂²

yes

characteristic

C₂×M₁₆×C₂₆₃

C₂²

yes

—

#101×C₂₆₃

—

#100×C₂₆₃

—

C₂×Dic₅₂₆

—

C₂×Dic₅₂₆

—

C₂×Dic₅₂₆

—

C₂×Dic₅₂₆

—

C₂²×Dic₂₆₃

C₂×C₄

yes

—

#16862360

—

#16862360

—

#16862360

D₄

yes

—

C₂×C₂₆₃⋊C₈

—

C₂×C₂₆₃⋊C₈

—

C₄⋊Dic₂₆₃

—

Dic₂₆₃⋊C₄

—

C₂²⋊Dic₂₆₃

—

C₂²⋊Dic₂₆₃

C₂×C₄

yes

—

C₄×Dic₂₆₃

—

#41×C₂₆₃

—

#41×C₂₆₃

—

C₂×Q₈×C₂₆₃

D₄

yes

characteristic

C₂×D₄×C₂₆₃

C₂×C₄

yes

—

C₂×D₄×C₂₆₃

D₄

yes

characteristic

C₂²×C₄×C₂₆₃

D₄

yes

—

C₂²×C₄×C₂₆₃

C₂³

yes

characteristic

M₁₆×C₂₆₃

—

M₁₆×C₂₆₃

—

C₂×C₈×C₂₆₃

—

Dic₅₂₆

—

Dic₅₂₆

—

Dic₅₂₆

—

C₂×Dic₂₆₃

—

C₂×Dic₂₆₃

—

C₂×Dic₂₆₃

—

C₂₆₃⋊C₈

—

C₂³×C₂₆₃

C₂²⋊C₄

yes

—

C₂³×C₂₆₃

C₂²×C₄

yes

characteristic

Q₈×C₂₆₃

—

D₄×C₂₆₃

—

D₄×C₂₆₃

—

C₂×C₄×C₂₆₃

—

C₂×C₄×C₂₆₃

—

C₂×C₄×C₂₆₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₆₃

C₂×D₄

yes

—

C₂×C₄×C₂₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₃

#41

yes

—

C₈×C₂₆₃

—

C₂²×C₂₆₃

—

C₂²×C₂₆₃

—

C₂²×C₂₆₃

—

C₂²×C₂₆₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₆₃

C₄×D₄

yes

—

C₄×C₂₆₃

—

C₄×C₂₆₃

#120

yes

characteristic

C₄×C₂₆₃

#121

yes

—

C₄×C₂₆₃

#123

yes

characteristic

Dic₂₆₃

—

Dic₂₆₃

—

C₂×C₂₆₃

—

C₂×C₂₆₃

—

C₂×C₂₆₃

—

C₂×C₂₆₃

#386

yes

characteristic

C₂₆₃

#1915

yes

characteristic, complemented, Sylow

#1915

—

263

maximal, Sylow

#568

—

263

—

#536

—

263

—

#413

D₂₆₃

yes

characteristic

C₂×#101

—

263

—

#344

—

263

—

C₂×#41

C₂×D₂₆₃

yes

characteristic

C₂²×Q₈

—

263

—

#131

—

526

—

C₂×M₁₆

—

263

—

C₂×M₁₆

—

526

—

C₂×M₁₆

C₂×D₂₆₃

yes

—

C₄⋊Q₈

—

526

—

#124

—

526

—

#123

—

263

—

C₂²⋊Q₈

—

526

—

#117

—

263

—

#101

—

#101

—

526

—

100

#100

—

101

#41

—

102

#41

—

103

C₂×Q₈

—

526

—

104

C₂×Q₈

—

526

—

105

C₂×Q₈

—

526

—

106

C₂×Q₈

—

526

—

107

C₂×Q₈

C₂²⋊D₂₆₃

yes

characteristic

108

C₂×D₄

C₄×D₂₆₃

yes

—

109

C₂×D₄

C₂²⋊D₂₆₃

yes

characteristic

110

C₂²×C₄

—

263

—

111

C₂²×C₄

D₁₀₅₂

yes

—

112

C₂²×C₄

C₂²×D₂₆₃

yes

characteristic

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

263

—

116

M₁₆

—

526

—

117

M₁₆

—

526

—

118

C₂×C₈

—

119

C₂×C₈

—

526

—

120

C₂×C₈

—

526

—

121

C₄⋊C₄

—

526

—

122

C₄⋊C₄

—

526

—

123

C₂²⋊C₄

—

263

—

124

C₂²⋊C₄

—

526

—

125

C₄²

—

526

—

126

C₂³

(C₂×D₂₆₃)⋊C₄

yes

—

127

C₂³

C₂×C₄×D₂₆₃

yes

characteristic

128

Q₈

—

129

Q₈

—

1052

—

130

Q₈

—

1052

—

131

Q₈

—

1052

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

526

—

137

C₂×C₄

—

526

—

138

C₂×C₄

—

526

—

139

C₂×C₄

(C₂×D₂₆₃)⋊C₄

yes

—

140

C₂×C₄

D₄×D₂₆₃

yes

—

141

C₂×C₄

C₂×C₂²⋊D₂₆₃

yes

characteristic

142

C₂×C₄

C₂×D₁₀₅₂

yes

characteristic

143

C₂×C₄

#16862375

yes

—

144

C₈

—

145

C₈

—

526

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

C₂×(C₂×D₂₆₃)⋊C₄

yes

characteristic

150

C₂²

(C₂²⋊D₂₆₃)⋊C₄

yes

—

151

C₄

—

152

C₄

—

1052

—

153

C₄

—

1052

—

154

C₄

#22554440

yes

characteristic

155

C₄

#22554442

yes

—

156

C₄

#22554454

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

—

160

C₂

#30681815

yes

characteristic

	subgroup	G/H
center	C₂	#30681815
commutator	C₂³×C₂₆₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₆₃
Fitting	#413×C₂₆₃	C₂
socle	C₂×C₂₆₃	#386
prime radical	C₂×#41×C₂₆₃	C₂²
cosocle	C₂²×C₄×C₂₆₃	C₂³
hypercenter	#413	D₂₆₃
hypocenter	C₂₆₃	#1915
LCS	G ▷ C₂³×C₂₆₃ ▷ C₂×C₂₆₃ ▷ C₂₆₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #413

subgroup

G/H

center

C₂

#30681815

commutator

C₂³×C₂₆₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₆₃

Fitting

#413×C₂₆₃

C₂

socle

C₂×C₂₆₃

#386

prime radical

C₂×#41×C₂₆₃

C₂²

cosocle

C₂²×C₄×C₂₆₃

C₂³

D₂₆₃

C₂₆₃

G ▷ C₂³×C₂₆₃ ▷ C₂×C₂₆₃ ▷ C₂₆₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #413

N	H ≅ G/N	type
3. C₂×#22554388	C₂	semidirect
4. #30681709	C₂	semidirect
80. C₂₆₃	#1915	semidirect

H ≅ G/N

type

3. C₂×#22554388

C₂

semidirect

4. #30681709

C₂

semidirect

80. C₂₆₃

#1915

semidirect

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

subgroup	index
#30682705	2
#30682509	2
C₂×#22554388	2
#30681709	2
#413×C₂₆₃	2
#1915	263

subgroup

index

#413×C₂₆₃

263

Subgroups — #46783297

Conjugacy classes of subgroups

Complements

Maximal subgroups