Group #20276654 — subgroups

Subgroups — #20276654

Order 6784.

220 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 51 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×#13133240	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133160	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810861	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810427	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810425	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810402	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×Q₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719949	—	2	1	no	—
9	#13133240	—	2	2	no	—
10	#13133240	—	2	2	no	—
11	C₂×#11719924	—	2	1	no	—
12	C₂×D₄₂₄	C₂²	1	1	yes	characteristic
13	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
14	#13133193	—	2	1	no	—
15	#13133160	—	2	2	no	—
16	C₂×#11719908	C₂²	1	1	yes	characteristic
17	#13133151	C₂²	1	1	yes	characteristic
18	#13133141	—	2	1	no	—
19	C₂×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
21	Q₃₂×C₅₃	—	2	2	no	—
22	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₅₃	—	2	1	no	—
24	C₂×Q₈×C₅₃	—	2	1	no	—
25	#11719949	—	2	4	no	—
26	#11719949	—	4	2	no	—
27	C₂×D₂₁₂	—	2	1	no	—
28	C₂×D₂₁₂	—	4	1	no	—
29	C₂×C₄×D₅₃	—	4	1	no	—
30	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
31	Q₁₆×C₅₃	—	2	1	no	—
32	Q₁₆×C₅₃	—	2	1	no	—
33	Q₁₆×C₅₃	—	2	1	no	—
34	Q₁₆×C₅₃	D₄	1	2	yes	—
35	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	#11719924	—	2	4	no	—
37	#11719921	—	4	1	no	—
38	C₄⋊Dic₅₃	—	2	1	no	—
39	#11719919	—	4	1	no	—
40	C₂×#88322	D₄	1	1	yes	characteristic
41	D₄₂₄	—	2	1	no	—
42	D₄₂₄	D₄	1	2	yes	—
43	C₈×D₅₃	—	2	1	no	—
44	C₈×D₅₃	—	2	1	no	—
45	C₁₆×C₅₃	—	2	1	no	—
46	#11719908	—	2	1	no	—
47	C₂²×D₅₃	—	4	1	no	—
48	C₂²×D₅₃	D₈	1	1	yes	characteristic
49	Q₈×C₅₃	—	2	2	no	—
50	Q₈×C₅₃	—	4	1	no	—
51	Q₈×C₅₃	—	4	1	no	—
52	Q₈×C₅₃	—	4	1	no	—
53	C₂×C₄×C₅₃	—	4	1	no	—
54	C₂×C₄×C₅₃	—	4	1	no	—
55	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂×Dic₅₃	—	4	1	no	—
57	C₂×Dic₅₃	D₈	1	1	yes	characteristic
58	D₂₁₂	—	2	2	no	—
59	D₂₁₂	—	4	4	no	—
60	D₂₁₂	—	4	1	no	—
61	C₄×D₅₃	—	2	1	no	—
62	C₄×D₅₃	—	2	1	no	—
63	C₄×D₅₃	—	4	4	no	—
64	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
66	#88322	—	2	1	no	—
67	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
68	C₂×D₅₃	—	2	1	no	—
69	C₂×D₅₃	—	2	1	no	—
70	C₂×D₅₃	—	2	1	no	—
71	C₂×D₅₃	—	4	2	no	—
72	C₂×D₅₃	—	8	1	no	—
73	C₂×D₅₃	—	8	1	no	—
74	C₄×C₅₃	—	4	2	no	—
75	C₄×C₅₃	—	8	1	no	—
76	C₄×C₅₃	#120	1	1	yes	characteristic
77	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
78	Dic₅₃	—	2	1	no	—
79	Dic₅₃	—	8	1	no	—
80	#2189	—	53	1	no	maximal, Sylow
81	C₂×C₅₃	#447	1	1	yes	characteristic
82	C₂×C₅₃	#508	1	1	yes	characteristic
83	C₂×C₅₃	#510	1	1	yes	characteristic
84	D₅₃	—	4	1	no	—
85	D₅₃	—	8	2	no	—
86	C₂×#135	—	53	1	no	—
87	C₂×Q₃₂	D₅₃	1	1	yes	characteristic, complemented
88	C₂×SD₃₂	—	53	1	no	—
89	#467	—	53	1	no	—
90	#358	—	53	1	no	—
91	#357	—	53	1	no	—
92	C₂²⋊C₁₆	—	53	1	no	—
93	C₅₃	#2189	1	1	yes	characteristic, complemented, Sylow
94	C₂×#41	—	106	1	no	—
95	#135	—	106	2	no	—
96	#135	—	106	2	no	—
97	C₂×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
98	C₂×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
99	C₂×SD₁₆	—	106	1	no	—
100	C₂×D₈	—	53	1	no	—
101	C₂²×C₈	—	53	1	no	—
102	C₂²⋊Q₈	—	106	1	no	—
103	Q₃₂	—	2	2	no	—
104	SD₃₂	—	106	2	no	—
105	C₂×C₁₆	—	53	1	no	—
106	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
107	#107	—	53	1	no	—
108	#103	—	106	1	no	—
109	#41	—	106	4	no	—
110	#41	—	212	2	no	—
111	C₂×Q₈	—	2	1	no	—
112	C₂×Q₈	—	2	1	no	—
113	C₂×D₄	—	106	1	no	—
114	C₂×D₄	—	212	1	no	—
115	C₂²×C₄	—	53	1	no	—
116	C₂²×C₄	—	212	1	no	—
117	Q₁₆	—	2	1	no	—
118	Q₁₆	—	2	1	no	—
119	Q₁₆	—	2	1	no	—
120	Q₁₆	C₂²⋊D₅₃	1	2	yes	—
121	SD₁₆	—	106	4	no	—
122	D₈	—	53	2	no	—
123	D₈	—	106	1	no	—
124	C₂×C₈	—	53	1	no	—
125	C₂×C₈	—	106	1	no	—
126	C₂×C₈	—	106	1	no	—
127	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
128	C₄⋊C₄	—	106	1	no	—
129	C₄⋊C₄	—	212	1	no	—
130	C₂²⋊C₄	—	212	1	no	—
131	C₁₆	—	2	1	no	—
132	C₁₆	—	106	1	no	—
133	C₂³	—	53	1	no	—
134	C₂³	—	212	1	no	—
135	Q₈	—	2	2	no	—
136	Q₈	—	4	1	no	—
137	Q₈	—	4	1	no	—
138	Q₈	—	4	1	no	—
139	D₄	—	106	2	no	—
140	D₄	—	212	4	no	—
141	D₄	—	212	1	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	4	1	no	—
144	C₂×C₄	—	53	1	no	—
145	C₂×C₄	—	106	1	no	—
146	C₂×C₄	—	106	1	no	—
147	C₂×C₄	—	212	4	no	—
148	C₂×C₄	—	212	1	no	—
149	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
150	C₈	—	106	1	no	—
151	C₈	D₄×D₅₃	1	1	yes	characteristic
152	C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
153	C₂²	—	106	1	no	—
154	C₂²	—	106	1	no	—
155	C₂²	—	106	1	no	—
156	C₂²	—	424	1	no	—
157	C₂²	—	424	1	no	—
158	C₂²	—	212	2	no	—
159	C₂²	D₈×D₅₃	1	1	yes	characteristic
160	C₄	—	4	2	no	—
161	C₄	—	8	1	no	—
162	C₄	—	106	1	no	—
163	C₄	—	424	1	no	—
164	C₄	#13133185	1	1	yes	characteristic
165	C₄	#13133241	1	1	yes	characteristic
166	C₂	—	424	2	no	—
167	C₂	—	212	1	no	—
168	C₂	#16810755	1	1	yes	characteristic
169	C₂	#16811056	1	1	yes	characteristic
170	C₂	#16811065	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133240

C₂

yes

maximal, characteristic

C₂×#13133160

C₂

yes

maximal, characteristic, complemented

#16810861

C₂

yes

maximal, characteristic, complemented

#16810427

C₂

yes

maximal, characteristic, complemented

#16810425

C₂

yes

maximal, characteristic, complemented

#16810402

C₂

yes

maximal, characteristic, complemented

C₂×Q₃₂×C₅₃

C₂

yes

maximal, characteristic, complemented

—

—

—

—

C₂×D₄₂₄

C₂²

yes

characteristic

C₂×C₈×D₅₃

C₂²

yes

—

—

C₂²

yes

characteristic

#13133151

C₂²

yes

characteristic

#13133141

—

C₂×Q₁₆×C₅₃

C₂²

yes

characteristic

C₂×Q₁₆×C₅₃

C₂²

yes

characteristic

Q₃₂×C₅₃

—

C₂×C₁₆×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

#11719949

—

#11719949

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

D₄

yes

characteristic

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

#11719924

—

#11719921

—

C₄⋊Dic₅₃

—

#11719919

—

C₂×#88322

D₄

yes

characteristic

D₄₂₄

—

D₄₂₄

D₄

yes

—

C₈×D₅₃

—

C₈×D₅₃

—

C₁₆×C₅₃

—

#11719908

—

C₂²×D₅₃

—

C₂²×D₅₃

D₈

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

D₈

yes

characteristic

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

C₂×D₄

yes

characteristic

C₈×C₅₃

C₂×D₄

yes

characteristic

#88322

—

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

D₅₃

—

C₂×#135

—

C₂×Q₃₂

D₅₃

yes

characteristic, complemented

C₂×SD₃₂

—

#467

—

#358

—

#357

—

C₂²⋊C₁₆

—

C₅₃

#2189

yes

characteristic, complemented, Sylow

C₂×#41

—

106

—

#135

—

106

—

#135

—

106

—

C₂×Q₁₆

C₂×D₅₃

yes

characteristic

C₂×Q₁₆

C₂×D₅₃

yes

characteristic

C₂×SD₁₆

—

106

—

100

C₂×D₈

—

101

C₂²×C₈

—

102

C₂²⋊Q₈

—

106

—

103

Q₃₂

—

104

SD₃₂

—

106

—

105

C₂×C₁₆

—

106

C₂×C₁₆

C₂×D₅₃

yes

characteristic

107

#107

—

108

#103

—

106

—

109

#41

—

106

—

110

#41

—

212

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂×D₄

—

106

—

114

C₂×D₄

—

212

—

115

C₂²×C₄

—

116

C₂²×C₄

—

212

—

117

Q₁₆

—

118

Q₁₆

—

119

Q₁₆

—

120

Q₁₆

C₂²⋊D₅₃

yes

—

121

SD₁₆

—

106

—

122

D₈

—

123

D₈

—

106

—

124

C₂×C₈

—

125

C₂×C₈

—

106

—

126

C₂×C₈

—

106

—

127

C₂×C₈

C₂²×D₅₃

yes

characteristic

128

C₄⋊C₄

—

106

—

129

C₄⋊C₄

—

212

—

130

C₂²⋊C₄

—

212

—

131

C₁₆

—

132

C₁₆

—

106

—

133

C₂³

—

134

C₂³

—

212

—

135

Q₈

—

136

Q₈

—

137

Q₈

—

138

Q₈

—

139

D₄

—

106

—

140

D₄

—

212

—

141

D₄

—

212

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

106

—

146

C₂×C₄

—

106

—

147

C₂×C₄

—

212

—

148

C₂×C₄

—

212

—

149

C₂×C₄

D₄×D₅₃

yes

characteristic

150

C₈

—

106

—

151

C₈

D₄×D₅₃

yes

characteristic

152

C₈

C₂×C₂²⋊D₅₃

yes

characteristic

153

C₂²

—

106

—

154

C₂²

—

106

—

155

C₂²

—

106

—

156

C₂²

—

424

—

157

C₂²

—

424

—

158

C₂²

—

212

—

159

C₂²

D₈×D₅₃

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

106

—

163

C₄

—

424

—

164

C₄

#13133185

yes

characteristic

165

C₄

#13133241

yes

characteristic

166

C₂

—

424

—

167

C₂

—

212

—

168

C₂

#16810755

yes

characteristic

169

C₂

#16811056

yes

characteristic

170

C₂

#16811065

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₅₃
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₂×Q₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	C₂×Q₃₂	D₅₃
hypocenter	C₅₃	#2189
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

subgroup

G/H

center

C₂²

D₈×D₅₃

commutator

C₂×C₈×C₅₃

C₂³

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₂×Q₃₂×C₅₃

C₂

socle

C₂²×C₅₃

C₂×D₈

hypercenter

C₂×Q₃₂

D₅₃

hypocenter

C₅₃

#2189

LCS

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

N	H ≅ G/N	type
2. C₂×#13133160	C₂	semidirect
3. #16810861	C₂	semidirect
4. #16810427	C₂	semidirect
5. #16810425	C₂	semidirect
6. #16810402	C₂	semidirect
7. C₂×Q₃₂×C₅₃	C₂	semidirect
87. C₂×Q₃₂	D₅₃	semidirect
93. C₅₃	#2189	semidirect

H ≅ G/N

type

2. C₂×#13133160

C₂

semidirect

3. #16810861

C₂

semidirect

4. #16810427

C₂

semidirect

5. #16810425

C₂

semidirect

6. #16810402

C₂

semidirect

7. C₂×Q₃₂×C₅₃

C₂

semidirect

87. C₂×Q₃₂

D₅₃

semidirect

93. C₅₃

#2189

semidirect

Subgroups — #20276654

Conjugacy classes of subgroups

Complements