Group #20276421 — subgroups

Subgroups — #20276421

Order 6784.

228 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 69 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×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810660	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	#16810423	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#358	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719949	—	2	1	no	—
9	#13133240	—	2	2	no	—
10	#13133240	C₄	1	4	yes	complemented
11	C₂×#11719924	—	2	1	no	—
12	C₂×D₄₂₄	C₂²	1	1	yes	characteristic
13	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
14	#13133175	—	2	1	no	—
15	C₂×#11719908	C₂²	1	1	yes	characteristic
16	C₁₆×D₅₃	—	2	2	no	—
17	#13133151	C₂²	1	1	yes	characteristic
18	#13133148	—	2	1	no	—
19	C₂×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
21	#107×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×Q₈×C₅₃	—	2	1	no	—
23	#11719949	—	2	4	no	—
24	#11719949	—	4	2	no	—
25	C₂×D₂₁₂	—	2	1	no	—
26	C₂×D₂₁₂	—	4	1	no	—
27	C₂×C₄×D₅₃	—	4	1	no	—
28	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
29	Q₁₆×C₅₃	—	2	1	no	—
30	Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₅₃	—	2	1	no	—
33	#11719924	—	2	4	no	—
34	#11719921	—	4	1	no	—
35	C₄⋊Dic₅₃	—	2	1	no	—
36	C₄×Dic₅₃	—	4	1	no	—
37	C₂×#88322	D₄	1	1	yes	characteristic
38	D₄₂₄	—	2	1	no	—
39	D₄₂₄	C₂×C₄	1	2	yes	—
40	C₈×D₅₃	C₂×C₄	1	2	yes	—
41	C₈×D₅₃	D₄	1	2	yes	—
42	C₁₆×C₅₃	—	2	1	no	—
43	#11719908	—	2	1	no	—
44	C₂²×D₅₃	—	4	1	no	—
45	C₂²×D₅₃	D₈	1	1	yes	characteristic
46	Q₈×C₅₃	—	2	2	no	—
47	Q₈×C₅₃	—	4	1	no	—
48	C₂×C₄×C₅₃	—	4	1	no	—
49	C₂×C₄×C₅₃	—	4	1	no	—
50	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₅₃	—	4	1	no	—
52	C₂×Dic₅₃	D₈	1	1	yes	characteristic
53	D₂₁₂	—	2	2	no	—
54	D₂₁₂	—	4	4	no	—
55	D₂₁₂	—	4	1	no	—
56	C₄×D₅₃	—	4	4	no	—
57	C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
58	C₄×D₅₃	SD₁₆	1	2	yes	—
59	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
61	#88322	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
63	C₂×D₅₃	—	2	1	no	—
64	C₂×D₅₃	—	2	1	no	—
65	C₂×D₅₃	—	4	2	no	—
66	C₂×D₅₃	—	8	1	no	—
67	C₂×D₅₃	—	8	1	no	—
68	C₂×D₅₃	#102	1	2	yes	—
69	C₄×C₅₃	—	4	2	no	—
70	C₄×C₅₃	—	8	1	no	—
71	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
73	Dic₅₃	—	8	1	no	—
74	Dic₅₃	#102	1	2	yes	—
75	#2138	—	53	1	no	maximal, Sylow
76	C₂×C₅₃	C₂×#102	1	1	yes	characteristic
77	C₂×C₅₃	#508	1	1	yes	characteristic
78	C₂×C₅₃	#508	1	1	yes	characteristic
79	D₅₃	—	2	2	no	—
80	D₅₃	—	8	2	no	—
81	C₂×#135	—	53	1	no	—
82	C₂²×C₁₆	—	53	1	no	—
83	#427	—	53	1	no	—
84	#358	—	53	1	no	—
85	#358	D₅₃	1	1	yes	characteristic, complemented
86	#357	—	53	1	no	—
87	#357	—	53	1	no	—
88	C₅₃	#2138	1	1	yes	characteristic, complemented, Sylow
89	C₂×#41	—	106	1	no	—
90	#135	—	53	4	no	—
91	#135	—	106	2	no	—
92	C₂×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
93	C₂×SD₁₆	—	106	1	no	—
94	C₂×D₈	—	53	1	no	—
95	C₂²×C₈	—	53	1	no	—
96	#117	—	106	1	no	—
97	C₂×C₁₆	—	53	1	no	—
98	C₂×C₁₆	—	106	2	no	—
99	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
100	#107	—	53	1	no	—
101	#107	C₂×D₅₃	1	1	yes	characteristic
102	#106	—	106	1	no	—
103	#41	—	106	4	no	—
104	#41	—	212	2	no	—
105	C₂×Q₈	—	2	1	no	—
106	C₂×D₄	—	106	1	no	—
107	C₂×D₄	—	212	1	no	—
108	C₂²×C₄	—	53	1	no	—
109	C₂²×C₄	—	212	1	no	—
110	Q₁₆	—	2	1	no	—
111	Q₁₆	C₄×D₅₃	1	2	yes	—
112	SD₁₆	—	106	4	no	—
113	D₈	—	53	2	no	—
114	D₈	—	106	1	no	—
115	C₂×C₈	—	53	1	no	—
116	C₂×C₈	—	53	2	no	—
117	C₂×C₈	—	53	2	no	—
118	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	106	1	no	—
121	C₂²⋊C₄	—	212	1	no	—
122	C₄²	—	212	1	no	—
123	C₁₆	—	2	1	no	—
124	C₁₆	—	106	1	no	—
125	C₂³	—	53	1	no	—
126	C₂³	—	212	1	no	—
127	Q₈	—	2	2	no	—
128	Q₈	—	4	1	no	—
129	D₄	—	106	2	no	—
130	D₄	—	212	4	no	—
131	D₄	—	212	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	53	1	no	—
135	C₂×C₄	—	53	2	no	—
136	C₂×C₄	—	53	2	no	—
137	C₂×C₄	—	212	4	no	—
138	C₂×C₄	—	212	1	no	—
139	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
140	C₈	—	53	2	no	—
141	C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
142	C₈	D₄×D₅₃	1	1	yes	characteristic
143	C₂²	—	53	2	no	—
144	C₂²	—	106	1	no	—
145	C₂²	—	106	1	no	—
146	C₂²	—	424	1	no	—
147	C₂²	—	424	1	no	—
148	C₂²	—	212	2	no	—
149	C₂²	D₈×D₅₃	1	1	yes	characteristic
150	C₄	—	4	2	no	—
151	C₄	—	8	1	no	—
152	C₄	—	53	2	no	—
153	C₄	—	424	1	no	—
154	C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
155	C₄	SD₁₆×D₅₃	1	1	yes	characteristic
156	C₂	—	106	2	no	—
157	C₂	—	424	2	no	—
158	C₂	#102×D₅₃	1	1	yes	characteristic
159	C₂	#16811054	1	1	yes	characteristic
160	C₂	#16811056	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133240

C₂

yes

maximal, characteristic

C₂×C₁₆×D₅₃

C₂

yes

maximal, characteristic, complemented

#16810660

C₂

yes

maximal, characteristic, complemented

#16810427

C₂

yes

maximal, characteristic, complemented

#16810425

C₂

yes

maximal, characteristic, complemented

#16810423

C₂

yes

maximal, characteristic, complemented

C₅₃×#358

C₂

yes

maximal, characteristic, complemented

C₂×#11719949

—

#13133240

—

#13133240

C₄

yes

complemented

C₂×#11719924

—

C₂×D₄₂₄

C₂²

yes

characteristic

C₂×C₈×D₅₃

C₂²

yes

characteristic

#13133175

—

C₂×#11719908

C₂²

yes

characteristic

C₁₆×D₅₃

—

#13133151

C₂²

yes

characteristic

#13133148

—

C₂×Q₁₆×C₅₃

C₂²

yes

characteristic

C₂×C₁₆×C₅₃

C₂²

yes

characteristic

#107×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

#11719949

—

#11719949

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

D₄

yes

characteristic

Q₁₆×C₅₃

—

Q₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

#11719924

—

#11719921

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

D₄

yes

characteristic

D₄₂₄

—

D₄₂₄

C₂×C₄

yes

—

C₈×D₅₃

C₂×C₄

yes

—

C₈×D₅₃

D₄

yes

—

C₁₆×C₅₃

—

#11719908

—

C₂²×D₅₃

—

C₂²×D₅₃

D₈

yes

characteristic

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

C₂²⋊C₄

yes

—

C₄×D₅₃

SD₁₆

yes

—

C₈×C₅₃

C₂²×C₄

yes

characteristic

C₈×C₅₃

C₂×D₄

yes

characteristic

#88322

C₂²⋊C₄

yes

—

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

#102

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

C₂×SD₁₆

yes

characteristic

Dic₅₃

—

Dic₅₃

#102

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

D₅₃

—

C₂×#135

—

C₂²×C₁₆

—

#427

—

#358

—

#358

D₅₃

yes

characteristic, complemented

#357

—

#357

—

C₅₃

#2138

yes

characteristic, complemented, Sylow

C₂×#41

—

106

—

#135

—

#135

—

106

—

C₂×Q₁₆

C₂×D₅₃

yes

characteristic

C₂×SD₁₆

—

106

—

C₂×D₈

—

C₂²×C₈

—

#117

—

106

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₂×C₁₆

C₂×D₅₃

yes

characteristic

100

#107

—

101

#107

C₂×D₅₃

yes

characteristic

102

#106

—

106

—

103

#41

—

106

—

104

#41

—

212

—

105

C₂×Q₈

—

106

C₂×D₄

—

106

—

107

C₂×D₄

—

212

—

108

C₂²×C₄

—

109

C₂²×C₄

—

212

—

110

Q₁₆

—

111

Q₁₆

C₄×D₅₃

yes

—

112

SD₁₆

—

106

—

113

D₈

—

114

D₈

—

106

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

C₂²×D₅₃

yes

characteristic

119

C₄⋊C₄

—

120

C₄⋊C₄

—

106

—

121

C₂²⋊C₄

—

212

—

122

C₄²

—

212

—

123

C₁₆

—

124

C₁₆

—

106

—

125

C₂³

—

126

C₂³

—

212

—

127

Q₈

—

128

Q₈

—

129

D₄

—

106

—

130

D₄

—

212

—

131

D₄

—

212

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

212

—

138

C₂×C₄

—

212

—

139

C₂×C₄

D₄×D₅₃

yes

characteristic

140

C₈

—

141

C₈

C₂×C₄×D₅₃

yes

characteristic

142

C₈

D₄×D₅₃

yes

characteristic

143

C₂²

—

144

C₂²

—

106

—

145

C₂²

—

106

—

146

C₂²

—

424

—

147

C₂²

—

424

—

148

C₂²

—

212

—

149

C₂²

D₈×D₅₃

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

424

—

154

C₄

C₂²⋊C₄×D₅₃

yes

characteristic

155

C₄

SD₁₆×D₅₃

yes

characteristic

156

C₂

—

106

—

157

C₂

—

424

—

158

C₂

#102×D₅₃

yes

characteristic

159

C₂

#16811054

yes

characteristic

160

C₂

#16811056

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₅₃
commutator	C₈×C₅₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#358	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	#358	D₅₃
hypocenter	C₅₃	#2138
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₅₃

commutator

C₈×C₅₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#358

C₂

socle

C₂²×C₅₃

C₂×D₈

D₅₃

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
2. C₂×C₁₆×D₅₃	C₂	semidirect
3. #16810660	C₂	semidirect
4. #16810427	C₂	semidirect
5. #16810425	C₂	semidirect
6. #16810423	C₂	semidirect
7. C₅₃×#358	C₂	semidirect
10. #13133240	C₄	semidirect
85. #358	D₅₃	semidirect
88. C₅₃	#2138	semidirect

H ≅ G/N

type

2. C₂×C₁₆×D₅₃

C₂

semidirect

3. #16810660

C₂

semidirect

4. #16810427

C₂

semidirect

5. #16810425

C₂

semidirect

6. #16810423

C₂

semidirect

7. C₅₃×#358

C₂

semidirect

10. #13133240

C₄

semidirect

85. #358

D₅₃

semidirect

88. C₅₃

#2138

semidirect

Subgroups — #20276421

Conjugacy classes of subgroups

Complements