Group #20276433 — subgroups

Subgroups — #20276433

Order 6784.

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

normal subgroups only

This group has 220 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×Q₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133156	C₂	1	1	yes	maximal, characteristic
3	#16810660	C₂	1	1	yes	maximal, characteristic
4	#16810428	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810427	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810426	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#358	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₅₃	—	2	1	no	—
9	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic
10	C₂×#11719925	—	2	1	no	—
11	Q₁₆×D₅₃	—	2	2	no	—
12	Q₁₆×D₅₃	C₄	1	4	yes	complemented
13	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
14	#13133175	—	2	1	no	—
15	#13133156	—	2	2	no	—
16	C₂×#11719908	C₂²	1	1	yes	characteristic
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	Q₈×D₅₃	—	2	4	no	—
24	Q₈×D₅₃	—	4	2	no	—
25	C₂×C₄×D₅₃	—	4	1	no	—
26	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
27	C₂×Dic₁₀₆	—	2	1	no	—
28	C₂×Dic₁₀₆	—	4	1	no	—
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	#11719925	—	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	Dic₂₁₂	—	2	1	no	—
39	Dic₂₁₂	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	D₈	1	1	yes	characteristic
45	Q₈×C₅₃	—	2	2	no	—
46	Q₈×C₅₃	—	4	1	no	—
47	C₂×C₄×C₅₃	—	4	1	no	—
48	C₂×C₄×C₅₃	—	4	1	no	—
49	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₅₃	—	4	1	no	—
51	C₂×Dic₅₃	—	4	1	no	—
52	C₂×Dic₅₃	D₈	1	1	yes	characteristic
53	C₄×D₅₃	—	4	4	no	—
54	C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
55	C₄×D₅₃	SD₁₆	1	2	yes	—
56	Dic₁₀₆	—	2	2	no	—
57	Dic₁₀₆	—	4	1	no	—
58	Dic₁₀₆	—	4	4	no	—
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	#102	1	2	yes	—
66	C₄×C₅₃	—	4	2	no	—
67	C₄×C₅₃	—	8	1	no	—
68	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
70	Dic₅₃	—	4	2	no	—
71	Dic₅₃	—	8	1	no	—
72	Dic₅₃	#102	1	2	yes	—
73	#2141	—	53	1	no	maximal, Sylow
74	C₂×C₅₃	C₂×#102	1	1	yes	characteristic
75	C₂×C₅₃	#510	1	1	yes	characteristic
76	C₂×C₅₃	#510	1	1	yes	characteristic
77	D₅₃	—	2	2	no	—
78	C₂²×Q₁₆	—	53	1	no	—
79	C₂×M₃₂	—	53	1	no	—
80	#427	—	53	1	no	—
81	#358	—	53	1	no	—
82	#358	—	53	1	no	—
83	#358	—	53	1	no	—
84	#358	D₅₃	1	1	yes	characteristic, complemented
85	C₅₃	#2141	1	1	yes	characteristic, complemented, Sylow
86	C₂²×Q₈	—	106	1	no	—
87	C₂×Q₁₆	—	53	4	no	—
88	C₂×Q₁₆	—	53	1	no	—
89	C₂×Q₁₆	—	106	2	no	—
90	C₂×Q₁₆	—	106	1	no	—
91	C₂×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
92	C₂²×C₈	—	53	1	no	—
93	#117	—	106	1	no	—
94	M₃₂	—	106	2	no	—
95	C₂×C₁₆	—	53	1	no	—
96	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
97	#107	—	53	1	no	—
98	#107	C₂×D₅₃	1	1	yes	characteristic
99	#106	—	106	1	no	—
100	C₂×Q₈	—	2	1	no	—
101	C₂×Q₈	—	106	4	no	—
102	C₂×Q₈	—	106	1	no	—
103	C₂×Q₈	—	212	1	no	—
104	C₂×Q₈	—	212	2	no	—
105	C₂²×C₄	—	53	1	no	—
106	C₂²×C₄	—	212	1	no	—
107	Q₁₆	—	2	1	no	—
108	Q₁₆	—	53	2	no	—
109	Q₁₆	—	106	4	no	—
110	Q₁₆	—	106	1	no	—
111	Q₁₆	C₄×D₅₃	1	2	yes	—
112	C₂×C₈	—	53	1	no	—
113	C₂×C₈	—	53	2	no	—
114	C₂×C₈	—	53	2	no	—
115	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	106	1	no	—
118	C₂²⋊C₄	—	212	1	no	—
119	C₄²	—	212	1	no	—
120	C₁₆	—	2	1	no	—
121	C₁₆	—	106	1	no	—
122	C₂³	—	53	1	no	—
123	Q₈	—	2	2	no	—
124	Q₈	—	4	1	no	—
125	Q₈	—	106	2	no	—
126	Q₈	—	212	4	no	—
127	Q₈	—	212	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	53	1	no	—
131	C₂×C₄	—	53	2	no	—
132	C₂×C₄	—	53	2	no	—
133	C₂×C₄	—	212	4	no	—
134	C₂×C₄	—	212	1	no	—
135	C₂×C₄	—	212	1	no	—
136	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
137	C₈	—	53	2	no	—
138	C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
139	C₈	D₄×D₅₃	1	1	yes	characteristic
140	C₂²	—	53	2	no	—
141	C₂²	—	106	1	no	—
142	C₂²	—	106	1	no	—
143	C₂²	D₈×D₅₃	1	1	yes	characteristic
144	C₄	—	4	2	no	—
145	C₄	—	8	1	no	—
146	C₄	—	53	2	no	—
147	C₄	—	424	1	no	—
148	C₄	—	212	2	no	—
149	C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
150	C₄	SD₁₆×D₅₃	1	1	yes	characteristic
151	C₂	—	106	2	no	—
152	C₂	#102×D₅₃	1	1	yes	characteristic
153	C₂	#16811062	1	1	yes	characteristic
154	C₂	#16811065	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₁₆×D₅₃

C₂

yes

maximal, characteristic

C₂×#13133156

C₂

yes

maximal, characteristic

#16810660

C₂

yes

maximal, characteristic

#16810428

C₂

yes

maximal, characteristic, complemented

#16810427

C₂

yes

maximal, characteristic, complemented

#16810426

C₂

yes

maximal, characteristic, complemented

C₅₃×#358

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₅₃

—

C₂×Dic₂₁₂

C₂²

yes

characteristic

C₂×#11719925

—

Q₁₆×D₅₃

—

Q₁₆×D₅₃

C₄

yes

complemented

C₂×C₈×D₅₃

C₂²

yes

—

—

C₂²

yes

characteristic

#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

—

Q₈×D₅₃

—

Q₈×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

#11719925

—

#11719921

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

D₄

yes

characteristic

Dic₂₁₂

—

Dic₂₁₂

C₂×C₄

yes

—

C₈×D₅₃

C₂×C₄

yes

—

C₈×D₅₃

D₄

yes

—

C₁₆×C₅₃

—

#11719908

—

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

—

C₂×Dic₅₃

D₈

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

C₂²⋊C₄

yes

—

C₄×D₅₃

SD₁₆

yes

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

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

#102

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

C₂×SD₁₆

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#102

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

C₂²×Q₁₆

—

C₂×M₃₂

—

—

—

—

—

D₅₃

yes

characteristic, complemented

C₅₃

#2141

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

106

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

106

—

C₂×Q₁₆

—

106

—

C₂×Q₁₆

C₂×D₅₃

yes

characteristic

C₂²×C₈

—

#117

—

106

—

M₃₂

—

106

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₃

yes

characteristic

#107

—

#107

C₂×D₅₃

yes

characteristic

#106

—

106

—

100

C₂×Q₈

—

101

C₂×Q₈

—

106

—

102

C₂×Q₈

—

106

—

103

C₂×Q₈

—

212

—

104

C₂×Q₈

—

212

—

105

C₂²×C₄

—

106

C₂²×C₄

—

212

—

107

Q₁₆

—

108

Q₁₆

—

109

Q₁₆

—

106

—

110

Q₁₆

—

106

—

111

Q₁₆

C₄×D₅₃

yes

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

C₂²×D₅₃

yes

characteristic

116

C₄⋊C₄

—

117

C₄⋊C₄

—

106

—

118

C₂²⋊C₄

—

212

—

119

C₄²

—

212

—

120

C₁₆

—

121

C₁₆

—

106

—

122

C₂³

—

123

Q₈

—

124

Q₈

—

125

Q₈

—

106

—

126

Q₈

—

212

—

127

Q₈

—

212

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

212

—

134

C₂×C₄

—

212

—

135

C₂×C₄

—

212

—

136

C₂×C₄

D₄×D₅₃

yes

characteristic

137

C₈

—

138

C₈

C₂×C₄×D₅₃

yes

characteristic

139

C₈

D₄×D₅₃

yes

characteristic

140

C₂²

—

141

C₂²

—

106

—

142

C₂²

—

106

—

143

C₂²

D₈×D₅₃

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

424

—

148

C₄

—

212

—

149

C₄

C₂²⋊C₄×D₅₃

yes

characteristic

150

C₄

SD₁₆×D₅₃

yes

characteristic

151

C₂

—

106

—

152

C₂

#102×D₅₃

yes

characteristic

153

C₂

#16811062

yes

characteristic

154

C₂

#16811065

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	#2141
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
4. #16810428	C₂	semidirect
5. #16810427	C₂	semidirect
6. #16810426	C₂	semidirect
7. C₅₃×#358	C₂	semidirect
12. Q₁₆×D₅₃	C₄	semidirect
84. #358	D₅₃	semidirect
85. C₅₃	#2141	semidirect

H ≅ G/N

type

4. #16810428

C₂

semidirect

5. #16810427

C₂

semidirect

6. #16810426

C₂

semidirect

7. C₅₃×#358

C₂

semidirect

12. Q₁₆×D₅₃

C₄

semidirect

84. #358

D₅₃

semidirect

85. C₅₃

#2141

semidirect

Subgroups — #20276433

Conjugacy classes of subgroups

Complements