Group #20273488 — subgroups

Subgroups — #20273488

Order 6784.

168 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 73 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	#16810674	C₂	1	1	yes	maximal, characteristic
2	#16810645	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#16810376	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810375	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810375	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719912	—	2	1	no	—
9	C₂×C₈×D₅₃	—	2	1	no	—
10	C₄²×D₅₃	C₂²	1	1	yes	characteristic
11	C₄⋊#88322	C₂²	1	1	yes	characteristic
12	C₄⋊#88322	C₂²	1	1	yes	characteristic
13	#13133146	—	2	1	no	—
14	#13133146	—	2	1	no	—
15	#13133146	C₄	1	1	yes	characteristic
16	#13133146	C₄	1	1	yes	characteristic
17	#13133129	C₄	1	1	yes	characteristic
18	#13133129	C₄	1	1	yes	characteristic
19	C₄×#88322	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
25	C₂×C₄×D₅₃	Q₈	1	1	yes	characteristic
26	C₂×C₈×C₅₃	—	2	1	no	—
27	C₂×C₈×C₅₃	—	2	1	no	—
28	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄×Dic₅₃	D₄	1	1	yes	characteristic
33	C₄×Dic₅₃	Q₈	1	1	yes	characteristic
34	C₂×#88322	—	2	1	no	—
35	C₂×#88322	—	2	1	no	—
36	C₂×#88322	C₂×C₄	1	1	yes	characteristic
37	C₂×#88322	C₂×C₄	1	1	yes	characteristic
38	#11719912	—	4	2	no	—
39	C₈×D₅₃	—	4	2	no	—
40	C₂²×D₅₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
47	C₄×D₅₃	—	2	1	no	—
48	C₄×D₅₃	—	2	1	no	—
49	C₄×D₅₃	—	4	1	no	—
50	C₄×D₅₃	D₈	1	2	yes	—
51	C₄×D₅₃	Q₁₆	1	2	yes	—
52	C₈×C₅₃	—	2	1	no	—
53	C₈×C₅₃	—	2	1	no	—
54	C₈×C₅₃	—	4	1	no	—
55	C₈×C₅₃	—	4	1	no	—
56	#88322	—	2	1	no	—
57	#88322	—	2	1	no	—
58	#88322	—	4	1	no	—
59	#88322	—	4	1	no	—
60	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×D₅₃	—	2	1	no	—
62	C₂×D₅₃	—	2	1	no	—
63	C₂×D₅₃	#107	1	2	yes	—
64	C₄×C₅₃	—	2	1	no	—
65	C₄×C₅₃	#131	1	1	yes	characteristic
66	C₄×C₅₃	#131	1	1	yes	characteristic
67	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
68	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
69	Dic₅₃	—	2	1	no	—
70	Dic₅₃	—	2	1	no	—
71	Dic₅₃	#107	1	2	yes	—
72	#1571	—	53	1	no	maximal, Sylow
73	C₂×C₅₃	#424	1	1	yes	characteristic
74	C₂×C₅₃	C₂×#107	1	1	yes	characteristic
75	C₂×C₅₃	#430	1	1	yes	characteristic
76	D₅₃	—	2	2	no	—
77	#431	—	53	1	no	—
78	#423	—	53	1	no	—
79	C₂×C₄⋊C₈	—	53	1	no	—
80	#335	—	53	1	no	—
81	#335	—	53	1	no	—
82	#335	—	53	1	no	—
83	#335	D₅₃	1	1	yes	characteristic, complemented
84	C₅₃	#1571	1	1	yes	characteristic, complemented, Sylow
85	C₂×M₁₆	—	106	1	no	—
86	C₂²×C₈	—	106	1	no	—
87	C₂×C₄²	—	53	1	no	—
88	C₄⋊C₈	—	53	1	no	—
89	C₄⋊C₈	—	53	1	no	—
90	C₄⋊C₈	—	53	1	no	—
91	C₄⋊C₈	—	53	1	no	—
92	C₄⋊C₈	—	106	1	no	—
93	C₄⋊C₈	—	106	1	no	—
94	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
95	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
96	C₂²⋊C₈	—	53	1	no	—
97	C₂²⋊C₈	—	53	1	no	—
98	C₄×C₈	—	53	1	no	—
99	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
100	C₂²×C₄	—	53	1	no	—
101	C₂²×C₄	—	53	1	no	—
102	C₂²×C₄	—	53	1	no	—
103	M₁₆	—	212	2	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	53	1	no	—
107	C₂×C₈	—	53	1	no	—
108	C₂×C₈	—	106	1	no	—
109	C₂×C₈	—	106	1	no	—
110	C₂×C₈	—	212	2	no	—
111	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
112	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
113	C₄²	—	53	1	no	—
114	C₄²	—	53	1	no	—
115	C₄²	—	53	1	no	—
116	C₄²	C₂²×D₅₃	1	1	yes	characteristic
117	C₂³	—	53	1	no	—
118	C₂×C₄	—	53	1	no	—
119	C₂×C₄	—	53	2	no	—
120	C₂×C₄	—	53	2	no	—
121	C₂×C₄	—	53	1	no	—
122	C₂×C₄	—	53	1	no	—
123	C₂×C₄	—	106	1	no	—
124	C₂×C₄	—	106	1	no	—
125	C₂×C₄	—	212	1	no	—
126	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
127	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
128	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
129	C₈	—	2	1	no	—
130	C₈	—	2	1	no	—
131	C₈	—	4	1	no	—
132	C₈	—	4	1	no	—
133	C₈	—	106	1	no	—
134	C₈	—	106	1	no	—
135	C₈	—	212	1	no	—
136	C₈	—	212	1	no	—
137	C₂²	—	53	2	no	—
138	C₂²	—	106	1	no	—
139	C₂²	—	106	1	no	—
140	C₂²	C₄⋊C₄×D₅₃	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	53	2	no	—
143	C₄	—	106	1	no	—
144	C₄	—	106	1	no	—
145	C₄	#13133223	1	1	yes	characteristic
146	C₄	#13133225	1	1	yes	characteristic
147	C₄	D₈×D₅₃	1	1	yes	characteristic
148	C₄	Q₁₆×D₅₃	1	1	yes	characteristic
149	C₂	—	106	2	no	—
150	C₂	#16810648	1	1	yes	characteristic
151	C₂	#107×D₅₃	1	1	yes	characteristic
152	C₂	#16810670	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810674

C₂

yes

maximal, characteristic

#16810645

C₂

yes

maximal, characteristic

C₄⋊C₈×D₅₃

C₂

yes

maximal, characteristic

#16810376

C₂

yes

maximal, characteristic, complemented

#16810375

C₂

yes

maximal, characteristic, complemented

#16810375

C₂

yes

maximal, characteristic, complemented

C₅₃×#335

C₂

yes

maximal, characteristic, complemented

C₂×#11719912

—

C₂×C₈×D₅₃

—

C₄²×D₅₃

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

—

—

C₄

yes

characteristic

#13133146

C₄

yes

characteristic

#13133129

C₄

yes

characteristic

#13133129

C₄

yes

characteristic

C₄×#88322

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

Q₈

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

Q₈

yes

—

—

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

#11719912

—

C₈×D₅₃

—

C₂²×D₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×Dic₅₃

C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

C₄⋊C₄

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

D₈

yes

—

C₄×D₅₃

Q₁₆

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

—

—

—

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

#107

yes

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₂×D₈

yes

characteristic

C₄×C₅₃

C₂×Q₁₆

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#107

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

#431

—

#423

—

C₂×C₄⋊C₈

—

—

—

—

D₅₃

yes

characteristic, complemented

C₅₃

#1571

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₂²×C₈

—

106

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

106

—

C₄⋊C₈

C₂×D₅₃

yes

characteristic

C₄⋊C₈

C₂×D₅₃

yes

characteristic

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₅₃

yes

characteristic

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

M₁₆

—

212

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

106

—

109

C₂×C₈

—

106

—

110

C₂×C₈

—

212

—

111

C₂×C₈

C₄×D₅₃

yes

characteristic

112

C₂×C₈

C₄×D₅₃

yes

characteristic

113

C₄²

—

114

C₄²

—

115

C₄²

—

116

C₄²

C₂²×D₅₃

yes

characteristic

117

C₂³

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

106

—

124

C₂×C₄

—

106

—

125

C₂×C₄

—

212

—

126

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

127

C₂×C₄

D₄×D₅₃

yes

characteristic

128

C₂×C₄

Q₈×D₅₃

yes

characteristic

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

106

—

134

C₈

—

106

—

135

C₈

—

212

—

136

C₈

—

212

—

137

C₂²

—

138

C₂²

—

106

—

139

C₂²

—

106

—

140

C₂²

C₄⋊C₄×D₅₃

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

106

—

144

C₄

—

106

—

145

C₄

#13133223

yes

characteristic

146

C₄

#13133225

yes

characteristic

147

C₄

D₈×D₅₃

yes

characteristic

148

C₄

Q₁₆×D₅₃

yes

characteristic

149

C₂

—

106

—

150

C₂

#16810648

yes

characteristic

151

C₂

#107×D₅₃

yes

characteristic

152

C₂

#16810670

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#335	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	#335	D₅₃
hypocenter	C₅₃	#1571
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

subgroup

G/H

center

C₂×C₄

D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#335

C₂

socle

C₂²×C₅₃

C₂×C₄⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

N	H ≅ G/N	type
4. #16810376	C₂	semidirect
5. #16810375	C₂	semidirect
6. #16810375	C₂	semidirect
7. C₅₃×#335	C₂	semidirect
83. #335	D₅₃	semidirect
84. C₅₃	#1571	semidirect

H ≅ G/N

type

4. #16810376

C₂

semidirect

5. #16810375

C₂

semidirect

6. #16810375

C₂

semidirect

7. C₅₃×#335

C₂

semidirect

83. #335

D₅₃

semidirect

84. C₅₃

#1571

semidirect

Subgroups — #20273488

Conjugacy classes of subgroups

Complements