Group #20273477 — subgroups

Subgroups — #20273477

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	#16810679	C₂	1	2	yes	maximal
2	C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#16810376	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810373	C₂	1	2	yes	maximal, complemented
5	C₅₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₅₃	C₄	1	1	yes	characteristic
7	C₂×C₈×D₅₃	C₄	1	1	yes	characteristic
8	C₄²×D₅₃	C₂²	1	1	yes	characteristic
9	C₄⋊#88322	C₂²	1	2	yes	—
10	#13133129	—	2	2	no	—
11	#13133127	—	2	2	no	—
12	C₄×#88322	C₂²	1	1	yes	characteristic
13	C₈×Dic₅₃	C₄	1	1	yes	characteristic
14	C₈×Dic₅₃	C₄	1	1	yes	characteristic
15	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
16	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
19	C₂×C₄×D₅₃	Q₈	1	1	yes	characteristic
20	C₂×C₈×C₅₃	—	2	2	no	—
21	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₅₃	C₂³	1	1	yes	characteristic
24	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
25	C₄×Dic₅₃	D₄	1	1	yes	characteristic
26	C₄×Dic₅₃	Q₈	1	1	yes	characteristic
27	C₂×#88322	—	2	2	no	—
28	C₂×#88322	C₂×C₄	1	1	yes	characteristic
29	C₂×#88322	C₂×C₄	1	1	yes	characteristic
30	C₈×D₅₃	—	2	2	no	—
31	C₈×D₅₃	—	2	2	no	—
32	C₂²×D₅₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
36	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₄×D₅₃	—	2	2	no	—
40	C₄×D₅₃	—	2	1	no	—
41	C₄×D₅₃	—	2	1	no	—
42	C₄×D₅₃	—	2	1	no	—
43	C₄×D₅₃	—	2	1	no	—
44	C₈×C₅₃	—	2	1	no	—
45	C₈×C₅₃	—	4	2	no	—
46	C₈×C₅₃	M₁₆	1	2	yes	—
47	#88322	—	2	1	no	—
48	#88322	—	4	2	no	—
49	#88322	M₁₆	1	2	yes	—
50	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂×D₅₃	—	2	1	no	—
52	C₂×D₅₃	—	2	1	no	—
53	C₂×D₅₃	#108	1	2	yes	—
54	C₄×C₅₃	—	2	1	no	—
55	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
56	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
57	C₄×C₅₃	#135	1	1	yes	characteristic
58	C₄×C₅₃	#135	1	1	yes	characteristic
59	Dic₅₃	—	2	1	no	—
60	Dic₅₃	—	2	1	no	—
61	Dic₅₃	—	2	1	no	—
62	#1568	—	53	1	no	maximal, Sylow
63	C₂×C₅₃	#423	1	1	yes	characteristic
64	C₂×C₅₃	#427	1	1	yes	characteristic
65	C₂×C₅₃	C₂×#108	1	1	yes	characteristic
66	D₅₃	—	2	2	no	—
67	#432	—	53	2	no	—
68	C₂×C₄×C₈	—	53	1	no	—
69	#335	—	53	1	no	—
70	#335	D₅₃	1	1	yes	characteristic, complemented
71	#334	—	53	2	no	—
72	C₅₃	#1568	1	1	yes	characteristic, complemented, Sylow
73	C₂²×C₈	—	53	1	no	—
74	C₂²×C₈	—	53	1	no	—
75	C₂×C₄²	—	53	1	no	—
76	C₄⋊C₈	—	53	2	no	—
77	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
78	C₂²⋊C₈	—	106	2	no	—
79	#97	—	106	2	no	—
80	C₄×C₈	—	53	1	no	—
81	C₄×C₈	—	53	1	no	—
82	C₄×C₈	—	53	1	no	—
83	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
84	C₂²×C₄	—	53	1	no	—
85	C₂²×C₄	—	53	1	no	—
86	C₂²×C₄	—	53	1	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	53	1	no	—
89	C₂×C₈	—	53	1	no	—
90	C₂×C₈	—	106	2	no	—
91	C₂×C₈	—	106	2	no	—
92	C₂×C₈	—	106	2	no	—
93	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
94	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
95	C₄²	—	53	1	no	—
96	C₄²	—	53	1	no	—
97	C₄²	—	53	1	no	—
98	C₄²	C₂²×D₅₃	1	1	yes	characteristic
99	C₂³	—	53	1	no	—
100	C₂×C₄	—	53	1	no	—
101	C₂×C₄	—	53	1	no	—
102	C₂×C₄	—	53	1	no	—
103	C₂×C₄	—	106	2	no	—
104	C₂×C₄	—	106	1	no	—
105	C₂×C₄	—	106	1	no	—
106	C₂×C₄	—	106	1	no	—
107	C₂×C₄	—	106	1	no	—
108	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
109	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
110	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
111	C₈	—	2	1	no	—
112	C₈	—	4	2	no	—
113	C₈	—	53	2	no	—
114	C₈	—	106	1	no	—
115	C₈	—	212	2	no	—
116	C₈	#11719912	1	2	yes	—
117	C₂²	—	53	2	no	—
118	C₂²	—	106	1	no	—
119	C₂²	—	106	1	no	—
120	C₂²	C₄⋊C₄×D₅₃	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	106	1	no	—
123	C₄	—	106	1	no	—
124	C₄	—	106	1	no	—
125	C₄	M₁₆×D₅₃	1	1	yes	characteristic
126	C₄	C₂×#11719912	1	1	yes	characteristic
127	C₄	#13133236	1	1	yes	characteristic
128	C₄	#13133240	1	1	yes	characteristic
129	C₂	—	106	2	no	—
130	C₂	#16810645	1	1	yes	characteristic
131	C₂	#16810660	1	1	yes	characteristic
132	C₂	#108×D₅₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810679

C₂

yes

maximal

C₄×C₈×D₅₃

C₂

yes

maximal, characteristic

#16810376

C₂

yes

maximal, characteristic, complemented

#16810373

C₂

yes

maximal, complemented

C₅₃×#335

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₅₃

C₄

yes

characteristic

C₂×C₈×D₅₃

C₄

yes

characteristic

C₄²×D₅₃

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

—

#13133129

—

#13133127

—

C₄×#88322

C₂²

yes

characteristic

C₈×Dic₅₃

C₄

yes

characteristic

C₈×Dic₅₃

C₄

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

—

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×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

characteristic

C₂×#88322

—

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₈×D₅₃

—

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

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

M₁₆

yes

—

#88322

—

#88322

—

#88322

M₁₆

yes

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

#108

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₄×C₅₃

C₂×M₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

#432

—

C₂×C₄×C₈

—

#335

—

#335

D₅₃

yes

characteristic, complemented

#334

—

C₅₃

#1568

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅₃

yes

—

C₂²⋊C₈

—

106

—

#97

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

C₄×D₅₃

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

106

—

104

C₂×C₄

—

106

—

105

C₂×C₄

—

106

—

106

C₂×C₄

—

106

—

107

C₂×C₄

—

106

—

108

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

109

C₂×C₄

D₄×D₅₃

yes

characteristic

110

C₂×C₄

Q₈×D₅₃

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

—

106

—

115

C₈

—

212

—

116

C₈

#11719912

yes

—

117

C₂²

—

118

C₂²

—

106

—

119

C₂²

—

106

—

120

C₂²

C₄⋊C₄×D₅₃

yes

characteristic

121

C₄

—

122

C₄

—

106

—

123

C₄

—

106

—

124

C₄

—

106

—

125

C₄

M₁₆×D₅₃

yes

characteristic

126

C₄

C₂×#11719912

yes

characteristic

127

C₄

#13133236

yes

characteristic

128

C₄

#13133240

yes

characteristic

129

C₂

—

106

—

130

C₂

#16810645

yes

characteristic

131

C₂

#16810660

yes

characteristic

132

C₂

#108×D₅₃

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	#1568
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
3. #16810376	C₂	semidirect
4. #16810373	C₂	semidirect
5. C₅₃×#335	C₂	semidirect
70. #335	D₅₃	semidirect
72. C₅₃	#1568	semidirect

H ≅ G/N

type

3. #16810376

C₂

semidirect

4. #16810373

C₂

semidirect

5. C₅₃×#335

C₂

semidirect

70. #335

D₅₃

semidirect

72. C₅₃

#1568

semidirect

Subgroups — #20273477

Conjugacy classes of subgroups

Complements