Group #20290655 — subgroups

Subgroups — #20290655

Order 6784.

220 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 90 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×#13133288	C₂	1	2	yes	maximal
2	C₂×#13133287	C₂	1	2	yes	maximal
3	#16811628	C₂	1	2	yes	maximal
4	C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719936	C₂²	1	4	yes	—
6	#13133288	—	2	4	no	—
7	#13133287	—	2	4	no	—
8	#13133284	—	2	4	no	—
9	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
11	C₄²×D₅₃	C₂²	1	1	yes	characteristic
12	C₄×#88322	C₄	1	1	yes	characteristic
13	C₈×Dic₅₃	C₄	1	1	yes	characteristic
14	C₈×Dic₅₃	C₄	1	1	yes	characteristic
15	C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
16	C₂×C₄×D₅₃	Q₈	1	2	yes	—
17	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
18	C₂×#88324	—	2	4	no	—
19	#11719936	—	2	4	no	—
20	#11719936	—	2	4	no	—
21	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
25	C₄×Dic₅₃	D₄	1	2	yes	—
26	C₂×#88322	C₂×C₄	1	1	yes	characteristic
27	C₂×#88322	C₂×C₄	1	1	yes	characteristic
28	C₈×D₅₃	D₄	1	2	yes	—
29	C₈×D₅₃	D₄	1	2	yes	—
30	C₈×D₅₃	Q₈	1	2	yes	—
31	C₈×D₅₃	Q₈	1	2	yes	—
32	C₂²×D₅₃	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
34	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×Dic₅₃	C₄⋊C₄	1	2	yes	—
36	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
37	C₄×D₅₃	—	2	4	no	—
38	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
39	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
40	C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
41	C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
42	#88324	—	4	4	no	—
43	C₈×C₅₃	C₄⋊C₄	1	2	yes	—
44	C₈×C₅₃	C₄⋊C₄	1	2	yes	—
45	#88322	C₄⋊C₄	1	2	yes	—
46	#88322	C₄⋊C₄	1	2	yes	—
47	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×D₅₃	—	2	2	no	—
49	C₂×D₅₃	C₄⋊Q₈	1	1	yes	characteristic
50	C₂×D₅₃	C₄⋊Q₈	1	1	yes	characteristic
51	C₄×C₅₃	—	2	2	no	—
52	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	Dic₅₃	#108	1	4	yes	complemented
55	Dic₅₃	#127	1	1	yes	characteristic
56	Dic₅₃	C₄⋊Q₈	1	1	yes	characteristic
57	#1848	—	53	1	no	maximal, Sylow
58	C₂×C₅₃	#384	1	1	yes	characteristic
59	C₂×C₅₃	C₂×#108	1	2	yes	—
60	D₅₃	—	2	1	no	—
61	D₅₃	—	2	1	no	—
62	C₂×#108	—	53	2	no	—
63	C₂×#108	—	53	2	no	—
64	#423	—	53	2	no	—
65	C₂×C₄×C₈	—	53	1	no	—
66	C₅₃	#1848	1	1	yes	characteristic, complemented, Sylow
67	C₂×M₁₆	—	53	4	no	—
68	C₂²×C₈	—	53	1	no	—
69	C₂²×C₈	—	53	1	no	—
70	C₂×C₄²	—	53	1	no	—
71	#108	—	106	4	no	—
72	#108	—	106	4	no	—
73	C₄⋊C₈	—	106	4	no	—
74	C₄×C₈	—	53	1	no	—
75	C₄×C₈	—	53	1	no	—
76	C₄×C₈	—	53	1	no	—
77	C₄×C₈	#6157	1	1	yes	characteristic
78	C₂²×C₄	—	53	1	no	—
79	C₂²×C₄	—	53	2	no	—
80	M₁₆	—	106	4	no	—
81	M₁₆	—	106	4	no	—
82	C₂×C₈	—	53	1	no	—
83	C₂×C₈	—	53	1	no	—
84	C₂×C₈	—	53	2	no	—
85	C₂×C₈	—	53	2	no	—
86	C₂×C₈	—	53	2	no	—
87	C₂×C₈	—	53	2	no	—
88	C₂×C₈	—	106	4	no	—
89	C₂×C₈	C₂×#6157	1	1	yes	characteristic
90	C₂×C₈	C₂×#6157	1	1	yes	characteristic
91	C₄²	—	53	2	no	—
92	C₄²	—	53	1	no	—
93	C₄²	C₂×#6157	1	1	yes	characteristic
94	C₂³	—	53	1	no	—
95	C₂×C₄	—	53	2	no	—
96	C₂×C₄	—	53	1	no	—
97	C₂×C₄	—	53	1	no	—
98	C₂×C₄	—	53	1	no	—
99	C₂×C₄	—	53	1	no	—
100	C₂×C₄	—	53	1	no	—
101	C₂×C₄	—	106	4	no	—
102	C₂×C₄	C₄⋊#6157	1	2	yes	—
103	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
104	C₈	—	53	2	no	—
105	C₈	—	53	2	no	—
106	C₈	—	212	4	no	—
107	C₈	C₄⋊#6157	1	2	yes	—
108	C₈	C₄⋊#6157	1	2	yes	—
109	C₂²	—	53	1	no	—
110	C₂²	—	53	1	no	—
111	C₂²	—	106	2	no	—
112	C₂²	C₂×C₄⋊#6157	1	1	yes	characteristic
113	C₄	—	2	2	no	—
114	C₄	—	53	1	no	—
115	C₄	—	53	1	no	—
116	C₄	—	53	4	no	—
117	C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
118	C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
119	C₂	—	106	1	no	—
120	C₂	—	106	1	no	—
121	C₂	#16811569	1	1	yes	characteristic
122	C₂	D₅₃⋊#108	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133288

C₂

yes

maximal

C₂×#13133287

C₂

yes

maximal

#16811628

C₂

yes

maximal

C₄×C₈×D₅₃

C₂

yes

maximal, characteristic

C₂×#11719936

C₂²

yes

—

#13133288

—

#13133287

—

#13133284

—

C₂×C₈×D₅₃

C₂²

yes

characteristic

C₂×C₈×D₅₃

C₂²

yes

characteristic

C₄²×D₅₃

C₂²

yes

characteristic

C₄×#88322

C₄

yes

characteristic

C₈×Dic₅₃

C₄

yes

characteristic

C₈×Dic₅₃

C₄

yes

characteristic

C₄×C₈×C₅₃

C₄

yes

characteristic

C₂×C₄×D₅₃

Q₈

yes

—

C₂×C₄×D₅₃

C₂³

yes

—

—

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

—

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₈×D₅₃

D₄

yes

—

C₈×D₅₃

D₄

yes

—

C₈×D₅₃

Q₈

yes

—

C₈×D₅₃

Q₈

yes

—

C₂²×D₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₄⋊C₄

yes

—

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

C₂×D₄

yes

characteristic

C₄×D₅₃

C₂×D₄

yes

characteristic

C₄×D₅₃

C₂×Q₈

yes

characteristic

C₄×D₅₃

C₂×Q₈

yes

characteristic

#88324

—

C₈×C₅₃

C₄⋊C₄

yes

—

C₈×C₅₃

C₄⋊C₄

yes

—

#88322

C₄⋊C₄

yes

—

#88322

C₄⋊C₄

yes

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

C₄⋊Q₈

yes

characteristic

C₂×D₅₃

C₄⋊Q₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₂×C₄⋊C₄

yes

Dic₅₃

yes

Dic₅₃

yes

Dic₅₃

C₄⋊Q₈

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

—

D₅₃

—

C₂×#108

—

C₂×#108

—

#423

—

C₂×C₄×C₈

—

C₅₃

#1848

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

#108

—

106

—

#108

—

106

—

C₄⋊C₈

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#6157

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

106

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

C₄²

—

C₄²

C₂×#6157

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

106

—

102

C₂×C₄

C₄⋊#6157

yes

—

103

C₂×C₄

C₂²×#6157

yes

characteristic

104

C₈

—

105

C₈

—

106

C₈

—

212

—

107

C₈

C₄⋊#6157

yes

—

108

C₈

C₄⋊#6157

yes

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

106

—

112

C₂²

C₂×C₄⋊#6157

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

C₂×C₄⋊#6157

yes

characteristic

118

C₄

C₂×C₄⋊#6157

yes

characteristic

119

C₂

—

106

—

120

C₂

—

106

—

121

C₂

#16811569

yes

characteristic

122

C₂

D₅₃⋊#108

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	C₄×C₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	C₄×C₈	#6157
hypocenter	C₅₃	#1848
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂²

C₂×C₄⋊#6157

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6157

Fitting

C₄×C₈×C₅₃

C₄

socle

C₂²×C₅₃

C₂×C₄⋊C₄

C₄×C₈

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
54. Dic₅₃	#108	semidirect
66. C₅₃	#1848	semidirect

H ≅ G/N

type

54. Dic₅₃

#108

semidirect

66. C₅₃

#1848

semidirect

Subgroups — #20290655

Conjugacy classes of subgroups

Complements