Group #20289779 — subgroups

Subgroups — #20289779

Order 6784.

136 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 50 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	D₅₃⋊M₃₂	C₂	1	1	yes	maximal, characteristic
2	D₅₃⋊#108	C₂	1	1	yes	maximal, characteristic
3	#106×D₅₃	C₂	1	1	yes	maximal, characteristic
4	D₅₃⋊M₁₆	—	2	1	no	—
5	#13133292	—	2	1	no	—
6	#13133291	—	2	1	no	—
7	D₅₃⋊C₁₆	C₄	1	2	yes	complemented
8	#13133288	—	2	1	no	—
9	#13133287	C₄	1	2	yes	complemented
10	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
11	C₄⋊C₄×D₅₃	—	2	1	no	—
12	#13133149	C₄	1	1	yes	characteristic
13	#13133148	—	2	1	no	—
14	#106×C₅₃	C₄	1	1	yes	characteristic
15	C₂×C₄×D₅₃	—	4	1	no	—
16	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
17	#11719940	—	4	1	no	—
18	#11719936	—	2	2	no	—
19	D₅₃⋊C₈	—	4	1	no	—
20	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₄⋊C₄×C₅₃	—	2	1	no	—
22	C₄⋊Dic₅₃	—	2	1	no	—
23	#11719919	—	4	1	no	—
24	C₂×#88322	D₄	1	1	yes	characteristic
25	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
26	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
27	C₈×D₅₃	D₄	1	1	yes	characteristic
28	C₈×D₅₃	Q₈	1	1	yes	characteristic
29	#11719910	—	2	2	no	—
30	C₂²×D₅₃	SD₁₆	1	1	yes	characteristic
31	C₂×C₄×C₅₃	—	4	1	no	—
32	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×Dic₅₃	—	4	1	no	—
34	C₂×Dic₅₃	D₈	1	1	yes	characteristic
35	C₄×D₅₃	—	8	2	no	—
36	C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
38	C₄×D₅₃	SD₁₆	1	1	yes	characteristic
39	C₄×D₅₃	Q₁₆	1	1	yes	characteristic
40	#88324	—	4	2	no	—
41	C₈×C₅₃	C₄²	1	1	yes	characteristic
42	C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
43	#88322	C₂²⋊C₄	1	1	yes	characteristic
44	#88322	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₅₃	#102	1	1	yes	characteristic
46	C₂×D₅₃	—	2	2	no	—
47	C₂×D₅₃	#103	1	1	yes	characteristic
48	C₂×D₅₃	#106	1	1	yes	characteristic
49	C₄×C₅₃	—	8	1	no	—
50	C₄×C₅₃	#95	1	1	yes	characteristic
51	C₄×C₅₃	#103	1	1	yes	characteristic
52	Dic₅₃	—	8	1	no	—
53	Dic₅₃	#102	1	1	yes	characteristic
54	Dic₅₃	#107	1	1	yes	characteristic
55	#1383	—	53	1	no	maximal, Sylow
56	C₂×C₅₃	—	2	1	no	—
57	C₂×C₅₃	#340	1	1	yes	characteristic
58	D₅₃	—	2	1	no	—
59	D₅₃	#365	1	2	yes	—
60	C₂×M₃₂	—	53	1	no	—
61	C₂×#108	—	53	1	no	—
62	C₂×#106	—	53	1	no	—
63	C₅₃	#1383	1	1	yes	characteristic, complemented, Sylow
64	C₂×M₁₆	—	106	1	no	—
65	C₂²×C₈	—	53	1	no	—
66	C₂×C₄⋊C₄	—	106	1	no	—
67	M₃₂	—	106	1	no	—
68	M₃₂	—	106	1	no	—
69	C₂×C₁₆	—	53	2	no	—
70	#108	—	53	2	no	—
71	#108	—	106	1	no	—
72	#106	—	53	1	no	—
73	#106	—	106	1	no	—
74	#106	#6157	1	1	yes	characteristic
75	C₂²×C₄	—	53	1	no	—
76	C₂²×C₄	—	212	1	no	—
77	M₁₆	—	106	2	no	—
78	M₁₆	—	212	1	no	—
79	C₂×C₈	—	53	1	no	—
80	C₂×C₈	—	53	1	no	—
81	C₂×C₈	—	53	1	no	—
82	C₂×C₈	—	53	1	no	—
83	C₂×C₈	—	53	1	no	—
84	C₂×C₈	—	212	1	no	—
85	C₂×C₈	C₂×#6157	1	1	yes	characteristic
86	C₄⋊C₄	—	2	1	no	—
87	C₄⋊C₄	—	106	1	no	—
88	C₄⋊C₄	—	212	1	no	—
89	C₁₆	—	106	2	no	—
90	C₂³	—	53	1	no	—
91	C₂×C₄	—	4	1	no	—
92	C₂×C₄	—	53	1	no	—
93	C₂×C₄	—	53	1	no	—
94	C₂×C₄	—	53	1	no	—
95	C₂×C₄	—	53	1	no	—
96	C₂×C₄	—	53	1	no	—
97	C₂×C₄	—	424	2	no	—
98	C₂×C₄	—	212	1	no	—
99	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
100	C₈	—	53	1	no	—
101	C₈	—	53	1	no	—
102	C₈	—	212	2	no	—
103	C₈	C₄×#6157	1	1	yes	characteristic
104	C₈	C₄⋊#6157	1	1	yes	characteristic
105	C₂²	—	53	1	no	—
106	C₂²	—	53	1	no	—
107	C₂²	—	106	2	no	—
108	C₂²	#13133279	1	1	yes	characteristic
109	C₄	—	8	1	no	—
110	C₄	—	53	1	no	—
111	C₄	—	53	1	no	—
112	C₄	—	424	1	no	—
113	C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
114	C₄	#13133280	1	1	yes	characteristic
115	C₂	—	2	1	no	—
116	C₂	—	53	2	no	—
117	C₂	—	106	1	no	—
118	C₂	#16811496	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₅₃⋊M₃₂

C₂

yes

maximal, characteristic

D₅₃⋊#108

C₂

yes

maximal, characteristic

#106×D₅₃

C₂

yes

maximal, characteristic

D₅₃⋊M₁₆

—

#13133292

—

#13133291

—

D₅₃⋊C₁₆

C₄

yes

complemented

#13133288

—

#13133287

C₄

yes

complemented

C₂×C₈×D₅₃

C₂²

yes

characteristic

C₄⋊C₄×D₅₃

—

#13133149

C₄

yes

characteristic

#13133148

—

#106×C₅₃

C₄

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

D₄

yes

characteristic

#11719940

—

#11719936

—

D₅₃⋊C₈

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

C₂×#88322

D₄

yes

characteristic

C₈×D₅₃

C₂×C₄

yes

characteristic

C₈×D₅₃

C₂×C₄

yes

characteristic

C₈×D₅₃

D₄

yes

characteristic

C₈×D₅₃

Q₈

yes

characteristic

#11719910

—

C₂²×D₅₃

SD₁₆

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

D₈

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

C₂²⋊C₄

yes

characteristic

C₄×D₅₃

C₄⋊C₄

yes

characteristic

C₄×D₅₃

SD₁₆

yes

characteristic

C₄×D₅₃

Q₁₆

yes

characteristic

#88324

—

C₈×C₅₃

C₄²

yes

characteristic

C₈×C₅₃

C₄⋊C₄

yes

characteristic

#88322

C₂²⋊C₄

yes

characteristic

#88322

C₄⋊C₄

yes

characteristic

C₂²×C₅₃

#102

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

yes

C₂×D₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

yes

Dic₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

#340

yes

characteristic

D₅₃

—

D₅₃

#365

yes

—

C₂×M₃₂

—

C₂×#108

—

C₂×#106

—

C₅₃

#1383

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

106

—

M₃₂

—

106

—

M₃₂

—

106

—

C₂×C₁₆

—

#108

—

#108

—

106

—

#106

—

#106

—

106

—

#106

#6157

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

212

—

M₁₆

—

106

—

M₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

212

—

C₂×C₈

C₂×#6157

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₁₆

—

106

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

424

—

C₂×C₄

—

212

—

C₂×C₄

C₂²⋊#6157

yes

characteristic

100

C₈

—

101

C₈

—

102

C₈

—

212

—

103

C₈

C₄×#6157

yes

characteristic

104

C₈

C₄⋊#6157

yes

characteristic

105

C₂²

—

106

C₂²

—

107

C₂²

—

106

—

108

C₂²

#13133279

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

424

—

113

C₄

(C₂×#6157)⋊C₄

yes

characteristic

114

C₄

#13133280

yes

characteristic

115

C₂

—

116

C₂

—

117

C₂

—

106

—

118

C₂

#16811496

yes

characteristic

	subgroup	G/H
center	C₂	#16811496
commutator	C₈×C₅₃	C₄²
Frattini	C₂×C₈	C₂×#6157
Fitting	#106×C₅₃	C₄
socle	C₂×C₅₃	#340
hypercenter	#106	#6157
hypocenter	C₅₃	#1383
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #106

subgroup

G/H

center

C₂

#16811496

commutator

C₈×C₅₃

C₄²

Frattini

C₂×C₈

C₂×#6157

Fitting

#106×C₅₃

C₄

C₂×C₅₃

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #106

N	H ≅ G/N	type
7. D₅₃⋊C₁₆	C₄	semidirect
9. #13133287	C₄	semidirect
63. C₅₃	#1383	semidirect

H ≅ G/N

type

7. D₅₃⋊C₁₆

C₄

semidirect

9. #13133287

C₄

semidirect

63. C₅₃

#1383

semidirect

Subgroups — #20289779

Conjugacy classes of subgroups

Complements