Group #20273628 — subgroups

Subgroups — #20273628

Order 6784.

176 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 63 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	C₄×D₄₂₄	C₂	1	1	yes	maximal, characteristic
2	#16810700	C₂	1	2	yes	maximal, complemented
3	#16810350	C₂	1	2	yes	maximal, complemented
4	#16810344	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#334	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄₂₄	C₄	1	1	yes	characteristic
7	C₂×#11719912	—	2	2	no	—
8	C₄×D₂₁₂	C₂²	1	2	yes	—
9	#13133151	C₄	1	1	yes	characteristic
10	#13133139	C₄	1	2	yes	—
11	#13133129	—	2	2	no	—
12	C₄×#88322	C₂²	1	1	yes	characteristic
13	C₄×#88322	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
15	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×D₂₁₂	C₂×C₄	1	2	yes	—
17	C₂×C₄×D₅₃	—	2	2	no	—
18	C₂×C₈×C₅₃	—	2	2	no	—
19	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₄²×C₅₃	C₂³	1	1	yes	characteristic
22	#11719921	—	2	2	no	—
23	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
24	C₂×#88322	—	2	1	no	—
25	C₂×#88322	D₄	1	2	yes	—
26	D₄₂₄	—	2	2	no	—
27	#11719912	—	4	4	no	—
28	C₂²×D₅₃	—	2	2	no	—
29	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
32	C₂×Dic₅₃	—	2	2	no	—
33	D₂₁₂	—	2	2	no	—
34	D₂₁₂	—	2	2	no	—
35	C₄×D₅₃	—	4	2	no	—
36	C₄×D₅₃	—	4	2	no	—
37	C₈×C₅₃	—	2	1	no	—
38	C₈×C₅₃	—	4	2	no	—
39	C₈×C₅₃	M₁₆	1	2	yes	complemented
40	#88322	—	2	2	no	—
41	#88322	—	4	1	no	—
42	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
43	C₂×D₅₃	—	4	2	no	—
44	C₂×D₅₃	—	4	2	no	—
45	C₂×D₅₃	—	4	2	no	—
46	C₄×C₅₃	—	2	1	no	—
47	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
48	C₄×C₅₃	#131	1	1	yes	characteristic
49	C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
50	Dic₅₃	—	4	2	no	—
51	#1594	—	53	1	no	maximal, Sylow
52	C₂×C₅₃	#436	1	1	yes	characteristic
53	C₂×C₅₃	#442	1	1	yes	characteristic
54	C₂×C₅₃	#444	1	1	yes	characteristic
55	D₅₃	—	8	2	no	—
56	C₄×D₈	—	53	1	no	—
57	#436	—	53	2	no	—
58	#334	D₅₃	1	1	yes	characteristic, complemented
59	D₄⋊C₈	—	53	2	no	—
60	#322	—	53	1	no	—
61	C₅₃	#1594	1	1	yes	characteristic, complemented, Sylow
62	C₂×D₈	—	53	1	no	—
63	C₂×M₁₆	—	106	2	no	—
64	C₄×D₄	—	53	2	no	—
65	#107	—	53	1	no	—
66	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
67	#102	—	53	2	no	—
68	C₂²⋊C₈	—	106	2	no	—
69	C₄×C₈	—	53	1	no	—
70	C₄×C₈	—	53	1	no	—
71	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
72	C₂×D₄	—	53	2	no	—
73	C₂²×C₄	—	106	2	no	—
74	D₈	—	106	2	no	—
75	M₁₆	—	212	4	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	53	2	no	—
78	C₂×C₈	—	106	1	no	—
79	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
80	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
81	C₄⋊C₄	—	53	2	no	—
82	C₂²⋊C₄	—	106	2	no	—
83	C₄²	C₂²×D₅₃	1	1	yes	characteristic
84	C₂³	—	106	2	no	—
85	D₄	—	106	2	no	—
86	D₄	—	106	2	no	—
87	C₂×C₄	—	106	2	no	—
88	C₂×C₄	—	212	2	no	—
89	C₂×C₄	—	212	2	no	—
90	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
91	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
92	C₂×C₄	#11719949	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	4	2	no	—
95	C₈	—	106	2	no	—
96	C₈	—	212	1	no	—
97	C₈	#11719912	1	2	yes	complemented
98	C₂²	—	212	2	no	—
99	C₂²	—	212	2	no	—
100	C₂²	—	212	2	no	—
101	C₂²	#13133179	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	212	2	no	—
104	C₄	C₂×#11719912	1	1	yes	characteristic
105	C₄	#13133225	1	1	yes	characteristic
106	C₄	#13133244	1	2	yes	—
107	C₂	—	424	2	no	—
108	C₂	#16810700	1	1	yes	characteristic
109	C₂	#16810734	1	1	yes	characteristic
110	C₂	#16810744	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₄₂₄

C₂

yes

maximal, characteristic

#16810700

C₂

yes

maximal, complemented

#16810350

C₂

yes

maximal, complemented

#16810344

C₂

yes

maximal, characteristic, complemented

C₅₃×#334

C₂

yes

maximal, characteristic, complemented

C₂×D₄₂₄

C₄

yes

characteristic

C₂×#11719912

—

C₄×D₂₁₂

C₂²

yes

—

#13133151

C₄

yes

characteristic

#13133139

C₄

yes

—

#13133129

—

C₄×#88322

C₂²

yes

characteristic

C₄×#88322

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×D₂₁₂

C₂×C₄

yes

—

C₂×C₄×D₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

#11719921

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

C₂×#88322

—

C₂×#88322

D₄

yes

—

D₄₂₄

—

#11719912

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

M₁₆

yes

complemented

#88322

—

#88322

—

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×M₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

C₄×D₈

—

#436

—

#334

D₅₃

yes

characteristic, complemented

D₄⋊C₈

—

#322

—

C₅₃

#1594

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×M₁₆

—

106

—

C₄×D₄

—

#107

—

C₄⋊C₈

C₂×D₅₃

yes

—

#102

—

C₂²⋊C₈

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

106

—

D₈

—

106

—

M₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

C₄×D₅₃

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

106

—

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

—

106

—

D₄

—

106

—

D₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

C₂×C₄

D₄×D₅₃

yes

characteristic

C₂×C₄

#11719949

yes

characteristic

C₈

—

C₈

—

C₈

—

106

—

C₈

—

212

—

C₈

#11719912

yes

complemented

C₂²

—

212

—

C₂²

—

212

—

100

C₂²

—

212

—

101

C₂²

#13133179

yes

characteristic

102

C₄

—

103

C₄

—

212

—

104

C₄

C₂×#11719912

yes

characteristic

105

C₄

#13133225

yes

characteristic

106

C₄

#13133244

yes

—

107

C₂

—

424

—

108

C₂

#16810700

yes

characteristic

109

C₂

#16810734

yes

characteristic

110

C₂

#16810744

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#334	C₂
socle	C₂²×C₅₃	C₄×D₄
hypercenter	#334	D₅₃
hypocenter	C₅₃	#1594
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

subgroup

G/H

center

C₂×C₄

D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#334

C₂

socle

C₂²×C₅₃

C₄×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

N	H ≅ G/N	type
2. #16810700	C₂	semidirect
3. #16810350	C₂	semidirect
4. #16810344	C₂	semidirect
5. C₅₃×#334	C₂	semidirect
39. C₈×C₅₃	M₁₆	semidirect
58. #334	D₅₃	semidirect
61. C₅₃	#1594	semidirect
97. C₈	#11719912	semidirect

H ≅ G/N

type

2. #16810700

C₂

semidirect

3. #16810350

C₂

semidirect

4. #16810344

C₂

semidirect

5. C₅₃×#334

C₂

semidirect

39. C₈×C₅₃

M₁₆

semidirect

58. #334

D₅₃

semidirect

61. C₅₃

#1594

semidirect

97. C₈

#11719912

semidirect

Subgroups — #20273628

Conjugacy classes of subgroups

Complements