Group #20273540 — subgroups

Subgroups — #20273540

Order 6784.

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

normal subgroups only

This group has 192 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	C₄×D₄₂₄	C₂	1	1	yes	maximal, characteristic
2	C₈×D₂₁₂	C₂	1	2	yes	maximal, complemented
3	#16810374	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810352	C₂	1	2	yes	maximal, complemented
5	C₅₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄₂₄	C₄	1	1	yes	characteristic
7	C₂×C₈×D₅₃	—	2	2	no	—
8	C₄×D₂₁₂	C₂²	1	2	yes	—
9	#13133151	C₄	1	1	yes	characteristic
10	C₄⋊#88322	C₂²	1	2	yes	—
11	#13133139	C₄	1	2	yes	—
12	#13133129	—	2	2	no	—
13	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
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	1	no	—
19	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₅₃	D₄	1	2	yes	—
22	C₄²×C₅₃	C₂³	1	1	yes	characteristic
23	#11719921	—	2	2	no	—
24	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
25	C₂×#88322	—	2	2	no	—
26	D₄₂₄	C₈	1	4	yes	complemented
27	C₈×D₅₃	—	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	C₂×C₈	1	4	yes	complemented
35	C₄×D₅₃	—	4	2	no	—
36	C₄×D₅₃	—	4	2	no	—
37	C₈×C₅₃	—	2	1	no	—
38	C₈×C₅₃	—	2	2	no	—
39	C₈×C₅₃	—	4	1	no	—
40	C₈×C₅₃	C₂×C₈	1	2	yes	complemented
41	#88322	—	4	2	no	—
42	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
43	C₂×D₅₃	—	2	4	no	—
44	C₂×D₅₃	—	4	2	no	—
45	C₂×D₅₃	—	4	2	no	—
46	C₄×C₅₃	—	2	1	no	—
47	C₄×C₅₃	C₂²×C₈	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	#1580	—	53	1	no	maximal, Sylow
52	C₂×C₅₃	C₈×D₄	1	1	yes	characteristic
53	C₂×C₅₃	#442	1	1	yes	characteristic
54	C₂×C₅₃	#444	1	1	yes	characteristic
55	D₅₃	—	4	4	no	—
56	C₄×D₈	—	53	1	no	—
57	C₈×D₄	—	53	2	no	—
58	#334	—	53	1	no	—
59	D₄⋊C₈	—	53	2	no	—
60	#322	D₅₃	1	1	yes	characteristic, complemented
61	C₅₃	#1580	1	1	yes	characteristic, complemented, Sylow
62	C₂×D₈	—	53	1	no	—
63	C₂²×C₈	—	106	2	no	—
64	C₄×D₄	—	53	2	no	—
65	#107	—	53	1	no	—
66	C₄⋊C₈	—	53	2	no	—
67	#102	—	53	2	no	—
68	C₂²⋊C₈	—	106	2	no	—
69	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
70	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
71	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
72	C₂×D₄	—	53	2	no	—
73	C₂²×C₄	—	106	2	no	—
74	D₈	—	53	4	no	—
75	C₂×C₈	—	2	1	no	—
76	C₂×C₈	—	106	2	no	—
77	C₂×C₈	—	212	4	no	—
78	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
79	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
80	C₂×C₈	D₂₁₂	1	2	yes	—
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	—	53	4	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	C₂×D₂₁₂	1	1	yes	characteristic
92	C₂×C₄	#11719945	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	2	2	no	—
95	C₈	—	4	1	no	—
96	C₈	—	212	2	no	—
97	C₈	C₈×D₅₃	1	2	yes	complemented
98	C₂²	—	106	4	no	—
99	C₂²	—	212	2	no	—
100	C₂²	—	212	2	no	—
101	C₂²	C₄×D₂₁₂	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	212	2	no	—
104	C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
105	C₄	#13133223	1	1	yes	characteristic
106	C₄	#13133245	1	2	yes	—
107	C₂	—	212	4	no	—
108	C₂	C₈×D₂₁₂	1	1	yes	characteristic
109	C₂	#16810736	1	1	yes	characteristic
110	C₂	#16810746	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₄₂₄

C₂

yes

maximal, characteristic

C₈×D₂₁₂

C₂

yes

maximal, complemented

#16810374

C₂

yes

maximal, characteristic, complemented

#16810352

C₂

yes

maximal, complemented

C₅₃×#322

C₂

yes

maximal, characteristic, complemented

C₂×D₄₂₄

C₄

yes

characteristic

C₂×C₈×D₅₃

—

C₄×D₂₁₂

C₂²

yes

—

#13133151

C₄

yes

characteristic

C₄⋊#88322

C₂²

yes

—

#13133139

C₄

yes

—

#13133129

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

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

C₂×C₈×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

#11719921

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

C₂×#88322

—

D₄₂₄

C₈

yes

complemented

C₈×D₅₃

—

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

C₂×C₈

yes

complemented

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₂×C₈

yes

complemented

#88322

—

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²×C₈

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

Dic₅₃

—

#1580

—

maximal, Sylow

C₂×C₅₃

C₈×D₄

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

C₄×D₈

—

C₈×D₄

—

#334

—

D₄⋊C₈

—

#322

D₅₃

yes

characteristic, complemented

C₅₃

#1580

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂²×C₈

—

106

—

C₄×D₄

—

#107

—

C₄⋊C₈

—

#102

—

C₂²⋊C₈

—

106

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₄×C₈

C₂×D₅₃

yes

characteristic

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

106

—

D₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

212

—

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

D₂₁₂

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

106

—

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

—

106

—

D₄

—

D₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

C₂×C₄

C₂×D₂₁₂

yes

characteristic

C₂×C₄

#11719945

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

212

—

C₈

C₈×D₅₃

yes

complemented

C₂²

—

106

—

C₂²

—

212

—

100

C₂²

—

212

—

101

C₂²

C₄×D₂₁₂

yes

characteristic

102

C₄

—

103

C₄

—

212

—

104

C₄

C₂×C₈×D₅₃

yes

characteristic

105

C₄

#13133223

yes

characteristic

106

C₄

#13133245

yes

—

107

C₂

—

212

—

108

C₂

C₈×D₂₁₂

yes

characteristic

109

C₂

#16810736

yes

characteristic

110

C₂

#16810746

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₁₂
commutator	C₄×C₅₃	C₂²×C₈
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#322	C₂
socle	C₂²×C₅₃	C₄×D₄
hypercenter	#322	D₅₃
hypocenter	C₅₃	#1580
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

subgroup

G/H

center

C₂×C₄

C₂×D₂₁₂

commutator

C₄×C₅₃

C₂²×C₈

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#322

C₂

socle

C₂²×C₅₃

C₄×D₄

D₅₃

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

N	H ≅ G/N	type
2. C₈×D₂₁₂	C₂	semidirect
3. #16810374	C₂	semidirect
4. #16810352	C₂	semidirect
5. C₅₃×#322	C₂	semidirect
26. D₄₂₄	C₈	semidirect
34. D₂₁₂	C₂×C₈	semidirect
40. C₈×C₅₃	C₂×C₈	semidirect
60. #322	D₅₃	semidirect
61. C₅₃	#1580	semidirect
97. C₈	C₈×D₅₃	semidirect

H ≅ G/N

type

2. C₈×D₂₁₂

C₂

semidirect

3. #16810374

C₂

semidirect

4. #16810352

C₂

semidirect

5. C₅₃×#322

C₂

semidirect

26. D₄₂₄

C₈

semidirect

34. D₂₁₂

C₂×C₈

semidirect

40. C₈×C₅₃

C₂×C₈

semidirect

60. #322

D₅₃

semidirect

61. C₅₃

#1580

semidirect

97. C₈

C₈×D₅₃

semidirect

Subgroups — #20273540

Conjugacy classes of subgroups

Complements