Group #20272759 — subgroups

Subgroups — #20272759

Order 6784.

102 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 102 conjugacy classes of subgroups — too many to draw. Only its 39 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	#16811007	C₂	1	1	yes	maximal, characteristic
2	#16810374	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×Dic₂₁₂	C₄	1	1	yes	characteristic
5	C₂×#11719913	—	2	1	no	—
6	C₂×D₄₂₄	C₄	1	1	yes	characteristic
7	#13133205	—	2	1	no	—
8	C₄⋊#88322	—	2	1	no	—
9	C₂×C₁₆×C₅₃	—	2	1	no	—
10	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×D₂₁₂	—	2	1	no	—
12	C₂×Dic₁₀₆	—	2	1	no	—
13	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
15	C₄²×C₅₃	D₄	1	1	yes	characteristic
16	#11719921	—	4	1	no	—
17	#11719921	—	4	1	no	—
18	C₂×#88322	—	4	1	no	—
19	Dic₂₁₂	—	4	1	no	—
20	D₄₂₄	—	4	1	no	—
21	#11719913	—	4	1	no	—
22	#11719913	—	4	1	no	—
23	C₁₆×C₅₃	—	4	1	no	—
24	C₂²×D₅₃	—	4	1	no	—
25	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
27	C₂×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
28	C₂×Dic₅₃	—	4	1	no	—
29	D₂₁₂	—	4	1	no	—
30	D₂₁₂	—	4	1	no	—
31	Dic₁₀₆	—	4	1	no	—
32	Dic₁₀₆	—	4	1	no	—
33	C₈×C₅₃	—	2	1	no	—
34	C₈×C₅₃	D₈	1	1	yes	characteristic
35	C₈×C₅₃	SD₁₆	1	1	yes	characteristic
36	#88322	—	8	1	no	—
37	C₂²×C₅₃	#102	1	1	yes	characteristic
38	C₂×D₅₃	—	8	1	no	—
39	C₂×D₅₃	—	8	1	no	—
40	C₂×D₅₃	—	8	1	no	—
41	C₄×C₅₃	—	2	1	no	—
42	C₄×C₅₃	—	2	1	no	—
43	C₄×C₅₃	#100	1	1	yes	characteristic
44	C₄×C₅₃	#102	1	1	yes	characteristic
45	Dic₅₃	—	8	1	no	—
46	#1360	—	53	1	no	maximal, Sylow
47	C₂×C₅₃	#331	1	1	yes	characteristic
48	C₂×C₅₃	#359	1	1	yes	characteristic
49	C₂×C₅₃	#360	1	1	yes	characteristic
50	D₅₃	—	16	1	no	—
51	#495	—	53	1	no	—
52	C₄⋊C₁₆	D₅₃	1	1	yes	characteristic, complemented
53	#334	—	53	1	no	—
54	C₅₃	#1360	1	1	yes	characteristic, complemented, Sylow
55	C₂×Q₁₆	—	53	1	no	—
56	C₂×SD₁₆	—	106	1	no	—
57	C₂×D₈	—	53	1	no	—
58	#124	—	106	1	no	—
59	C₂×C₁₆	—	2	1	no	—
60	C₄⋊C₈	—	106	1	no	—
61	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
62	C₂×Q₈	—	106	1	no	—
63	C₂×D₄	—	106	1	no	—
64	Q₁₆	—	212	1	no	—
65	SD₁₆	—	212	1	no	—
66	SD₁₆	—	212	1	no	—
67	D₈	—	212	1	no	—
68	C₂×C₈	—	212	1	no	—
69	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
70	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
71	C₂²⋊C₄	—	212	1	no	—
72	C₂²⋊C₄	—	212	1	no	—
73	C₄²	D₂₁₂	1	1	yes	characteristic
74	C₁₆	—	4	1	no	—
75	C₂³	—	212	1	no	—
76	Q₈	—	212	1	no	—
77	Q₈	—	212	1	no	—
78	D₄	—	212	1	no	—
79	D₄	—	212	1	no	—
80	C₂×C₄	—	212	1	no	—
81	C₂×C₄	#11719913	1	1	yes	characteristic
82	C₂×C₄	D₄₂₄	1	1	yes	characteristic
83	C₂×C₄	#11719921	1	1	yes	characteristic
84	C₈	—	2	1	no	—
85	C₈	—	424	1	no	—
86	C₈	#11719922	1	1	yes	characteristic
87	C₈	#11719924	1	1	yes	characteristic
88	C₂²	—	424	1	no	—
89	C₂²	—	424	1	no	—
90	C₂²	—	424	1	no	—
91	C₂²	#13133139	1	1	yes	characteristic
92	C₄	—	2	1	no	—
93	C₄	—	2	1	no	—
94	C₄	—	424	1	no	—
95	C₄	#13133133	1	1	yes	characteristic
96	C₄	#13133137	1	1	yes	characteristic
97	C₂	—	848	1	no	—
98	C₂	#16810366	1	1	yes	characteristic
99	C₂	#16810431	1	1	yes	characteristic
100	C₂	#16810434	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811007

C₂

yes

maximal, characteristic

#16810374

C₂

yes

maximal, characteristic, complemented

C₅₃×C₄⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₁₂

C₄

yes

characteristic

C₂×#11719913

—

C₂×D₄₂₄

C₄

yes

characteristic

#13133205

—

C₄⋊#88322

—

C₂×C₁₆×C₅₃

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×D₂₁₂

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

—

—

—

Dic₂₁₂

—

D₄₂₄

—

#11719913

—

#11719913

—

C₁₆×C₅₃

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

SD₁₆

yes

characteristic

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

D₈

yes

characteristic

C₈×C₅₃

SD₁₆

yes

characteristic

#88322

—

C₂²×C₅₃

#102

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

#495

—

C₄⋊C₁₆

D₅₃

yes

characteristic, complemented

#334

—

C₅₃

#1360

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×SD₁₆

—

106

—

C₂×D₈

—

#124

—

106

—

C₂×C₁₆

—

C₄⋊C₈

—

106

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

106

—

C₂×D₄

—

106

—

Q₁₆

—

212

—

SD₁₆

—

212

—

SD₁₆

—

212

—

D₈

—

212

—

C₂×C₈

—

212

—

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂²⋊C₄

—

212

—

C₂²⋊C₄

—

212

—

C₄²

D₂₁₂

yes

characteristic

C₁₆

—

C₂³

—

212

—

Q₈

—

212

—

Q₈

—

212

—

D₄

—

212

—

D₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

#11719913

yes

characteristic

C₂×C₄

D₄₂₄

yes

characteristic

C₂×C₄

#11719921

yes

characteristic

C₈

—

C₈

—

424

—

C₈

yes

C₈

yes

C₂²

—

424

—

C₂²

—

424

—

C₂²

—

424

—

C₂²

#13133139

yes

characteristic

C₄

—

C₄

—

C₄

—

424

—

C₄

yes

C₄

yes

C₂

—

848

—

C₂

yes

C₂

yes

100

C₂

#16810434

yes

characteristic

	subgroup	G/H
center	C₂²	#13133139
commutator	C₂×C₈×C₅₃	C₂×C₄
Frattini	C₄×C₈	C₂×D₅₃
Fitting	C₅₃×C₄⋊C₁₆	C₂
socle	C₂²×C₅₃	#102
hypercenter	C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	#1360
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center

C₂²

#13133139

commutator

C₂×C₈×C₅₃

C₂×C₄

Frattini

C₄×C₈

C₂×D₅₃

Fitting

C₅₃×C₄⋊C₁₆

C₂

socle

C₂²×C₅₃

#102

hypercenter

C₄⋊C₁₆

D₅₃

hypocenter

C₅₃

#1360

LCS

G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
2. #16810374	C₂	semidirect
3. C₅₃×C₄⋊C₁₆	C₂	semidirect
52. C₄⋊C₁₆	D₅₃	semidirect
54. C₅₃	#1360	semidirect

H ≅ G/N

type

2. #16810374

C₂

semidirect

3. C₅₃×C₄⋊C₁₆

C₂

semidirect

52. C₄⋊C₁₆

D₅₃

semidirect

54. C₅₃

#1360

semidirect

Subgroups — #20272759

Conjugacy classes of subgroups

Complements