Group #20272758 — subgroups

Subgroups — #20272758

Order 6784.

102 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 102 conjugacy classes of subgroups — too many to draw. Only its 41 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⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810374	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#495	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719908	—	2	1	no	—
5	C₄⋊#88322	—	2	1	no	—
6	C₂×Q₁₆×C₅₃	C₄	1	1	yes	characteristic
7	C₂×SD₁₆×C₅₃	—	2	1	no	—
8	C₂×D₈×C₅₃	C₄	1	1	yes	characteristic
9	#124×C₅₃	—	2	1	no	—
10	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×Q₈×C₅₃	—	2	1	no	—
12	C₂×D₄×C₅₃	—	2	1	no	—
13	Q₁₆×C₅₃	—	4	1	no	—
14	SD₁₆×C₅₃	—	4	1	no	—
15	SD₁₆×C₅₃	—	4	1	no	—
16	D₈×C₅₃	—	4	1	no	—
17	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
19	C₂²⋊C₄×C₅₃	—	4	1	no	—
20	C₂²⋊C₄×C₅₃	—	4	1	no	—
21	C₄²×C₅₃	D₄	1	1	yes	characteristic
22	C₂×#88322	—	4	1	no	—
23	#11719908	—	4	1	no	—
24	C₂³×C₅₃	—	4	1	no	—
25	Q₈×C₅₃	—	4	1	no	—
26	Q₈×C₅₃	—	4	1	no	—
27	D₄×C₅₃	—	4	1	no	—
28	D₄×C₅₃	—	4	1	no	—
29	C₂×C₄×C₅₃	—	4	1	no	—
30	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
32	C₂×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
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	—	8	1	no	—
38	C₂²×C₅₃	—	8	1	no	—
39	C₂²×C₅₃	—	8	1	no	—
40	C₂²×C₅₃	#102	1	1	yes	characteristic
41	C₄×C₅₃	—	2	1	no	—
42	C₄×C₅₃	—	2	1	no	—
43	C₄×C₅₃	—	8	1	no	—
44	C₄×C₅₃	#100	1	1	yes	characteristic
45	C₄×C₅₃	#102	1	1	yes	characteristic
46	#1360	—	53	1	no	maximal, Sylow
47	C₂×C₅₃	—	16	1	no	—
48	C₂×C₅₃	#331	1	1	yes	characteristic
49	C₂×C₅₃	#359	1	1	yes	characteristic
50	C₂×C₅₃	#360	1	1	yes	characteristic
51	#495	D₅₃	1	1	yes	characteristic
52	C₄⋊C₁₆	—	53	1	no	—
53	#334	—	53	1	no	—
54	C₅₃	#1360	1	1	yes	characteristic, complemented, Sylow
55	C₂×Q₁₆	Dic₅₃	1	1	yes	characteristic
56	C₂×SD₁₆	—	2	1	no	—
57	C₂×D₈	Dic₅₃	1	1	yes	characteristic
58	#124	—	2	1	no	—
59	C₂×C₁₆	—	106	1	no	—
60	C₄⋊C₈	—	106	1	no	—
61	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
62	C₂×Q₈	—	2	1	no	—
63	C₂×D₄	—	2	1	no	—
64	Q₁₆	—	4	1	no	—
65	SD₁₆	—	4	1	no	—
66	SD₁₆	—	4	1	no	—
67	D₈	—	4	1	no	—
68	C₂×C₈	—	212	1	no	—
69	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
70	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
71	C₂²⋊C₄	—	4	1	no	—
72	C₂²⋊C₄	—	4	1	no	—
73	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
74	C₁₆	—	212	1	no	—
75	C₂³	—	4	1	no	—
76	Q₈	—	4	1	no	—
77	Q₈	—	4	1	no	—
78	D₄	—	4	1	no	—
79	D₄	—	4	1	no	—
80	C₂×C₄	—	4	1	no	—
81	C₂×C₄	#11719922	1	1	yes	characteristic
82	C₂×C₄	#11719923	1	1	yes	characteristic
83	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
84	C₈	—	2	1	no	—
85	C₈	—	424	1	no	—
86	C₈	#11719922	1	1	yes	characteristic
87	C₈	#11719923	1	1	yes	characteristic
88	C₂²	—	8	1	no	—
89	C₂²	—	8	1	no	—
90	C₂²	—	8	1	no	—
91	C₂²	#13133138	1	1	yes	characteristic
92	C₄	—	2	1	no	—
93	C₄	—	2	1	no	—
94	C₄	—	8	1	no	—
95	C₄	#13133134	1	1	yes	characteristic
96	C₄	#13133138	1	1	yes	characteristic
97	C₂	—	16	1	no	—
98	C₂	#16810367	1	1	yes	characteristic
99	C₂	#16810430	1	1	yes	characteristic
100	C₂	#16810433	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊#11719908

C₂

yes

maximal, characteristic, complemented

#16810374

C₂

yes

maximal, characteristic, complemented

C₅₃×#495

C₂

yes

maximal, characteristic

C₂×#11719908

—

C₄⋊#88322

—

C₂×Q₁₆×C₅₃

C₄

yes

characteristic

C₂×SD₁₆×C₅₃

—

C₂×D₈×C₅₃

C₄

yes

characteristic

#124×C₅₃

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

—

Q₁₆×C₅₃

—

SD₁₆×C₅₃

—

SD₁₆×C₅₃

—

D₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

D₄

yes

characteristic

C₂×#88322

—

#11719908

—

C₂³×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

SD₁₆

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

D₈

yes

characteristic

C₈×C₅₃

SD₁₆

yes

characteristic

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#102

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

yes

characteristic

C₄⋊C₁₆

—

#334

—

C₅₃

#1360

yes

characteristic, complemented, Sylow

C₂×Q₁₆

Dic₅₃

yes

characteristic

C₂×SD₁₆

—

C₂×D₈

Dic₅₃

yes

characteristic

#124

—

C₂×C₁₆

—

106

—

C₄⋊C₈

—

106

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

Q₁₆

—

SD₁₆

—

SD₁₆

—

D₈

—

C₂×C₈

—

212

—

C₂×C₈

C₂×Dic₅₃

yes

characteristic

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂²⋊D₅₃

yes

characteristic

C₁₆

—

212

—

C₂³

—

Q₈

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

C₈

—

C₈

—

424

—

C₈

yes

C₈

yes

C₂²

—

C₂²

—

C₂²

—

C₂²

#13133138

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₂

—

C₂

yes

C₂

yes

100

C₂

#16810433

yes

characteristic

	subgroup	G/H
center	C₂²	#13133138
commutator	C₂×C₈×C₅₃	C₂×C₄
Frattini	C₄×C₈	C₂×D₅₃
Fitting	C₅₃×#495	C₂
socle	C₂²×C₅₃	#102
hypercenter	#495	D₅₃
hypocenter	C₅₃	#1360
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #495

subgroup

G/H

center

C₂²

#13133138

commutator

C₂×C₈×C₅₃

C₂×C₄

Frattini

C₄×C₈

C₂×D₅₃

Fitting

C₅₃×#495

C₂

C₂²×C₅₃

D₅₃

C₅₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #495

N	H ≅ G/N	type
1. C₄⋊#11719908	C₂	semidirect
2. #16810374	C₂	semidirect
54. C₅₃	#1360	semidirect

H ≅ G/N

type

1. C₄⋊#11719908

C₂

semidirect

2. #16810374

C₂

semidirect

54. C₅₃

#1360

semidirect

Subgroups — #20272758

Conjugacy classes of subgroups

Complements