Group #20273624 — subgroups

Subgroups — #20273624

Order 6784.

148 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 71 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	#16810754	C₂	1	2	yes	maximal
2	#16810374	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#88322	C₂	1	2	yes	maximal
4	#16810344	C₂	1	1	yes	maximal, characteristic
5	C₄×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#88322	—	2	2	no	—
7	C₄⋊#88322	C₂²	1	2	yes	—
8	#13133128	—	2	2	no	—
9	C₄×#88322	C₂²	1	1	yes	characteristic
10	C₄×#88322	C₂²	1	1	yes	characteristic
11	C₂×Q₁₆×C₅₃	C₄	1	1	yes	characteristic
12	C₄×Q₈×C₅₃	C₂²	1	2	yes	—
13	#107×C₅₃	C₄	1	1	yes	characteristic
14	#103×C₅₃	C₄	1	2	yes	—
15	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
17	Q₁₆×C₅₃	—	2	2	no	—
18	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₄⋊C₄×C₅₃	—	2	2	no	—
21	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
22	C₄²×C₅₃	—	2	2	no	—
23	C₄²×C₅₃	C₂³	1	1	yes	characteristic
24	C₂×#88322	—	2	2	no	—
25	C₂×#88322	—	2	1	no	—
26	C₂×#88322	D₄	1	2	yes	—
27	Q₈×C₅₃	—	2	2	no	—
28	Q₈×C₅₃	—	2	2	no	—
29	C₂×C₄×C₅₃	—	2	2	no	—
30	C₂×C₄×C₅₃	—	2	2	no	—
31	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
34	C₈×C₅₃	—	2	1	no	—
35	C₈×C₅₃	M₁₆	1	2	yes	—
36	#88322	—	2	2	no	—
37	#88322	—	4	1	no	—
38	#88322	—	4	2	no	—
39	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
40	C₄×C₅₃	—	2	1	no	—
41	C₄×C₅₃	—	4	2	no	—
42	C₄×C₅₃	—	4	2	no	—
43	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
44	C₄×C₅₃	#131	1	1	yes	characteristic
45	C₄×C₅₃	#137	1	2	yes	—
46	#1593	—	53	1	no	maximal, Sylow
47	C₂×C₅₃	#436	1	1	yes	characteristic
48	C₂×C₅₃	#441	1	1	yes	characteristic
49	C₂×C₅₃	#444	1	1	yes	characteristic
50	#446	—	53	2	no	—
51	C₄×Q₁₆	D₅₃	1	1	yes	characteristic
52	#334	—	53	1	no	—
53	Q₈⋊C₈	—	53	2	no	—
54	#322	—	53	1	no	—
55	C₅₃	#1593	1	1	yes	characteristic, complemented, Sylow
56	C₂×Q₁₆	Dic₅₃	1	1	yes	characteristic
57	C₄×Q₈	C₂×D₅₃	1	2	yes	—
58	#107	Dic₅₃	1	1	yes	characteristic
59	C₄⋊C₈	—	53	2	no	—
60	C₄⋊C₈	—	106	2	no	—
61	#103	Dic₅₃	1	2	yes	—
62	#97	—	106	2	no	—
63	C₄×C₈	—	53	1	no	—
64	C₄×C₈	—	53	1	no	—
65	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
66	C₂×Q₈	C₂×Dic₅₃	1	2	yes	—
67	Q₁₆	—	2	2	no	—
68	C₂×C₈	—	53	2	no	—
69	C₂×C₈	—	106	2	no	—
70	C₂×C₈	—	106	1	no	—
71	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
72	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
75	C₄²	—	2	2	no	—
76	C₄²	C₂²×D₅₃	1	1	yes	characteristic
77	Q₈	—	2	2	no	—
78	Q₈	—	2	2	no	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
82	C₂×C₄	#11719947	1	1	yes	characteristic
83	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
84	C₈	—	2	1	no	—
85	C₈	—	106	2	no	—
86	C₈	—	212	1	no	—
87	C₈	—	212	2	no	—
88	C₈	#11719917	1	2	yes	—
89	C₂²	D₄×Dic₅₃	1	1	yes	characteristic
90	C₄	—	2	1	no	—
91	C₄	—	4	2	no	—
92	C₄	—	4	2	no	—
93	C₄	C₂×#11719917	1	1	yes	characteristic
94	C₄	#13133224	1	1	yes	characteristic
95	C₄	#13133249	1	2	yes	—
96	C₂	#16810701	1	1	yes	characteristic
97	C₂	#16810730	1	1	yes	characteristic
98	C₂	#16810745	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810754

C₂

yes

maximal

#16810374

C₂

yes

maximal, characteristic

Q₈⋊#88322

C₂

yes

maximal

#16810344

C₂

yes

maximal, characteristic

C₄×Q₁₆×C₅₃

C₂

yes

maximal, characteristic

C₄⋊#88322

—

C₄⋊#88322

C₂²

yes

—

#13133128

—

C₄×#88322

C₂²

yes

characteristic

C₄×#88322

C₂²

yes

characteristic

C₂×Q₁₆×C₅₃

C₄

yes

characteristic

C₄×Q₈×C₅₃

C₂²

yes

—

#107×C₅₃

C₄

yes

characteristic

#103×C₅₃

C₄

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

C₂×C₄

yes

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂³

yes

—

—

D₄

yes

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

M₁₆

yes

—

#88322

—

#88322

—

#88322

—

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×M₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₄×Q₁₆

D₅₃

yes

characteristic

#334

—

Q₈⋊C₈

—

#322

—

C₅₃

#1593

yes

characteristic, complemented, Sylow

C₂×Q₁₆

Dic₅₃

yes

characteristic

C₄×Q₈

C₂×D₅₃

yes

—

#107

Dic₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

106

—

#103

Dic₅₃

yes

—

#97

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

C₂×Dic₅₃

yes

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

C₂×Dic₅₃

yes

characteristic

C₂×C₈

C₂×Dic₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₅₃

yes

—

C₄²

—

C₄²

C₂²×D₅₃

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₄×D₅₃

yes

characteristic

C₂×C₄

#11719947

yes

characteristic

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

C₈

—

C₈

—

106

—

C₈

—

212

—

C₈

—

212

—

C₈

#11719917

yes

—

C₂²

D₄×Dic₅₃

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₄

yes

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂×C₄	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₄×Q₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₄×D₄
hypercenter	C₄×Q₁₆	D₅₃
hypocenter	C₅₃	#1593
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

subgroup

G/H

center

C₂×C₄

D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₄×Q₁₆×C₅₃

C₂

socle

C₂²×C₅₃

C₄×D₄

hypercenter

C₄×Q₁₆

D₅₃

hypocenter

C₅₃

#1593

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

N	H ≅ G/N	type
55. C₅₃	#1593	semidirect

H ≅ G/N

type

55. C₅₃

#1593

semidirect

Subgroups — #20273624

Conjugacy classes of subgroups

Complements