Group #20274865 — subgroups

Subgroups — #20274865

Order 6784.

162 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 65 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	M₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
2	#16810395	C₂	1	2	yes	maximal
3	#16810384	C₂	1	2	yes	maximal
4	#16810380	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#338	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	C₂²	1	1	yes	characteristic
7	#13133176	C₂²	1	2	yes	—
8	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
9	#13133127	C₄	1	2	yes	complemented
10	C₈×Dic₅₃	C₄	1	2	yes	complemented
11	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
12	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
14	C₂²×Dic₅₃	—	2	1	no	—
15	M₁₆×C₅₃	—	2	2	no	—
16	M₁₆×C₅₃	—	2	2	no	—
17	M₁₆×C₅₃	—	2	2	no	—
18	C₂×C₈×C₅₃	—	2	2	no	—
19	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
20	C₂²⋊Dic₅₃	—	2	2	no	—
21	C₄⋊Dic₅₃	—	2	2	no	—
22	C₄×Dic₅₃	—	2	1	no	—
23	C₄×Dic₅₃	—	2	2	no	—
24	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
25	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
26	#11719917	—	2	2	no	—
27	C₂×#88322	C₂×C₄	1	2	yes	—
28	C₂³×C₅₃	#41	1	1	yes	characteristic
29	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₅₃	#41	1	2	yes	—
31	C₂×C₄×C₅₃	#41	1	2	yes	—
32	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
33	C₂×Dic₅₃	—	2	1	no	—
34	C₂×Dic₅₃	—	2	1	no	—
35	C₂×Dic₅₃	—	4	2	no	—
36	C₂×Dic₅₃	—	4	2	no	—
37	C₈×C₅₃	—	2	2	no	—
38	C₈×C₅₃	—	4	2	no	—
39	#88322	—	2	2	no	—
40	C₂²×C₅₃	—	4	1	no	—
41	C₂²×C₅₃	#117	1	1	yes	characteristic
42	C₂²×C₅₃	#126	1	2	yes	—
43	C₄×C₅₃	#117	1	1	yes	characteristic
44	C₄×C₅₃	#117	1	1	yes	characteristic
45	C₄×C₅₃	#126	1	2	yes	—
46	Dic₅₃	—	2	2	no	—
47	Dic₅₃	—	4	1	no	—
48	Dic₅₃	—	8	2	no	—
49	#1841	—	53	1	no	maximal, Sylow
50	C₂×C₅₃	—	2	1	no	—
51	C₂×C₅₃	—	2	2	no	—
52	C₂×C₅₃	#383	1	1	yes	characteristic
53	C₄×M₁₆	—	53	1	no	—
54	#344	—	53	2	no	—
55	#339	—	53	2	no	—
56	#338	D₅₃	1	1	yes	characteristic
57	H(3,ℤ/4)	—	53	1	no	—
58	C₅₃	#1841	1	1	yes	characteristic, complemented, Sylow
59	C₂×M₁₆	—	53	1	no	—
60	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
61	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
62	#117	—	53	2	no	—
63	C₂×C₄²	—	53	1	no	—
64	#97	—	53	2	no	—
65	C₄×C₈	—	53	2	no	—
66	C₂²×C₄	—	106	1	no	—
67	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	2	2	no	—
70	M₁₆	—	2	2	no	—
71	M₁₆	—	106	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	53	2	no	—
74	C₂×C₈	C₄×D₅₃	1	2	yes	—
75	C₄⋊C₄	—	106	2	no	—
76	C₂²⋊C₄	—	106	2	no	—
77	C₄²	—	53	1	no	—
78	C₄²	—	53	1	no	—
79	C₄²	—	106	1	no	—
80	C₄²	—	106	2	no	—
81	C₂³	#11719949	1	1	yes	characteristic
82	C₂×C₄	—	106	1	no	—
83	C₂×C₄	—	106	1	no	—
84	C₂×C₄	—	212	2	no	—
85	C₂×C₄	—	212	2	no	—
86	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
87	C₂×C₄	#11719945	1	2	yes	—
88	C₂×C₄	#11719947	1	2	yes	—
89	C₂×C₄	#11719947	1	1	yes	characteristic
90	C₈	—	2	2	no	—
91	C₈	—	4	2	no	—
92	C₈	—	106	2	no	—
93	C₂²	—	4	1	no	—
94	C₂²	#13133175	1	1	yes	characteristic
95	C₂²	#13133212	1	2	yes	—
96	C₄	—	106	2	no	—
97	C₄	—	424	2	no	—
98	C₄	—	212	1	no	—
99	C₄	#13133173	1	1	yes	characteristic
100	C₄	#13133174	1	1	yes	characteristic
101	C₄	#13133210	1	2	yes	—
102	C₂	—	2	1	no	—
103	C₂	—	2	2	no	—
104	C₂	#16810490	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

M₁₆×Dic₅₃

C₂

yes

maximal, characteristic

#16810395

C₂

yes

maximal

#16810384

C₂

yes

maximal

#16810380

C₂

yes

maximal, characteristic

C₅₃×#338

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

characteristic

#13133176

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

#13133127

C₄

yes

complemented

C₈×Dic₅₃

C₄

yes

complemented

C₂×M₁₆×C₅₃

C₂²

yes

—

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

characteristic

#11719917

—

C₂×#88322

C₂×C₄

yes

—

C₂³×C₅₃

#41

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

#117

yes

characteristic

C₂²×C₅₃

#126

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#1841

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#383

yes

characteristic

C₄×M₁₆

—

#344

—

#339

—

#338

D₅₃

yes

characteristic

H(3,ℤ/4)

—

C₅₃

#1841

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

—

C₂×M₁₆

C₂×D₅₃

yes

characteristic

#117

—

C₂×C₄²

—

#97

—

C₄×C₈

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₅₃

yes

—

C₄⋊C₄

—

106

—

C₂²⋊C₄

—

106

—

C₄²

—

C₄²

—

C₄²

—

106

—

C₄²

—

106

—

C₂³

#11719949

yes

characteristic

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

C₂×C₄

#11719945

yes

—

C₂×C₄

#11719947

yes

—

C₂×C₄

#11719947

yes

characteristic

C₈

—

C₈

—

C₈

—

106

—

C₂²

—

C₂²

#13133175

yes

characteristic

C₂²

#13133212

yes

—

C₄

—

106

—

C₄

—

424

—

C₄

—

212

—

C₄

#13133173

yes

characteristic

100

C₄

#13133174

yes

characteristic

101

C₄

#13133210

yes

—

102

C₂

—

103

C₂

—

104

C₂

#16810490

yes

characteristic

	subgroup	G/H
center	C₄	#13133174
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#338	C₂
socle	C₂×C₅₃	#383
hypercenter	#338	D₅₃
hypocenter	C₅₃	#1841
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

subgroup

G/H

center

C₄

#13133174

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#338

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

N	H ≅ G/N	type
9. #13133127	C₄	semidirect
10. C₈×Dic₅₃	C₄	semidirect
58. C₅₃	#1841	semidirect

H ≅ G/N

type

9. #13133127

C₄

semidirect

10. C₈×Dic₅₃

C₄

semidirect

58. C₅₃

#1841

semidirect

Subgroups — #20274865

Conjugacy classes of subgroups

Complements