Group #20290271 — subgroups

Subgroups — #20290271

Order 6784.

184 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 78 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	#16811627	C₂	1	2	yes	maximal
2	#16811485	C₂	1	4	yes	maximal, complemented
3	C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719936	—	2	2	no	—
5	C₄⋊#88324	—	2	2	no	—
6	C₄⋊#88324	C₂²	1	4	yes	—
7	C₂×C₈×D₅₃	C₄	1	2	yes	—
8	C₄²×D₅₃	C₂²	1	1	yes	characteristic
9	C₄×#88322	C₄	1	1	yes	characteristic
10	C₈×Dic₅₃	C₂²	1	2	yes	—
11	C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
12	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
14	C₂×C₄×D₅₃	Q₈	1	1	yes	characteristic
15	C₂×#88324	—	2	4	no	—
16	#11719936	—	4	4	no	—
17	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
18	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₄×Dic₅₃	D₄	1	1	yes	characteristic
20	C₄×Dic₅₃	Q₈	1	1	yes	characteristic
21	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
22	C₂×#88322	C₂×C₄	1	2	yes	—
23	C₈×D₅₃	—	2	4	no	—
24	C₂²×D₅₃	C₄⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
28	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
29	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
30	C₂×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
31	C₄×D₅₃	—	2	2	no	—
32	C₄×D₅₃	—	2	2	no	—
33	C₄×D₅₃	SD₁₆	1	4	yes	—
34	#88324	—	4	4	no	—
35	C₈×C₅₃	—	2	2	no	—
36	#88322	M₁₆	1	4	yes	complemented
37	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂×D₅₃	—	2	1	no	—
39	C₂×D₅₃	—	2	1	no	—
40	C₂×D₅₃	#106	1	2	yes	—
41	C₄×C₅₃	#106	1	2	yes	—
42	C₄×C₅₃	#108	1	2	yes	—
43	C₄×C₅₃	C₂×M₁₆	1	2	yes	—
44	Dic₅₃	—	2	1	no	—
45	Dic₅₃	—	2	1	no	—
46	Dic₅₃	C₂×SD₁₆	1	2	yes	—
47	#1566	—	53	1	no	maximal, Sylow
48	C₂×C₅₃	#423	1	1	yes	characteristic
49	C₂×C₅₃	C₂×#106	1	1	yes	characteristic
50	C₂×C₅₃	C₂×#108	1	1	yes	characteristic
51	D₅₃	—	2	2	no	—
52	#423	—	53	2	no	—
53	C₂×C₄×C₈	—	53	1	no	—
54	#334	—	53	4	no	—
55	C₅₃	#1566	1	1	yes	characteristic, complemented, Sylow
56	C₂×M₁₆	—	106	2	no	—
57	C₂²×C₈	—	53	2	no	—
58	C₂×C₄²	—	53	1	no	—
59	C₄⋊C₈	—	53	4	no	—
60	C₄⋊C₈	—	106	2	no	—
61	C₄×C₈	—	53	2	no	—
62	C₄×C₈	—	53	1	no	—
63	C₄×C₈	#6157	1	1	yes	characteristic
64	C₂²×C₄	—	53	1	no	—
65	C₂²×C₄	—	53	1	no	—
66	C₂²×C₄	—	53	1	no	—
67	M₁₆	—	212	4	no	—
68	C₂×C₈	—	53	2	no	—
69	C₂×C₈	—	106	4	no	—
70	C₂×C₈	—	106	4	no	—
71	C₂×C₈	C₂×#6157	1	2	yes	—
72	C₄²	—	53	1	no	—
73	C₄²	—	53	1	no	—
74	C₄²	—	53	1	no	—
75	C₄²	C₂×#6157	1	1	yes	characteristic
76	C₂³	—	53	1	no	—
77	C₂×C₄	—	53	1	no	—
78	C₂×C₄	—	53	4	no	—
79	C₂×C₄	—	53	1	no	—
80	C₂×C₄	—	53	1	no	—
81	C₂×C₄	—	106	2	no	—
82	C₂×C₄	—	106	2	no	—
83	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
84	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
85	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
86	C₈	—	2	2	no	—
87	C₈	—	53	4	no	—
88	C₈	—	212	4	no	—
89	C₂²	—	53	2	no	—
90	C₂²	—	106	1	no	—
91	C₂²	—	106	1	no	—
92	C₂²	C₂×C₄⋊#6157	1	1	yes	characteristic
93	C₄	—	53	2	no	—
94	C₄	—	106	1	no	—
95	C₄	—	106	1	no	—
96	C₄	#13133285	1	2	yes	—
97	C₄	#13133288	1	2	yes	—
98	C₄	D₅₃⋊M₁₆	1	2	yes	—
99	C₂	—	106	2	no	—
100	C₂	#16811628	1	1	yes	characteristic
101	C₂	C₂×#13133285	1	1	yes	characteristic
102	C₂	C₂×#13133288	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811627

C₂

yes

maximal

#16811485

C₂

yes

maximal, complemented

C₄×C₈×D₅₃

C₂

yes

maximal, characteristic

C₂×#11719936

—

C₄⋊#88324

—

C₄⋊#88324

C₂²

yes

—

C₂×C₈×D₅₃

C₄

yes

—

C₄²×D₅₃

C₂²

yes

characteristic

C₄×#88322

C₄

yes

characteristic

C₈×Dic₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₄

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

Q₈

yes

characteristic

C₂×#88324

—

#11719936

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

Q₈

yes

characteristic

C₄×Dic₅₃

C₂³

yes

characteristic

C₂×#88322

C₂×C₄

yes

—

C₈×D₅₃

—

C₂²×D₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

C₂×Q₈

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

SD₁₆

yes

—

#88324

—

C₈×C₅₃

—

#88322

M₁₆

yes

complemented

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

#106

yes

—

C₄×C₅₃

#106

yes

—

C₄×C₅₃

#108

yes

—

C₄×C₅₃

C₂×M₁₆

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₂×SD₁₆

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

#423

—

C₂×C₄×C₈

—

#334

—

C₅₃

#1566

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#6157

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

C₂×#6157

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×#6157

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²×#6157

yes

characteristic

C₈

—

C₈

—

C₈

—

212

—

C₂²

—

C₂²

—

106

—

C₂²

—

106

—

C₂²

C₂×C₄⋊#6157

yes

characteristic

C₄

—

C₄

—

106

—

C₄

—

106

—

C₄

#13133285

yes

—

C₄

#13133288

yes

—

C₄

D₅₃⋊M₁₆

yes

—

C₂

—

106

—

100

C₂

#16811628

yes

characteristic

101

C₂

C₂×#13133285

yes

characteristic

102

C₂

C₂×#13133288

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	C₄×C₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	C₄×C₈	#6157
hypocenter	C₅₃	#1566
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂²

C₂×C₄⋊#6157

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6157

Fitting

C₄×C₈×C₅₃

C₄

socle

C₂²×C₅₃

C₂×C₄⋊C₄

C₄×C₈

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
2. #16811485	C₂	semidirect
36. #88322	M₁₆	semidirect
55. C₅₃	#1566	semidirect

H ≅ G/N

type

2. #16811485

C₂

semidirect

36. #88322

M₁₆

semidirect

55. C₅₃

#1566

semidirect

Subgroups — #20290271

Conjugacy classes of subgroups

Complements