Group #20274620 — subgroups

Subgroups — #20274620

Order 6784.

220 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 105 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	C₂×#13133152	C₂	1	2	yes	maximal
2	M₁₆×Dic₅₃	C₂	1	2	yes	maximal
3	C₂×#13133127	C₂	1	1	yes	maximal, characteristic
4	#16810388	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#341	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	C₂²	1	2	yes	—
7	C₂²×#88322	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
9	#13133152	C₄	1	8	yes	complemented
10	#13133127	—	2	2	no	—
11	#13133127	C₄	1	4	yes	—
12	C₈×Dic₅₃	—	2	2	no	—
13	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
14	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
16	C₂²×Dic₅₃	Q₈	1	2	yes	—
17	M₁₆×C₅₃	—	2	2	no	—
18	M₁₆×C₅₃	C₂×C₄	1	4	yes	—
19	C₂×C₈×C₅₃	—	2	2	no	—
20	C₂×C₈×C₅₃	—	2	2	no	—
21	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
23	C₄×Dic₅₃	D₄	1	2	yes	—
24	#11719917	—	2	2	no	—
25	#11719917	C₂×C₄	1	4	yes	—
26	C₂×#88322	—	2	2	no	—
27	C₂×#88322	C₂×C₄	1	2	yes	—
28	C₂×#88322	C₂×C₄	1	4	yes	—
29	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
32	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₅₃	#41	1	2	yes	—
34	C₂×Dic₅₃	—	4	2	no	—
35	C₂×Dic₅₃	C₄⋊C₄	1	4	yes	—
36	C₈×C₅₃	—	2	2	no	—
37	C₈×C₅₃	—	2	4	no	—
38	#88322	—	2	4	no	—
39	#88322	C₄²	1	4	yes	—
40	C₂²×C₅₃	—	2	2	no	—
41	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₅₃	C₄×Q₈	1	2	yes	—
43	C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
44	C₄×C₅₃	#117	1	1	yes	characteristic
45	C₄×C₅₃	C₄×D₄	1	2	yes	—
46	Dic₅₃	—	2	4	no	—
47	#1778	—	53	1	no	maximal, Sylow
48	C₂×C₅₃	—	2	2	no	—
49	C₂×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₅₃	#430	1	2	yes	—
51	C₂×#108	—	53	2	no	—
52	C₄×M₁₆	—	53	2	no	—
53	C₂×#97	—	53	1	no	—
54	#341	—	53	1	no	—
55	#341	D₅₃	1	1	yes	characteristic
56	C₅₃	#1778	1	1	yes	characteristic, complemented, Sylow
57	C₂×M₁₆	—	53	2	no	—
58	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
59	C₂²×C₈	—	53	1	no	—
60	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
61	C₂×C₄²	—	53	1	no	—
62	#108	—	53	8	no	—
63	#97	—	53	4	no	—
64	#97	—	106	2	no	—
65	C₄×C₈	—	106	2	no	—
66	C₂²×C₄	—	53	2	no	—
67	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	53	4	no	—
70	M₁₆	—	106	2	no	—
71	M₁₆	C₄×D₅₃	1	4	yes	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	53	2	no	—
75	C₂×C₈	—	53	4	no	—
76	C₂×C₈	—	106	2	no	—
77	C₂×C₈	C₄×D₅₃	1	2	yes	—
78	C₄²	—	53	2	no	—
79	C₄²	—	53	2	no	—
80	C₂³	Q₈×D₅₃	1	1	yes	characteristic
81	C₂×C₄	—	53	4	no	—
82	C₂×C₄	—	212	2	no	—
83	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
84	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
85	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
86	C₂×C₄	#11719947	1	2	yes	—
87	C₈	—	2	2	no	—
88	C₈	—	2	4	no	—
89	C₈	—	53	4	no	—
90	C₈	—	106	4	no	—
91	C₂²	—	2	2	no	—
92	C₂²	C₄⋊C₄×D₅₃	1	1	yes	characteristic
93	C₂²	#13133182	1	2	yes	—
94	C₄	—	106	4	no	—
95	C₄	C₄²×D₅₃	1	1	yes	characteristic
96	C₄	#13133173	1	1	yes	characteristic
97	C₄	#13133177	1	2	yes	—
98	C₂	—	2	2	no	—
99	C₂	#16810474	1	1	yes	characteristic
100	C₂	#16810670	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133152

C₂

yes

maximal

M₁₆×Dic₅₃

C₂

yes

maximal

C₂×#13133127

C₂

yes

maximal, characteristic

#16810388

C₂

yes

maximal, characteristic

C₅₃×#341

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

C₂²×#88322

C₂²

yes

characteristic

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

#13133152

C₄

yes

complemented

#13133127

—

#13133127

C₄

yes

—

C₈×Dic₅₃

—

C₂×M₁₆×C₅₃

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

Q₈

yes

—

M₁₆×C₅₃

—

M₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

D₄

yes

—

#11719917

—

#11719917

C₂×C₄

yes

—

C₂×#88322

—

C₂×#88322

C₂×C₄

yes

—

C₂×#88322

C₂×C₄

yes

—

C₂³×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₄⋊C₄

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

C₄²

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

C₄×Q₈

yes

—

C₄×C₅₃

C₂×C₄²

yes

characteristic

C₄×C₅₃

#117

yes

characteristic

C₄×C₅₃

C₄×D₄

yes

—

Dic₅₃

—

#1778

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₄×C₄⋊C₄

yes

characteristic

C₂×C₅₃

#430

yes

—

C₂×#108

—

C₄×M₁₆

—

C₂×#97

—

#341

—

#341

D₅₃

yes

characteristic

C₅₃

#1778

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂×C₄²

—

#108

—

#97

—

#97

—

106

—

C₄×C₈

—

106

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

106

—

M₁₆

C₄×D₅₃

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

C₄×D₅₃

yes

—

C₄²

—

C₄²

—

C₂³

Q₈×D₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

212

—

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

C₂×C₄

C₂×C₄×D₅₃

yes

—

C₂×C₄

D₄×D₅₃

yes

characteristic

C₂×C₄

#11719947

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

106

—

C₂²

—

C₂²

C₄⋊C₄×D₅₃

yes

characteristic

C₂²

#13133182

yes

—

C₄

—

106

—

C₄

C₄²×D₅₃

yes

C₄

yes

C₄

yes

—

C₂

—

C₂

#16810474

yes

characteristic

100

C₂

#16810670

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₅₃
commutator	C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#341	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	#341	D₅₃
hypocenter	C₅₃	#1778
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₅₃

commutator

C₄×C₅₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#341

C₂

socle

C₂²×C₅₃

C₂×C₄⋊C₄

D₅₃

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
9. #13133152	C₄	semidirect
56. C₅₃	#1778	semidirect

H ≅ G/N

type

9. #13133152

C₄

semidirect

56. C₅₃

#1778

semidirect

Subgroups — #20274620

Conjugacy classes of subgroups

Complements