Group #20272729 — subgroups

Subgroups — #20272729

Order 6784.

114 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 114 conjugacy classes of subgroups — too many to draw. Only its 35 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	#16810859	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#341	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₄₂₄	C₄	1	1	yes	characteristic
5	#13133191	—	2	1	no	—
6	C₂×#11719908	—	2	1	no	—
7	#13133151	C₄	1	1	yes	characteristic
8	#13133139	—	2	1	no	—
9	C₂×M₁₆×C₅₃	—	2	1	no	—
10	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
12	C₂×C₂²⋊D₅₃	—	4	1	no	—
13	C₂×D₂₁₂	—	2	1	no	—
14	M₁₆×C₅₃	—	2	2	no	—
15	M₁₆×C₅₃	—	4	1	no	—
16	C₂×C₈×C₅₃	—	4	1	no	—
17	C₂×C₈×C₅₃	—	4	1	no	—
18	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
20	#11719921	—	4	1	no	—
21	C₄⋊Dic₅₃	—	2	1	no	—
22	D₄₂₄	—	4	1	no	—
23	#11719908	—	4	1	no	—
24	C₂³×C₅₃	SD₁₆	1	1	yes	characteristic
25	C₂²×D₅₃	—	4	1	no	—
26	C₂×C₄×C₅₃	—	2	1	no	—
27	C₂×C₄×C₅₃	—	2	1	no	—
28	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
30	C₂²⋊D₅₃	—	8	2	no	—
31	C₂×Dic₅₃	—	4	1	no	—
32	D₂₁₂	—	4	1	no	—
33	D₂₁₂	—	4	1	no	—
34	C₈×C₅₃	—	2	1	no	—
35	C₈×C₅₃	—	2	1	no	—
36	C₈×C₅₃	—	4	2	no	—
37	C₂²×C₅₃	—	2	1	no	—
38	C₂²×C₅₃	—	2	1	no	—
39	C₂²×C₅₃	—	2	1	no	—
40	C₂²×C₅₃	#102	1	1	yes	characteristic
41	C₂×D₅₃	—	8	1	no	—
42	C₂×D₅₃	—	8	1	no	—
43	C₂×D₅₃	—	8	1	no	—
44	C₄×C₅₃	—	2	1	no	—
45	C₄×C₅₃	#99	1	1	yes	characteristic
46	C₄×C₅₃	#104	1	1	yes	characteristic
47	Dic₅₃	—	8	1	no	—
48	#1349	—	53	1	no	maximal, Sylow
49	C₂×C₅₃	—	4	1	no	—
50	C₂×C₅₃	#327	1	1	yes	characteristic
51	C₂×C₅₃	#359	1	1	yes	characteristic
52	C₂×C₅₃	#361	1	1	yes	characteristic
53	D₅₃	—	16	1	no	—
54	#466	—	53	1	no	—
55	C₂²⋊C₁₆	—	53	1	no	—
56	#341	D₅₃	1	1	yes	characteristic, complemented
57	C₅₃	#1349	1	1	yes	characteristic, complemented, Sylow
58	C₂×D₈	—	53	1	no	—
59	C₂×M₁₆	—	2	1	no	—
60	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
61	#121	—	106	1	no	—
62	C₂×C₁₆	—	106	1	no	—
63	#107	—	53	1	no	—
64	#102	—	106	1	no	—
65	C₂×D₄	—	106	1	no	—
66	C₂×D₄	—	212	1	no	—
67	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
68	D₈	—	212	1	no	—
69	M₁₆	—	2	2	no	—
70	M₁₆	—	4	1	no	—
71	C₂×C₈	—	4	1	no	—
72	C₂×C₈	—	4	1	no	—
73	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
74	C₂×C₈	D₂₁₂	1	1	yes	characteristic
75	C₄⋊C₄	—	106	1	no	—
76	C₂²⋊C₄	—	212	1	no	—
77	C₁₆	—	212	1	no	—
78	C₂³	—	212	1	no	—
79	C₂³	#11719924	1	1	yes	characteristic
80	D₄	—	424	2	no	—
81	D₄	—	212	1	no	—
82	D₄	—	212	1	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	212	1	no	—
86	C₂×C₄	#11719921	1	1	yes	characteristic
87	C₂×C₄	#11719922	1	1	yes	characteristic
88	C₈	—	2	1	no	—
89	C₈	—	2	1	no	—
90	C₈	—	4	2	no	—
91	C₂²	—	2	1	no	—
92	C₂²	—	2	1	no	—
93	C₂²	—	2	1	no	—
94	C₂²	—	424	1	no	—
95	C₂²	—	424	1	no	—
96	C₂²	—	424	1	no	—
97	C₂²	#13133137	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	424	1	no	—
100	C₄	#13133131	1	1	yes	characteristic
101	C₄	#13133145	1	1	yes	characteristic
102	C₂	—	4	1	no	—
103	C₂	—	848	1	no	—
104	C₂	#16810356	1	1	yes	characteristic
105	C₂	#16810429	1	1	yes	characteristic
106	C₂	#16810435	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810859

C₂

yes

maximal, characteristic

C₂²⋊#11719908

C₂

yes

maximal, characteristic, complemented

C₅₃×#341

C₂

yes

maximal, characteristic, complemented

C₂×D₄₂₄

C₄

yes

—

—

C₄

yes

characteristic

#13133139

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂×C₂²⋊D₅₃

—

C₂×D₂₁₂

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

#11719921

—

C₄⋊Dic₅₃

—

D₄₂₄

—

#11719908

—

C₂³×C₅₃

SD₁₆

yes

characteristic

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

D₈

yes

characteristic

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#102

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#1349

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

#466

—

C₂²⋊C₁₆

—

#341

D₅₃

yes

characteristic, complemented

C₅₃

#1349

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

#121

—

106

—

C₂×C₁₆

—

106

—

#107

—

#102

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

212

—

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

D₈

—

212

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

D₂₁₂

yes

characteristic

C₄⋊C₄

—

106

—

C₂²⋊C₄

—

212

—

C₁₆

—

212

—

C₂³

—

212

—

C₂³

#11719924

yes

characteristic

D₄

—

424

—

D₄

—

212

—

D₄

—

212

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

212

—

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

424

—

C₂²

—

424

—

C₂²

—

424

—

C₂²

#13133137

yes

characteristic

C₄

—

C₄

—

424

—

100

C₄

#13133131

yes

characteristic

101

C₄

#13133145

yes

characteristic

102

C₂

—

103

C₂

—

848

—

104

C₂

#16810356

yes

characteristic

105

C₂

#16810429

yes

characteristic

106

C₂

#16810435

yes

characteristic

	subgroup	G/H
center	C₂²	#13133137
commutator	C₂×C₈×C₅₃	C₂×C₄
Frattini	C₂²×C₈	C₂×D₅₃
Fitting	C₅₃×#341	C₂
socle	C₂²×C₅₃	#102
hypercenter	#341	D₅₃
hypocenter	C₅₃	#1349
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #341

subgroup

G/H

center

C₂²

#13133137

commutator

C₂×C₈×C₅₃

C₂×C₄

Frattini

C₂²×C₈

C₂×D₅₃

Fitting

C₅₃×#341

C₂

C₂²×C₅₃

D₅₃

C₅₃

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #341

N	H ≅ G/N	type
2. C₂²⋊#11719908	C₂	semidirect
3. C₅₃×#341	C₂	semidirect
56. #341	D₅₃	semidirect
57. C₅₃	#1349	semidirect

H ≅ G/N

type

2. C₂²⋊#11719908

C₂

semidirect

3. C₅₃×#341

C₂

semidirect

56. #341

D₅₃

semidirect

57. C₅₃

#1349

semidirect

Subgroups — #20272729

Conjugacy classes of subgroups

Complements