Group #20274849 — subgroups

Subgroups — #20274849

Order 6784.

224 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 109 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	#16810493	C₂	1	1	yes	maximal, characteristic
2	#16810489	C₂	1	1	yes	maximal, characteristic
3	#16810386	C₂	1	4	yes	maximal
4	C₂×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Dic₅₃	—	2	2	no	—
6	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
8	#13133124	—	2	2	no	—
9	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
10	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	#97×C₅₃	C₄	1	4	yes	complemented
12	C₂²×C₄×C₅₃	D₄	1	2	yes	—
13	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
14	C₂²×Dic₅₃	—	2	2	no	—
15	C₂²×Dic₅₃	—	2	2	no	—
16	C₂×C₈×C₅₃	—	2	2	no	—
17	C₂×C₈×C₅₃	—	2	2	no	—
18	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
19	C₄²×C₅₃	C₂×C₄	1	2	yes	—
20	C₄²×C₅₃	Q₈	1	2	yes	—
21	C₄⋊Dic₅₃	—	2	4	no	—
22	C₄⋊Dic₅₃	—	2	4	no	—
23	C₄⋊Dic₅₃	—	4	4	no	—
24	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
25	C₂×C₄×C₅₃	—	2	2	no	—
26	C₂×C₄×C₅₃	—	2	2	no	—
27	C₂×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
28	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₄×C₅₃	#41	1	2	yes	—
31	C₂×C₄×C₅₃	#41	1	2	yes	—
32	C₂×Dic₅₃	—	4	2	no	—
33	C₂×Dic₅₃	—	4	2	no	—
34	C₂×Dic₅₃	—	4	2	no	—
35	C₂×Dic₅₃	—	4	4	no	—
36	C₂×Dic₅₃	—	4	2	no	—
37	C₈×C₅₃	—	2	4	no	—
38	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₅₃	#124	1	1	yes	characteristic
40	C₂²×C₅₃	#124	1	1	yes	characteristic
41	C₂²×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
42	C₂²×C₅₃	#137	1	2	yes	—
43	C₄×C₅₃	—	2	4	no	—
44	C₄×C₅₃	#117	1	2	yes	—
45	C₄×C₅₃	#125	1	2	yes	—
46	Dic₅₃	—	8	2	no	—
47	Dic₅₃	—	8	2	no	—
48	#1836	—	53	1	no	maximal, Sylow
49	C₂×C₅₃	#382	1	1	yes	characteristic
50	C₂×C₅₃	#428	1	2	yes	—
51	C₂×C₅₃	#489	1	2	yes	—
52	C₂×C₅₃	#490	1	1	yes	characteristic
53	C₂×C₅₃	#491	1	1	yes	characteristic
54	C₂×#97	D₅₃	1	1	yes	characteristic
55	#384	—	53	1	no	—
56	#382	—	53	1	no	—
57	#340	—	53	4	no	—
58	C₅₃	#1836	1	1	yes	characteristic, complemented, Sylow
59	C₂²×C₈	C₂×D₅₃	1	2	yes	—
60	C₂×C₄⋊C₄	—	53	2	no	—
61	C₂×C₄⋊C₄	—	53	2	no	—
62	C₂×C₄⋊C₄	—	106	2	no	—
63	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
64	#97	Dic₅₃	1	4	yes	complemented
65	#95	—	106	2	no	—
66	C₂²×C₄	—	106	2	no	—
67	C₂²×C₄	—	106	2	no	—
68	C₂²×C₄	D₂₁₂	1	2	yes	—
69	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
70	C₂×C₈	—	2	2	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	C₂×Dic₅₃	1	4	yes	—
73	C₄⋊C₄	—	106	4	no	—
74	C₄⋊C₄	—	106	4	no	—
75	C₄⋊C₄	—	212	4	no	—
76	C₄²	Dic₁₀₆	1	2	yes	—
77	C₄²	C₂×Dic₅₃	1	2	yes	—
78	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	212	2	no	—
82	C₂×C₄	—	212	2	no	—
83	C₂×C₄	—	212	4	no	—
84	C₂×C₄	—	212	2	no	—
85	C₂×C₄	—	212	2	no	—
86	C₂×C₄	C₄⋊Dic₅₃	1	4	yes	—
87	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
88	C₂×C₄	#11719945	1	2	yes	—
89	C₂×C₄	#11719945	1	2	yes	—
90	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
91	C₈	—	2	4	no	—
92	C₂²	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
93	C₂²	#13133205	1	1	yes	characteristic
94	C₂²	#13133205	1	1	yes	characteristic
95	C₂²	#13133245	1	2	yes	—
96	C₂²	#13133250	1	2	yes	—
97	C₄	—	2	4	no	—
98	C₄	—	424	2	no	—
99	C₄	—	424	2	no	—
100	C₄	#13133176	1	2	yes	—
101	C₄	#13133209	1	2	yes	—
102	C₂	#16810489	1	1	yes	characteristic
103	C₂	#16810665	1	2	yes	—
104	C₂	#16810983	1	2	yes	—
105	C₂	#16810988	1	1	yes	characteristic
106	C₂	#16810992	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810493

C₂

yes

maximal, characteristic

#16810489

C₂

yes

maximal, characteristic

#16810386

C₂

yes

maximal

C₂×#97×C₅₃

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

#13133124

—

C₂²×C₈×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#97×C₅₃

C₄

yes

complemented

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

Q₈

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#137

yes

—

C₄×C₅₃

—

C₄×C₅₃

#117

yes

—

C₄×C₅₃

#125

yes

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#489

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×#97

D₅₃

yes

—

—

—

C₅₃

#1836

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×D₅₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#97

Dic₅₃

yes

complemented

#95

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

D₂₁₂

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₅₃

yes

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₄²

Dic₁₀₆

yes

—

C₄²

C₂×Dic₅₃

yes

—

C₂³

C₂×D₂₁₂

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

C₄⋊Dic₅₃

yes

—

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

C₂×C₄

#11719945

yes

—

C₂×C₄

#11719945

yes

—

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

C₈

—

C₂²

C₂×C₄⋊Dic₅₃

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#13133250

yes

—

C₄

—

C₄

—

424

—

C₄

—

424

—

100

C₄

#13133176

yes

—

101

C₄

#13133209

yes

—

102

C₂

#16810489

yes

characteristic

103

C₂

#16810665

yes

—

104

C₂

#16810983

yes

—

105

C₂

#16810988

yes

characteristic

106

C₂

#16810992

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×#97×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×#97	D₅₃
hypocenter	C₅₃	#1836
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×#97×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

hypercenter

C₂×#97

D₅₃

hypocenter

C₅₃

#1836

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
11. #97×C₅₃	C₄	semidirect
58. C₅₃	#1836	semidirect
64. #97	Dic₅₃	semidirect

H ≅ G/N

type

11. #97×C₅₃

C₄

semidirect

58. C₅₃

#1836

semidirect

64. #97

Dic₅₃

semidirect

Subgroups — #20274849

Conjugacy classes of subgroups

Complements