Group #20274879 — subgroups

Subgroups — #20274879

Order 6784.

248 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 141 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×#13133151	C₂	1	2	yes	maximal
2	C₂×#13133149	C₂	1	2	yes	maximal
3	#16810489	C₂	1	2	yes	maximal
4	C₂×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Dic₅₃	C₂²	1	4	yes	—
6	#13133151	—	2	4	no	—
7	#13133149	—	2	4	no	—
8	#13133124	—	2	4	no	—
9	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	C₄×C₈×C₅₃	C₄	1	4	yes	complemented
13	C₂²×C₄×C₅₃	D₄	1	2	yes	—
14	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
15	C₂²×Dic₅₃	—	2	4	no	—
16	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
18	C₂×C₈×C₅₃	D₄	1	2	yes	—
19	C₂×C₈×C₅₃	D₄	1	2	yes	—
20	C₂×C₈×C₅₃	Q₈	1	2	yes	—
21	C₂×C₈×C₅₃	Q₈	1	2	yes	—
22	C₄²×C₅₃	C₂×C₄	1	2	yes	—
23	C₄²×C₅₃	Q₈	1	2	yes	—
24	C₄⋊Dic₅₃	—	2	4	no	—
25	C₄⋊Dic₅₃	—	2	4	no	—
26	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
27	C₂×C₄×C₅₃	—	2	4	no	—
28	C₂×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
29	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
35	C₂×Dic₅₃	—	4	4	no	—
36	C₂×Dic₅₃	—	4	8	no	—
37	C₈×C₅₃	C₄⋊C₄	1	4	yes	—
38	C₈×C₅₃	C₄⋊C₄	1	4	yes	—
39	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₅₃	#127	1	1	yes	characteristic
41	C₂²×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
42	C₂²×C₅₃	#135	1	4	yes	—
43	C₄×C₅₃	—	2	4	no	—
44	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
47	C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
48	Dic₅₃	—	8	4	no	—
49	#1846	—	53	1	no	maximal, Sylow
50	C₂×C₅₃	#384	1	1	yes	characteristic
51	C₂×C₅₃	#427	1	2	yes	—
52	C₂×C₅₃	#495	1	2	yes	—
53	C₂×C₅₃	#499	1	2	yes	—
54	C₂×#107	—	53	2	no	—
55	C₂×#106	—	53	2	no	—
56	C₂×C₄×C₈	D₅₃	1	1	yes	characteristic
57	#382	—	53	2	no	—
58	C₅₃	#1846	1	1	yes	characteristic, complemented, Sylow
59	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
60	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
61	C₂×C₄⋊C₄	—	53	4	no	—
62	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
63	#107	—	106	4	no	—
64	#106	—	106	4	no	—
65	C₄×C₈	Dic₅₃	1	4	yes	complemented
66	#95	—	106	4	no	—
67	C₂²×C₄	—	106	4	no	—
68	C₂²×C₄	D₂₁₂	1	2	yes	—
69	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
70	C₂×C₈	Dic₁₀₆	1	2	yes	—
71	C₂×C₈	Dic₁₀₆	1	2	yes	—
72	C₂×C₈	D₂₁₂	1	2	yes	—
73	C₂×C₈	D₂₁₂	1	2	yes	—
74	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
75	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
76	C₄⋊C₄	—	106	4	no	—
77	C₄⋊C₄	—	106	4	no	—
78	C₄²	Dic₁₀₆	1	2	yes	—
79	C₄²	C₂×Dic₅₃	1	2	yes	—
80	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	212	4	no	—
83	C₂×C₄	—	212	8	no	—
84	C₂×C₄	C₄⋊Dic₅₃	1	4	yes	—
85	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
86	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
87	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
88	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
89	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
90	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
91	C₈	C₄⋊Dic₅₃	1	4	yes	—
92	C₈	C₄⋊Dic₅₃	1	4	yes	—
93	C₂²	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
94	C₂²	#13133214	1	1	yes	characteristic
95	C₂²	#13133217	1	1	yes	characteristic
96	C₂²	#13133239	1	4	yes	—
97	C₄	—	2	4	no	—
98	C₄	—	424	4	no	—
99	C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
100	C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
101	C₄	#13133217	1	1	yes	characteristic
102	C₄	#13133217	1	1	yes	characteristic
103	C₂	#16810493	1	1	yes	characteristic
104	C₂	#16810661	1	2	yes	—
105	C₂	#16811007	1	2	yes	—
106	C₂	#16811025	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133151

C₂

yes

maximal

C₂×#13133149

C₂

yes

maximal

#16810489

C₂

yes

maximal

C₂×C₄×C₈×C₅₃

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

—

#13133151

—

#13133149

—

#13133124

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₄

yes

complemented

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

Q₈

yes

—

C₂×C₈×C₅₃

Q₈

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

Q₈

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

C₄⋊C₄

yes

—

C₈×C₅₃

C₄⋊C₄

yes

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

#127

yes

characteristic

C₂²×C₅₃

C₄⋊Q₈

yes

characteristic

C₂²×C₅₃

#135

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₄⋊Q₈

yes

characteristic

C₄×C₅₃

C₄⋊Q₈

yes

characteristic

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#495

yes

—

C₂×C₅₃

#499

yes

—

C₂×#107

—

C₂×#106

—

C₂×C₄×C₈

D₅₃

yes

characteristic

#382

—

C₅₃

#1846

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#107

—

106

—

#106

—

106

—

C₄×C₈

Dic₅₃

yes

complemented

#95

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

D₂₁₂

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

Dic₁₀₆

yes

—

C₂×C₈

Dic₁₀₆

yes

—

C₂×C₈

D₂₁₂

yes

—

C₂×C₈

D₂₁₂

yes

—

C₂×C₈

C₂×Dic₅₃

yes

—

C₂×C₈

C₂×Dic₅₃

yes

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄²

Dic₁₀₆

yes

—

C₄²

C₂×Dic₅₃

yes

—

C₂³

C₂×D₂₁₂

yes

characteristic

C₂×C₄

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

C₄⋊Dic₅₃

yes

—

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

C₂×C₄

C₂×D₂₁₂

yes

characteristic

C₂×C₄

C₂×D₂₁₂

yes

characteristic

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

C₈

C₄⋊Dic₅₃

yes

—

C₈

C₄⋊Dic₅₃

yes

—

C₂²

C₂×C₄⋊Dic₅₃

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₄

—

C₄

—

424

—

C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

100

C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

101

C₄

#13133217

yes

characteristic

102

C₄

#13133217

yes

characteristic

103

C₂

#16810493

yes

characteristic

104

C₂

#16810661

yes

—

105

C₂

#16811007

yes

—

106

C₂

#16811025

yes

—

	subgroup	G/H
center	C₂³	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×C₄×C₈	D₅₃
hypocenter	C₅₃	#1846
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×C₄×C₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

hypercenter

C₂×C₄×C₈

D₅₃

hypocenter

C₅₃

#1846

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
12. C₄×C₈×C₅₃	C₄	semidirect
58. C₅₃	#1846	semidirect
65. C₄×C₈	Dic₅₃	semidirect

H ≅ G/N

type

12. C₄×C₈×C₅₃

C₄

semidirect

58. C₅₃

#1846

semidirect

65. C₄×C₈

Dic₅₃

semidirect

Subgroups — #20274879

Conjugacy classes of subgroups

Complements