Group #20274850 — subgroups

Subgroups — #20274850

Order 6784.

224 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 101 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×#13133128	C₂	1	1	yes	maximal, characteristic
2	#16810489	C₂	1	1	yes	maximal, characteristic
3	#16810385	C₂	1	4	yes	maximal
4	C₅₃×#384	C₂	1	1	yes	maximal, characteristic
5	C₂²×#88322	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
7	#13133128	C₄	1	4	yes	complemented
8	#13133124	—	2	2	no	—
9	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
10	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
11	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₅₃	—	2	2	no	—
13	C₂²×C₄×C₅₃	D₄	1	2	yes	—
14	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
15	C₂²×Dic₅₃	—	2	2	no	—
16	C₄⋊C₄×C₅₃	—	2	4	no	—
17	C₄⋊C₄×C₅₃	—	4	4	no	—
18	C₄²×C₅₃	C₂×C₄	1	2	yes	—
19	C₄²×C₅₃	Q₈	1	2	yes	—
20	C₄⋊Dic₅₃	—	2	4	no	—
21	C₂×#88322	—	2	2	no	—
22	C₂×#88322	—	2	2	no	—
23	C₂×#88322	C₂×C₄	1	4	yes	—
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	—	4	2	no	—
28	C₂×C₄×C₅₃	—	4	4	no	—
29	C₂×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
30	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₄×C₅₃	#41	1	2	yes	—
33	C₂×C₄×C₅₃	#41	1	2	yes	—
34	C₂×Dic₅₃	—	4	2	no	—
35	C₂×Dic₅₃	—	4	2	no	—
36	C₂×Dic₅₃	—	4	2	no	—
37	#88322	—	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	—	8	2	no	—
45	C₄×C₅₃	#117	1	2	yes	—
46	C₄×C₅₃	#125	1	2	yes	—
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	—	53	1	no	—
55	#384	D₅₃	1	1	yes	characteristic
56	#382	—	53	1	no	—
57	#340	—	53	4	no	—
58	C₅₃	#1836	1	1	yes	characteristic, complemented, Sylow
59	C₂²×C₈	—	53	2	no	—
60	C₂×C₄⋊C₄	—	2	2	no	—
61	C₂×C₄⋊C₄	—	53	2	no	—
62	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
63	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
64	#97	—	53	4	no	—
65	#95	—	106	2	no	—
66	C₂²×C₄	—	2	2	no	—
67	C₂²×C₄	—	106	2	no	—
68	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
69	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
70	C₂×C₈	—	53	4	no	—
71	C₂×C₈	—	106	2	no	—
72	C₂×C₈	—	106	2	no	—
73	C₄⋊C₄	—	2	4	no	—
74	C₄⋊C₄	—	4	4	no	—
75	C₄⋊C₄	—	106	4	no	—
76	C₄²	Dic₁₀₆	1	2	yes	—
77	C₄²	C₄×D₅₃	1	2	yes	—
78	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	4	4	no	—
83	C₂×C₄	—	212	2	no	—
84	C₂×C₄	—	212	2	no	—
85	C₂×C₄	—	212	2	no	—
86	C₂×C₄	#11719919	1	4	yes	—
87	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
88	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
89	C₂×C₄	#11719947	1	2	yes	—
90	C₂×C₄	#11719949	1	2	yes	—
91	C₈	—	106	4	no	—
92	C₂²	C₂×#11719919	1	1	yes	characteristic
93	C₂²	#13133204	1	1	yes	characteristic
94	C₂²	#13133206	1	1	yes	characteristic
95	C₂²	#13133243	1	2	yes	—
96	C₂²	#13133248	1	2	yes	—
97	C₄	—	2	4	no	—
98	C₄	—	8	2	no	—
99	C₄	—	424	2	no	—
100	C₄	#13133175	1	2	yes	—
101	C₄	#13133208	1	2	yes	—
102	C₂	#16810488	1	1	yes	characteristic
103	C₂	#16810663	1	2	yes	—
104	C₂	#16810985	1	2	yes	—
105	C₂	#16810987	1	1	yes	characteristic
106	C₂	#16810991	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133128

C₂

yes

maximal, characteristic

#16810489

C₂

yes

maximal, characteristic

#16810385

C₂

yes

maximal

C₅₃×#384

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

#13133128

C₄

yes

complemented

#13133124

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

Q₈

yes

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

C₂×C₄

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

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

—

#88322

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#137

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#117

yes

—

C₄×C₅₃

#125

yes

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#489

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×#97

—

#384

D₅₃

yes

characteristic

#382

—

#340

—

C₅₃

#1836

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#97

—

#95

—

106

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄²

Dic₁₀₆

yes

—

C₄²

C₄×D₅₃

yes

—

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

#11719919

yes

—

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

C₂×C₄

#11719947

yes

—

C₂×C₄

#11719949

yes

—

C₈

—

106

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#13133248

yes

—

C₄

—

C₄

—

C₄

—

424

—

100

C₄

#13133175

yes

—

101

C₄

#13133208

yes

—

102

C₂

#16810488

yes

characteristic

103

C₂

#16810663

yes

—

104

C₂

#16810985

yes

—

105

C₂

#16810987

yes

characteristic

106

C₂

#16810991

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#384	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	#384	D₅₃
hypocenter	C₅₃	#1836
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#384

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
7. #13133128	C₄	semidirect
58. C₅₃	#1836	semidirect

H ≅ G/N

type

7. #13133128

C₄

semidirect

58. C₅₃

#1836

semidirect

Subgroups — #20274850

Conjugacy classes of subgroups

Complements