Group #20273059 — subgroups

Subgroups — #20273059

Order 6784.

236 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 89 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	#16810680	C₂	1	2	yes	maximal
2	#16810349	C₂	1	4	yes	maximal
3	C₂×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#88322	—	2	2	no	—
5	C₂²⋊#88322	C₂²	1	4	yes	—
6	#13133128	—	2	2	no	—
7	C₂²×Q₈×C₅₃	C₄	1	1	yes	characteristic
8	C₄×Q₈×C₅₃	—	2	2	no	—
9	C₄×Q₈×C₅₃	—	2	2	no	—
10	C₂×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
11	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
12	C₂×C₄²×C₅₃	C₄	1	2	yes	—
13	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×Q₈×C₅₃	—	2	2	no	—
15	C₂×Q₈×C₅₃	—	2	4	no	—
16	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
21	C₄⋊C₄×C₅₃	—	2	2	no	—
22	C₄⋊C₄×C₅₃	—	2	4	no	—
23	C₄²×C₅₃	—	2	2	no	—
24	C₄²×C₅₃	—	2	2	no	—
25	C₄²×C₅₃	D₄	1	2	yes	—
26	C₄²×C₅₃	D₄	1	2	yes	—
27	C₂×#88322	—	2	4	no	—
28	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
29	Q₈×C₅₃	—	2	8	no	—
30	C₂×C₄×C₅₃	—	2	2	no	—
31	C₂×C₄×C₅₃	—	2	2	no	—
32	C₂×C₄×C₅₃	—	2	2	no	—
33	C₂×C₄×C₅₃	—	2	2	no	—
34	C₂×C₄×C₅₃	—	2	2	no	—
35	C₂×C₄×C₅₃	—	4	1	no	—
36	C₂×C₄×C₅₃	—	4	2	no	—
37	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₅₃	M₁₆	1	4	yes	—
41	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
42	#88322	—	4	4	no	—
43	C₂²×C₅₃	—	2	1	no	—
44	C₂²×C₅₃	—	2	1	no	—
45	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₅₃	C₂×M₁₆	1	2	yes	—
47	C₄×C₅₃	—	2	2	no	—
48	C₄×C₅₃	—	2	4	no	—
49	C₄×C₅₃	—	2	2	no	—
50	C₄×C₅₃	—	2	1	no	—
51	C₄×C₅₃	—	4	2	no	—
52	C₄×C₅₃	#101	1	2	yes	—
53	#1467	—	53	1	no	maximal, Sylow
54	C₂×C₅₃	—	2	2	no	—
55	C₂×C₅₃	#407	1	1	yes	characteristic
56	C₂×C₅₃	#410	1	1	yes	characteristic
57	C₂×C₅₃	C₂×#101	1	1	yes	characteristic
58	C₂×C₄×Q₈	D₅₃	1	1	yes	characteristic
59	#432	—	53	2	no	—
60	#324	—	53	4	no	—
61	C₅₃	#1467	1	1	yes	characteristic, complemented, Sylow
62	C₂²×Q₈	Dic₅₃	1	1	yes	characteristic
63	C₄×Q₈	—	2	2	no	—
64	C₄×Q₈	—	2	2	no	—
65	C₂×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
66	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
67	C₂×C₄²	Dic₅₃	1	2	yes	—
68	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
69	C₄⋊C₈	—	106	2	no	—
70	C₂²⋊C₈	—	53	4	no	—
71	#97	—	106	2	no	—
72	C₂×Q₈	—	2	4	no	—
73	C₂×Q₈	—	2	2	no	—
74	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
75	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
76	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
77	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
78	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
79	C₂×C₈	—	106	4	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄²	—	2	2	no	—
83	C₄²	—	2	2	no	—
84	C₄²	C₂²⋊D₅₃	1	2	yes	—
85	C₄²	C₂²⋊D₅₃	1	2	yes	—
86	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
87	Q₈	—	2	8	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	4	1	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	#11719917	1	4	yes	—
96	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
97	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
98	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
99	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
100	C₈	—	212	4	no	—
101	C₂²	—	2	1	no	—
102	C₂²	—	2	1	no	—
103	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
104	C₂²	C₂×#11719917	1	2	yes	—
105	C₄	—	2	4	no	—
106	C₄	—	2	2	no	—
107	C₄	—	2	2	no	—
108	C₄	—	2	1	no	—
109	C₄	—	4	2	no	—
110	C₄	#13133136	1	2	yes	—
111	C₂	—	2	2	no	—
112	C₂	#16810587	1	1	yes	characteristic
113	C₂	#16810597	1	1	yes	characteristic
114	C₂	C₂×#13133136	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810680

C₂

yes

maximal

#16810349

C₂

yes

maximal

C₂×C₄×Q₈×C₅₃

C₂

yes

maximal, characteristic

C₄⋊#88322

—

C₂²⋊#88322

C₂²

yes

—

#13133128

—

C₂²×Q₈×C₅₃

C₄

yes

characteristic

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₄

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₄

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

D₄

yes

—

C₂×#88322

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

M₁₆

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

C₂×M₁₆

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#101

yes

—

#1467

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄×Q₈

D₅₃

yes

characteristic

#432

—

#324

—

C₅₃

#1467

yes

characteristic, complemented, Sylow

C₂²×Q₈

Dic₅₃

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

Dic₅₃

yes

characteristic

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

Dic₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

#97

—

106

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²⋊D₅₃

yes

—

C₄²

C₂²⋊D₅₃

yes

—

C₂³

C₂²×Dic₅₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#11719917

yes

—

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

C₂×C₄

C₂²⋊Dic₅₃

yes

—

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

100

C₈

—

212

—

101

C₂²

—

102

C₂²

—

103

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

104

C₂²

C₂×#11719917

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

#13133136

yes

—

111

C₂

—

112

C₂

#16810587

yes

characteristic

113

C₂

#16810597

yes

characteristic

114

C₂

C₂×#13133136

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₅₃
commutator	C₂³×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×C₄×Q₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	C₂×C₄×Q₈	D₅₃
hypocenter	C₅₃	#1467
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₅₃

commutator

C₂³×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×C₄×Q₈×C₅₃

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

hypercenter

C₂×C₄×Q₈

D₅₃

hypocenter

C₅₃

#1467

LCS

G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
61. C₅₃	#1467	semidirect

H ≅ G/N

type

61. C₅₃

#1467

semidirect

Subgroups — #20273059

Conjugacy classes of subgroups

Complements