Group #20272645 — subgroups

Subgroups — #20272645

Order 6784.

150 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 41 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	#16810347	C₂	1	2	yes	maximal
2	C₅₃×#381	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊#88322	—	2	2	no	—
4	C₂³×C₄×C₅₃	C₄	1	1	yes	characteristic
5	C₂×C₂²⋊C₄×C₅₃	C₄	1	1	yes	characteristic
6	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
7	#95×C₅₃	—	2	2	no	—
8	C₂⁴×C₅₃	C₂×C₄	1	1	yes	characteristic
9	C₂²×C₄×C₅₃	—	2	1	no	—
10	C₂²×C₄×C₅₃	—	2	1	no	—
11	C₂²×C₄×C₅₃	—	2	1	no	—
12	C₂²×C₄×C₅₃	—	4	1	no	—
13	C₂²×C₄×C₅₃	—	4	1	no	—
14	C₂²×C₄×C₅₃	C₈	1	2	yes	complemented
15	C₂²×C₄×C₅₃	D₄	1	2	yes	—
16	C₂²⋊C₄×C₅₃	—	4	1	no	—
17	C₂²⋊C₄×C₅₃	—	4	1	no	—
18	C₂×#88322	—	4	2	no	—
19	C₂³×C₅₃	—	2	1	no	—
20	C₂³×C₅₃	—	2	1	no	—
21	C₂³×C₅₃	—	4	2	no	—
22	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₂³×C₅₃	C₂×C₈	1	1	yes	characteristic
24	C₂³×C₅₃	M₁₆	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	1	no	—
29	C₂×C₄×C₅₃	—	4	1	no	—
30	C₂×C₄×C₅₃	—	4	2	no	—
31	C₂×C₄×C₅₃	—	4	2	no	—
32	C₂×C₄×C₅₃	—	4	2	no	—
33	C₂×C₄×C₅₃	—	4	1	no	—
34	#88322	—	8	2	no	—
35	C₂²×C₅₃	—	2	2	no	—
36	C₂²×C₅₃	—	2	1	no	—
37	C₂²×C₅₃	—	2	1	no	—
38	C₂²×C₅₃	—	4	1	no	—
39	C₂²×C₅₃	—	4	1	no	—
40	C₂²×C₅₃	—	4	2	no	—
41	C₂²×C₅₃	—	4	2	no	—
42	C₂²×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
43	C₂²×C₅₃	#99	1	1	yes	characteristic
44	C₂²×C₅₃	#100	1	1	yes	characteristic
45	C₄×C₅₃	—	4	2	no	—
46	C₄×C₅₃	—	4	2	no	—
47	C₄×C₅₃	—	8	1	no	—
48	#1317	—	53	1	no	maximal, Sylow
49	C₂×C₅₃	—	2	1	no	—
50	C₂×C₅₃	—	2	1	no	—
51	C₂×C₅₃	—	4	1	no	—
52	C₂×C₅₃	—	4	1	no	—
53	C₂×C₅₃	#323	1	1	yes	characteristic
54	C₂×C₅₃	#352	1	2	yes	—
55	#381	D₅₃	1	1	yes	characteristic
56	#323	—	53	2	no	—
57	C₅₃	#1317	1	1	yes	characteristic, complemented, Sylow
58	C₂³×C₄	Dic₅₃	1	1	yes	characteristic
59	C₂×C₂²⋊C₄	Dic₅₃	1	1	yes	characteristic
60	C₂×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic
61	C₂²⋊C₈	—	106	2	no	—
62	#95	—	2	2	no	—
63	C₂⁴	C₂×Dic₅₃	1	1	yes	characteristic
64	C₂²×C₄	—	2	1	no	—
65	C₂²×C₄	—	2	1	no	—
66	C₂²×C₄	—	2	1	no	—
67	C₂²×C₄	—	4	1	no	—
68	C₂²×C₄	—	4	1	no	—
69	C₂²×C₄	#88322	1	2	yes	complemented
70	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
71	C₂×C₈	—	212	2	no	—
72	C₂²⋊C₄	—	4	1	no	—
73	C₂²⋊C₄	—	4	1	no	—
74	C₂³	—	2	1	no	—
75	C₂³	—	2	1	no	—
76	C₂³	—	4	2	no	—
77	C₂³	C₂×#88322	1	1	yes	characteristic
78	C₂³	#11719917	1	1	yes	characteristic
79	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	4	1	no	—
84	C₂×C₄	—	4	1	no	—
85	C₂×C₄	—	4	2	no	—
86	C₂×C₄	—	4	2	no	—
87	C₂×C₄	—	4	2	no	—
88	C₂×C₄	—	4	1	no	—
89	C₈	—	424	2	no	—
90	C₂²	—	2	2	no	—
91	C₂²	—	2	1	no	—
92	C₂²	—	2	1	no	—
93	C₂²	—	4	1	no	—
94	C₂²	—	4	1	no	—
95	C₂²	—	4	2	no	—
96	C₂²	—	4	2	no	—
97	C₂²	C₂²⋊#88322	1	1	yes	characteristic
98	C₂²	#13133132	1	1	yes	characteristic
99	C₂²	#13133134	1	1	yes	characteristic
100	C₄	—	4	2	no	—
101	C₄	—	4	2	no	—
102	C₄	—	8	1	no	—
103	C₂	—	2	1	no	—
104	C₂	—	2	1	no	—
105	C₂	—	4	1	no	—
106	C₂	—	4	1	no	—
107	C₂	#16810347	1	1	yes	characteristic
108	C₂	#16810413	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16810347

C₂

yes

maximal

C₅₃×#381

C₂

yes

maximal, characteristic

C₂²⋊#88322

—

C₂³×C₄×C₅₃

C₄

yes

characteristic

C₂×C₂²⋊C₄×C₅₃

C₄

yes

characteristic

C₂×C₂²⋊C₄×C₅₃

C₂²

yes

characteristic

#95×C₅₃

—

C₂⁴×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₈

yes

complemented

C₂²×C₄×C₅₃

D₄

yes

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

C₂³×C₅₃

C₂×C₈

yes

characteristic

C₂³×C₅₃

M₁₆

yes

characteristic

C₂×C₄×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

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²⋊C₈

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

#1317

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#323

yes

characteristic

C₂×C₅₃

#352

yes

—

#381

D₅₃

yes

characteristic

#323

—

C₅₃

#1317

yes

characteristic, complemented, Sylow

C₂³×C₄

Dic₅₃

yes

characteristic

C₂×C₂²⋊C₄

Dic₅₃

yes

characteristic

C₂×C₂²⋊C₄

C₂×D₅₃

yes

characteristic

C₂²⋊C₈

—

106

—

#95

—

C₂⁴

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#88322

yes

complemented

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂×C₈

—

212

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂³

—

C₂³

—

C₂³

—

C₂³

yes

C₂³

yes

C₂³

C₂²⋊Dic₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₈

—

424

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

yes

C₂²

yes

C₂²

yes

100

C₄

—

101

C₄

—

102

C₄

—

103

C₂

—

104

C₂

—

105

C₂

—

106

C₂

—

107

C₂

#16810347

yes

characteristic

108

C₂

#16810413

yes

—

	subgroup	G/H
center	C₂²	#13133132
commutator	C₂³×C₅₃	C₂×C₈
Frattini	C₂×C₂²⋊C₄	C₂×D₅₃
Fitting	C₅₃×#381	C₂
socle	C₂²×C₅₃	#99
hypercenter	#381	D₅₃
hypocenter	C₅₃	#1317
LCS	G ▷ C₂³×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₂²⋊C₄ ◁ #381

subgroup

G/H

center

C₂²

#13133132

commutator

C₂³×C₅₃

C₂×C₈

Frattini

C₂×C₂²⋊C₄

C₂×D₅₃

Fitting

C₅₃×#381

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₂²⋊C₄ ◁ #381

N	H ≅ G/N	type
14. C₂²×C₄×C₅₃	C₈	semidirect
57. C₅₃	#1317	semidirect
69. C₂²×C₄	#88322	semidirect

H ≅ G/N

type

14. C₂²×C₄×C₅₃

C₈

semidirect

57. C₅₃

#1317

semidirect

69. C₂²×C₄

#88322

semidirect

Subgroups — #20272645

Conjugacy classes of subgroups

Complements