Group #20272647 — subgroups

Subgroups — #20272647

Order 6784.

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

normal subgroups only

This group has 170 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	#16810349	C₂	1	2	yes	maximal, complemented
2	C₅₃×#390	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊#88322	—	2	2	no	—
4	C₂²×D₄×C₅₃	C₄	1	1	yes	characteristic
5	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
6	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
7	C₂×C₄²×C₅₃	C₄	1	1	yes	characteristic
8	C₂⁴×C₅₃	—	2	1	no	—
9	C₂×D₄×C₅₃	—	2	1	no	—
10	C₂×D₄×C₅₃	—	4	1	no	—
11	C₂×D₄×C₅₃	—	4	1	no	—
12	C₂×D₄×C₅₃	C₈	1	2	yes	complemented
13	C₂²×C₄×C₅₃	—	2	1	no	—
14	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	D₄	1	2	yes	—
16	C₄⋊C₄×C₅₃	—	4	1	no	—
17	C₂²⋊C₄×C₅₃	—	4	4	no	—
18	C₄²×C₅₃	—	4	1	no	—
19	C₂×#88322	—	4	2	no	—
20	C₂³×C₅₃	—	2	2	no	—
21	C₂³×C₅₃	—	4	2	no	—
22	C₂³×C₅₃	—	4	2	no	—
23	C₂³×C₅₃	—	4	1	no	—
24	C₂³×C₅₃	—	4	1	no	—
25	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
26	D₄×C₅₃	—	4	2	no	—
27	D₄×C₅₃	—	4	2	no	—
28	C₂×C₄×C₅₃	—	2	1	no	—
29	C₂×C₄×C₅₃	—	2	1	no	—
30	C₂×C₄×C₅₃	—	2	2	no	—
31	C₂×C₄×C₅₃	—	2	2	no	—
32	C₂×C₄×C₅₃	—	4	1	no	—
33	C₂×C₄×C₅₃	—	4	1	no	—
34	C₂×C₄×C₅₃	—	4	2	no	—
35	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
36	C₂×C₄×C₅₃	M₁₆	1	1	yes	characteristic
37	#88322	—	8	2	no	—
38	C₂²×C₅₃	—	2	1	no	—
39	C₂²×C₅₃	—	2	1	no	—
40	C₂²×C₅₃	—	4	4	no	—
41	C₂²×C₅₃	—	4	2	no	—
42	C₂²×C₅₃	—	8	2	no	—
43	C₂²×C₅₃	—	8	1	no	—
44	C₂²×C₅₃	—	8	1	no	—
45	C₂²×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
46	C₂²×C₅₃	#99	1	1	yes	characteristic
47	C₂²×C₅₃	#100	1	1	yes	characteristic
48	C₄×C₅₃	—	2	1	no	—
49	C₄×C₅₃	—	2	1	no	—
50	C₄×C₅₃	—	4	2	no	—
51	C₄×C₅₃	—	4	2	no	—
52	#1318	—	53	1	no	maximal, Sylow
53	C₂×C₅₃	—	2	1	no	—
54	C₂×C₅₃	—	2	1	no	—
55	C₂×C₅₃	—	8	2	no	—
56	C₂×C₅₃	#323	1	1	yes	characteristic
57	C₂×C₅₃	#353	1	1	yes	characteristic
58	C₂×C₅₃	#355	1	1	yes	characteristic
59	#390	D₅₃	1	1	yes	characteristic
60	#324	—	53	2	no	—
61	C₅₃	#1318	1	1	yes	characteristic, complemented, Sylow
62	C₂²×D₄	Dic₅₃	1	1	yes	characteristic
63	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
64	C₂×C₂²⋊C₄	—	2	2	no	—
65	C₂×C₄²	Dic₅₃	1	1	yes	characteristic
66	C₂²⋊C₈	—	106	2	no	—
67	C₂⁴	—	2	1	no	—
68	C₂×D₄	—	2	1	no	—
69	C₂×D₄	—	4	1	no	—
70	C₂×D₄	—	4	1	no	—
71	C₂×D₄	#88322	1	2	yes	complemented
72	C₂²×C₄	—	2	1	no	—
73	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
74	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
75	C₂×C₈	—	212	2	no	—
76	C₄⋊C₄	—	4	1	no	—
77	C₂²⋊C₄	—	4	4	no	—
78	C₄²	—	4	1	no	—
79	C₂³	—	2	2	no	—
80	C₂³	—	4	2	no	—
81	C₂³	—	4	2	no	—
82	C₂³	—	4	1	no	—
83	C₂³	—	4	1	no	—
84	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
85	D₄	—	4	2	no	—
86	D₄	—	4	2	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	4	1	no	—
92	C₂×C₄	—	4	1	no	—
93	C₂×C₄	—	4	2	no	—
94	C₂×C₄	C₂×#88322	1	1	yes	characteristic
95	C₂×C₄	#11719917	1	1	yes	characteristic
96	C₈	—	424	2	no	—
97	C₂²	—	2	1	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	4	4	no	—
100	C₂²	—	4	2	no	—
101	C₂²	—	8	2	no	—
102	C₂²	—	8	1	no	—
103	C₂²	—	8	1	no	—
104	C₂²	C₂²⋊#88322	1	1	yes	characteristic
105	C₂²	#13133132	1	1	yes	characteristic
106	C₂²	#13133134	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	—	4	2	no	—
110	C₄	—	4	2	no	—
111	C₂	—	2	1	no	—
112	C₂	—	2	1	no	—
113	C₂	—	8	2	no	—
114	C₂	#16810347	1	1	yes	characteristic
115	C₂	#16810416	1	1	yes	characteristic
116	C₂	#16810420	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810349

C₂

yes

maximal, complemented

C₅₃×#390

C₂

yes

maximal, characteristic

C₂²⋊#88322

—

C₂²×D₄×C₅₃

C₄

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₄

yes

characteristic

C₂⁴×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₈

yes

complemented

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

D₄×C₅₃

—

D₄×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

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

M₁₆

yes

characteristic

#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

—

C₄×C₅₃

—

#1318

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

yes

characteristic

#324

—

C₅₃

#1318

yes

characteristic, complemented, Sylow

C₂²×D₄

Dic₅₃

yes

characteristic

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

Dic₅₃

yes

characteristic

C₂²⋊C₈

—

106

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

#88322

yes

complemented

C₂²×C₄

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂×C₈

—

212

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊Dic₅₃

yes

characteristic

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₈

—

424

—

C₂²

—

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

C₂²⋊#88322

yes

characteristic

105

C₂²

#13133132

yes

characteristic

106

C₂²

#13133134

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₂

—

112

C₂

—

113

C₂

—

114

C₂

#16810347

yes

characteristic

115

C₂

#16810416

yes

characteristic

116

C₂

#16810420

yes

characteristic

	subgroup	G/H
center	C₂²	#13133132
commutator	C₂×C₄×C₅₃	C₂×C₈
Frattini	C₂×C₄⋊C₄	C₂×D₅₃
Fitting	C₅₃×#390	C₂
socle	C₂²×C₅₃	#99
hypercenter	#390	D₅₃
hypocenter	C₅₃	#1318
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #390

subgroup

G/H

center

C₂²

#13133132

commutator

C₂×C₄×C₅₃

C₂×C₈

Frattini

C₂×C₄⋊C₄

C₂×D₅₃

Fitting

C₅₃×#390

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #390

N	H ≅ G/N	type
1. #16810349	C₂	semidirect
12. C₂×D₄×C₅₃	C₈	semidirect
61. C₅₃	#1318	semidirect
71. C₂×D₄	#88322	semidirect

H ≅ G/N

type

1. #16810349

C₂

semidirect

12. C₂×D₄×C₅₃

C₈

semidirect

61. C₅₃

#1318

semidirect

71. C₂×D₄

#88322

semidirect

Subgroups — #20272647

Conjugacy classes of subgroups

Complements