Group #20272552 — subgroups

Subgroups — #20272552

Order 6784.

172 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 85 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	#16810676	C₂	1	1	yes	maximal, characteristic
2	C₄×#11719917	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719917	C₄	1	2	yes	—
5	C₄⋊#88322	C₄	1	2	yes	—
6	C₂²⋊#88322	—	2	1	no	—
7	#13133128	—	2	1	no	—
8	C₄×#88322	—	2	1	no	—
9	C₄×#88322	—	2	1	no	—
10	C₂²×C₈×C₅₃	—	2	1	no	—
11	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	C₄×C₈×C₅₃	—	2	1	no	—
13	C₄×C₈×C₅₃	C₄	1	2	yes	—
14	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₅₃	Q₈	1	1	yes	characteristic
17	C₂×C₈×C₅₃	—	2	4	no	—
18	C₂×C₈×C₅₃	—	2	2	no	—
19	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₄²×C₅₃	D₄	1	1	yes	characteristic
22	C₄²×C₅₃	D₄	1	1	yes	characteristic
23	#11719917	—	2	2	no	—
24	#11719917	C₈	1	4	yes	complemented
25	C₂×#88322	—	2	2	no	—
26	C₂×#88322	—	2	2	no	—
27	C₂³×C₅₃	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₅₃	—	2	2	no	—
29	C₂×C₄×C₅₃	—	2	2	no	—
30	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
31	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
37	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
38	C₈×C₅₃	—	2	4	no	—
39	#88322	—	2	4	no	—
40	#88322	—	4	2	no	—
41	C₂²×C₅₃	—	2	1	no	—
42	C₂²×C₅₃	—	2	1	no	—
43	C₂²×C₅₃	#95	1	1	yes	characteristic
44	C₂²×C₅₃	C₄⋊C₈	1	2	yes	—
45	C₄×C₅₃	—	2	1	no	—
46	C₄×C₅₃	—	2	1	no	—
47	C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
48	C₄×C₅₃	#97	1	1	yes	characteristic
49	C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
50	C₄×C₅₃	#104	1	2	yes	—
51	C₄×C₅₃	#108	1	2	yes	—
52	#1278	—	53	1	no	maximal, Sylow
53	C₂×C₅₃	—	2	2	no	—
54	C₂×C₅₃	#336	1	1	yes	characteristic
55	C₂×C₅₃	#339	1	1	yes	characteristic
56	C₂×C₅₃	#341	1	1	yes	characteristic
57	#431	—	53	1	no	—
58	C₄×M₁₆	—	53	1	no	—
59	C₂×C₄×C₈	D₅₃	1	1	yes	characteristic
60	C₅₃	#1278	1	1	yes	characteristic, complemented, Sylow
61	C₂×M₁₆	—	53	2	no	—
62	C₂²×C₈	—	2	1	no	—
63	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
64	C₄⋊C₈	—	53	2	no	—
65	C₂²⋊C₈	—	106	1	no	—
66	#97	—	106	1	no	—
67	C₄×C₈	—	2	1	no	—
68	C₄×C₈	—	106	1	no	—
69	C₄×C₈	—	106	1	no	—
70	C₄×C₈	Dic₅₃	1	2	yes	—
71	C₂²×C₄	Dic₁₀₆	1	1	yes	characteristic
72	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
73	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
74	M₁₆	—	53	4	no	—
75	M₁₆	—	106	2	no	—
76	C₂×C₈	—	2	4	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	106	2	no	—
79	C₂×C₈	—	106	2	no	—
80	C₄²	C₄×D₅₃	1	1	yes	characteristic
81	C₄²	D₂₁₂	1	1	yes	characteristic
82	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
83	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
84	C₂³	#11719919	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	C₈×D₅₃	1	2	yes	—
88	C₂×C₄	#11719912	1	2	yes	—
89	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
90	C₂×C₄	#11719919	1	1	yes	characteristic
91	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
92	C₂×C₄	#11719921	1	1	yes	characteristic
93	C₂×C₄	#11719921	1	1	yes	characteristic
94	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
95	C₈	—	2	4	no	—
96	C₈	—	106	4	no	—
97	C₈	—	212	2	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	#13133124	1	1	yes	characteristic
101	C₂²	#13133146	1	2	yes	—
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	C₈×Dic₅₃	1	1	yes	characteristic
105	C₄	#13133127	1	1	yes	characteristic
106	C₄	#13133129	1	2	yes	—
107	C₄	#13133145	1	2	yes	—
108	C₄	#13133153	1	2	yes	—
109	C₂	—	2	2	no	—
110	C₂	#16810377	1	1	yes	characteristic
111	C₂	#16810383	1	1	yes	characteristic
112	C₂	#16810388	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810676

C₂

yes

maximal, characteristic

C₄×#11719917

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

C₄

yes

—

C₄⋊#88322

C₄

yes

—

—

—

—

—

C₂²×C₈×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

Q₈

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

#11719917

—

#11719917

C₈

yes

complemented

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

M₁₆

yes

—

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#95

yes

characteristic

C₂²×C₅₃

C₄⋊C₈

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×C₈

yes

characteristic

C₄×C₅₃

#97

yes

characteristic

C₄×C₅₃

C₂²⋊C₈

yes

—

C₄×C₅₃

#104

yes

—

C₄×C₅₃

#108

yes

—

#1278

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₄×M₁₆

—

C₂×C₄×C₈

D₅₃

yes

characteristic

C₅₃

#1278

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

106

—

#97

—

106

—

C₄×C₈

—

C₄×C₈

—

106

—

C₄×C₈

—

106

—

C₄×C₈

Dic₅₃

yes

—

C₂²×C₄

Dic₁₀₆

yes

characteristic

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄²

C₄×D₅₃

yes

characteristic

C₄²

D₂₁₂

yes

characteristic

C₄²

C₂×Dic₅₃

yes

characteristic

C₄²

C₂²⋊D₅₃

yes

characteristic

C₂³

#11719919

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₈×D₅₃

yes

—

C₂×C₄

#11719912

yes

—

C₂×C₄

C₄×Dic₅₃

yes

characteristic

C₂×C₄

#11719919

yes

characteristic

C₂×C₄

C₄⋊Dic₅₃

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

C₈

—

C₈

—

106

—

C₈

—

212

—

C₂²

—

C₂²

—

100

C₂²

#13133124

yes

characteristic

101

C₂²

#13133146

yes

—

102

C₄

—

103

C₄

—

104

C₄

C₈×Dic₅₃

yes

characteristic

105

C₄

#13133127

yes

characteristic

106

C₄

#13133129

yes

—

107

C₄

#13133145

yes

—

108

C₄

#13133153

yes

—

109

C₂

—

110

C₂

#16810377

yes

characteristic

111

C₂

#16810383

yes

characteristic

112

C₂

#16810388

yes

characteristic

	subgroup	G/H
center	C₄²	D₂₁₂
commutator	C₄×C₅₃	C₄×C₈
Frattini	C₂×C₄²	C₂×D₅₃
Fitting	C₂×C₄×C₈×C₅₃	C₂
socle	C₂²×C₅₃	#95
hypercenter	C₂×C₄×C₈	D₅₃
hypocenter	C₅₃	#1278
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₄²

D₂₁₂

commutator

C₄×C₅₃

C₄×C₈

Frattini

C₂×C₄²

C₂×D₅₃

Fitting

C₂×C₄×C₈×C₅₃

C₂

socle

C₂²×C₅₃

#95

hypercenter

C₂×C₄×C₈

D₅₃

hypocenter

C₅₃

#1278

LCS

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
24. #11719917	C₈	semidirect
60. C₅₃	#1278	semidirect

H ≅ G/N

type

24. #11719917

C₈

semidirect

60. C₅₃

#1278

semidirect

Subgroups — #20272552

Conjugacy classes of subgroups

Complements