Group #20272581 — subgroups

Subgroups — #20272581

Order 6784.

152 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 57 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	C₄×#11719917	C₂	1	1	yes	maximal, characteristic
2	#16810491	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#432	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719917	—	2	1	no	—
5	#13133128	—	2	1	no	—
6	C₄×#88322	C₄	1	2	yes	complemented
7	#13133124	—	2	1	no	—
8	#13133124	—	2	1	no	—
9	#13133124	C₄	1	2	yes	—
10	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₅₃	—	2	1	no	—
12	C₂²⋊C₈×C₅₃	—	2	1	no	—
13	#97×C₅₃	C₄	1	2	yes	complemented
14	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
17	C₂²×Dic₅₃	—	2	2	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	Q₈	1	1	yes	characteristic
23	#11719917	—	2	4	no	—
24	C₂×#88322	—	2	2	no	—
25	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₅₃	—	2	2	no	—
27	C₂×C₄×C₅₃	—	4	1	no	—
28	C₂×C₄×C₅₃	—	4	1	no	—
29	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
30	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×Dic₅₃	—	4	2	no	—
36	C₂×Dic₅₃	—	4	2	no	—
37	C₂×Dic₅₃	—	4	2	no	—
38	C₈×C₅₃	—	4	2	no	—
39	#88322	—	2	4	no	—
40	C₂²×C₅₃	—	2	1	no	—
41	C₂²×C₅₃	—	2	1	no	—
42	C₂²×C₅₃	#95	1	1	yes	characteristic
43	C₂²×C₅₃	#104	1	2	yes	—
44	C₄×C₅₃	—	2	1	no	—
45	C₄×C₅₃	—	2	1	no	—
46	C₄×C₅₃	—	2	1	no	—
47	C₄×C₅₃	—	2	1	no	—
48	C₄×C₅₃	—	2	1	no	—
49	C₄×C₅₃	—	2	1	no	—
50	Dic₅₃	—	8	2	no	—
51	#1288	—	53	1	no	maximal, Sylow
52	C₂×C₅₃	—	2	2	no	—
53	C₂×C₅₃	H(3,ℤ/4)	1	1	yes	characteristic
54	C₂×C₅₃	#339	1	1	yes	characteristic
55	C₂×C₅₃	#344	1	1	yes	characteristic
56	#432	D₅₃	1	1	yes	characteristic
57	C₄×M₁₆	—	53	1	no	—
58	#383	—	53	1	no	—
59	C₅₃	#1288	1	1	yes	characteristic, complemented, Sylow
60	C₂×M₁₆	—	106	1	no	—
61	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
62	C₄⋊C₈	—	2	1	no	—
63	C₂²⋊C₈	—	2	1	no	—
64	#97	—	106	1	no	—
65	#97	Dic₅₃	1	2	yes	complemented
66	C₄×C₈	—	53	2	no	—
67	#95	—	53	2	no	—
68	#95	—	106	1	no	—
69	#95	—	106	1	no	—
70	C₂²×C₄	—	106	2	no	—
71	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
72	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
73	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
74	M₁₆	—	106	4	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	106	2	no	—
77	C₄²	Dic₁₀₆	1	1	yes	characteristic
78	C₄²	C₄×D₅₃	1	1	yes	characteristic
79	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
80	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
81	C₂³	#11719921	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	4	1	no	—
84	C₂×C₄	—	4	1	no	—
85	C₂×C₄	—	212	2	no	—
86	C₂×C₄	—	212	2	no	—
87	C₂×C₄	—	212	2	no	—
88	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
89	C₂×C₄	#11719919	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	C₂²⋊Dic₅₃	1	1	yes	characteristic
94	C₈	—	4	2	no	—
95	C₈	—	106	4	no	—
96	C₂²	—	2	1	no	—
97	C₂²	—	2	1	no	—
98	C₂²	#13133124	1	1	yes	characteristic
99	C₂²	#13133143	1	2	yes	—
100	C₄	—	2	1	no	—
101	C₄	—	2	1	no	—
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	—	2	1	no	—
106	C₄	—	424	2	no	—
107	C₂	—	2	2	no	—
108	C₂	#16810379	1	1	yes	characteristic
109	C₂	#16810384	1	1	yes	characteristic
110	C₂	#16810395	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#11719917

C₂

yes

maximal, characteristic

#16810491

C₂

yes

maximal, characteristic

C₅₃×#432

C₂

yes

maximal, characteristic

C₂×#11719917

—

#13133128

—

C₄×#88322

C₄

yes

—

—

C₄

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

#97×C₅₃

C₄

yes

complemented

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

Q₈

yes

characteristic

#11719917

—

C₂×#88322

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

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

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#95

yes

characteristic

C₂²×C₅₃

#104

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

#1288

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

yes

characteristic

C₄×M₁₆

—

#383

—

C₅₃

#1288

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

106

—

#97

Dic₅₃

yes

complemented

C₄×C₈

—

#95

—

#95

—

106

—

#95

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

D₂₁₂

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

106

—

C₄²

Dic₁₀₆

yes

characteristic

C₄²

C₄×D₅₃

yes

characteristic

C₄²

C₂×Dic₅₃

yes

characteristic

C₄²

C₂²⋊D₅₃

yes

characteristic

C₂³

#11719921

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

C₄×Dic₅₃

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₄⋊Dic₅₃

yes

characteristic

C₂×C₄

#11719921

yes

characteristic

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

C₈

—

C₈

—

106

—

C₂²

—

C₂²

—

C₂²

#13133124

yes

characteristic

C₂²

#13133143

yes

—

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

424

—

107

C₂

—

108

C₂

#16810379

yes

characteristic

109

C₂

#16810384

yes

characteristic

110

C₂

#16810395

yes

characteristic

	subgroup	G/H
center	C₂²	#13133124
commutator	C₂×C₄×C₅₃	C₄²
Frattini	C₂×C₄²	C₂×D₅₃
Fitting	C₅₃×#432	C₂
socle	C₂²×C₅₃	#95
hypercenter	#432	D₅₃
hypocenter	C₅₃	#1288
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

subgroup

G/H

center

C₂²

#13133124

commutator

C₂×C₄×C₅₃

C₄²

Frattini

C₂×C₄²

C₂×D₅₃

Fitting

C₅₃×#432

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

N	H ≅ G/N	type
6. C₄×#88322	C₄	semidirect
13. #97×C₅₃	C₄	semidirect
59. C₅₃	#1288	semidirect
65. #97	Dic₅₃	semidirect

H ≅ G/N

type

6. C₄×#88322

C₄

semidirect

13. #97×C₅₃

C₄

semidirect

59. C₅₃

#1288

semidirect

65. #97

Dic₅₃

semidirect

Subgroups — #20272581

Conjugacy classes of subgroups

Complements