Group #20274417 — subgroups

Subgroups — #338×D₅₃ #20274417

Order 6784.

372 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 135 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	C₂×M₁₆×D₅₃	C₂	1	3	yes	maximal
2	#16810381	C₂	1	3	yes	maximal, complemented
3	C₅₃×#338	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
5	C₂×#11719917	C₂²	1	3	yes	—
6	C₂×#11719912	C₄	1	6	yes	—
7	M₁₆×D₅₃	—	2	6	no	—
8	M₁₆×D₅₃	—	2	6	no	—
9	C₂×C₈×D₅₃	C₄	1	6	yes	—
10	C₂×M₁₆×C₅₃	C₂²	1	3	yes	—
11	C₂³×D₅₃	Q₈	1	1	yes	characteristic
12	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
13	C₂²×Dic₅₃	Q₈	1	1	yes	characteristic
14	C₂×C₄×D₅₃	C₂×C₄	1	3	yes	—
15	C₂×C₄×D₅₃	C₂×C₄	1	3	yes	—
16	C₂×C₄×D₅₃	D₄	1	3	yes	—
17	C₂×C₄×D₅₃	D₄	1	3	yes	—
18	M₁₆×C₅₃	—	2	6	no	—
19	C₂×C₈×C₅₃	C₂×C₄	1	6	yes	—
20	#11719917	—	2	6	no	—
21	C₂×#88322	C₂×C₄	1	6	yes	—
22	#11719912	—	2	12	no	—
23	C₈×D₅₃	—	2	6	no	—
24	C₈×D₅₃	—	2	6	no	—
25	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
26	C₂²×D₅₃	—	4	2	no	—
27	C₂²×D₅₃	C₄⋊C₄	1	3	yes	—
28	C₂²×D₅₃	C₄⋊C₄	1	3	yes	—
29	C₂×C₄×C₅₃	C₂²×C₄	1	3	yes	—
30	C₂×C₄×C₅₃	C₂×D₄	1	3	yes	—
31	C₂×Dic₅₃	C₄⋊C₄	1	3	yes	—
32	C₂×Dic₅₃	C₄⋊C₄	1	3	yes	—
33	C₄×D₅₃	C₄²	1	3	yes	—
34	C₄×D₅₃	C₄²	1	1	yes	characteristic
35	C₄×D₅₃	C₂²⋊C₄	1	3	yes	—
36	C₄×D₅₃	C₂²⋊C₄	1	3	yes	—
37	C₄×D₅₃	C₂²⋊C₄	1	6	yes	—
38	C₈×C₅₃	—	2	6	no	—
39	#88322	—	2	6	no	—
40	C₂²×C₅₃	—	4	1	no	—
41	C₂²×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
42	C₂×D₅₃	—	2	6	no	—
43	C₂×D₅₃	—	4	3	no	—
44	C₂×D₅₃	#95	1	3	yes	—
45	C₂×D₅₃	#95	1	1	yes	characteristic
46	C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
47	C₄×C₅₃	C₂×C₂²⋊C₄	1	3	yes	—
48	Dic₅₃	#95	1	1	yes	characteristic
49	Dic₅₃	#95	1	3	yes	—
50	C₂×#338	—	53	1	no	maximal, Sylow
51	C₂×C₅₃	—	2	3	no	—
52	C₂×C₅₃	C₂×#95	1	1	yes	characteristic
53	D₅₃	—	2	3	no	—
54	D₅₃	#338	1	2	yes	direct product factor
55	C₂²×M₁₆	—	53	3	no	—
56	#338	—	53	3	no	—
57	#338	D₅₃	1	1	yes	characteristic, direct product factor
58	C₅₃	C₂×#338	1	1	yes	characteristic, complemented, Sylow
59	C₂³×C₄	—	53	1	no	—
60	C₂×M₁₆	—	53	3	no	—
61	C₂×M₁₆	—	53	6	no	—
62	C₂×M₁₆	—	106	6	no	—
63	C₂×M₁₆	—	106	6	no	—
64	C₂×M₁₆	C₂×D₅₃	1	3	yes	—
65	C₂²×C₈	—	53	6	no	—
66	C₂⁴	—	53	1	no	—
67	C₂²×C₄	—	53	1	no	—
68	C₂²×C₄	—	53	3	no	—
69	C₂²×C₄	—	53	3	no	—
70	C₂²×C₄	—	53	3	no	—
71	C₂²×C₄	—	53	3	no	—
72	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
73	M₁₆	—	2	6	no	—
74	M₁₆	—	106	12	no	—
75	M₁₆	—	106	6	no	—
76	C₂×C₈	—	53	6	no	—
77	C₂×C₈	—	106	6	no	—
78	C₂×C₈	—	106	6	no	—
79	C₂×C₈	C₄×D₅₃	1	6	yes	—
80	C₂³	—	53	3	no	—
81	C₂³	—	53	3	no	—
82	C₂³	—	212	2	no	—
83	C₂³	Q₈×D₅₃	1	1	yes	characteristic
84	C₂×C₄	—	53	3	no	—
85	C₂×C₄	—	53	3	no	—
86	C₂×C₄	—	53	1	no	—
87	C₂×C₄	—	53	3	no	—
88	C₂×C₄	—	53	3	no	—
89	C₂×C₄	—	53	3	no	—
90	C₂×C₄	—	53	6	no	—
91	C₂×C₄	C₂×C₄×D₅₃	1	3	yes	—
92	C₂×C₄	D₄×D₅₃	1	3	yes	—
93	C₈	—	2	6	no	—
94	C₈	—	106	6	no	—
95	C₂²	—	4	1	no	—
96	C₂²	—	53	1	no	—
97	C₂²	—	53	3	no	—
98	C₂²	—	106	6	no	—
99	C₂²	—	212	3	no	—
100	C₂²	C₄⋊C₄×D₅₃	1	3	yes	—
101	C₄	—	53	1	no	—
102	C₄	—	53	3	no	—
103	C₄	C₄²×D₅₃	1	1	yes	characteristic
104	C₄	C₂²⋊C₄×D₅₃	1	3	yes	—
105	C₂	—	2	3	no	—
106	C₂	—	53	2	no	—
107	C₂	—	106	3	no	—
108	C₂	#95×D₅₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×M₁₆×D₅₃

C₂

yes

maximal

#16810381

C₂

yes

maximal, complemented

C₅₃×#338

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₅₃

C₂²

yes

characteristic

C₂×#11719917

C₂²

yes

—

C₂×#11719912

C₄

yes

—

M₁₆×D₅₃

—

M₁₆×D₅₃

—

C₂×C₈×D₅₃

C₄

yes

—

C₂×M₁₆×C₅₃

C₂²

yes

—

C₂³×D₅₃

Q₈

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

Q₈

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

D₄

yes

—

C₂×C₄×D₅₃

D₄

yes

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

#11719917

—

C₂×#88322

C₂×C₄

yes

—

#11719912

—

C₈×D₅₃

—

C₈×D₅₃

—

C₂³×C₅₃

C₂×Q₈

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

C₄⋊C₄

yes

—

C₂²×D₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

C₄⋊C₄

yes

—

C₂×Dic₅₃

C₄⋊C₄

yes

—

C₄×D₅₃

C₄²

yes

—

C₄×D₅₃

C₄²

yes

characteristic

C₄×D₅₃

C₂²⋊C₄

yes

—

C₄×D₅₃

C₂²⋊C₄

yes

—

C₄×D₅₃

C₂²⋊C₄

yes

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

#95

yes

—

C₂×D₅₃

#95

yes

characteristic

C₄×C₅₃

C₂×C₄²

yes

characteristic

C₄×C₅₃

C₂×C₂²⋊C₄

yes

—

Dic₅₃

#95

yes

characteristic

Dic₅₃

#95

yes

—

C₂×#338

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×#95

yes

characteristic

D₅₃

—

D₅₃

#338

yes

direct product factor

C₂²×M₁₆

—

#338

—

#338

D₅₃

yes

characteristic, direct product factor

C₅₃

C₂×#338

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

106

—

C₂×M₁₆

—

106

—

C₂×M₁₆

C₂×D₅₃

yes

—

C₂²×C₈

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

106

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

C₄×D₅₃

yes

—

C₂³

—

C₂³

—

C₂³

—

212

—

C₂³

Q₈×D₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄×D₅₃

yes

—

C₂×C₄

D₄×D₅₃

yes

—

C₈

—

C₈

—

106

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

106

—

C₂²

—

212

—

100

C₂²

C₄⋊C₄×D₅₃

yes

—

101

C₄

—

102

C₄

—

103

C₄

C₄²×D₅₃

yes

characteristic

104

C₄

C₂²⋊C₄×D₅₃

yes

—

105

C₂

—

106

C₂

—

107

C₂

—

106

—

108

C₂

#95×D₅₃

yes

characteristic

	subgroup	G/H
center	C₄	C₄²×D₅₃
commutator	C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#338	C₂
socle	C₂×C₅₃	C₂×#95
hypercenter	#338	D₅₃
hypocenter	C₅₃	C₂×#338
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ #338

subgroup

G/H

center

C₄

C₄²×D₅₃

commutator

C₄×C₅₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#338

C₂

socle

C₂×C₅₃

C₂×#95

D₅₃

C₅₃

G ▷ C₄×C₅₃ ▷ C₅₃

C₁ ◁ C₄ ◁ #338

N	H ≅ G/N	type
2. #16810381	C₂	semidirect
3. C₅₃×#338	C₂	semidirect
54. D₅₃	#338	direct (N × H)
57. #338	D₅₃	direct (N × H)
58. C₅₃	C₂×#338	semidirect

H ≅ G/N

type

2. #16810381

C₂

semidirect

3. C₅₃×#338

C₂

semidirect

54. D₅₃

#338

direct (N × H)

57. #338

D₅₃

direct (N × H)

58. C₅₃

C₂×#338

semidirect

Subgroups — #338×D₅₃ #20274417

Conjugacy classes of subgroups

Complements

Subgroups — #338×D53 #20274417

Conjugacy classes of subgroups

Complements

Subgroups — #338×D₅₃ #20274417