Group #20280434 — subgroups

Subgroups — #20280434

Order 6784.

782 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 181 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×#13133200	C₂	1	3	yes	maximal
2	#16810542	C₂	1	4	yes	maximal, complemented
3	#16810534	C₂	1	4	yes	maximal, complemented
4	#16810486	C₂	1	3	yes	maximal, complemented
5	C₂³×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
7	#13133200	—	2	12	no	—
8	C₂×#11719919	C₂²	1	6	yes	—
9	C₂×C₂²⋊Dic₅₃	C₂²	1	3	yes	—
10	C₂×#11719921	C₂²	1	6	yes	—
11	#13133124	C₂²	1	12	yes	—
12	C₂²×Q₈×C₅₃	—	2	12	no	—
13	C₂²×Q₈×C₅₃	C₂²	1	4	yes	—
14	C₂³×C₄×C₅₃	C₂²	1	3	yes	—
15	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
16	C₂³×D₅₃	C₂³	1	1	yes	characteristic
17	C₂×Q₈×C₅₃	—	2	32	no	—
18	C₂×Q₈×C₅₃	—	2	24	no	—
19	C₂²×C₄×C₅₃	—	2	12	no	—
20	C₂²×C₄×C₅₃	D₄	1	12	yes	—
21	C₂²×C₄×C₅₃	C₂³	1	6	yes	—
22	C₂×C₂²⋊D₅₃	—	2	6	no	—
23	C₂²×Dic₅₃	C₂³	1	6	yes	—
24	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
25	C₂²⋊Dic₅₃	—	2	6	no	—
26	#11719921	—	2	12	no	—
27	#11719919	—	2	12	no	—
28	C₂³×C₅₃	—	2	4	no	—
29	C₂³×C₅₃	C₂×D₄	1	6	yes	—
30	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
31	C₂²×D₅₃	—	2	3	no	—
32	C₂²×D₅₃	—	2	4	no	—
33	Q₈×C₅₃	—	2	32	no	—
34	C₂×C₄×C₅₃	—	2	24	no	—
35	C₂×C₄×C₅₃	—	2	12	no	—
36	C₂×C₄×C₅₃	C₂×D₄	1	12	yes	—
37	C₂²⋊D₅₃	—	4	4	no	—
38	C₂×Dic₅₃	—	2	3	no	—
39	C₂×Dic₅₃	—	2	6	no	—
40	C₂×Dic₅₃	—	2	12	no	—
41	C₂²×C₅₃	—	2	12	no	—
42	C₂²×C₅₃	#120	1	4	yes	—
43	C₂²×C₅₃	C₂²×D₄	1	3	yes	—
44	C₂²×C₅₃	2₋¹⁺⁴	1	4	yes	—
45	C₂×D₅₃	—	4	1	no	—
46	C₂×D₅₃	—	4	6	no	—
47	C₄×C₅₃	—	2	12	no	—
48	Dic₅₃	—	4	1	no	—
49	Dic₅₃	—	4	6	no	—
50	#2614	—	53	1	no	maximal, Sylow
51	C₂×C₅₃	—	2	4	no	—
52	C₂×C₅₃	C₂×#120	1	1	yes	characteristic
53	C₂×C₅₃	#536	1	6	yes	—
54	D₅₃	—	8	1	no	—
55	C₂³×Q₈	D₅₃	1	1	yes	characteristic, complemented
56	C₂×#123	—	53	3	no	—
57	#395	—	53	4	no	—
58	#393	—	53	4	no	—
59	#381	—	53	3	no	—
60	C₅₃	#2614	1	1	yes	characteristic, complemented, Sylow
61	C₂²×Q₈	—	2	12	no	—
62	C₂²×Q₈	C₂×D₅₃	1	4	yes	—
63	C₂²×D₄	—	53	1	no	—
64	C₂³×C₄	C₂×D₅₃	1	3	yes	—
65	#123	—	106	12	no	—
66	C₂×C₄⋊C₄	—	53	6	no	—
67	C₂×C₂²⋊C₄	—	53	3	no	—
68	C₂×C₂²⋊C₄	—	53	6	no	—
69	#95	—	53	12	no	—
70	C₂⁴	—	53	1	no	—
71	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
72	C₂×Q₈	—	2	32	no	—
73	C₂×Q₈	—	2	24	no	—
74	C₂×D₄	—	106	6	no	—
75	C₂²×C₄	—	2	12	no	—
76	C₂²×C₄	—	53	6	no	—
77	C₂²×C₄	—	53	1	no	—
78	C₂²×C₄	C₂²⋊D₅₃	1	12	yes	—
79	C₂²×C₄	C₂²×D₅₃	1	6	yes	—
80	C₄⋊C₄	—	106	12	no	—
81	C₂²⋊C₄	—	106	6	no	—
82	C₂²⋊C₄	—	106	12	no	—
83	C₂³	—	2	4	no	—
84	C₂³	—	106	3	no	—
85	C₂³	—	106	4	no	—
86	C₂³	C₂×C₂²⋊D₅₃	1	6	yes	—
87	C₂³	C₂³×D₅₃	1	1	yes	characteristic
88	Q₈	—	2	32	no	—
89	D₄	—	212	4	no	—
90	C₂×C₄	—	2	24	no	—
91	C₂×C₄	—	2	12	no	—
92	C₂×C₄	—	106	6	no	—
93	C₂×C₄	—	106	3	no	—
94	C₂×C₄	—	106	12	no	—
95	C₂×C₄	C₂×C₂²⋊D₅₃	1	12	yes	—
96	C₂²	—	2	12	no	—
97	C₂²	—	212	1	no	—
98	C₂²	—	212	6	no	—
99	C₂²	#13133186	1	4	yes	—
100	C₂²	C₂²×C₂²⋊D₅₃	1	3	yes	—
101	C₂²	#13133265	1	4	yes	—
102	C₄	—	2	12	no	—
103	C₄	—	212	1	no	—
104	C₄	—	212	6	no	—
105	C₂	—	2	4	no	—
106	C₂	—	424	1	no	—
107	C₂	C₂×#13133186	1	1	yes	characteristic
108	C₂	#16811197	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133200

C₂

yes

maximal

#16810542

C₂

yes

maximal, complemented

#16810534

C₂

yes

maximal, complemented

#16810486

C₂

yes

maximal, complemented

C₂³×Q₈×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂²×C₂²⋊D₅₃

C₂²

yes

characteristic

#13133200

—

C₂×#11719919

C₂²

yes

—

C₂×C₂²⋊Dic₅₃

C₂²

yes

—

C₂×#11719921

C₂²

yes

—

#13133124

C₂²

yes

—

C₂²×Q₈×C₅₃

—

C₂²×Q₈×C₅₃

C₂²

yes

—

C₂³×C₄×C₅₃

C₂²

yes

—

C₂⁴×C₅₃

C₂³

yes

characteristic

C₂³×D₅₃

C₂³

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂×C₂²⋊D₅₃

—

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

characteristic

C₂²⋊Dic₅₃

—

#11719921

—

#11719919

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂×D₄

yes

—

C₂³×C₅₃

C₂⁴

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#120

yes

—

C₂²×C₅₃

C₂²×D₄

yes

—

C₂²×C₅₃

2₋¹⁺⁴

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

Dic₅₃

—

Dic₅₃

—

#2614

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×#120

yes

characteristic

C₂×C₅₃

#536

yes

—

D₅₃

—

C₂³×Q₈

D₅₃

yes

characteristic, complemented

—

—

—

—

C₅₃

#2614

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

C₂×D₅₃

yes

—

C₂²×D₄

—

C₂³×C₄

C₂×D₅₃

yes

—

#123

—

106

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

#95

—

C₂⁴

—

C₂⁴

C₂²×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

106

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₄⋊C₄

—

106

—

C₂²⋊C₄

—

106

—

C₂²⋊C₄

—

106

—

C₂³

—

C₂³

—

106

—

C₂³

—

106

—

C₂³

C₂×C₂²⋊D₅₃

yes

—

C₂³

C₂³×D₅₃

yes

characteristic

Q₈

—

D₄

—

212

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

C₂²

—

C₂²

—

212

—

C₂²

—

212

—

C₂²

#13133186

yes

—

100

C₂²

C₂²×C₂²⋊D₅₃

yes

—

101

C₂²

#13133265

yes

—

102

C₄

—

103

C₄

—

212

—

104

C₄

—

212

—

105

C₂

—

106

C₂

—

424

—

107

C₂

C₂×#13133186

yes

characteristic

108

C₂

#16811197

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂⁴
Fitting	C₂³×Q₈×C₅₃	C₂
hypercenter	C₂³×Q₈	D₅₃
hypocenter	C₅₃	#2614
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₃

commutator, socle

C₂³×C₅₃

C₂⁴

Fitting

C₂³×Q₈×C₅₃

C₂

hypercenter

C₂³×Q₈

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ C₂³×Q₈

N	H ≅ G/N	type
2. #16810542	C₂	semidirect
3. #16810534	C₂	semidirect
4. #16810486	C₂	semidirect
5. C₂³×Q₈×C₅₃	C₂	semidirect
55. C₂³×Q₈	D₅₃	semidirect
60. C₅₃	#2614	semidirect

H ≅ G/N

type

2. #16810542

C₂

semidirect

3. #16810534

C₂

semidirect

4. #16810486

C₂

semidirect

5. C₂³×Q₈×C₅₃

C₂

semidirect

55. C₂³×Q₈

D₅₃

semidirect

60. C₅₃

#2614

semidirect

Subgroups — #20280434

Conjugacy classes of subgroups

Complements