Group #20280665 — subgroups

Subgroups — #20280665

Order 6784.

416 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 149 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×#13133209	C₂	1	1	yes	maximal, characteristic
2	#16810561	C₂	1	8	yes	maximal
3	#16810530	C₂	1	4	yes	maximal
4	#16810469	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	#13133209	—	2	4	no	—
7	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
8	C₂×#11719919	C₂²	1	4	yes	—
9	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
10	#13133124	C₂²	1	16	yes	—
11	#13133124	C₂²	1	4	yes	—
12	C₂²×Q₈×C₅₃	C₂²	1	2	yes	—
13	C₄⋊Q₈×C₅₃	—	2	4	no	—
14	C₂×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
15	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₅₃	—	2	4	no	—
17	C₂×Q₈×C₅₃	—	2	8	no	—
18	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
19	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
20	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
21	C₂²×Dic₅₃	C₂³	1	4	yes	—
22	C₂²×Dic₅₃	C₂³	1	4	yes	—
23	C₄⋊C₄×C₅₃	—	2	8	no	—
24	C₄²×C₅₃	D₄	1	4	yes	—
25	C₄⋊Dic₅₃	—	2	4	no	—
26	#11719919	—	2	8	no	—
27	C₄×Dic₅₃	—	2	4	no	—
28	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
29	Q₈×C₅₃	—	4	8	no	—
30	C₂×C₄×C₅₃	—	2	4	no	—
31	C₂×C₄×C₅₃	—	2	4	no	—
32	C₂×C₄×C₅₃	—	2	8	no	—
33	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
34	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
35	C₂×C₄×C₅₃	#41	1	4	yes	—
36	C₂×Dic₅₃	—	2	4	no	—
37	C₂×Dic₅₃	—	2	4	no	—
38	C₂×Dic₅₃	—	2	8	no	—
39	C₂×Dic₅₃	—	2	8	no	—
40	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
41	C₂²×C₅₃	C₂×#41	1	2	yes	—
42	C₂²×C₅₃	2₋¹⁺⁴	1	4	yes	—
43	C₄×C₅₃	—	2	4	no	—
44	C₄×C₅₃	—	4	4	no	—
45	C₄×C₅₃	#124	1	4	yes	—
46	Dic₅₃	—	4	4	no	—
47	Dic₅₃	—	4	4	no	—
48	#2642	—	53	1	no	maximal, Sylow
49	C₂×C₅₃	C₂×#124	1	1	yes	characteristic
50	C₂×C₅₃	#536	1	2	yes	—
51	C₂×C₅₃	#541	1	4	yes	—
52	C₂×C₄⋊Q₈	D₅₃	1	1	yes	characteristic
53	C₂×#125	—	53	1	no	—
54	#400	—	53	8	no	—
55	#391	—	53	4	no	—
56	#376	—	53	1	no	—
57	C₅₃	#2642	1	1	yes	characteristic, complemented, Sylow
58	C₂²×Q₈	C₂×D₅₃	1	2	yes	—
59	C₄⋊Q₈	—	2	4	no	—
60	#125	—	106	4	no	—
61	C₂×C₄⋊C₄	—	53	4	no	—
62	C₂×C₄⋊C₄	—	53	2	no	—
63	C₂×C₄⋊C₄	C₂×D₅₃	1	4	yes	—
64	C₂×C₄²	—	53	2	no	—
65	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
66	#95	—	53	4	no	—
67	#95	—	53	16	no	—
68	C₂×Q₈	—	2	8	no	—
69	C₂×Q₈	—	2	4	no	—
70	C₂²×C₄	—	53	4	no	—
71	C₂²×C₄	—	53	4	no	—
72	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
73	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
74	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
75	C₄⋊C₄	—	2	8	no	—
76	C₄⋊C₄	—	106	8	no	—
77	C₄⋊C₄	—	106	4	no	—
78	C₄²	—	106	4	no	—
79	C₄²	C₂²⋊D₅₃	1	4	yes	—
80	C₂³	C₂³×D₅₃	1	1	yes	characteristic
81	Q₈	—	4	8	no	—
82	C₂×C₄	—	2	4	no	—
83	C₂×C₄	—	2	4	no	—
84	C₂×C₄	—	2	8	no	—
85	C₂×C₄	—	106	4	no	—
86	C₂×C₄	—	106	4	no	—
87	C₂×C₄	—	106	8	no	—
88	C₂×C₄	—	106	8	no	—
89	C₂×C₄	#11719947	1	4	yes	—
90	C₂×C₄	C₂×C₂²⋊D₅₃	1	4	yes	—
91	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
92	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
93	C₂²	C₂×#11719947	1	2	yes	—
94	C₂²	#13133265	1	4	yes	—
95	C₄	—	2	4	no	—
96	C₄	—	4	4	no	—
97	C₄	—	212	4	no	—
98	C₄	—	212	4	no	—
99	C₄	#13133204	1	4	yes	—
100	C₂	C₂×#13133204	1	1	yes	characteristic
101	C₂	#16811197	1	2	yes	—
102	C₂	#16811234	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133209

C₂

yes

maximal, characteristic

#16810561

C₂

yes

maximal

#16810530

C₂

yes

maximal

#16810469

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₅₃

C₂

yes

maximal, characteristic

#13133209

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719919

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

#13133124

C₂²

yes

—

#13133124

C₂²

yes

—

C₂²×Q₈×C₅₃

C₂²

yes

—

C₄⋊Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

D₄

yes

—

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

—

C₂³×C₅₃

C₂⁴

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂²×C₅₃

C₂×#41

yes

—

C₂²×C₅₃

2₋¹⁺⁴

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#124

yes

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#541

yes

—

C₂×C₄⋊Q₈

D₅₃

yes

—

—

—

—

C₅₃

#2642

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₅₃

yes

—

C₄⋊Q₈

—

#125

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#95

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄²

—

106

—

C₄²

C₂²⋊D₅₃

yes

—

C₂³

C₂³×D₅₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

#11719947

yes

—

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

C₂²

C₂×#11719947

yes

—

C₂²

#13133265

yes

—

C₄

—

C₄

—

C₄

—

212

—

C₄

—

212

—

C₄

#13133204

yes

—

100

C₂

C₂×#13133204

yes

characteristic

101

C₂

#16811197

yes

—

102

C₂

#16811234

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂⁴
Fitting	C₂×C₄⋊Q₈×C₅₃	C₂
hypercenter	C₂×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	#2642
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₃

commutator, socle

C₂³×C₅₃

C₂⁴

Fitting

C₂×C₄⋊Q₈×C₅₃

C₂

hypercenter

C₂×C₄⋊Q₈

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
57. C₅₃	#2642	semidirect

H ≅ G/N

type

57. C₅₃

#2642

semidirect

Subgroups — #20280665

Conjugacy classes of subgroups

Complements