Group #20274792 — subgroups

Subgroups — #20274792

Order 6784.

300 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 93 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	#16810486	C₂	1	2	yes	maximal
2	#16810390	C₂	1	4	yes	maximal
3	C₂×#117×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂²⋊Dic₅₃	C₂²	1	4	yes	—
5	#13133124	—	2	4	no	—
6	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
7	#117×C₅₃	—	2	4	no	—
8	C₂×C₄⋊C₄×C₅₃	C₄	1	2	yes	complemented
9	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
10	C₂×C₄²×C₅₃	C₄	1	2	yes	complemented
11	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
12	C₂²×C₄×C₅₃	—	2	4	no	—
13	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	C₂×C₄	1	4	yes	—
16	C₂²×C₄×C₅₃	D₄	1	4	yes	—
17	C₂²×Dic₅₃	—	2	4	no	—
18	C₄⋊C₄×C₅₃	—	2	4	no	—
19	C₂²⋊C₄×C₅₃	—	2	4	no	—
20	C₄²×C₅₃	—	2	4	no	—
21	C₂²⋊Dic₅₃	—	2	8	no	—
22	C₂³×C₅₃	—	4	2	no	—
23	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
24	C₂³×C₅₃	C₂×D₄	1	2	yes	—
25	C₂³×C₅₃	#41	1	4	yes	—
26	C₂×C₄×C₅₃	—	2	2	no	—
27	C₂×C₄×C₅₃	—	2	2	no	—
28	C₂×C₄×C₅₃	—	2	8	no	—
29	C₂×C₄×C₅₃	—	2	4	no	—
30	C₂×C₄×C₅₃	—	4	4	no	—
31	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
32	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
33	C₂×Dic₅₃	—	4	8	no	—
34	C₂×Dic₅₃	—	4	4	no	—
35	C₂²×C₅₃	—	2	4	no	—
36	C₂²×C₅₃	—	2	2	no	—
37	C₂²×C₅₃	—	4	4	no	—
38	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₅₃	#117	1	2	yes	—
40	C₂²×C₅₃	#123	1	4	yes	—
41	C₄×C₅₃	—	2	2	no	—
42	C₄×C₅₃	—	2	2	no	—
43	C₄×C₅₃	—	4	4	no	—
44	Dic₅₃	—	8	4	no	—
45	#1820	—	53	1	no	maximal, Sylow
46	C₂×C₅₃	—	2	4	no	—
47	C₂×C₅₃	—	2	2	no	—
48	C₂×C₅₃	#381	1	1	yes	characteristic
49	C₂×C₅₃	#410	1	2	yes	—
50	C₂×#117	D₅₃	1	1	yes	characteristic
51	#381	—	53	2	no	—
52	#342	—	53	4	no	—
53	C₅₃	#1820	1	1	yes	characteristic, complemented, Sylow
54	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
55	#117	—	2	4	no	—
56	C₂×C₄⋊C₄	Dic₅₃	1	2	yes	complemented
57	C₂×C₂²⋊C₄	—	53	4	no	—
58	C₂×C₂²⋊C₄	C₂×D₅₃	1	2	yes	—
59	C₂×C₄²	Dic₅₃	1	2	yes	complemented
60	#95	—	106	4	no	—
61	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
62	C₂²×C₄	—	2	4	no	—
63	C₂²×C₄	—	106	4	no	—
64	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
65	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
66	C₂²×C₄	C₂×Dic₅₃	1	4	yes	—
67	C₂²×C₄	C₂²⋊D₅₃	1	4	yes	—
68	C₄⋊C₄	—	2	4	no	—
69	C₂²⋊C₄	—	2	4	no	—
70	C₂²⋊C₄	—	106	8	no	—
71	C₄²	—	2	4	no	—
72	C₂³	—	4	2	no	—
73	C₂³	#11719945	1	4	yes	—
74	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
75	C₂³	C₂×C₂²⋊D₅₃	1	2	yes	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	—	2	8	no	—
79	C₂×C₄	—	2	4	no	—
80	C₂×C₄	—	4	4	no	—
81	C₂×C₄	—	212	8	no	—
82	C₂×C₄	—	212	4	no	—
83	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
84	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
85	C₂²	—	2	2	no	—
86	C₂²	—	2	4	no	—
87	C₂²	—	4	4	no	—
88	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
89	C₂²	#13133176	1	2	yes	—
90	C₂²	#13133200	1	4	yes	—
91	C₄	—	2	2	no	—
92	C₄	—	2	2	no	—
93	C₄	—	4	4	no	—
94	C₄	—	424	4	no	—
95	C₂	—	2	4	no	—
96	C₂	—	2	2	no	—
97	C₂	#16810486	1	1	yes	characteristic
98	C₂	#16810597	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16810486

C₂

yes

maximal

#16810390

C₂

yes

maximal

C₂×#117×C₅₃

C₂

yes

maximal, characteristic

C₂×C₂²⋊Dic₅₃

C₂²

yes

—

#13133124

—

C₂³×C₄×C₅₃

C₂²

yes

characteristic

#117×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₄

yes

complemented

C₂×C₂²⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₄

yes

complemented

C₂⁴×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×Dic₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₂²⋊Dic₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂³×C₅₃

C₂×D₄

yes

—

C₂³×C₅₃

#41

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

#117

yes

—

C₂²×C₅₃

#123

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

#1820

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#381

yes

characteristic

C₂×C₅₃

#410

yes

—

C₂×#117

D₅₃

yes

characteristic

#381

—

#342

—

C₅₃

#1820

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₅₃

yes

characteristic

#117

—

C₂×C₄⋊C₄

Dic₅₃

yes

complemented

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

Dic₅₃

yes

complemented

#95

—

106

—

C₂⁴

C₂²×D₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

106

—

C₄²

—

C₂³

—

C₂³

#11719945

yes

—

C₂³

C₂²×Dic₅₃

yes

characteristic

C₂³

C₂×C₂²⋊D₅₃

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

C₂²⋊Dic₅₃

yes

—

C₂×C₄

C₂²⋊Dic₅₃

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

C₂²

#13133176

yes

—

C₂²

#13133200

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

424

—

C₂

—

C₂

—

C₂

#16810486

yes

characteristic

C₂

#16810597

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₅₃
commutator	C₂³×C₅₃	C₂²×C₄
Frattini	C₂⁴	C₂²×D₅₃
Fitting	C₂×#117×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	C₂×#117	D₅₃
hypocenter	C₅₃	#1820
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#117

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₅₃

commutator

C₂³×C₅₃

C₂²×C₄

Frattini

C₂⁴

C₂²×D₅₃

Fitting

C₂×#117×C₅₃

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#117

N	H ≅ G/N	type
8. C₂×C₄⋊C₄×C₅₃	C₄	semidirect
10. C₂×C₄²×C₅₃	C₄	semidirect
53. C₅₃	#1820	semidirect
56. C₂×C₄⋊C₄	Dic₅₃	semidirect
59. C₂×C₄²	Dic₅₃	semidirect

H ≅ G/N

type

8. C₂×C₄⋊C₄×C₅₃

C₄

semidirect

10. C₂×C₄²×C₅₃

C₄

semidirect

53. C₅₃

#1820

semidirect

56. C₂×C₄⋊C₄

Dic₅₃

semidirect

59. C₂×C₄²

Dic₅₃

semidirect

Subgroups — #20274792

Conjugacy classes of subgroups

Complements