Group #20272623 — subgroups

Subgroups — #20272623

Order 6784.

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

normal subgroups only

This group has 192 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	#16810587	C₂	1	2	yes	maximal
2	C₅₃×#381	C₂	1	1	yes	maximal, characteristic
3	C₂×#11719917	—	2	2	no	—
4	C₂²⋊#88322	—	2	2	no	—
5	C₂²⋊#88322	C₄	1	4	yes	complemented
6	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊C₄×C₅₃	C₄	1	2	yes	—
8	#95×C₅₃	—	2	2	no	—
9	C₂⁴×C₅₃	C₂×C₄	1	1	yes	characteristic
10	C₂²×C₄×C₅₃	—	2	2	no	—
11	C₂²×C₄×C₅₃	—	4	2	no	—
12	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
13	C₂²×C₄×C₅₃	D₄	1	2	yes	—
14	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	Q₈	1	1	yes	characteristic
16	C₂²⋊C₄×C₅₃	—	4	2	no	—
17	#11719917	—	4	4	no	—
18	C₂×#88322	—	2	4	no	—
19	C₂³×C₅₃	—	2	2	no	—
20	C₂³×C₅₃	—	4	2	no	—
21	C₂³×C₅₃	C₄²	1	1	yes	characteristic
22	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₂³×C₅₃	C₄⋊C₄	1	1	yes	characteristic
24	C₂×C₄×C₅₃	—	2	4	no	—
25	C₂×C₄×C₅₃	—	4	2	no	—
26	C₂×C₄×C₅₃	—	4	1	no	—
27	C₂×C₄×C₅₃	—	4	1	no	—
28	C₂×C₄×C₅₃	—	4	4	no	—
29	C₂×C₄×C₅₃	—	4	2	no	—
30	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
31	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
32	#88322	—	4	4	no	—
33	C₂²×C₅₃	—	2	2	no	—
34	C₂²×C₅₃	—	2	1	no	—
35	C₂²×C₅₃	—	2	1	no	—
36	C₂²×C₅₃	—	2	2	no	—
37	C₂²×C₅₃	—	2	2	no	—
38	C₂²×C₅₃	—	4	4	no	—
39	C₂²×C₅₃	#95	1	1	yes	characteristic
40	C₂²×C₅₃	#99	1	2	yes	—
41	C₄×C₅₃	—	2	2	no	—
42	C₄×C₅₃	—	2	2	no	—
43	C₄×C₅₃	—	8	2	no	—
44	#1306	—	53	1	no	maximal, Sylow
45	C₂×C₅₃	—	2	2	no	—
46	C₂×C₅₃	—	4	1	no	—
47	C₂×C₅₃	—	4	1	no	—
48	C₂×C₅₃	#342	1	1	yes	characteristic
49	C₂×C₅₃	#344	1	2	yes	—
50	#407	—	53	2	no	—
51	#381	D₅₃	1	1	yes	characteristic
52	C₅₃	#1306	1	1	yes	characteristic, complemented, Sylow
53	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
54	C₂×M₁₆	—	106	2	no	—
55	C₂×C₂²⋊C₄	Dic₅₃	1	2	yes	—
56	C₂²⋊C₈	—	53	4	no	—
57	C₂²⋊C₈	—	106	2	no	—
58	#95	—	2	2	no	—
59	C₂⁴	C₂×Dic₅₃	1	1	yes	characteristic
60	C₂²×C₄	—	2	2	no	—
61	C₂²×C₄	—	4	2	no	—
62	C₂²×C₄	Dic₁₀₆	1	1	yes	characteristic
63	C₂²×C₄	C₄×D₅₃	1	2	yes	—
64	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
65	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
66	M₁₆	—	212	4	no	—
67	C₂×C₈	—	106	4	no	—
68	C₂²⋊C₄	—	4	2	no	—
69	C₂³	—	2	2	no	—
70	C₂³	—	4	2	no	—
71	C₂³	C₄×Dic₅₃	1	1	yes	characteristic
72	C₂³	C₄⋊Dic₅₃	1	1	yes	characteristic
73	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	—	4	1	no	—
76	C₂×C₄	—	4	2	no	—
77	C₂×C₄	—	4	1	no	—
78	C₂×C₄	—	4	4	no	—
79	C₂×C₄	—	4	2	no	—
80	C₂×C₄	#11719919	1	2	yes	—
81	C₂×C₄	#11719921	1	2	yes	—
82	C₈	—	212	4	no	—
83	C₂²	—	2	2	no	—
84	C₂²	—	2	1	no	—
85	C₂²	—	2	1	no	—
86	C₂²	—	2	2	no	—
87	C₂²	—	2	2	no	—
88	C₂²	—	4	4	no	—
89	C₂²	#13133124	1	1	yes	characteristic
90	C₂²	#13133132	1	2	yes	—
91	C₄	—	2	2	no	—
92	C₄	—	2	2	no	—
93	C₄	—	8	2	no	—
94	C₂	—	2	2	no	—
95	C₂	—	4	1	no	—
96	C₂	—	4	1	no	—
97	C₂	#16810390	1	1	yes	characteristic
98	C₂	#16810393	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16810587

C₂

yes

maximal

C₅₃×#381

C₂

yes

maximal, characteristic

C₂×#11719917

—

C₂²⋊#88322

—

C₂²⋊#88322

C₄

yes

complemented

C₂³×C₄×C₅₃

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₅₃

C₄

yes

—

#95×C₅₃

—

C₂⁴×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

Q₈

yes

characteristic

C₂²⋊C₄×C₅₃

—

#11719917

—

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₄²

yes

characteristic

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

C₂³×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#95

yes

characteristic

C₂²×C₅₃

#99

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

#1306

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#342

yes

characteristic

C₂×C₅₃

#344

yes

—

#407

—

#381

D₅₃

yes

characteristic

C₅₃

#1306

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₅₃

yes

characteristic

C₂×M₁₆

—

106

—

C₂×C₂²⋊C₄

Dic₅₃

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

106

—

#95

—

C₂⁴

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₁₀₆

yes

characteristic

C₂²×C₄

C₄×D₅₃

yes

—

C₂²×C₄

D₂₁₂

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

—

M₁₆

—

212

—

C₂×C₈

—

106

—

C₂²⋊C₄

—

C₂³

—

C₂³

—

C₂³

C₄×Dic₅₃

yes

characteristic

C₂³

C₄⋊Dic₅₃

yes

characteristic

C₂³

C₂²⋊Dic₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#11719919

yes

—

C₂×C₄

#11719921

yes

—

C₈

—

212

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#13133124

yes

characteristic

C₂²

#13133132

yes

—

C₄

—

C₄

—

C₄

—

C₂

—

C₂

—

C₂

—

C₂

#16810390

yes

characteristic

C₂

#16810393

yes

—

	subgroup	G/H
center	C₂²	#13133124
commutator	C₂³×C₅₃	C₄²
Frattini	C₂³×C₄	C₂×D₅₃
Fitting	C₅₃×#381	C₂
socle	C₂²×C₅₃	#95
hypercenter	#381	D₅₃
hypocenter	C₅₃	#1306
LCS	G ▷ C₂³×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #381

subgroup

G/H

center

C₂²

#13133124

commutator

C₂³×C₅₃

C₄²

Frattini

C₂³×C₄

C₂×D₅₃

Fitting

C₅₃×#381

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #381

N	H ≅ G/N	type
5. C₂²⋊#88322	C₄	semidirect
52. C₅₃	#1306	semidirect

H ≅ G/N

type

5. C₂²⋊#88322

C₄

semidirect

52. C₅₃

#1306

semidirect

Subgroups — #20272623

Conjugacy classes of subgroups

Complements