Group #20272550 — subgroups

Subgroups — #20272550

Order 6784.

180 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 113 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×C₄⋊#88322	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×#88322	—	2	2	no	—
4	C₄⋊#88322	—	2	2	no	—
5	C₄⋊#88322	C₄	1	4	yes	—
6	C₂²×C₈×C₅₃	C₄	1	2	yes	—
7	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
8	C₄×C₈×C₅₃	—	2	2	no	—
9	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
10	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
11	C₂²×C₄×C₅₃	Q₈	1	1	yes	characteristic
12	C₂×C₈×C₅₃	—	2	2	no	—
13	C₂×C₈×C₅₃	—	2	2	no	—
14	C₂×C₈×C₅₃	C₈	1	4	yes	complemented
15	C₄²×C₅₃	C₂×C₄	1	2	yes	—
16	C₄²×C₅₃	D₄	1	2	yes	—
17	C₂×#88322	—	2	4	no	—
18	C₂×#88322	—	4	2	no	—
19	C₂×#88322	—	4	2	no	—
20	C₂³×C₅₃	C₄⋊C₄	1	1	yes	characteristic
21	C₂×C₄×C₅₃	—	2	2	no	—
22	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
23	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
24	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
26	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
27	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
28	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
29	C₂×C₄×C₅₃	SD₁₆	1	2	yes	—
30	C₂×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
31	C₈×C₅₃	—	2	4	no	—
32	#88322	—	4	4	no	—
33	C₂²×C₅₃	#95	1	1	yes	characteristic
34	C₂²×C₅₃	C₄⋊C₈	1	2	yes	—
35	C₂²×C₅₃	#106	1	1	yes	characteristic
36	C₂²×C₅₃	#107	1	1	yes	characteristic
37	C₂²×C₅₃	#108	1	2	yes	—
38	C₄×C₅₃	—	2	2	no	—
39	C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
40	C₄×C₅₃	#97	1	1	yes	characteristic
41	C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
42	C₄×C₅₃	#102	1	2	yes	—
43	C₄×C₅₃	#103	1	2	yes	—
44	#1277	—	53	1	no	maximal, Sylow
45	C₂×C₅₃	#334	1	2	yes	—
46	C₂×C₅₃	#335	1	2	yes	—
47	C₂×C₅₃	#336	1	1	yes	characteristic
48	C₂×C₅₃	#340	1	1	yes	characteristic
49	C₂×C₅₃	#341	1	1	yes	characteristic
50	C₂×C₄⋊C₈	—	53	2	no	—
51	C₂×C₄×C₈	D₅₃	1	1	yes	characteristic
52	C₅₃	#1277	1	1	yes	characteristic, complemented, Sylow
53	C₂²×C₈	—	106	2	no	—
54	C₂²×C₈	Dic₅₃	1	2	yes	—
55	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
56	C₄⋊C₈	—	53	4	no	—
57	C₄⋊C₈	—	106	2	no	—
58	C₄×C₈	—	2	2	no	—
59	C₂²×C₄	Dic₁₀₆	1	1	yes	characteristic
60	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
61	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	106	4	no	—
65	C₂×C₈	—	212	2	no	—
66	C₂×C₈	—	212	2	no	—
67	C₂×C₈	#88322	1	4	yes	complemented
68	C₄²	C₄×D₅₃	1	2	yes	—
69	C₄²	C₂²⋊D₅₃	1	2	yes	—
70	C₂³	C₄⋊Dic₅₃	1	1	yes	characteristic
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	#11719913	1	2	yes	—
73	C₂×C₄	D₄₂₄	1	1	yes	characteristic
74	C₂×C₄	Dic₂₁₂	1	1	yes	characteristic
75	C₂×C₄	C₂×#88322	1	2	yes	—
76	C₂×C₄	#11719917	1	2	yes	—
77	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
78	C₂×C₄	#11719919	1	2	yes	—
79	C₂×C₄	#11719921	1	2	yes	—
80	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
81	C₈	—	2	4	no	—
82	C₈	—	212	4	no	—
83	C₂²	#13133124	1	1	yes	characteristic
84	C₂²	C₄⋊#88322	1	2	yes	—
85	C₂²	#13133149	1	1	yes	characteristic
86	C₂²	#13133151	1	1	yes	characteristic
87	C₂²	#13133153	1	2	yes	—
88	C₄	—	2	2	no	—
89	C₄	C₄×#88322	1	1	yes	characteristic
90	C₄	#13133128	1	1	yes	characteristic
91	C₄	C₂²⋊#88322	1	2	yes	—
92	C₄	#13133139	1	2	yes	—
93	C₄	#13133142	1	2	yes	—
94	C₂	#16810374	1	2	yes	—
95	C₂	#16810376	1	2	yes	—
96	C₂	#16810378	1	1	yes	characteristic
97	C₂	#16810386	1	1	yes	characteristic
98	C₂	#16810388	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#88322

C₂

yes

maximal

C₂×C₄×C₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

—

C₄⋊#88322

—

C₄⋊#88322

C₄

yes

—

C₂²×C₈×C₅₃

C₄

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

Q₈

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₈

yes

complemented

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

D₄

yes

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

M₁₆

yes

—

C₂×C₄×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

SD₁₆

yes

—

C₂×C₄×C₅₃

Q₁₆

yes

characteristic

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

#95

yes

characteristic

C₂²×C₅₃

C₄⋊C₈

yes

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₄×C₈

yes

characteristic

C₄×C₅₃

#97

yes

characteristic

C₄×C₅₃

C₂²⋊C₈

yes

—

C₄×C₅₃

#102

yes

—

C₄×C₅₃

#103

yes

—

#1277

—

maximal, Sylow

C₂×C₅₃

#334

yes

—

C₂×C₅₃

#335

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄⋊C₈

—

C₂×C₄×C₈

D₅₃

yes

characteristic

C₅₃

#1277

yes

characteristic, complemented, Sylow

C₂²×C₈

—

106

—

C₂²×C₈

Dic₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₄×C₈

—

C₂²×C₄

Dic₁₀₆

yes

characteristic

C₂²×C₄

D₂₁₂

yes

characteristic

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

212

—

C₂×C₈

—

212

—

C₂×C₈

#88322

yes

complemented

C₄²

C₄×D₅₃

yes

—

C₄²

C₂²⋊D₅₃

yes

—

C₂³

C₄⋊Dic₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

#11719913

yes

—

C₂×C₄

D₄₂₄

yes

characteristic

C₂×C₄

Dic₂₁₂

yes

characteristic

C₂×C₄

C₂×#88322

yes

—

C₂×C₄

#11719917

yes

—

C₂×C₄

C₄×Dic₅₃

yes

characteristic

C₂×C₄

#11719919

yes

—

C₂×C₄

#11719921

yes

—

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

C₈

—

C₈

—

212

—

C₂²

#13133124

yes

characteristic

C₂²

C₄⋊#88322

yes

—

C₂²

yes

C₂²

yes

C₂²

yes

—

C₄

—

C₄

yes

C₄

yes

C₄

yes

—

C₄

#13133139

yes

—

C₄

#13133142

yes

—

C₂

#16810374

yes

—

C₂

#16810376

yes

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂²×C₄	D₂₁₂
commutator	C₄×C₅₃	C₄×C₈
Frattini	C₂×C₄²	C₂×D₅₃
Fitting	C₂×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₄⋊C₄
hypercenter	C₂×C₄×C₈	D₅₃
hypocenter	C₅₃	#1277
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂²×C₄

D₂₁₂

commutator

C₄×C₅₃

C₄×C₈

Frattini

C₂×C₄²

C₂×D₅₃

Fitting

C₂×C₄×C₈×C₅₃

C₂

socle

C₂³×C₅₃

C₄⋊C₄

hypercenter

C₂×C₄×C₈

D₅₃

hypocenter

C₅₃

#1277

LCS

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
14. C₂×C₈×C₅₃	C₈	semidirect
52. C₅₃	#1277	semidirect
67. C₂×C₈	#88322	semidirect

H ≅ G/N

type

14. C₂×C₈×C₅₃

C₈

semidirect

52. C₅₃

#1277

semidirect

67. C₂×C₈

#88322

semidirect

Subgroups — #20272550

Conjugacy classes of subgroups

Complements