Group #20290651 — subgroups

Subgroups — #20290651

Order 6784.

280 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 114 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×#13133286	C₂	1	4	yes	maximal
2	#16811569	C₂	1	2	yes	maximal
3	C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#6157	—	2	4	no	—
5	C₂×C₄⋊#6157	C₂²	1	4	yes	—
6	#13133286	—	2	8	no	—
7	C₂×C₈×D₅₃	C₂²	1	2	yes	—
8	C₄²×D₅₃	C₂²	1	1	yes	characteristic
9	C₄×#88322	C₄	1	1	yes	characteristic, complemented
10	C₈×Dic₅₃	C₄	1	2	yes	complemented
11	C₄×C₈×C₅₃	C₄	1	1	yes	characteristic, complemented
12	C₂²×#6157	—	2	4	no	—
13	C₂×C₄×D₅₃	D₄	1	2	yes	—
14	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
15	C₄⋊#6157	—	2	8	no	—
16	C₄⋊#6157	—	4	8	no	—
17	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
18	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
20	C₄×Dic₅₃	Q₈	1	2	yes	—
21	C₂×#88322	C₂×C₄	1	2	yes	—
22	C₈×D₅₃	D₄	1	4	yes	—
23	C₈×D₅₃	Q₈	1	4	yes	—
24	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
25	C₂×#6157	—	4	4	no	—
26	C₂×#6157	—	4	8	no	—
27	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
28	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
29	C₂×Dic₅₃	C₄⋊C₄	1	2	yes	—
30	C₂×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
31	C₄×D₅₃	D₈	1	4	yes	—
32	C₄×D₅₃	Q₁₆	1	4	yes	—
33	C₄×D₅₃	C₂×D₄	1	2	yes	—
34	C₄×D₅₃	C₂×Q₈	1	2	yes	—
35	C₈×C₅₃	C₄⋊C₄	1	4	yes	complemented
36	#88322	C₄⋊C₄	1	4	yes	complemented
37	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂×D₅₃	#127	1	1	yes	characteristic
39	C₂×D₅₃	C₄⋊Q₈	1	1	yes	characteristic
40	C₂×D₅₃	C₂×D₈	1	2	yes	—
41	C₂×D₅₃	C₂×Q₁₆	1	2	yes	—
42	#6157	—	8	4	no	—
43	C₄×C₅₃	#107	1	4	yes	complemented
44	C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
45	Dic₅₃	#107	1	4	yes	complemented
46	Dic₅₃	C₄⋊Q₈	1	2	yes	—
47	#1845	—	53	1	no	maximal, Sylow
48	C₂×C₅₃	#384	1	1	yes	characteristic
49	C₂×C₅₃	C₂×#107	1	2	yes	—
50	D₅₃	#493	1	1	yes	characteristic
51	D₅₃	#494	1	1	yes	characteristic
52	D₅₃	#500	1	2	yes	—
53	C₂×#107	—	53	4	no	—
54	C₂×C₄×C₈	—	53	1	no	—
55	#384	—	53	2	no	—
56	C₅₃	#1845	1	1	yes	characteristic, complemented, Sylow
57	C₂²×C₈	—	53	2	no	—
58	C₂×C₄⋊C₄	—	53	4	no	—
59	C₂×C₄⋊C₄	—	106	4	no	—
60	C₂×C₄²	—	53	1	no	—
61	#107	—	106	8	no	—
62	C₄×C₈	—	53	2	no	—
63	C₄×C₈	—	53	1	no	—
64	C₄×C₈	#6157	1	1	yes	characteristic, complemented
65	C₂²×C₄	—	53	2	no	—
66	C₂²×C₄	—	53	1	no	—
67	C₂²×C₄	—	106	4	no	—
68	C₂×C₈	—	53	2	no	—
69	C₂×C₈	—	53	4	no	—
70	C₂×C₈	—	53	4	no	—
71	C₂×C₈	C₂×#6157	1	2	yes	—
72	C₄⋊C₄	—	106	8	no	—
73	C₄⋊C₄	—	212	8	no	—
74	C₄²	—	53	2	no	—
75	C₄²	—	53	1	no	—
76	C₄²	C₂×#6157	1	1	yes	characteristic
77	C₂³	—	53	1	no	—
78	C₂×C₄	—	53	2	no	—
79	C₂×C₄	—	53	1	no	—
80	C₂×C₄	—	53	4	no	—
81	C₂×C₄	—	53	4	no	—
82	C₂×C₄	—	53	2	no	—
83	C₂×C₄	—	53	2	no	—
84	C₂×C₄	—	212	4	no	—
85	C₂×C₄	—	212	8	no	—
86	C₂×C₄	C₄⋊#6157	1	2	yes	—
87	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
88	C₈	—	53	4	no	—
89	C₈	C₄⋊#6157	1	4	yes	complemented
90	C₂²	—	53	1	no	—
91	C₂²	—	53	1	no	—
92	C₂²	—	53	2	no	—
93	C₂²	—	53	2	no	—
94	C₂²	C₂×C₄⋊#6157	1	1	yes	characteristic
95	C₄	—	53	2	no	—
96	C₄	—	53	4	no	—
97	C₄	—	424	4	no	—
98	C₄	#13133286	1	4	yes	complemented
99	C₄	C₂×C₄⋊#6157	1	2	yes	—
100	C₂	—	53	1	no	—
101	C₂	—	53	1	no	—
102	C₂	—	53	2	no	—
103	C₂	#16811569	1	1	yes	characteristic
104	C₂	C₂×#13133286	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133286

C₂

yes

maximal

#16811569

C₂

yes

maximal

C₄×C₈×D₅₃

C₂

yes

maximal, characteristic

C₂×C₄⋊#6157

—

C₂×C₄⋊#6157

C₂²

yes

—

#13133286

—

C₂×C₈×D₅₃

C₂²

yes

—

C₄²×D₅₃

C₂²

yes

characteristic

C₄×#88322

C₄

yes

characteristic, complemented

C₈×Dic₅₃

C₄

yes

complemented

C₄×C₈×C₅₃

C₄

yes

characteristic, complemented

C₂²×#6157

—

C₂×C₄×D₅₃

D₄

yes

—

C₂×C₄×D₅₃

C₂³

yes

characteristic

C₄⋊#6157

—

C₄⋊#6157

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

Q₈

yes

—

C₂×#88322

C₂×C₄

yes

—

C₈×D₅₃

D₄

yes

—

C₈×D₅₃

Q₈

yes

—

C₂²×D₅₃

C₂×D₄

yes

characteristic

C₂×#6157

—

C₂×#6157

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₄⋊C₄

yes

—

C₂×Dic₅₃

C₂×Q₈

yes

characteristic

C₄×D₅₃

D₈

yes

—

C₄×D₅₃

Q₁₆

yes

—

C₄×D₅₃

C₂×D₄

yes

—

C₄×D₅₃

C₂×Q₈

yes

—

C₈×C₅₃

C₄⋊C₄

yes

complemented

#88322

C₄⋊C₄

yes

complemented

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×D₅₃

#127

yes

characteristic

C₂×D₅₃

C₄⋊Q₈

yes

characteristic

C₂×D₅₃

C₂×D₈

yes

—

C₂×D₅₃

C₂×Q₁₆

yes

—

#6157

—

C₄×C₅₃

#107

yes

complemented

C₄×C₅₃

C₂×C₄⋊C₄

yes

—

Dic₅₃

#107

yes

complemented

Dic₅₃

C₄⋊Q₈

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

yes

D₅₃

yes

D₅₃

yes

—

C₂×#107

—

C₂×C₄×C₈

—

#384

—

C₅₃

#1845

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

—

#107

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#6157

yes

characteristic, complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#6157

yes

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₄²

—

C₄²

—

C₄²

C₂×#6157

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

C₄⋊#6157

yes

—

C₂×C₄

C₂²×#6157

yes

characteristic

C₈

—

C₈

C₄⋊#6157

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂×C₄⋊#6157

yes

characteristic

C₄

—

C₄

—

C₄

—

424

—

C₄

#13133286

yes

complemented

C₄

C₂×C₄⋊#6157

yes

—

100

C₂

—

101

C₂

—

102

C₂

—

103

C₂

#16811569

yes

characteristic

104

C₂

C₂×#13133286

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	C₄×C₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	C₄×C₈	#6157
hypocenter	C₅₃	#1845
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂²

C₂×C₄⋊#6157

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6157

Fitting

C₄×C₈×C₅₃

C₄

socle

C₂²×C₅₃

C₂×C₄⋊C₄

C₄×C₈

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
9. C₄×#88322	C₄	semidirect
10. C₈×Dic₅₃	C₄	semidirect
11. C₄×C₈×C₅₃	C₄	semidirect
35. C₈×C₅₃	C₄⋊C₄	semidirect
36. #88322	C₄⋊C₄	semidirect
43. C₄×C₅₃	#107	semidirect
45. Dic₅₃	#107	semidirect
56. C₅₃	#1845	semidirect
64. C₄×C₈	#6157	semidirect
89. C₈	C₄⋊#6157	semidirect
98. C₄	#13133286	semidirect

H ≅ G/N

type

9. C₄×#88322

C₄

semidirect

10. C₈×Dic₅₃

C₄

semidirect

11. C₄×C₈×C₅₃

C₄

semidirect

35. C₈×C₅₃

C₄⋊C₄

semidirect

36. #88322

C₄⋊C₄

semidirect

43. C₄×C₅₃

#107

semidirect

45. Dic₅₃

#107

semidirect

56. C₅₃

#1845

semidirect

64. C₄×C₈

#6157

semidirect

89. C₈

C₄⋊#6157

semidirect

98. C₄

#13133286

semidirect

Subgroups — #20290651

Conjugacy classes of subgroups

Complements