Group #20289669 — subgroups

Subgroups — #20289669

Order 6784.

110 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 110 conjugacy classes of subgroups — too many to draw. Only its 40 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	#16811509	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811486	C₂	1	1	yes	maximal, characteristic, complemented
3	D₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719910	—	2	1	no	—
5	C₄⋊#88324	—	2	1	no	—
6	D₄×Dic₅₃	—	2	1	no	—
7	#13133151	C₄	1	1	yes	characteristic
8	#13133138	—	2	1	no	—
9	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
10	C₂×D₈×C₅₃	C₄	1	1	yes	characteristic
11	C₂×D₄×C₅₃	—	2	1	no	—
12	C₂²×Dic₅₃	—	4	1	no	—
13	C₂×#88324	—	4	1	no	—
14	D₈×C₅₃	—	2	1	no	—
15	D₈×C₅₃	C₈	1	2	yes	complemented
16	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂²⋊Dic₅₃	—	4	1	no	—
18	C₄⋊Dic₅₃	—	2	1	no	—
19	C₄×Dic₅₃	D₄	1	1	yes	characteristic
20	C₂×#88322	D₄	1	1	yes	characteristic
21	#11719910	—	2	2	no	—
22	C₂³×C₅₃	—	4	1	no	—
23	D₄×C₅₃	—	2	2	no	—
24	D₄×C₅₃	—	4	1	no	—
25	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×Dic₅₃	—	4	1	no	—
27	C₂×Dic₅₃	—	8	1	no	—
28	C₂×Dic₅₃	—	8	1	no	—
29	C₂×Dic₅₃	D₈	1	1	yes	characteristic
30	C₂×Dic₅₃	SD₁₆	1	1	yes	characteristic
31	#88324	—	8	1	no	—
32	C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
33	C₈×C₅₃	M₁₆	1	1	yes	characteristic
34	#88322	—	2	1	no	—
35	C₂²×C₅₃	—	4	2	no	—
36	C₂²×C₅₃	—	8	1	no	—
37	C₂²×C₅₃	—	8	1	no	—
38	C₂²×C₅₃	#102	1	1	yes	characteristic
39	C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
40	C₄×C₅₃	#104	1	1	yes	characteristic
41	Dic₅₃	—	2	1	no	—
42	Dic₅₃	—	8	1	no	—
43	Dic₅₃	D₁₆	1	1	yes	characteristic
44	Dic₅₃	SD₃₂	1	1	yes	characteristic
45	#1331	—	53	1	no	maximal, Sylow
46	C₂×C₅₃	—	8	2	no	—
47	C₂×C₅₃	D₄⋊C₈	1	1	yes	characteristic
48	C₂×C₅₃	#357	1	1	yes	characteristic
49	C₂×C₅₃	#359	1	1	yes	characteristic
50	C₄×D₈	—	53	1	no	—
51	C₄×C₁₆	—	53	1	no	—
52	#335	—	53	1	no	—
53	C₅₃	#1331	1	1	yes	characteristic, complemented, Sylow
54	C₂×D₈	#6157	1	1	yes	characteristic
55	C₄×D₄	—	106	1	no	—
56	C₂×C₁₆	—	106	1	no	—
57	#107	—	53	1	no	—
58	C₄⋊C₈	—	106	1	no	—
59	#102	—	106	1	no	—
60	C₄×C₈	—	53	1	no	—
61	C₂×D₄	—	2	1	no	—
62	C₂²×C₄	—	212	1	no	—
63	D₈	—	2	1	no	—
64	D₈	#88324	1	2	yes	complemented
65	C₂×C₈	—	53	1	no	—
66	C₂×C₈	—	212	1	no	—
67	C₂×C₈	C₂×#6157	1	1	yes	characteristic
68	C₄⋊C₄	—	106	1	no	—
69	C₂²⋊C₄	—	212	1	no	—
70	C₄²	—	53	1	no	—
71	C₁₆	—	106	2	no	—
72	C₂³	—	4	1	no	—
73	D₄	—	2	2	no	—
74	D₄	—	4	1	no	—
75	C₂×C₄	—	53	1	no	—
76	C₂×C₄	—	53	1	no	—
77	C₂×C₄	—	424	1	no	—
78	C₂×C₄	—	424	1	no	—
79	C₂×C₄	—	212	1	no	—
80	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
81	C₈	—	106	1	no	—
82	C₈	—	424	1	no	—
83	C₈	C₂×#88324	1	1	yes	characteristic
84	C₈	#11719940	1	1	yes	characteristic
85	C₂²	—	4	2	no	—
86	C₂²	—	8	1	no	—
87	C₂²	—	8	1	no	—
88	C₂²	#13133279	1	1	yes	characteristic
89	C₄	—	53	1	no	—
90	C₄	—	53	1	no	—
91	C₄	—	106	1	no	—
92	C₄	—	424	1	no	—
93	C₄	C₂²⋊#88324	1	1	yes	characteristic
94	C₄	#13133281	1	1	yes	characteristic
95	C₂	—	8	2	no	—
96	C₂	D₄⋊#88324	1	1	yes	characteristic
97	C₂	#16811535	1	1	yes	characteristic
98	C₂	#16811537	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811509

C₂

yes

maximal, characteristic, complemented

#16811486

C₂

yes

maximal, characteristic, complemented

D₈×Dic₅₃

C₂

yes

maximal, characteristic

C₂×#11719910

—

C₄⋊#88324

—

D₄×Dic₅₃

—

#13133151

C₄

yes

characteristic

#13133138

—

C₈×Dic₅₃

C₂²

yes

characteristic

C₂×D₈×C₅₃

C₄

yes

characteristic

C₂×D₄×C₅₃

—

C₂²×Dic₅₃

—

C₂×#88324

—

D₈×C₅₃

—

D₈×C₅₃

C₈

yes

complemented

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

D₄

yes

characteristic

C₂×#88322

D₄

yes

characteristic

#11719910

—

C₂³×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

D₈

yes

characteristic

C₂×Dic₅₃

SD₁₆

yes

characteristic

#88324

—

C₈×C₅₃

C₂×C₈

yes

characteristic

C₈×C₅₃

M₁₆

yes

characteristic

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#102

yes

characteristic

C₄×C₅₃

C₂²⋊C₈

yes

characteristic

C₄×C₅₃

#104

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

D₁₆

yes

characteristic

Dic₅₃

SD₃₂

yes

characteristic

#1331

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

D₄⋊C₈

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₄×D₈

—

C₄×C₁₆

—

#335

—

C₅₃

#1331

yes

characteristic, complemented, Sylow

C₂×D₈

#6157

yes

characteristic

C₄×D₄

—

106

—

C₂×C₁₆

—

106

—

#107

—

C₄⋊C₈

—

106

—

#102

—

106

—

C₄×C₈

—

C₂×D₄

—

C₂²×C₄

—

212

—

D₈

—

D₈

#88324

yes

complemented

C₂×C₈

—

C₂×C₈

—

212

—

C₂×C₈

C₂×#6157

yes

characteristic

C₄⋊C₄

—

106

—

C₂²⋊C₄

—

212

—

C₄²

—

C₁₆

—

106

—

C₂³

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

424

—

C₂×C₄

—

424

—

C₂×C₄

—

212

—

C₂×C₄

C₂²⋊#6157

yes

characteristic

C₈

—

106

—

C₈

—

424

—

C₈

yes

C₈

yes

C₂²

—

C₂²

—

C₂²

—

C₂²

#13133279

yes

characteristic

C₄

—

C₄

—

C₄

—

106

—

C₄

—

424

—

C₄

yes

C₄

yes

C₂

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂²	#13133279
commutator	C₈×C₅₃	C₂×C₈
Frattini	C₂×C₈	C₂×#6157
Fitting	C₂×D₈×C₅₃	C₄
socle	C₂²×C₅₃	#102
hypercenter	C₂×D₈	#6157
hypocenter	C₅₃	#1331
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₈

subgroup

G/H

center

C₂²

#13133279

commutator

C₈×C₅₃

C₂×C₈

Frattini

C₂×C₈

C₂×#6157

Fitting

C₂×D₈×C₅₃

C₄

C₂²×C₅₃

C₂×D₈

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₈

N	H ≅ G/N	type
1. #16811509	C₂	semidirect
2. #16811486	C₂	semidirect
15. D₈×C₅₃	C₈	semidirect
53. C₅₃	#1331	semidirect
64. D₈	#88324	semidirect

H ≅ G/N

type

1. #16811509

C₂

semidirect

2. #16811486

C₂

semidirect

15. D₈×C₅₃

C₈

semidirect

53. C₅₃

#1331

semidirect

64. D₈

#88324

semidirect

Subgroups — #20289669

Conjugacy classes of subgroups

Complements