Group #20276561 — subgroups

Subgroups — #20276561

Order 6784.

174 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 59 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×D₈₄₈	C₂	1	1	yes	maximal, characteristic
2	#16810705	C₂	1	2	yes	maximal, complemented
3	#16810423	C₂	1	2	yes	maximal, complemented
4	C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#366	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄₂₄	C₂²	1	2	yes	—
7	#13133179	—	2	2	no	—
8	D₈₄₈	C₄	1	4	yes	complemented
9	C₂×#11719908	C₂²	1	1	yes	characteristic
10	#13133137	—	2	2	no	—
11	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	#107×C₅₃	C₂²	1	2	yes	—
14	C₂×D₂₁₂	—	2	2	no	—
15	C₂×C₄×D₅₃	—	4	2	no	—
16	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
17	C₄⋊C₄×C₅₃	—	2	2	no	—
18	#11719921	—	4	2	no	—
19	C₄×Dic₅₃	D₄	1	1	yes	characteristic
20	C₂×#88322	D₄	1	1	yes	characteristic
21	D₄₂₄	—	2	2	no	—
22	D₄₂₄	C₂×C₄	1	4	yes	complemented
23	C₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
24	#11719908	—	2	1	no	—
25	C₂²×D₅₃	—	4	2	no	—
26	C₂×C₄×C₅₃	—	4	2	no	—
27	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
28	C₂×Dic₅₃	D₈	1	1	yes	characteristic
29	C₂×Dic₅₃	D₈	1	1	yes	characteristic
30	D₂₁₂	—	2	4	no	—
31	D₂₁₂	—	4	2	no	—
32	C₄×D₅₃	—	8	4	no	—
33	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₈×C₅₃	#41	1	1	yes	characteristic
35	#88322	—	2	1	no	—
36	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
37	C₂×D₅₃	—	4	4	no	—
38	C₂×D₅₃	—	8	2	no	—
39	C₂×D₅₃	—	8	2	no	—
40	C₄×C₅₃	—	8	2	no	—
41	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
42	C₄×C₅₃	#135	1	1	yes	characteristic
43	Dic₅₃	—	2	1	no	—
44	Dic₅₃	D₁₆	1	2	yes	complemented
45	C₄×D₁₆	—	53	1	no	maximal, Sylow
46	C₂×C₅₃	C₄×D₈	1	1	yes	characteristic
47	C₂×C₅₃	C₂×D₁₆	1	1	yes	characteristic
48	C₂×C₅₃	#508	1	1	yes	characteristic
49	D₅₃	—	8	4	no	—
50	C₂×D₁₆	—	53	1	no	—
51	C₄×D₈	—	53	2	no	—
52	#366	D₅₃	1	1	yes	characteristic, complemented
53	#357	—	53	2	no	—
54	C₄×C₁₆	—	53	1	no	—
55	C₅₃	C₄×D₁₆	1	1	yes	characteristic, complemented, Sylow
56	C₂×D₈	—	53	2	no	—
57	C₄×D₄	—	106	2	no	—
58	D₁₆	—	53	4	no	—
59	C₂×C₁₆	—	53	1	no	—
60	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
61	#107	C₂×D₅₃	1	2	yes	—
62	#102	—	106	2	no	—
63	C₄×C₈	—	53	1	no	—
64	C₂×D₄	—	106	2	no	—
65	C₂²×C₄	—	212	2	no	—
66	D₈	—	53	4	no	—
67	D₈	—	106	2	no	—
68	C₂×C₈	—	53	1	no	—
69	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
70	C₄⋊C₄	—	2	2	no	—
71	C₂²⋊C₄	—	212	2	no	—
72	C₄²	—	53	1	no	—
73	C₁₆	—	106	1	no	—
74	C₁₆	C₄×D₅₃	1	2	yes	complemented
75	C₂³	—	212	2	no	—
76	D₄	—	106	4	no	—
77	D₄	—	212	2	no	—
78	C₂×C₄	—	4	2	no	—
79	C₂×C₄	—	53	1	no	—
80	C₂×C₄	—	53	1	no	—
81	C₂×C₄	—	424	4	no	—
82	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
83	C₈	—	106	1	no	—
84	C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
85	C₈	#11719949	1	1	yes	characteristic
86	C₂²	—	424	2	no	—
87	C₂²	—	424	2	no	—
88	C₂²	—	212	4	no	—
89	C₂²	D₈×D₅₃	1	1	yes	characteristic
90	C₄	—	8	2	no	—
91	C₄	—	53	2	no	—
92	C₄	—	106	1	no	—
93	C₄	#13133179	1	1	yes	characteristic
94	C₄	#13133240	1	1	yes	characteristic
95	C₂	—	424	4	no	—
96	C₂	#16810705	1	1	yes	characteristic
97	C₂	D₁₆×D₅₃	1	1	yes	characteristic
98	C₂	#16811056	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₈₄₈

C₂

yes

maximal, characteristic

#16810705

C₂

yes

maximal, complemented

#16810423

C₂

yes

maximal, complemented

C₁₆×Dic₅₃

C₂

yes

maximal, characteristic, complemented

C₅₃×#366

C₂

yes

maximal, characteristic, complemented

C₂×D₄₂₄

C₂²

yes

—

#13133179

—

D₈₄₈

C₄

yes

complemented

C₂×#11719908

C₂²

yes

characteristic

#13133137

—

C₈×Dic₅₃

C₂²

yes

characteristic

C₂×C₁₆×C₅₃

C₂²

yes

characteristic

#107×C₅₃

C₂²

yes

—

C₂×D₂₁₂

—

C₂×C₄×D₅₃

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

#11719921

—

C₄×Dic₅₃

D₄

yes

characteristic

C₂×#88322

D₄

yes

characteristic

D₄₂₄

—

D₄₂₄

C₂×C₄

yes

complemented

C₁₆×C₅₃

C₂×C₄

yes

complemented

#11719908

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

D₈

yes

characteristic

C₂×Dic₅₃

D₈

yes

characteristic

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₈×C₅₃

C₂²×C₄

yes

C₈×C₅₃

yes

—

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×D₄

yes

characteristic

C₄×C₅₃

#135

yes

characteristic

Dic₅₃

—

Dic₅₃

D₁₆

yes

complemented

C₄×D₁₆

—

maximal, Sylow

C₂×C₅₃

C₄×D₈

yes

characteristic

C₂×C₅₃

C₂×D₁₆

yes

characteristic

C₂×C₅₃

#508

yes

characteristic

D₅₃

—

C₂×D₁₆

—

C₄×D₈

—

#366

D₅₃

yes

characteristic, complemented

#357

—

C₄×C₁₆

—

C₅₃

C₄×D₁₆

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₄×D₄

—

106

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₃

yes

characteristic

#107

C₂×D₅₃

yes

—

#102

—

106

—

C₄×C₈

—

C₂×D₄

—

106

—

C₂²×C₄

—

212

—

D₈

—

D₈

—

106

—

C₂×C₈

—

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

212

—

C₄²

—

C₁₆

—

106

—

C₁₆

C₄×D₅₃

yes

complemented

C₂³

—

212

—

D₄

—

106

—

D₄

—

212

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

424

—

C₂×C₄

D₄×D₅₃

yes

characteristic

C₈

—

106

—

C₈

C₂×C₄×D₅₃

yes

characteristic

C₈

#11719949

yes

characteristic

C₂²

—

424

—

C₂²

—

424

—

C₂²

—

212

—

C₂²

D₈×D₅₃

yes

characteristic

C₄

—

C₄

—

C₄

—

106

—

C₄

yes

C₄

yes

C₂

—

424

—

C₂

#16810705

yes

characteristic

C₂

D₁₆×D₅₃

yes

characteristic

C₂

#16811056

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₅₃
commutator	C₈×C₅₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#366	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	#366	D₅₃
hypocenter	C₅₃	C₄×D₁₆
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

subgroup

G/H

center

C₂²

D₈×D₅₃

commutator

C₈×C₅₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#366

C₂

socle

C₂²×C₅₃

C₂×D₈

hypercenter

#366

D₅₃

hypocenter

C₅₃

C₄×D₁₆

LCS

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

N	H ≅ G/N	type
2. #16810705	C₂	semidirect
3. #16810423	C₂	semidirect
4. C₁₆×Dic₅₃	C₂	semidirect
5. C₅₃×#366	C₂	semidirect
8. D₈₄₈	C₄	semidirect
22. D₄₂₄	C₂×C₄	semidirect
23. C₁₆×C₅₃	C₂×C₄	semidirect
44. Dic₅₃	D₁₆	semidirect
52. #366	D₅₃	semidirect
55. C₅₃	C₄×D₁₆	semidirect
74. C₁₆	C₄×D₅₃	semidirect

H ≅ G/N

type

2. #16810705

C₂

semidirect

3. #16810423

C₂

semidirect

4. C₁₆×Dic₅₃

C₂

semidirect

5. C₅₃×#366

C₂

semidirect

8. D₈₄₈

C₄

semidirect

22. D₄₂₄

C₂×C₄

semidirect

23. C₁₆×C₅₃

C₂×C₄

semidirect

44. Dic₅₃

D₁₆

semidirect

52. #366

D₅₃

semidirect

55. C₅₃

C₄×D₁₆

semidirect

74. C₁₆

C₄×D₅₃

semidirect

Subgroups — #20276561

Conjugacy classes of subgroups

Complements