Group #20276597 — subgroups

Subgroups — #20276597

Order 6784.

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

normal subgroups only

This group has 146 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	#16811055	C₂	1	1	yes	maximal, characteristic
2	#16810741	C₂	1	2	yes	maximal, complemented
3	#16810453	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810431	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#13133239	C₂²	1	2	yes	—
7	#13133223	—	2	2	no	—
8	Dic₄₂₄	C₄	1	1	yes	characteristic, complemented
9	#13133162	C₄	1	2	yes	complemented
10	D₈₄₈	C₄	1	1	yes	characteristic, complemented
11	#13133153	C₂²	1	2	yes	—
12	#13133145	—	2	2	no	—
13	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	#11719945	—	2	2	no	—
17	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
18	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
19	C₄²×C₅₃	D₄	1	1	yes	characteristic
20	#11719917	—	2	2	no	—
21	Dic₂₁₂	C₂×C₄	1	2	yes	—
22	D₄₂₄	C₂×C₄	1	2	yes	—
23	#11719913	—	2	2	no	—
24	#11719912	—	4	2	no	—
25	C₈×D₅₃	—	4	2	no	—
26	C₁₆×C₅₃	—	2	1	no	—
27	C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×C₅₃	—	2	1	no	—
30	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
31	C₂²⋊D₅₃	—	4	2	no	—
32	D₂₁₂	—	2	2	no	—
33	C₄×D₅₃	—	4	2	no	—
34	Dic₁₀₆	—	2	2	no	—
35	C₈×C₅₃	D₈	1	1	yes	characteristic
36	C₈×C₅₃	D₈	1	1	yes	characteristic
37	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₈×C₅₃	#41	1	1	yes	characteristic
39	#88322	—	4	2	no	—
40	C₂²×C₅₃	#135	1	1	yes	characteristic
41	C₂×D₅₃	—	4	2	no	—
42	C₄×C₅₃	—	2	2	no	—
43	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
44	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
45	Dic₅₃	—	4	2	no	—
46	#2178	—	53	1	no	maximal, Sylow
47	C₂×C₅₃	—	2	1	no	—
48	C₂×C₅₃	C₄×D₈	1	1	yes	characteristic
49	D₅₃	—	8	2	no	—
50	#508	—	53	1	no	—
51	#443	—	53	2	no	—
52	#368	—	53	1	no	—
53	#359	—	53	2	no	—
54	C₄×C₁₆	D₅₃	1	1	yes	characteristic, complemented
55	C₅₃	#2178	1	1	yes	characteristic, complemented, Sylow
56	#135	—	53	2	no	—
57	#131	—	106	2	no	—
58	Q₃₂	—	53	1	no	—
59	SD₃₂	—	53	2	no	—
60	D₁₆	—	53	1	no	—
61	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
62	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
63	#108	—	53	2	no	—
64	#104	—	106	2	no	—
65	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
66	#41	—	106	2	no	—
67	Q₁₆	—	53	2	no	—
68	SD₁₆	—	106	2	no	—
69	D₈	—	53	2	no	—
70	M₁₆	—	106	2	no	—
71	M₁₆	—	212	2	no	—
72	C₂×C₈	—	212	2	no	—
73	C₂×C₈	D₂₁₂	1	1	yes	characteristic
74	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
75	C₄²	D₂₁₂	1	1	yes	characteristic
76	C₁₆	—	2	1	no	—
77	C₁₆	C₄×D₅₃	1	1	yes	characteristic
78	C₁₆	C₄×D₅₃	1	1	yes	characteristic
79	Q₈	—	106	2	no	—
80	D₄	—	106	2	no	—
81	D₄	—	212	2	no	—
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	212	2	no	—
84	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
85	C₈	—	212	2	no	—
86	C₈	D₄₂₄	1	1	yes	characteristic
87	C₈	D₄₂₄	1	1	yes	characteristic
88	C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
89	C₈	#11719945	1	1	yes	characteristic
90	C₂²	—	212	2	no	—
91	C₂²	#13133239	1	1	yes	characteristic
92	C₄	—	2	2	no	—
93	C₄	—	212	2	no	—
94	C₄	C₄×D₂₁₂	1	1	yes	characteristic
95	C₄	C₂×D₄₂₄	1	1	yes	characteristic
96	C₂	—	2	1	no	—
97	C₂	—	424	2	no	—
98	C₂	C₄×D₄₂₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811055

C₂

yes

maximal, characteristic

#16810741

C₂

yes

maximal, complemented

#16810453

C₂

yes

maximal, characteristic, complemented

#16810431

C₂

yes

maximal, complemented

C₄×C₁₆×C₅₃

C₂

yes

maximal, characteristic, complemented

#13133239

C₂²

yes

—

#13133223

—

Dic₄₂₄

C₄

yes

characteristic, complemented

#13133162

C₄

yes

complemented

D₈₄₈

C₄

yes

characteristic, complemented

#13133153

C₂²

yes

—

#13133145

—

C₂×C₁₆×C₅₃

C₂²

yes

characteristic

C₂×C₁₆×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

#11719945

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

#11719917

—

Dic₂₁₂

C₂×C₄

yes

—

D₄₂₄

C₂×C₄

yes

—

#11719913

—

#11719912

—

C₈×D₅₃

—

C₁₆×C₅₃

—

C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²⋊D₅₃

—

D₂₁₂

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

D₈

yes

characteristic

C₈×C₅₃

D₈

yes

characteristic

C₈×C₅₃

C₂²×C₄

yes

C₈×C₅₃

yes

—

C₂²×C₅₃

#135

yes

characteristic

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×D₄

yes

characteristic

C₄×C₅₃

C₂×D₈

yes

characteristic

Dic₅₃

—

#2178

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₄×D₈

yes

characteristic

D₅₃

—

—

—

—

—

C₄×C₁₆

D₅₃

yes

characteristic, complemented

C₅₃

#2178

yes

characteristic, complemented, Sylow

#135

—

#131

—

106

—

Q₃₂

—

SD₃₂

—

D₁₆

—

C₂×C₁₆

C₂×D₅₃

yes

characteristic

C₂×C₁₆

C₂×D₅₃

yes

characteristic

#108

—

#104

—

106

—

C₄×C₈

C₂×D₅₃

yes

characteristic

#41

—

106

—

Q₁₆

—

SD₁₆

—

106

—

D₈

—

M₁₆

—

106

—

M₁₆

—

212

—

C₂×C₈

—

212

—

C₂×C₈

D₂₁₂

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄²

D₂₁₂

yes

characteristic

C₁₆

—

C₁₆

C₄×D₅₃

yes

characteristic

C₁₆

C₄×D₅₃

yes

characteristic

Q₈

—

106

—

D₄

—

106

—

D₄

—

212

—

C₂×C₄

—

C₂×C₄

—

212

—

C₂×C₄

C₂×D₂₁₂

yes

characteristic

C₈

—

212

—

C₈

D₄₂₄

yes

characteristic

C₈

D₄₂₄

yes

characteristic

C₈

C₂×C₄×D₅₃

yes

characteristic

C₈

#11719945

yes

characteristic

C₂²

—

212

—

C₂²

#13133239

yes

characteristic

C₄

—

C₄

—

212

—

C₄

C₄×D₂₁₂

yes

characteristic

C₄

C₂×D₄₂₄

yes

characteristic

C₂

—

C₂

—

424

—

C₂

C₄×D₄₂₄

yes

characteristic

	subgroup	G/H
center	C₈	D₄₂₄
commutator	C₈×C₅₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₄×C₁₆×C₅₃	C₂
socle	C₂×C₅₃	C₄×D₈
hypercenter	C₄×C₁₆	D₅₃
hypocenter	C₅₃	#2178
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄×C₁₆

subgroup

G/H

center

C₈

D₄₂₄

commutator

C₈×C₅₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₄×C₁₆×C₅₃

C₂

socle

C₂×C₅₃

C₄×D₈

hypercenter

C₄×C₁₆

D₅₃

hypocenter

C₅₃

#2178

LCS

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄×C₁₆

N	H ≅ G/N	type
2. #16810741	C₂	semidirect
3. #16810453	C₂	semidirect
4. #16810431	C₂	semidirect
5. C₄×C₁₆×C₅₃	C₂	semidirect
8. Dic₄₂₄	C₄	semidirect
9. #13133162	C₄	semidirect
10. D₈₄₈	C₄	semidirect
54. C₄×C₁₆	D₅₃	semidirect
55. C₅₃	#2178	semidirect

H ≅ G/N

type

2. #16810741

C₂

semidirect

3. #16810453

C₂

semidirect

4. #16810431

C₂

semidirect

5. C₄×C₁₆×C₅₃

C₂

semidirect

8. Dic₄₂₄

C₄

semidirect

9. #13133162

C₄

semidirect

10. D₈₄₈

C₄

semidirect

54. C₄×C₁₆

D₅₃

semidirect

55. C₅₃

#2178

semidirect

Subgroups — #20276597

Conjugacy classes of subgroups

Complements