Group #20276497 — subgroups

Subgroups — #20276497

Order 6784.

152 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₂²×#11719908	C₂	1	1	yes	maximal, characteristic
2	#16810661	C₂	1	1	yes	maximal, characteristic
3	#16810450	C₂	1	2	yes	maximal, complemented
4	#16810448	C₂	1	2	yes	maximal, complemented
5	C₅₃×#427	C₂	1	1	yes	maximal, characteristic
6	#13133176	—	2	1	no	—
7	C₂×#11719908	C₄	1	4	yes	complemented
8	C₂×#11719908	C₂²	1	1	yes	characteristic
9	C₂×#11719908	C₂²	1	1	yes	characteristic
10	#13133151	C₂²	1	2	yes	—
11	#13133149	—	2	1	no	—
12	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	#117×C₅₃	—	2	1	no	—
14	#107×C₅₃	C₂²	1	2	yes	—
15	#106×C₅₃	—	2	1	no	—
16	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
17	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
18	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
19	C₂×C₈×C₅₃	Q₈	1	2	yes	—
20	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
21	C₄⋊C₄×C₅₃	—	2	2	no	—
22	C₂²⋊C₄×C₅₃	—	4	1	no	—
23	C₄²×C₅₃	—	4	1	no	—
24	C₂²⋊Dic₅₃	—	4	1	no	—
25	C₄⋊Dic₅₃	—	2	2	no	—
26	C₄×Dic₅₃	—	4	1	no	—
27	#11719908	C₂×C₄	1	2	yes	—
28	#11719908	C₂×C₄	1	2	yes	—
29	C₂³×C₅₃	D₈	1	1	yes	characteristic
30	C₂×C₄×C₅₃	—	4	2	no	—
31	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
32	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
33	C₂×C₄×C₅₃	Q₁₆	1	2	yes	—
34	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂×Dic₅₃	—	4	2	no	—
36	C₈×C₅₃	C₄⋊C₄	1	2	yes	—
37	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₅₃	—	2	2	no	—
40	C₂²×C₅₃	#107	1	2	yes	—
41	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
42	C₄×C₅₃	—	8	2	no	—
43	C₄×C₅₃	#107	1	2	yes	—
44	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
46	Dic₅₃	—	8	2	no	—
47	#2158	—	53	1	no	maximal, Sylow
48	C₂×C₅₃	—	2	2	no	—
49	C₂×C₅₃	C₂×#107	1	1	yes	characteristic
50	C₂×C₅₃	#508	1	2	yes	—
51	C₂²×C₁₆	—	53	1	no	—
52	#427	—	53	1	no	—
53	#427	D₅₃	1	1	yes	characteristic
54	#367	—	53	2	no	—
55	#366	—	53	2	no	—
56	C₅₃	#2158	1	1	yes	characteristic, complemented, Sylow
57	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
58	#117	—	2	1	no	—
59	#117	—	106	1	no	—
60	C₂×C₁₆	—	53	4	no	—
61	C₂×C₁₆	—	53	1	no	—
62	C₂×C₁₆	—	53	1	no	—
63	#107	—	53	2	no	—
64	#107	C₂×D₅₃	1	2	yes	—
65	#106	—	2	1	no	—
66	#106	—	106	1	no	—
67	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
68	C₂×C₈	Dic₁₀₆	1	2	yes	—
69	C₂×C₈	C₄×D₅₃	1	2	yes	—
70	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
71	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
72	C₄⋊C₄	—	2	2	no	—
73	C₄⋊C₄	—	106	2	no	—
74	C₂²⋊C₄	—	4	1	no	—
75	C₂²⋊C₄	—	212	1	no	—
76	C₄²	—	4	1	no	—
77	C₄²	—	212	1	no	—
78	C₁₆	—	53	2	no	—
79	C₁₆	—	53	2	no	—
80	C₂³	#11719922	1	1	yes	characteristic
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	212	2	no	—
83	C₂×C₄	#11719919	1	2	yes	—
84	C₂×C₄	#11719922	1	1	yes	characteristic
85	C₂×C₄	#11719925	1	2	yes	—
86	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
87	C₈	#11719919	1	2	yes	—
88	C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
89	C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
90	C₂²	—	2	2	no	—
91	C₂²	#13133150	1	2	yes	—
92	C₂²	C₂×#11719922	1	1	yes	characteristic
93	C₄	—	8	2	no	—
94	C₄	—	424	2	no	—
95	C₄	#13133150	1	2	yes	—
96	C₄	C₂×#11719919	1	1	yes	characteristic
97	C₄	C₂×#11719925	1	1	yes	characteristic
98	C₂	—	2	2	no	—
99	C₂	C₂×#13133150	1	1	yes	characteristic
100	C₂	#16811053	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11719908

C₂

yes

maximal, characteristic

#16810661

C₂

yes

maximal, characteristic

#16810450

C₂

yes

maximal, complemented

#16810448

C₂

yes

maximal, complemented

C₅₃×#427

C₂

yes

maximal, characteristic

#13133176

—

C₂×#11719908

C₄

yes

complemented

C₂×#11719908

C₂²

yes

characteristic

C₂×#11719908

C₂²

yes

characteristic

#13133151

C₂²

yes

—

#13133149

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

#117×C₅₃

—

#107×C₅₃

C₂²

yes

—

#106×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

Q₈

yes

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

#11719908

C₂×C₄

yes

—

#11719908

C₂×C₄

yes

—

C₂³×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

Q₁₆

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₈×C₅₃

C₄⋊C₄

yes

—

C₈×C₅₃

C₂²×C₄

yes

characteristic

C₈×C₅₃

C₂×Q₈

yes

characteristic

C₂²×C₅₃

—

C₂²×C₅₃

#107

yes

—

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

#107

yes

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₂×Q₁₆

yes

characteristic

Dic₅₃

—

#2158

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×#107

yes

characteristic

C₂×C₅₃

#508

yes

—

C₂²×C₁₆

—

#427

—

#427

D₅₃

yes

characteristic

#367

—

#366

—

C₅₃

#2158

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×D₅₃

yes

characteristic

#117

—

#117

—

106

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

#107

C₂×D₅₃

yes

—

#106

—

#106

—

106

—

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

Dic₁₀₆

yes

—

C₂×C₈

C₄×D₅₃

yes

—

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₂²⋊C₄

—

C₂²⋊C₄

—

212

—

C₄²

—

C₄²

—

212

—

C₁₆

—

C₁₆

—

C₂³

#11719922

yes

characteristic

C₂×C₄

—

C₂×C₄

—

212

—

C₂×C₄

#11719919

yes

—

C₂×C₄

#11719922

yes

characteristic

C₂×C₄

#11719925

yes

—

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

C₈

#11719919

yes

—

C₈

C₂×Dic₁₀₆

yes

characteristic

C₈

C₂×C₄×D₅₃

yes

characteristic

C₂²

—

C₂²

#13133150

yes

—

C₂²

C₂×#11719922

yes

characteristic

C₄

—

C₄

—

424

—

C₄

#13133150

yes

—

C₄

yes

C₄

yes

C₂

—

C₂

C₂×#13133150

yes

characteristic

100

C₂

#16811053

yes

—

	subgroup	G/H
center	C₂×C₄	#11719922
commutator	C₈×C₅₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#427	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	#427	D₅₃
hypocenter	C₅₃	#2158
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

subgroup

G/H

center

C₂×C₄

#11719922

commutator

C₈×C₅₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#427

C₂

socle

C₂²×C₅₃

C₂×D₈

D₅₃

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

N	H ≅ G/N	type
3. #16810450	C₂	semidirect
4. #16810448	C₂	semidirect
7. C₂×#11719908	C₄	semidirect
56. C₅₃	#2158	semidirect

H ≅ G/N

type

3. #16810450

C₂

semidirect

4. #16810448

C₂

semidirect

7. C₂×#11719908

C₄

semidirect

56. C₅₃

#2158

semidirect

Subgroups — #20276497

Conjugacy classes of subgroups

Complements