Group #20276225 — subgroups

Subgroups — #20276225

Order 6784.

176 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 69 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	#16810666	C₂	1	2	yes	maximal
2	C₂×#13133149	C₂	1	1	yes	maximal, characteristic
3	#16810391	C₂	1	2	yes	maximal
4	#16810388	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#424	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
8	#13133150	—	2	2	no	—
9	#13133149	—	2	2	no	—
10	#13133148	—	2	2	no	—
11	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	#117×C₅₃	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₅₃	—	2	2	no	—
15	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
16	C₂²×Dic₅₃	—	2	2	no	—
17	M₁₆×C₅₃	—	2	2	no	—
18	C₂×C₈×C₅₃	—	2	2	no	—
19	C₂×C₈×C₅₃	—	2	1	no	—
20	C₂×C₈×C₅₃	D₄	1	2	yes	—
21	C₄⋊C₄×C₅₃	D₄	1	2	yes	—
22	C₂²⋊C₄×C₅₃	—	2	1	no	—
23	C₄²×C₅₃	—	2	1	no	—
24	C₄⋊Dic₅₃	—	2	2	no	—
25	C₄⋊Dic₅₃	—	2	2	no	—
26	#11719917	—	2	2	no	—
27	#11719917	Q₈	1	4	yes	—
28	C₂×#88322	—	2	2	no	—
29	C₂³×C₅₃	#41	1	1	yes	characteristic
30	C₂×C₄×C₅₃	—	2	2	no	—
31	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
34	C₂×C₄×C₅₃	#41	1	2	yes	—
35	C₂×Dic₅₃	—	4	2	no	—
36	C₂×Dic₅₃	—	8	2	no	—
37	C₈×C₅₃	—	2	2	no	—
38	C₈×C₅₃	—	4	1	no	—
39	#88322	—	2	4	no	—
40	C₂²×C₅₃	—	2	2	no	—
41	C₂²×C₅₃	#121	1	1	yes	characteristic
42	C₂²×C₅₃	#125	1	2	yes	—
43	C₄×C₅₃	—	4	2	no	—
44	C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
45	C₄×C₅₃	#123	1	1	yes	characteristic
46	C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
47	Dic₅₃	—	8	2	no	—
48	#2090	—	53	1	no	maximal, Sylow
49	C₂×C₅₃	—	2	2	no	—
50	C₂×C₅₃	#398	1	1	yes	characteristic
51	C₂×C₅₃	#471	1	2	yes	—
52	#428	—	53	2	no	—
53	C₂×#106	—	53	1	no	—
54	#424	D₅₃	1	1	yes	characteristic
55	#343	—	53	2	no	—
56	#341	—	53	1	no	—
57	C₅₃	#2090	1	1	yes	characteristic, complemented, Sylow
58	C₂×M₁₆	—	53	2	no	—
59	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
60	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
61	#117	C₂×D₅₃	1	1	yes	characteristic
62	C₂×C₄⋊C₄	—	53	2	no	—
63	#107	—	106	2	no	—
64	#106	—	106	2	no	—
65	#106	—	106	2	no	—
66	C₄⋊C₈	—	2	2	no	—
67	C₂²×C₄	—	106	2	no	—
68	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
69	M₁₆	—	2	2	no	—
70	M₁₆	—	53	4	no	—
71	M₁₆	—	106	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	106	2	no	—
75	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
76	C₄⋊C₄	—	106	2	no	—
77	C₄⋊C₄	—	106	2	no	—
78	C₄⋊C₄	D₂₁₂	1	2	yes	—
79	C₂²⋊C₄	—	2	1	no	—
80	C₄²	—	2	1	no	—
81	C₂³	#11719945	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	424	2	no	—
84	C₂×C₄	—	212	2	no	—
85	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
86	C₂×C₄	#11719947	1	2	yes	—
87	C₂×C₄	Q₈×D₅₃	1	2	yes	—
88	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
89	C₈	—	2	2	no	—
90	C₈	—	4	1	no	—
91	C₈	—	106	4	no	—
92	C₂²	—	2	2	no	—
93	C₂²	#13133191	1	1	yes	characteristic
94	C₂²	#13133207	1	2	yes	—
95	C₄	—	4	2	no	—
96	C₄	—	424	2	no	—
97	C₄	#13133197	1	2	yes	—
98	C₄	#13133198	1	1	yes	characteristic
99	C₄	#13133215	1	1	yes	characteristic
100	C₂	—	2	2	no	—
101	C₂	#16810553	1	1	yes	characteristic
102	C₂	#16810886	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16810666

C₂

yes

maximal

C₂×#13133149

C₂

yes

maximal, characteristic

#16810391

C₂

yes

maximal

#16810388

C₂

yes

maximal, characteristic

C₅₃×#424

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

#13133150

—

#13133149

—

#13133148

—

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

#117×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

D₄

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719917

—

#11719917

Q₈

yes

—

C₂×#88322

—

C₂³×C₅₃

#41

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

#121

yes

characteristic

C₂²×C₅₃

#125

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊Q₈

yes

—

C₄×C₅₃

#123

yes

characteristic

C₄×C₅₃

C₄⋊Q₈

yes

characteristic

Dic₅₃

—

#2090

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#398

yes

characteristic

C₂×C₅₃

#471

yes

—

#428

—

C₂×#106

—

#424

D₅₃

yes

characteristic

#343

—

#341

—

C₅₃

#2090

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

characteristic

C₂²×C₈

C₂×D₅₃

yes

characteristic

#117

C₂×D₅₃

yes

characteristic

C₂×C₄⋊C₄

—

#107

—

106

—

#106

—

106

—

#106

—

106

—

C₄⋊C₈

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

C₂²⋊D₅₃

yes

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

D₂₁₂

yes

—

C₂²⋊C₄

—

C₄²

—

C₂³

#11719945

yes

characteristic

C₂×C₄

—

C₂×C₄

—

424

—

C₂×C₄

—

212

—

C₂×C₄

C₂×D₂₁₂

yes

characteristic

C₂×C₄

#11719947

yes

—

C₂×C₄

Q₈×D₅₃

yes

—

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

C₈

—

C₈

—

C₈

—

106

—

C₂²

—

C₂²

#13133191

yes

characteristic

C₂²

#13133207

yes

—

C₄

—

C₄

—

424

—

C₄

#13133197

yes

—

C₄

yes

C₄

yes

100

C₂

—

101

C₂

#16810553

yes

characteristic

102

C₂

#16810886

yes

—

	subgroup	G/H
center	C₂²	#13133191
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#424	C₂
socle	C₂²×C₅₃	#121
hypercenter	#424	D₅₃
hypocenter	C₅₃	#2090
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₂²

#13133191

commutator

C₂²×C₄×C₅₃

C₂³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#424

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
57. C₅₃	#2090	semidirect

H ≅ G/N

type

57. C₅₃

#2090

semidirect

Subgroups — #20276225

Conjugacy classes of subgroups

Complements