Group #27277808 — subgroups

Subgroups — #27277808

Order 13568.

220 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 99 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	#20273478	C₂	1	2	yes	maximal
2	#20272577	C₂	1	4	yes	maximal
3	C₅₃×#1452	C₂	1	1	yes	maximal, characteristic
4	#16810680	C₂²	1	4	yes	—
5	#16810376	—	2	2	no	—
6	#16810374	—	2	2	no	—
7	C₅₃×#431	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#322	C₄	1	4	yes	—
10	C₄⋊#88322	—	2	4	no	—
11	C₂²⋊#88322	—	2	4	no	—
12	#13133128	—	2	4	no	—
13	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
14	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
15	C₄⋊C₈×C₅₃	—	2	2	no	—
16	C₂²⋊C₈×C₅₃	—	2	2	no	—
17	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
18	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₅₃	D₄	1	2	yes	—
20	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	#41	1	2	yes	—
22	C₂×C₈×C₅₃	—	2	4	no	—
23	C₂×C₈×C₅₃	—	4	2	no	—
24	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
25	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
26	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
27	C₄²×C₅₃	#41	1	2	yes	—
28	C₂×#88322	—	4	4	no	—
29	C₂³×C₅₃	#125	1	1	yes	characteristic
30	C₂×C₄×C₅₃	—	2	2	no	—
31	C₂×C₄×C₅₃	—	4	2	no	—
32	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₅₃	#117	1	2	yes	—
34	C₂×C₄×C₅₃	#124	1	2	yes	—
35	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
36	C₈×C₅₃	—	2	4	no	—
37	C₈×C₅₃	—	4	4	no	—
38	#88322	—	8	4	no	—
39	#8799	—	53	1	no	maximal, Sylow
40	C₂²×C₅₃	—	2	2	no	—
41	C₂²×C₅₃	—	2	1	no	—
42	C₂²×C₅₃	#382	1	1	yes	characteristic
43	C₄×C₅₃	—	2	2	no	—
44	C₄×C₅₃	—	2	2	no	—
45	C₄×C₅₃	—	2	1	no	—
46	C₄×C₅₃	#430	1	2	yes	—
47	#1568	—	53	2	no	—
48	#1452	D₅₃	1	1	yes	characteristic
49	#1287	—	53	4	no	—
50	C₂×C₅₃	—	4	1	no	—
51	C₂×C₅₃	#1833	1	2	yes	—
52	C₂×C₅₃	#1838	1	1	yes	characteristic
53	#432	—	53	4	no	—
54	#431	C₂×D₅₃	1	2	yes	—
55	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
56	#335	—	106	2	no	—
57	#334	—	106	2	no	—
58	#322	Dic₅₃	1	4	yes	—
59	C₅₃	#8799	1	1	yes	characteristic, complemented, Sylow
60	C₂²×C₈	Dic₁₀₆	1	2	yes	—
61	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
62	C₄⋊C₈	—	2	2	no	—
63	C₄⋊C₈	—	106	4	no	—
64	C₂²⋊C₈	—	2	2	no	—
65	C₂²⋊C₈	—	106	4	no	—
66	#97	—	106	4	no	—
67	C₄×C₈	D₂₁₂	1	2	yes	—
68	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
69	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
70	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
71	C₂²×C₄	#11719945	1	2	yes	—
72	C₂×C₈	—	2	4	no	—
73	C₂×C₈	—	4	2	no	—
74	C₂×C₈	—	212	4	no	—
75	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
76	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
77	C₄²	#11719945	1	2	yes	—
78	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
79	C₂³	#13133209	1	1	yes	characteristic
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
83	C₂×C₄	#13133176	1	2	yes	—
84	C₂×C₄	#13133205	1	2	yes	—
85	C₂×C₄	#13133209	1	1	yes	characteristic
86	C₈	—	2	4	no	—
87	C₈	—	4	4	no	—
88	C₈	—	424	4	no	—
89	C₂²	—	2	2	no	—
90	C₂²	—	2	1	no	—
91	C₂²	#16810489	1	1	yes	characteristic
92	C₄	—	2	2	no	—
93	C₄	—	2	2	no	—
94	C₄	—	2	1	no	—
95	C₄	#16810672	1	2	yes	—
96	C₂	—	4	1	no	—
97	C₂	#20274839	1	2	yes	—
98	C₂	#20274856	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20273478

C₂

yes

maximal

#20272577

C₂

yes

maximal

C₅₃×#1452

C₂

yes

maximal, characteristic

#16810680

C₂²

yes

—

#16810376

—

#16810374

—

C₅₃×#431

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#322

C₄

yes

—

C₄⋊#88322

—

C₂²⋊#88322

—

#13133128

—

C₂²×C₈×C₅₃

Q₈

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

#41

yes

—

C₂×#88322

—

C₂³×C₅₃

#125

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

#124

yes

—

C₂×C₄×C₅₃

#125

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#8799

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#382

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#430

yes

—

#1568

—

#1452

D₅₃

yes

characteristic

#1287

—

C₂×C₅₃

—

C₂×C₅₃

#1833

yes

—

C₂×C₅₃

yes

—

C₂×D₅₃

yes

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#335

—

106

—

#334

—

106

—

#322

Dic₅₃

yes

—

C₅₃

#8799

yes

characteristic, complemented, Sylow

C₂²×C₈

Dic₁₀₆

yes

—

C₂×C₄²

C₂²×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

C₂²⋊C₈

—

106

—

#97

—

106

—

C₄×C₈

D₂₁₂

yes

—

C₄×C₈

C₂×Dic₅₃

yes

—

C₄×C₈

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₁₀₆

yes

characteristic

C₂²×C₄

#11719945

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

212

—

C₂×C₈

C₄⋊Dic₅₃

yes

—

C₄²

C₂×D₂₁₂

yes

characteristic

C₄²

#11719945

yes

—

C₄²

C₂²×Dic₅₃

yes

characteristic

C₂³

#13133209

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

C₂×C₄

#13133176

yes

—

C₂×C₄

#13133205

yes

—

C₂×C₄

#13133209

yes

characteristic

C₈

—

C₈

—

C₈

—

424

—

C₂²

—

C₂²

—

C₂²

#16810489

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#16810672

yes

—

C₂

—

C₂

#20274839

yes

—

C₂

#20274856

yes

characteristic

	subgroup	G/H
center	C₂²	#16810489
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1452	C₂
socle	C₂²×C₅₃	#382
prime radical	C₂³×C₅₃	#125
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1452	D₅₃
hypocenter	C₅₃	#8799
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

subgroup

G/H

center

C₂²

#16810489

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1452

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

N	H ≅ G/N	type
59. C₅₃	#8799	semidirect

H ≅ G/N

type

59. C₅₃

#8799

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20273478	2
#20272577	2
C₅₃×#1452	2
#8799	53

subgroup

index

#20273478

#20272577

C₅₃×#1452

#8799

Subgroups — #27277808

Conjugacy classes of subgroups

Complements

Maximal subgroups