Group #27277766 — subgroups

Subgroups — #27277766

Order 13568.

322 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 147 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	#20274832	C₂	1	2	yes	maximal
2	#20272550	C₂	1	4	yes	maximal
3	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#88322	C₂²	1	4	yes	—
5	#16810378	—	2	4	no	—
6	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
7	C₄³×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×#336	C₄	1	4	yes	—
9	C₂²×#88322	—	2	4	no	—
10	C₄⋊#88322	—	2	8	no	—
11	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
12	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
13	C₂×C₄²×C₅₃	D₄	1	2	yes	—
14	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
15	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₅₃	—	2	4	no	—
17	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
18	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
21	C₂×C₈×C₅₃	—	2	4	no	—
22	C₂×C₈×C₅₃	—	2	8	no	—
23	C₄²×C₅₃	—	2	4	no	—
24	C₄²×C₅₃	—	2	4	no	—
25	C₄²×C₅₃	—	4	2	no	—
26	C₄²×C₅₃	#41	1	4	yes	—
27	C₂×#88322	—	4	4	no	—
28	C₂×#88322	—	4	8	no	—
29	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₅₃	—	2	2	no	—
31	C₂×C₄×C₅₃	—	2	4	no	—
32	C₂×C₄×C₅₃	—	2	4	no	—
33	C₂×C₄×C₅₃	—	2	4	no	—
34	C₂×C₄×C₅₃	#117	1	2	yes	—
35	C₂×C₄×C₅₃	#124	1	2	yes	—
36	C₂×C₄×C₅₃	#125	1	2	yes	—
37	C₂×C₄×C₅₃	#131	1	4	yes	—
38	C₂×C₄×C₅₃	#135	1	4	yes	—
39	C₈×C₅₃	—	4	4	no	—
40	#88322	—	8	4	no	—
41	#8788	—	53	1	no	maximal, Sylow
42	C₂²×C₅₃	#382	1	1	yes	characteristic
43	C₂²×C₅₃	#424	1	2	yes	—
44	C₂²×C₅₃	#427	1	2	yes	—
45	C₂²×C₅₃	#430	1	2	yes	—
46	C₄×C₅₃	—	2	2	no	—
47	C₄×C₅₃	—	4	4	no	—
48	C₄×C₅₃	#433	1	4	yes	—
49	C₄×C₅₃	#488	1	4	yes	—
50	#1831	—	53	2	no	—
51	#1744	D₅₃	1	1	yes	characteristic
52	#1277	—	53	4	no	—
53	C₂×C₅₃	#1572	1	4	yes	—
54	C₂×C₅₃	#1832	1	1	yes	characteristic
55	C₂×C₅₃	#1835	1	1	yes	characteristic
56	C₂×C₅₃	#1838	1	1	yes	characteristic
57	C₂×C₄⋊C₈	—	53	4	no	—
58	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
59	C₄³	C₂×D₅₃	1	1	yes	characteristic
60	#336	—	106	4	no	—
61	#336	Dic₅₃	1	4	yes	—
62	C₅₃	#8788	1	1	yes	characteristic, complemented, Sylow
63	C₂²×C₈	—	106	4	no	—
64	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
65	C₂×C₄²	Dic₁₀₆	1	2	yes	—
66	C₂×C₄²	D₂₁₂	1	2	yes	—
67	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
68	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
69	C₄⋊C₈	—	106	8	no	—
70	C₄×C₈	—	2	4	no	—
71	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
72	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
73	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
74	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
75	C₂×C₈	—	2	4	no	—
76	C₂×C₈	—	2	8	no	—
77	C₂×C₈	—	212	4	no	—
78	C₂×C₈	—	212	8	no	—
79	C₄²	—	2	4	no	—
80	C₄²	—	2	4	no	—
81	C₄²	—	4	2	no	—
82	C₄²	#11719945	1	4	yes	—
83	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	4	no	—
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	#13133176	1	2	yes	—
89	C₂×C₄	#13133205	1	2	yes	—
90	C₂×C₄	#13133209	1	2	yes	—
91	C₂×C₄	#13133224	1	4	yes	—
92	C₂×C₄	#13133239	1	4	yes	—
93	C₈	—	4	4	no	—
94	C₈	—	424	4	no	—
95	C₂²	#16810489	1	1	yes	characteristic
96	C₂²	#16810650	1	2	yes	—
97	C₂²	#16810661	1	2	yes	—
98	C₂²	#16810672	1	2	yes	—
99	C₄	—	2	2	no	—
100	C₄	—	4	4	no	—
101	C₄	#16810684	1	4	yes	—
102	C₄	#16810978	1	4	yes	—
103	C₂	#20273495	1	4	yes	—
104	C₂	#20274835	1	1	yes	characteristic
105	C₂	#20274846	1	1	yes	characteristic
106	C₂	#20274856	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274832

C₂

yes

maximal

#20272550

C₂

yes

maximal

C₅₃×#1744

C₂

yes

maximal, characteristic

C₂×C₄⋊#88322

C₂²

yes

—

#16810378

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

C₄³×C₅₃

C₂²

yes

characteristic

C₅₃×#336

C₄

yes

—

C₂²×#88322

—

C₄⋊#88322

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

Q₈

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

#41

yes

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

#124

yes

—

C₂×C₄×C₅₃

#125

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₂×C₄×C₅₃

#135

yes

—

C₈×C₅₃

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#427

yes

—

C₂²×C₅₃

#430

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#433

yes

—

C₄×C₅₃

#488

yes

—

#1831

—

#1744

D₅₃

yes

characteristic

#1277

—

C₂×C₅₃

#1572

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×D₅₃

yes

—

C₄³

C₂×D₅₃

yes

characteristic

#336

—

106

—

#336

Dic₅₃

yes

—

C₅₃

#8788

yes

characteristic, complemented, Sylow

C₂²×C₈

—

106

—

C₂²×C₈

C₂×Dic₅₃

yes

—

C₂×C₄²

Dic₁₀₆

yes

—

C₂×C₄²

D₂₁₂

yes

—

C₂×C₄²

C₂×Dic₅₃

yes

—

C₂×C₄²

C₂²×D₅₃

yes

characteristic

C₄⋊C₈

—

106

—

C₄×C₈

—

C₂²×C₄

C₄⋊Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₁₀₆

yes

characteristic

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

212

—

C₂×C₈

—

212

—

C₄²

—

C₄²

—

C₄²

—

C₄²

#11719945

yes

—

C₂³

C₂×C₄⋊Dic₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#13133176

yes

—

C₂×C₄

#13133205

yes

—

C₂×C₄

#13133209

yes

—

C₂×C₄

#13133224

yes

—

C₂×C₄

#13133239

yes

—

C₈

—

C₈

—

424

—

C₂²

#16810489

yes

characteristic

C₂²

#16810650

yes

—

C₂²

#16810661

yes

—

C₂²

#16810672

yes

—

C₄

—

100

C₄

—

101

C₄

#16810684

yes

—

102

C₄

#16810978

yes

—

103

C₂

#20273495

yes

—

104

C₂

#20274835

yes

characteristic

105

C₂

#20274846

yes

characteristic

106

C₂

#20274856

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1744	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1744	D₅₃
hypocenter	C₅₃	#8788
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

subgroup

G/H

center

C₂²×C₄

C₂×D₂₁₂

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1744

C₂

socle, prime radical

C₂³×C₅₃

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

N	H ≅ G/N	type
62. C₅₃	#8788	semidirect

H ≅ G/N

type

62. C₅₃

#8788

semidirect

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

subgroup	index
#20274832	2
#20272550	2
C₅₃×#1744	2
#8788	53

subgroup

index

#20274832

#20272550

C₅₃×#1744

#8788

Subgroups — #27277766

Conjugacy classes of subgroups

Complements

Maximal subgroups