Group #27277866 — subgroups

Subgroups — #27277866

Order 13568.

422 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 227 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	#20274872	C₂	1	2	yes	maximal
2	#20274843	C₂	1	4	yes	maximal
3	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133149	—	2	4	no	—
5	#16810493	C₂²	1	4	yes	—
6	#16810386	—	2	8	no	—
7	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₈²×C₅₃	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₅₃	—	2	8	no	—
11	C₂×C₄⋊Dic₅₃	—	2	4	no	—
12	#13133149	—	4	8	no	—
13	C₂²×C₈×C₅₃	D₄	1	4	yes	—
14	C₂²×C₈×C₅₃	D₄	1	2	yes	—
15	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
17	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₅₃	Q₈	1	4	yes	—
19	C₄×C₈×C₅₃	Q₈	1	2	yes	—
20	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
22	C₂²×Dic₅₃	—	4	4	no	—
23	C₂×C₈×C₅₃	—	2	8	no	—
24	C₂×C₈×C₅₃	C₄⋊C₄	1	8	yes	—
25	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
26	C₂×C₈×C₅₃	SD₁₆	1	8	yes	—
27	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
28	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
29	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
30	C₄⋊Dic₅₃	—	4	8	no	—
31	C₄⋊Dic₅₃	—	4	16	no	—
32	C₂³×C₅₃	#127	1	1	yes	characteristic
33	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
36	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
37	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
38	C₂×C₄×C₅₃	#135	1	8	yes	—
39	C₂×Dic₅₃	—	8	4	no	—
40	C₂×Dic₅₃	—	8	8	no	—
41	C₈×C₅₃	—	2	8	no	—
42	C₈×C₅₃	#106	1	8	yes	—
43	#8815	—	53	1	no	maximal, Sylow
44	C₂²×C₅₃	#384	1	1	yes	characteristic
45	C₂²×C₅₃	#492	1	2	yes	—
46	C₂²×C₅₃	#495	1	4	yes	—
47	C₄×C₅₃	C₂×#106	1	2	yes	—
48	C₄×C₅₃	#427	1	4	yes	—
49	C₄×C₅₃	#498	1	2	yes	—
50	C₄×C₅₃	#499	1	4	yes	—
51	Dic₅₃	—	16	4	no	—
52	#1844	—	53	2	no	—
53	#1834	—	53	4	no	—
54	C₂×C₈²	D₅₃	1	1	yes	characteristic
55	C₂×C₅₃	#1708	1	4	yes	—
56	C₂×C₅₃	#1844	1	1	yes	characteristic
57	C₂×C₅₃	#1846	1	2	yes	—
58	C₂×#106	—	106	4	no	—
59	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
60	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
61	#384	—	53	4	no	—
62	#340	—	106	8	no	—
63	C₈²	Dic₅₃	1	4	yes	complemented
64	C₅₃	#8815	1	1	yes	characteristic, complemented, Sylow
65	C₂²×C₈	D₂₁₂	1	4	yes	—
66	C₂²×C₈	D₂₁₂	1	2	yes	—
67	C₂×C₄⋊C₄	—	106	8	no	—
68	C₂×C₄⋊C₄	—	106	4	no	—
69	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
70	#106	—	212	8	no	—
71	C₄×C₈	Dic₁₀₆	1	4	yes	—
72	C₄×C₈	Dic₁₀₆	1	2	yes	—
73	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
74	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
75	C₂²×C₄	—	212	4	no	—
76	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
77	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
78	C₂×C₈	—	2	8	no	—
79	C₂×C₈	#11719913	1	8	yes	—
80	C₂×C₈	C₄⋊Dic₅₃	1	8	yes	—
81	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
82	C₄⋊C₄	—	212	8	no	—
83	C₄⋊C₄	—	212	16	no	—
84	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
85	C₄²	C₂×Dic₁₀₆	1	2	yes	—
86	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
87	C₂³	#13133214	1	1	yes	characteristic
88	C₂×C₄	—	424	4	no	—
89	C₂×C₄	—	424	8	no	—
90	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
91	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
92	C₂×C₄	#13133217	1	1	yes	characteristic
93	C₂×C₄	#13133217	1	2	yes	—
94	C₂×C₄	C₂×#11719913	1	4	yes	—
95	C₂×C₄	#13133239	1	8	yes	—
96	C₈	—	2	8	no	—
97	C₈	#13133149	1	8	yes	—
98	C₂²	#16810493	1	1	yes	characteristic
99	C₂²	#16810996	1	2	yes	—
100	C₂²	#16811007	1	4	yes	—
101	C₄	—	848	4	no	—
102	C₄	C₂×#13133149	1	2	yes	—
103	C₄	#16810661	1	4	yes	—
104	C₄	#16811021	1	2	yes	—
105	C₄	#16811025	1	4	yes	—
106	C₂	#20274316	1	4	yes	—
107	C₂	#20274872	1	1	yes	characteristic
108	C₂	#20274879	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20274872

C₂

yes

maximal

#20274843

C₂

yes

maximal

C₂×C₈²×C₅₃

C₂

yes

maximal, characteristic

C₂×#13133149

—

#16810493

C₂²

yes

—

#16810386

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₈²×C₅₃

C₄

yes

complemented

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

#13133149

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

Q₈

yes

—

C₄×C₈×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂³×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

—

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₂×C₄×C₅₃

#135

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#106

yes

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#495

yes

—

C₄×C₅₃

C₂×#106

yes

—

C₄×C₅₃

#427

yes

—

C₄×C₅₃

#498

yes

—

C₄×C₅₃

#499

yes

—

Dic₅₃

—

#1844

—

#1834

—

C₂×C₈²

D₅₃

yes

characteristic

C₂×C₅₃

#1708

yes

—

C₂×C₅₃

#1844

yes

characteristic

C₂×C₅₃

#1846

yes

—

C₂×#106

—

106

—

C₂×C₄×C₈

C₂×D₅₃

yes

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#384

—

#340

—

106

—

C₈²

Dic₅₃

yes

complemented

C₅₃

#8815

yes

characteristic, complemented, Sylow

C₂²×C₈

D₂₁₂

yes

—

C₂²×C₈

D₂₁₂

yes

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

C₂²×D₅₃

yes

characteristic

#106

—

212

—

C₄×C₈

Dic₁₀₆

yes

—

C₄×C₈

Dic₁₀₆

yes

—

C₄×C₈

C₂×Dic₅₃

yes

—

C₄×C₈

C₂×Dic₅₃

yes

—

C₂²×C₄

—

212

—

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

C₂²×C₄

C₂×D₂₁₂

yes

—

C₂×C₈

—

C₂×C₈

#11719913

yes

—

C₂×C₈

C₄⋊Dic₅₃

yes

—

C₂×C₈

C₄⋊Dic₅₃

yes

—

C₄⋊C₄

—

212

—

C₄⋊C₄

—

212

—

C₄²

C₂×Dic₁₀₆

yes

characteristic

C₄²

C₂×Dic₁₀₆

yes

—

C₄²

C₂²×Dic₅₃

yes

characteristic

C₂³

#13133214

yes

characteristic

C₂×C₄

—

424

—

C₂×C₄

—

424

—

C₂×C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

C₂×C₄

C₂×C₄⋊Dic₅₃

yes

—

C₂×C₄

#13133217

yes

characteristic

C₂×C₄

#13133217

yes

—

C₂×C₄

C₂×#11719913

yes

—

C₂×C₄

#13133239

yes

—

C₈

—

C₈

#13133149

yes

—

C₂²

#16810493

yes

characteristic

C₂²

#16810996

yes

—

100

C₂²

#16811007

yes

—

101

C₄

—

848

—

102

C₄

C₂×#13133149

yes

—

103

C₄

#16810661

yes

—

104

C₄

#16811021

yes

—

105

C₄

#16811025

yes

—

106

C₂

#20274316

yes

—

107

C₂

#20274872

yes

characteristic

108

C₂

#20274879

yes

—

	subgroup	G/H
center	C₂³	#13133214
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₈²×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	#127
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×C₈²	D₅₃
hypocenter	C₅₃	#8815
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂³

#13133214

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₂×C₈²×C₅₃

C₂

socle, prime radical

C₂³×C₅₃

#127

cosocle

C₂×C₄²×C₅₃

C₂³

hypercenter

C₂×C₈²

D₅₃

hypocenter

C₅₃

#8815

LCS

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
9. C₈²×C₅₃	C₄	semidirect
63. C₈²	Dic₅₃	semidirect
64. C₅₃	#8815	semidirect

H ≅ G/N

type

9. C₈²×C₅₃

C₄

semidirect

63. C₈²

Dic₅₃

semidirect

64. C₅₃

#8815

semidirect

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

subgroup	index
#20274872	2
#20274843	2
C₂×C₈²×C₅₃	2
#8815	53

subgroup

index

#20274872

#20274843

C₂×C₈²×C₅₃

#8815

Subgroups — #27277866

Conjugacy classes of subgroups

Complements

Maximal subgroups