Group #27298795 — subgroups

Subgroups — #27298795

Order 13568.

290 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 79 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	#20276938	C₂	1	4	yes	maximal, complemented
2	#20274325	C₂	1	1	yes	maximal, characteristic, complemented
3	#20272632	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1712	C₂	1	1	yes	maximal, characteristic
5	#16811028	C₂²	1	2	yes	—
6	#16810974	—	2	2	no	—
7	#16810424	—	2	8	no	—
8	C₄×#11719908	C₂²	1	2	yes	—
9	C₅₃×#493	—	2	2	no	—
10	C₅₃×#493	C₂²	1	2	yes	—
11	C₈²×C₅₃	C₂²	1	1	yes	characteristic
12	#13133217	—	2	2	no	—
13	C₂×#11719908	D₄	1	4	yes	—
14	#13133151	—	2	4	no	—
15	#13133142	—	4	4	no	—
16	C₂×D₈×C₅₃	—	2	4	no	—
17	C₂×D₈×C₅₃	—	4	4	no	—
18	#127×C₅₃	—	2	2	no	—
19	C₄×C₈×C₅₃	D₄	1	2	yes	—
20	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
21	C₂×D₄×C₅₃	—	4	2	no	—
22	C₂×D₄×C₅₃	—	4	4	no	—
23	C₂×Dic₁₀₆	—	4	2	no	—
24	D₈×C₅₃	—	4	8	no	—
25	D₈×C₅₃	—	8	8	no	—
26	C₂×C₈×C₅₃	D₈	1	4	yes	—
27	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
28	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
29	C₄⋊Dic₅₃	—	4	2	no	—
30	C₄⋊Dic₅₃	—	4	2	no	—
31	#11719908	—	2	4	no	—
32	C₂³×C₅₃	—	8	2	no	—
33	D₄×C₅₃	—	8	4	no	—
34	D₄×C₅₃	—	8	8	no	—
35	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
36	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
37	C₂×Dic₅₃	—	8	2	no	—
38	Dic₁₀₆	—	8	4	no	—
39	C₈×C₅₃	—	2	4	no	—
40	C₈×C₅₃	#135	1	4	yes	—
41	#12513	—	53	1	no	maximal, Sylow
42	C₂²×C₅₃	—	16	2	no	—
43	C₂²×C₅₃	—	16	4	no	—
44	C₂²×C₅₃	#493	1	1	yes	characteristic
45	C₄×C₅₃	#495	1	2	yes	—
46	C₄×C₅₃	#509	1	4	yes	—
47	Dic₅₃	—	16	2	no	—
48	#2240	—	53	4	no	—
49	#1712	D₅₃	1	1	yes	characteristic
50	#1710	—	53	1	no	—
51	#1311	—	53	1	no	—
52	C₂×C₅₃	—	32	2	no	—
53	C₂×C₅₃	#1709	1	1	yes	characteristic
54	C₂×C₅₃	#2250	1	2	yes	—
55	#500	—	53	2	no	—
56	#493	—	2	2	no	—
57	#493	C₂×D₅₃	1	2	yes	—
58	#487	—	106	2	no	—
59	#357	—	106	8	no	—
60	C₄×C₁₆	—	53	2	no	—
61	C₈²	C₂×D₅₃	1	1	yes	characteristic
62	C₅₃	#12513	1	1	yes	characteristic, complemented, Sylow
63	C₂×D₈	—	2	4	no	—
64	C₂×D₈	—	4	4	no	—
65	C₄⋊Q₈	—	106	2	no	—
66	#127	—	2	2	no	—
67	C₂×C₁₆	—	53	4	no	—
68	#107	—	106	4	no	—
69	#103	—	212	4	no	—
70	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
71	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
72	C₂×Q₈	—	212	2	no	—
73	C₂×D₄	—	4	2	no	—
74	C₂×D₄	—	4	4	no	—
75	D₈	—	4	8	no	—
76	D₈	—	8	8	no	—
77	C₂×C₈	#11719922	1	4	yes	—
78	C₂×C₈	D₄×D₅₃	1	2	yes	—
79	C₄⋊C₄	—	212	2	no	—
80	C₄⋊C₄	—	212	2	no	—
81	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
82	C₁₆	—	106	4	no	—
83	C₂³	—	8	2	no	—
84	Q₈	—	424	4	no	—
85	D₄	—	8	8	no	—
86	D₄	—	8	4	no	—
87	C₂×C₄	—	424	2	no	—
88	C₂×C₄	#13133213	1	1	yes	characteristic
89	C₂×C₄	C₂×#11719922	1	2	yes	—
90	C₈	—	2	4	no	—
91	C₈	#13133236	1	4	yes	—
92	C₂²	—	16	2	no	—
93	C₂²	—	16	4	no	—
94	C₂²	#16810999	1	1	yes	characteristic
95	C₄	—	848	2	no	—
96	C₄	#16811006	1	2	yes	—
97	C₄	#16811059	1	4	yes	—
98	C₂	—	32	2	no	—
99	C₂	#20274320	1	1	yes	characteristic
100	C₂	#20276979	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20276938

C₂

yes

maximal, complemented

#20274325

C₂

yes

maximal, characteristic, complemented

#20272632

C₂

yes

maximal, characteristic, complemented

C₅₃×#1712

C₂

yes

maximal, characteristic

#16811028

C₂²

yes

—

#16810974

—

#16810424

—

C₄×#11719908

C₂²

yes

—

C₅₃×#493

—

C₅₃×#493

C₂²

yes

—

C₈²×C₅₃

C₂²

yes

characteristic

#13133217

—

C₂×#11719908

D₄

yes

—

#13133151

—

#13133142

—

C₂×D₈×C₅₃

—

C₂×D₈×C₅₃

—

#127×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×Dic₁₀₆

—

D₈×C₅₃

—

D₈×C₅₃

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719908

—

C₂³×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

—

C₂×Dic₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

#135

yes

—

#12513

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#493

yes

characteristic

C₄×C₅₃

#495

yes

—

C₄×C₅₃

#509

yes

—

Dic₅₃

—

#2240

—

#1712

D₅₃

yes

characteristic

#1710

—

#1311

—

C₂×C₅₃

—

C₂×C₅₃

#1709

yes

characteristic

C₂×C₅₃

#2250

yes

—

#500

—

#493

—

#493

C₂×D₅₃

yes

—

#487

—

106

—

#357

—

106

—

C₄×C₁₆

—

C₈²

C₂×D₅₃

yes

characteristic

C₅₃

#12513

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

106

—

#127

—

C₂×C₁₆

—

#107

—

106

—

#103

—

212

—

C₄×C₈

C₂²⋊D₅₃

yes

—

C₄×C₈

C₂²×D₅₃

yes

characteristic

C₂×Q₈

—

212

—

C₂×D₄

—

C₂×D₄

—

D₈

—

D₈

—

C₂×C₈

#11719922

yes

—

C₂×C₈

D₄×D₅₃

yes

—

C₄⋊C₄

—

212

—

C₄⋊C₄

—

212

—

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

C₁₆

—

106

—

C₂³

—

Q₈

—

424

—

D₄

—

D₄

—

C₂×C₄

—

424

—

C₂×C₄

#13133213

yes

characteristic

C₂×C₄

C₂×#11719922

yes

—

C₈

—

C₈

#13133236

yes

—

C₂²

—

C₂²

—

C₂²

#16810999

yes

characteristic

C₄

—

848

—

C₄

#16811006

yes

—

C₄

#16811059

yes

—

C₂

—

C₂

#20274320

yes

characteristic

100

C₂

#20276979

yes

—

	subgroup	G/H
center	C₂²	#16810999
commutator, cosocle	C₄×C₈×C₅₃	C₂³
Frattini	C₄×C₈	C₂²×D₅₃
Fitting, prime radical	C₅₃×#1712	C₂
socle	C₂²×C₅₃	#493
hypercenter	#1712	D₅₃
hypocenter	C₅₃	#12513
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

subgroup

G/H

center

C₂²

#16810999

commutator, cosocle

C₄×C₈×C₅₃

C₂³

Frattini

C₄×C₈

C₂²×D₅₃

Fitting, prime radical

C₅₃×#1712

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

N	H ≅ G/N	type
1. #20276938	C₂	semidirect
2. #20274325	C₂	semidirect
3. #20272632	C₂	semidirect
62. C₅₃	#12513	semidirect

H ≅ G/N

type

1. #20276938

C₂

semidirect

2. #20274325

C₂

semidirect

3. #20272632

C₂

semidirect

62. C₅₃

#12513

semidirect

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

subgroup	index
#20276938	2
#20274325	2
#20272632	2
C₅₃×#1712	2
#12513	53

subgroup

index

C₅₃×#1712

Subgroups — #27298795

Conjugacy classes of subgroups

Complements

Maximal subgroups