Group #27220025 — subgroups

Subgroups — C₅₃×#12513 #27220025

Order 13568.

290 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 94 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	C₅₃×#2240	C₂	1	4	yes	maximal, complemented
2	C₅₃×#1712	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1710	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1311	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#500	C₂²	1	2	yes	—
6	C₅₃×#493	—	2	2	no	—
7	C₅₃×#493	C₂²	1	2	yes	—
8	C₅₃×#487	—	2	2	no	—
9	C₅₃×#357	—	2	8	no	—
10	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
11	C₈²×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×D₈×C₅₃	—	2	4	no	—
13	C₂×D₈×C₅₃	—	4	4	no	—
14	C₄⋊Q₈×C₅₃	—	2	2	no	—
15	#127×C₅₃	—	2	2	no	—
16	C₂×C₁₆×C₅₃	D₄	1	4	yes	—
17	#107×C₅₃	—	2	4	no	—
18	#103×C₅₃	—	4	4	no	—
19	C₄×C₈×C₅₃	D₄	1	2	yes	—
20	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
21	C₂×Q₈×C₅₃	—	4	2	no	—
22	C₂×D₄×C₅₃	—	4	2	no	—
23	C₂×D₄×C₅₃	—	4	4	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	C₄⋊C₄×C₅₃	—	4	2	no	—
29	C₄⋊C₄×C₅₃	—	4	2	no	—
30	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
31	C₁₆×C₅₃	—	2	4	no	—
32	C₂³×C₅₃	—	8	2	no	—
33	Q₈×C₅₃	—	8	4	no	—
34	D₄×C₅₃	—	8	4	no	—
35	D₄×C₅₃	—	8	8	no	—
36	C₂×C₄×C₅₃	—	8	2	no	—
37	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
38	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
39	C₈×C₅₃	—	2	4	no	—
40	C₈×C₅₃	#135	1	4	yes	—
41	#12513	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂²×C₅₃	—	16	2	no	—
43	C₂²×C₅₃	—	16	4	no	—
44	C₂²×C₅₃	#493	1	1	yes	characteristic
45	C₄×C₅₃	—	16	2	no	—
46	C₄×C₅₃	#495	1	2	yes	—
47	C₄×C₅₃	#509	1	4	yes	—
48	#2240	C₂×C₅₃	1	4	yes	complemented
49	#1712	C₂×C₅₃	1	1	yes	characteristic
50	#1710	C₂×C₅₃	1	1	yes	characteristic, complemented
51	#1311	C₂×C₅₃	1	1	yes	characteristic, complemented
52	C₂×C₅₃	—	32	2	no	—
53	C₂×C₅₃	#1709	1	1	yes	characteristic
54	C₂×C₅₃	#2250	1	2	yes	—
55	#500	C₂²×C₅₃	1	2	yes	—
56	#493	—	2	2	no	—
57	#493	C₂²×C₅₃	1	2	yes	—
58	#487	—	2	2	no	—
59	#357	—	2	8	no	—
60	C₄×C₁₆	C₂²×C₅₃	1	2	yes	—
61	C₈²	C₂²×C₅₃	1	1	yes	characteristic
62	C₅₃	#12513	1	1	yes	characteristic, direct product factor, Sylow
63	C₂×D₈	—	2	4	no	—
64	C₂×D₈	—	4	4	no	—
65	C₄⋊Q₈	—	2	2	no	—
66	#127	—	2	2	no	—
67	C₂×C₁₆	D₄×C₅₃	1	4	yes	—
68	#107	—	2	4	no	—
69	#103	—	4	4	no	—
70	C₄×C₈	D₄×C₅₃	1	2	yes	—
71	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
72	C₂×Q₈	—	4	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	D₈×C₅₃	1	4	yes	—
78	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
79	C₄⋊C₄	—	4	2	no	—
80	C₄⋊C₄	—	4	2	no	—
81	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
82	C₁₆	—	2	4	no	—
83	C₂³	—	8	2	no	—
84	Q₈	—	8	4	no	—
85	D₄	—	8	4	no	—
86	D₄	—	8	8	no	—
87	C₂×C₄	—	8	2	no	—
88	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
89	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
90	C₈	—	2	4	no	—
91	C₈	#135×C₅₃	1	4	yes	—
92	C₂²	—	16	2	no	—
93	C₂²	—	16	4	no	—
94	C₂²	C₅₃×#493	1	1	yes	characteristic
95	C₄	—	16	2	no	—
96	C₄	C₅₃×#495	1	2	yes	—
97	C₄	C₅₃×#509	1	4	yes	—
98	C₂	—	32	2	no	—
99	C₂	C₅₃×#1709	1	1	yes	characteristic
100	C₂	C₅₃×#2250	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#2240

C₂

yes

maximal, complemented

C₅₃×#1712

C₂

yes

maximal, characteristic

C₅₃×#1710

C₂

yes

maximal, characteristic, complemented

C₅₃×#1311

C₂

yes

maximal, characteristic, complemented

C₅₃×#500

C₂²

yes

—

C₅₃×#493

—

C₅₃×#493

C₂²

yes

—

C₅₃×#487

—

C₅₃×#357

—

C₄×C₁₆×C₅₃

C₂²

yes

—

C₈²×C₅₃

C₂²

yes

characteristic

C₂×D₈×C₅₃

—

C₂×D₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

#127×C₅₃

—

C₂×C₁₆×C₅₃

D₄

yes

—

#107×C₅₃

—

#103×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

D₈×C₅₃

—

D₈×C₅₃

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₁₆×C₅₃

—

C₂³×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

—

C₈×C₅₃

—

C₈×C₅₃

#135

yes

—

#12513

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#493

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

#495

yes

—

C₄×C₅₃

#509

yes

—

#2240

C₂×C₅₃

yes

complemented

#1712

C₂×C₅₃

yes

characteristic

#1710

C₂×C₅₃

yes

characteristic, complemented

#1311

C₂×C₅₃

yes

characteristic, complemented

C₂×C₅₃

—

C₂×C₅₃

#1709

yes

characteristic

C₂×C₅₃

#2250

yes

—

#500

C₂²×C₅₃

yes

—

#493

—

#493

C₂²×C₅₃

yes

—

#487

—

#357

—

C₄×C₁₆

C₂²×C₅₃

yes

—

C₈²

C₂²×C₅₃

yes

characteristic

C₅₃

#12513

yes

characteristic, direct product factor, Sylow

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

D₄×C₅₃

yes

—

#107

—

#103

—

C₄×C₈

D₄×C₅₃

yes

—

C₄×C₈

C₂³×C₅₃

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

D₈

—

D₈

—

C₂×C₈

D₈×C₅₃

yes

—

C₂×C₈

C₂×D₄×C₅₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×D₄×C₅₃

yes

characteristic

C₁₆

—

C₂³

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

#127×C₅₃

yes

characteristic

C₂×C₄

C₂×D₈×C₅₃

yes

—

C₈

—

C₈

#135×C₅₃

yes

—

C₂²

—

C₂²

—

C₂²

C₅₃×#493

yes

characteristic

C₄

—

C₄

C₅₃×#495

yes

—

C₄

C₅₃×#509

yes

—

C₂

—

C₂

C₅₃×#1709

yes

characteristic

100

C₂

C₅₃×#2250

yes

—

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

subgroup

G/H

center, socle

C₂²×C₅₃

#493

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₅₃

prime radical

C₅₃×#1712

C₂

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₅₃ ◁ C₄²×C₅₃ ◁ C₈²×C₅₃ ◁ G

N	H ≅ G/N	type
1. C₅₃×#2240	C₂	semidirect
3. C₅₃×#1710	C₂	semidirect
4. C₅₃×#1311	C₂	semidirect
41. #12513	C₅₃	direct (N × H)
48. #2240	C₂×C₅₃	semidirect
50. #1710	C₂×C₅₃	semidirect
51. #1311	C₂×C₅₃	semidirect
62. C₅₃	#12513	direct (N × H)

H ≅ G/N

type

1. C₅₃×#2240

C₂

semidirect

3. C₅₃×#1710

C₂

semidirect

4. C₅₃×#1311

C₂

semidirect

41. #12513

C₅₃

direct (N × H)

48. #2240

C₂×C₅₃

semidirect

50. #1710

C₂×C₅₃

semidirect

51. #1311

C₂×C₅₃

semidirect

62. C₅₃

#12513

direct (N × H)

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

subgroup	index
C₅₃×#2240	2
C₅₃×#1712	2
C₅₃×#1710	2
C₅₃×#1311	2
#12513	53

subgroup

index

C₅₃×#2240

C₅₃×#1712

C₅₃×#1710

C₅₃×#1311

#12513

Subgroups — C₅₃×#12513 #27220025

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#12513 #27220025

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12513 #27220025