Subgroups — C₅₃×#12515 #27220027

Order 13568.

398 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2246	C₂	1	4	yes	maximal, complemented
2	C₅₃×#1712	C₂	1	2	yes	maximal, complemented
3	C₈×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₆×C₅₃	—	2	8	no	—
5	C₅₃×#493	—	2	4	no	—
6	C₅₃×#493	C₂²	1	4	yes	—
7	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
8	C₈²×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×D₈×C₅₃	—	2	8	no	—
10	C₂×D₈×C₅₃	—	4	8	no	—
11	#127×C₅₃	—	2	4	no	—
12	D₁₆×C₅₃	—	4	16	no	—
13	C₂×C₁₆×C₅₃	D₄	1	4	yes	—
14	C₄×C₈×C₅₃	D₄	1	2	yes	—
15	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
16	C₂×D₄×C₅₃	—	4	4	no	—
17	C₂×D₄×C₅₃	—	4	8	no	—
18	D₈×C₅₃	—	4	16	no	—
19	D₈×C₅₃	—	8	16	no	—
20	C₂×C₈×C₅₃	D₈	1	4	yes	—
21	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
22	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
23	C₁₆×C₅₃	D₈	1	8	yes	complemented
24	C₂³×C₅₃	—	8	4	no	—
25	D₄×C₅₃	—	8	8	no	—
26	D₄×C₅₃	—	8	16	no	—
27	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
28	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
29	C₈×C₅₃	D₁₆	1	8	yes	complemented
30	C₈×C₅₃	C₂×D₈	1	4	yes	—
31	#12515	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂²×C₅₃	—	16	4	no	—
33	C₂²×C₅₃	—	16	8	no	—
34	C₂²×C₅₃	#493	1	1	yes	characteristic
35	C₄×C₅₃	#493	1	2	yes	—
36	C₄×C₅₃	C₂×D₁₆	1	4	yes	—
37	#2246	C₂×C₅₃	1	4	yes	complemented
38	#1712	C₂×C₅₃	1	2	yes	complemented
39	C₈×C₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
40	C₂×C₅₃	—	32	4	no	—
41	C₂×C₅₃	#1712	1	1	yes	characteristic
42	C₂×C₅₃	#2246	1	2	yes	—
43	C₂×D₁₆	—	2	8	no	—
44	#493	—	2	4	no	—
45	#493	C₂²×C₅₃	1	4	yes	—
46	C₄×C₁₆	C₂²×C₅₃	1	2	yes	—
47	C₈²	C₂²×C₅₃	1	1	yes	characteristic
48	C₅₃	#12515	1	1	yes	characteristic, direct product factor, Sylow
49	C₂×D₈	—	2	8	no	—
50	C₂×D₈	—	4	8	no	—
51	#127	—	2	4	no	—
52	D₁₆	—	4	16	no	—
53	C₂×C₁₆	D₄×C₅₃	1	4	yes	—
54	C₄×C₈	D₄×C₅₃	1	2	yes	—
55	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
56	C₂×D₄	—	4	4	no	—
57	C₂×D₄	—	4	8	no	—
58	D₈	—	4	16	no	—
59	D₈	—	8	16	no	—
60	C₂×C₈	D₈×C₅₃	1	4	yes	—
61	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
62	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
63	C₁₆	D₈×C₅₃	1	8	yes	complemented
64	C₂³	—	8	4	no	—
65	D₄	—	8	8	no	—
66	D₄	—	8	16	no	—
67	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
68	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
69	C₈	D₁₆×C₅₃	1	8	yes	complemented
70	C₈	C₂×D₈×C₅₃	1	4	yes	—
71	C₂²	—	16	4	no	—
72	C₂²	—	16	8	no	—
73	C₂²	C₅₃×#493	1	1	yes	characteristic
74	C₄	C₅₃×#493	1	2	yes	—
75	C₄	C₂×D₁₆×C₅₃	1	4	yes	—
76	C₂	—	32	4	no	—
77	C₂	C₅₃×#1712	1	1	yes	characteristic
78	C₂	C₅₃×#2246	1	2	yes	—

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

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₅₃×#2246	C₂	semidirect
2. C₅₃×#1712	C₂	semidirect
3. C₈×C₁₆×C₅₃	C₂	semidirect
23. C₁₆×C₅₃	D₈	semidirect
29. C₈×C₅₃	D₁₆	semidirect
31. #12515	C₅₃	direct (N × H)
37. #2246	C₂×C₅₃	semidirect
38. #1712	C₂×C₅₃	semidirect
39. C₈×C₁₆	C₂×C₅₃	semidirect
48. C₅₃	#12515	direct (N × H)
63. C₁₆	D₈×C₅₃	semidirect
69. C₈	D₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#2246	2
C₅₃×#1712	2
C₈×C₁₆×C₅₃	2
#12515	53

Subgroups — C53×#12515 #27220027

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12515 #27220027