Subgroups — C₅₃×#12516 #27220028

Order 13568.

310 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2247	C₂	1	4	yes	maximal
2	C₅₃×#1713	C₂	1	2	yes	maximal
3	C₈×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₃₂×C₅₃	—	2	8	no	—
5	C₅₃×#500	—	2	4	no	—
6	C₅₃×#494	C₂²	1	4	yes	—
7	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
8	C₈²×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×Q₁₆×C₅₃	—	2	8	no	—
10	C₄⋊Q₈×C₅₃	—	2	4	no	—
11	Q₃₂×C₅₃	—	4	16	no	—
12	C₂×C₁₆×C₅₃	D₄	1	4	yes	—
13	#107×C₅₃	—	4	8	no	—
14	C₄×C₈×C₅₃	D₄	1	2	yes	—
15	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
16	C₂×Q₈×C₅₃	—	4	4	no	—
17	Q₁₆×C₅₃	—	4	16	no	—
18	C₂×C₈×C₅₃	D₈	1	4	yes	—
19	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
20	C₄⋊C₄×C₅₃	—	4	8	no	—
21	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
22	C₁₆×C₅₃	D₈	1	8	yes	—
23	Q₈×C₅₃	—	8	8	no	—
24	C₂×C₄×C₅₃	—	8	4	no	—
25	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
26	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
27	C₈×C₅₃	Q₃₂	1	8	yes	complemented
28	C₈×C₅₃	C₂×D₈	1	4	yes	—
29	#12516	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂²×C₅₃	#493	1	1	yes	characteristic
31	C₄×C₅₃	—	16	4	no	—
32	C₄×C₅₃	#493	1	2	yes	—
33	C₄×C₅₃	C₂×Q₃₂	1	4	yes	—
34	#2247	C₂×C₅₃	1	4	yes	—
35	#1713	C₂×C₅₃	1	2	yes	—
36	C₈×C₁₆	C₂×C₅₃	1	1	yes	characteristic
37	C₂×C₅₃	#1712	1	1	yes	characteristic
38	C₂×C₅₃	#2247	1	2	yes	—
39	C₂×Q₃₂	—	2	8	no	—
40	#500	—	2	4	no	—
41	#494	C₂²×C₅₃	1	4	yes	—
42	C₄×C₁₆	C₂²×C₅₃	1	2	yes	—
43	C₈²	C₂²×C₅₃	1	1	yes	characteristic
44	C₅₃	#12516	1	1	yes	characteristic, direct product factor, Sylow
45	C₂×Q₁₆	—	2	8	no	—
46	C₄⋊Q₈	—	2	4	no	—
47	Q₃₂	—	4	16	no	—
48	C₂×C₁₆	D₄×C₅₃	1	4	yes	—
49	#107	—	4	8	no	—
50	C₄×C₈	D₄×C₅₃	1	2	yes	—
51	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
52	C₂×Q₈	—	4	4	no	—
53	Q₁₆	—	4	16	no	—
54	C₂×C₈	D₈×C₅₃	1	4	yes	—
55	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
56	C₄⋊C₄	—	4	8	no	—
57	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
58	C₁₆	D₈×C₅₃	1	8	yes	—
59	Q₈	—	8	8	no	—
60	C₂×C₄	—	8	4	no	—
61	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
62	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
63	C₈	Q₃₂×C₅₃	1	8	yes	complemented
64	C₈	C₂×D₈×C₅₃	1	4	yes	—
65	C₂²	C₅₃×#493	1	1	yes	characteristic
66	C₄	—	16	4	no	—
67	C₄	C₅₃×#493	1	2	yes	—
68	C₄	C₂×Q₃₂×C₅₃	1	4	yes	—
69	C₂	C₅₃×#1712	1	1	yes	characteristic
70	C₂	C₅₃×#2247	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	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
27. C₈×C₅₃	Q₃₂	semidirect
29. #12516	C₅₃	direct (N × H)
44. C₅₃	#12516	direct (N × H)
63. C₈	Q₃₂×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#2247	2
C₅₃×#1713	2
C₈×C₁₆×C₅₃	2
#12516	53

Subgroups — C53×#12516 #27220028

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12516 #27220028