Subgroups — C₅₃×#12521 #27220033

Order 13568.

262 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2252	C₂	1	4	yes	maximal
2	C₅₃×#1713	C₂	1	2	yes	maximal
3	C₈×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#500	—	2	2	no	—
5	C₅₃×#500	C₂²	1	4	yes	—
6	C₅₃×#494	—	2	2	no	—
7	C₅₃×#366	—	2	8	no	—
8	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
9	C₈²×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×Q₁₆×C₅₃	—	4	4	no	—
11	C₄⋊Q₈×C₅₃	—	2	4	no	—
12	C₂×C₁₆×C₅₃	D₄	1	4	yes	—
13	#107×C₅₃	—	2	8	no	—
14	#107×C₅₃	—	4	4	no	—
15	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
16	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
17	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
18	C₂×Q₈×C₅₃	—	4	4	no	—
19	Q₁₆×C₅₃	—	8	8	no	—
20	C₂×C₈×C₅₃	D₈	1	2	yes	—
21	C₂×C₈×C₅₃	D₈	1	2	yes	—
22	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
23	C₄⋊C₄×C₅₃	—	4	4	no	—
24	C₄⋊C₄×C₅₃	—	4	4	no	—
25	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
26	C₁₆×C₅₃	Q₁₆	1	8	yes	complemented
27	Q₈×C₅₃	—	8	8	no	—
28	C₂×C₄×C₅₃	—	8	4	no	—
29	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
30	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
31	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
32	C₈×C₅₃	D₁₆	1	4	yes	—
33	C₈×C₅₃	Q₃₂	1	4	yes	—
34	C₈×C₅₃	C₂×Q₁₆	1	4	yes	—
35	#12521	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂²×C₅₃	#493	1	1	yes	characteristic
37	C₄×C₅₃	—	16	4	no	—
38	C₄×C₅₃	#494	1	2	yes	—
39	C₄×C₅₃	C₂×D₁₆	1	2	yes	—
40	C₄×C₅₃	C₂×Q₃₂	1	2	yes	—
41	#2252	C₂×C₅₃	1	4	yes	—
42	#1713	C₂×C₅₃	1	2	yes	—
43	C₈×C₁₆	C₂×C₅₃	1	1	yes	characteristic
44	C₂×C₅₃	#1713	1	1	yes	characteristic
45	C₂×C₅₃	#2246	1	1	yes	characteristic
46	C₂×C₅₃	#2247	1	1	yes	characteristic
47	#500	—	2	2	no	—
48	#500	C₂²×C₅₃	1	4	yes	—
49	#494	—	2	2	no	—
50	#366	—	2	8	no	—
51	C₄×C₁₆	C₂²×C₅₃	1	2	yes	—
52	C₈²	C₂²×C₅₃	1	1	yes	characteristic
53	C₅₃	#12521	1	1	yes	characteristic, direct product factor, Sylow
54	C₂×Q₁₆	—	4	4	no	—
55	C₄⋊Q₈	—	2	4	no	—
56	C₂×C₁₆	D₄×C₅₃	1	4	yes	—
57	#107	—	2	8	no	—
58	#107	—	4	4	no	—
59	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
60	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
61	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
62	C₂×Q₈	—	4	4	no	—
63	Q₁₆	—	8	8	no	—
64	C₂×C₈	D₈×C₅₃	1	2	yes	—
65	C₂×C₈	D₈×C₅₃	1	2	yes	—
66	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
67	C₄⋊C₄	—	4	4	no	—
68	C₄⋊C₄	—	4	4	no	—
69	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
70	C₁₆	Q₁₆×C₅₃	1	8	yes	complemented
71	Q₈	—	8	8	no	—
72	C₂×C₄	—	8	4	no	—
73	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
74	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
75	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
76	C₈	D₁₆×C₅₃	1	4	yes	—
77	C₈	Q₃₂×C₅₃	1	4	yes	—
78	C₈	C₂×Q₁₆×C₅₃	1	4	yes	—
79	C₂²	C₅₃×#493	1	1	yes	characteristic
80	C₄	—	16	4	no	—
81	C₄	C₅₃×#494	1	2	yes	—
82	C₄	C₂×D₁₆×C₅₃	1	2	yes	—
83	C₄	C₂×Q₃₂×C₅₃	1	2	yes	—
84	C₂	C₅₃×#1713	1	1	yes	characteristic
85	C₂	C₅₃×#2246	1	1	yes	characteristic
86	C₂	C₅₃×#2247	1	1	yes	characteristic

	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
26. C₁₆×C₅₃	Q₁₆	semidirect
35. #12521	C₅₃	direct (N × H)
53. C₅₃	#12521	direct (N × H)
70. C₁₆	Q₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#2252	2
C₅₃×#1713	2
C₈×C₁₆×C₅₃	2
#12521	53

Subgroups — C53×#12521 #27220033

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12521 #27220033