Subgroups — C₅₃×#8813 #27216325

Order 13568.

486 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#1844	C₂	1	6	yes	maximal
2	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#106×C₅₃	—	2	12	no	—
4	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
5	C₅₃×#384	C₂²	1	4	yes	—
6	C₈²×C₅₃	C₄	1	4	yes	complemented
7	C₂²×C₈×C₅₃	D₄	1	6	yes	—
8	C₂×C₄⋊C₄×C₅₃	—	2	12	no	—
9	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
10	#106×C₅₃	—	4	24	no	—
11	C₄×C₈×C₅₃	C₂×C₄	1	6	yes	—
12	C₄×C₈×C₅₃	Q₈	1	6	yes	—
13	C₂²×C₄×C₅₃	—	4	4	no	—
14	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
15	C₂×C₈×C₅₃	C₄⋊C₄	1	12	yes	—
16	C₂×C₈×C₅₃	SD₁₆	1	24	yes	—
17	C₄⋊C₄×C₅₃	—	4	24	no	—
18	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
19	C₄²×C₅₃	C₂×Q₈	1	3	yes	—
20	C₂³×C₅₃	#127	1	1	yes	characteristic
21	C₂×C₄×C₅₃	—	8	12	no	—
22	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
23	C₂×C₄×C₅₃	C₄⋊Q₈	1	3	yes	—
24	C₂×C₄×C₅₃	C₂×SD₁₆	1	12	yes	—
25	C₈×C₅₃	#106	1	24	yes	complemented
26	#8813	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂²×C₅₃	#384	1	1	yes	characteristic
28	C₂²×C₅₃	#492	1	6	yes	—
29	C₄×C₅₃	—	16	4	no	—
30	C₄×C₅₃	C₂×#106	1	6	yes	—
31	C₄×C₅₃	#498	1	6	yes	—
32	#1844	C₂×C₅₃	1	6	yes	—
33	C₂×C₈²	C₂×C₅₃	1	1	yes	characteristic
34	C₂×C₅₃	#1705	1	4	yes	—
35	C₂×C₅₃	#1844	1	3	yes	—
36	C₂×#106	—	2	12	no	—
37	C₂×C₄×C₈	C₂²×C₅₃	1	3	yes	—
38	#384	C₂²×C₅₃	1	4	yes	—
39	C₈²	C₄×C₅₃	1	4	yes	complemented
40	C₅₃	#8813	1	1	yes	characteristic, direct product factor, Sylow
41	C₂²×C₈	D₄×C₅₃	1	6	yes	—
42	C₂×C₄⋊C₄	—	2	12	no	—
43	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
44	#106	—	4	24	no	—
45	C₄×C₈	C₂×C₄×C₅₃	1	6	yes	—
46	C₄×C₈	Q₈×C₅₃	1	6	yes	—
47	C₂²×C₄	—	4	4	no	—
48	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
49	C₂×C₈	C₄⋊C₄×C₅₃	1	12	yes	—
50	C₂×C₈	SD₁₆×C₅₃	1	24	yes	—
51	C₄⋊C₄	—	4	24	no	—
52	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
53	C₄²	C₂×Q₈×C₅₃	1	3	yes	—
54	C₂³	#127×C₅₃	1	1	yes	characteristic
55	C₂×C₄	—	8	12	no	—
56	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	3	yes	—
57	C₂×C₄	C₄⋊Q₈×C₅₃	1	3	yes	—
58	C₂×C₄	C₂×SD₁₆×C₅₃	1	12	yes	—
59	C₈	#106×C₅₃	1	24	yes	complemented
60	C₂²	C₅₃×#384	1	1	yes	characteristic
61	C₂²	C₅₃×#492	1	6	yes	—
62	C₄	—	16	4	no	—
63	C₄	C₂×#106×C₅₃	1	6	yes	—
64	C₄	C₅₃×#498	1	6	yes	—
65	C₂	C₅₃×#1705	1	4	yes	—
66	C₂	C₅₃×#1844	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	#127
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₈²×C₅₃	C₄	semidirect
25. C₈×C₅₃	#106	semidirect
26. #8813	C₅₃	direct (N × H)
39. C₈²	C₄×C₅₃	semidirect
40. C₅₃	#8813	direct (N × H)
59. C₈	#106×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#1844	2
C₂×C₈²×C₅₃	2
#8813	53

Subgroups — C53×#8813 #27216325

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8813 #27216325