Subgroups — C₅₃×#7333 #27214845

Order 13568.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#1398	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1398	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1313	C₂	1	1	yes	maximal, characteristic
4	C₅₃×M₆₄	C₄	1	2	yes	—
5	C₅₃×M₆₄	C₄	1	2	yes	—
6	C₅₃×#346	C₄	1	1	yes	characteristic
7	C₅₃×#346	C₂²	1	1	yes	characteristic
8	C₅₃×#322	C₄	1	1	yes	characteristic
9	C₂×C₁₆×C₅₃	—	2	1	no	—
10	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
11	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₄×C₈×C₅₃	—	2	1	no	—
13	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₃₂×C₅₃	—	2	2	no	—
15	C₃₂×C₅₃	C₈	1	4	yes	complemented
16	C₂×C₈×C₅₃	—	4	1	no	—
17	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
18	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
19	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
20	C₁₆×C₅₃	—	4	1	no	—
21	C₁₆×C₅₃	C₂×C₈	1	2	yes	—
22	C₁₆×C₅₃	M₁₆	1	2	yes	—
23	C₂×C₄×C₅₃	—	2	1	no	—
24	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
25	C₈×C₅₃	—	2	1	no	—
26	C₈×C₅₃	—	8	1	no	—
27	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
28	C₈×C₅₃	#97	1	1	yes	characteristic
29	#7333	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂²×C₅₃	#347	1	1	yes	characteristic
31	C₄×C₅₃	—	4	1	no	—
32	C₄×C₅₃	#322	1	1	yes	characteristic
33	C₄×C₅₃	#347	1	1	yes	characteristic
34	#1398	C₂×C₅₃	1	1	yes	characteristic
35	#1398	C₂×C₅₃	1	1	yes	characteristic
36	#1313	C₂×C₅₃	1	1	yes	characteristic
37	C₂×C₅₃	—	2	1	no	—
38	C₂×C₅₃	#1313	1	1	yes	characteristic
39	M₆₄	C₄×C₅₃	1	2	yes	—
40	M₆₄	C₄×C₅₃	1	2	yes	—
41	#346	C₄×C₅₃	1	1	yes	characteristic
42	#346	C₂²×C₅₃	1	1	yes	characteristic
43	#322	C₄×C₅₃	1	1	yes	characteristic
44	C₅₃	#7333	1	1	yes	characteristic, direct product factor, Sylow
45	C₂×C₁₆	—	2	1	no	—
46	C₂×C₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
47	C₂×C₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
48	C₄×C₈	—	2	1	no	—
49	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
50	C₃₂	—	2	2	no	—
51	C₃₂	C₈×C₅₃	1	4	yes	complemented
52	C₂×C₈	—	4	1	no	—
53	C₂×C₈	C₄²×C₅₃	1	1	yes	characteristic
54	C₂×C₈	M₁₆×C₅₃	1	1	yes	characteristic
55	C₄²	M₁₆×C₅₃	1	1	yes	characteristic
56	C₁₆	—	4	1	no	—
57	C₁₆	C₂×C₈×C₅₃	1	2	yes	—
58	C₁₆	M₁₆×C₅₃	1	2	yes	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	#97×C₅₃	1	1	yes	characteristic
61	C₈	—	2	1	no	—
62	C₈	—	8	1	no	—
63	C₈	C₄×C₈×C₅₃	1	1	yes	characteristic
64	C₈	#97×C₅₃	1	1	yes	characteristic
65	C₂²	C₅₃×#347	1	1	yes	characteristic
66	C₄	—	4	1	no	—
67	C₄	C₅₃×#322	1	1	yes	characteristic
68	C₄	C₅₃×#347	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	C₅₃×#1313	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅₃	#322
commutator	C₈	C₄×C₈×C₅₃
Frattini, cosocle	#346	C₂²×C₅₃
socle	C₂×C₅₃	#1313
prime radical	C₂²×C₅₃	#347
LCS	G ▷ C₈ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₅₃×#346 ◁ G

Complements

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

N	H ≅ G/N	type
15. C₃₂×C₅₃	C₈	semidirect
29. #7333	C₅₃	direct (N × H)
44. C₅₃	#7333	direct (N × H)
51. C₃₂	C₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#1398	2
C₅₃×#1313	2
#7333	53

Subgroups — C53×#7333 #27214845

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7333 #27214845