Subgroups — C₅₃×#1745 #20270682

Order 6784.

272 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#97×C₅₃	C₂	1	2	yes	maximal
2	C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#336	C₂	1	4	yes	maximal
4	C₂²×C₈×C₅₃	C₂²	1	4	yes	—
5	C₂×C₄²×C₅₃	C₄	1	4	yes	—
6	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
7	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
8	#97×C₅₃	C₄	1	8	yes	complemented
9	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
10	C₂²×C₄×C₅₃	C₂×C₄	1	4	yes	—
11	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
12	C₂×C₈×C₅₃	—	2	8	no	—
13	C₂×C₈×C₅₃	C₂×C₄	1	8	yes	—
14	C₄²×C₅₃	—	2	8	no	—
15	C₄²×C₅₃	—	2	4	no	—
16	C₄²×C₅₃	C₂×C₄	1	4	yes	—
17	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
18	C₂×C₄×C₅₃	—	2	4	no	—
19	C₂×C₄×C₅₃	—	2	4	no	—
20	C₂×C₄×C₅₃	—	2	4	no	—
21	C₂×C₄×C₅₃	C₄²	1	4	yes	—
22	C₂×C₄×C₅₃	M₁₆	1	8	yes	—
23	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
24	C₂×C₄×C₅₃	#41	1	4	yes	—
25	C₈×C₅₃	—	2	8	no	—
26	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
27	C₂²×C₅₃	#117	1	2	yes	—
28	C₂²×C₅₃	C₂×M₁₆	1	4	yes	—
29	C₄×C₅₃	—	2	4	no	—
30	C₄×C₅₃	—	2	8	no	—
31	C₄×C₅₃	#117	1	4	yes	—
32	#1745	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂×C₅₃	#376	1	1	yes	characteristic
34	C₂×C₅₃	C₄×M₁₆	1	2	yes	—
35	C₂×C₅₃	#432	1	4	yes	—
36	C₂×#97	C₂×C₅₃	1	2	yes	—
37	C₄³	C₂×C₅₃	1	1	yes	characteristic
38	#336	C₂×C₅₃	1	4	yes	—
39	C₅₃	#1745	1	1	yes	characteristic, direct product factor, Sylow
40	C₂²×C₈	C₂²×C₅₃	1	4	yes	—
41	C₂×C₄²	C₄×C₅₃	1	4	yes	—
42	C₂×C₄²	C₂²×C₅₃	1	2	yes	—
43	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
44	#97	C₄×C₅₃	1	8	yes	complemented
45	C₂²×C₄	C₂×C₄×C₅₃	1	2	yes	—
46	C₂²×C₄	C₂×C₄×C₅₃	1	4	yes	—
47	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
48	C₂×C₈	—	2	8	no	—
49	C₂×C₈	C₂×C₄×C₅₃	1	8	yes	—
50	C₄²	—	2	4	no	—
51	C₄²	—	2	8	no	—
52	C₄²	C₂×C₄×C₅₃	1	4	yes	—
53	C₂³	C₂²×C₄×C₅₃	1	1	yes	characteristic
54	C₂×C₄	—	2	4	no	—
55	C₂×C₄	—	2	4	no	—
56	C₂×C₄	—	2	4	no	—
57	C₂×C₄	C₄²×C₅₃	1	4	yes	—
58	C₂×C₄	M₁₆×C₅₃	1	8	yes	—
59	C₂×C₄	C₂²×C₄×C₅₃	1	2	yes	—
60	C₂×C₄	#41×C₅₃	1	4	yes	—
61	C₈	—	2	8	no	—
62	C₂²	C₂×C₄²×C₅₃	1	1	yes	characteristic
63	C₂²	#117×C₅₃	1	2	yes	—
64	C₂²	C₂×M₁₆×C₅₃	1	4	yes	—
65	C₄	—	2	4	no	—
66	C₄	—	2	8	no	—
67	C₄	#117×C₅₃	1	4	yes	—
68	C₂	C₅₃×#376	1	1	yes	characteristic
69	C₂	C₄×M₁₆×C₅₃	1	2	yes	—
70	C₂	C₅₃×#432	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
8. #97×C₅₃	C₄	semidirect
32. #1745	C₅₃	direct (N × H)
39. C₅₃	#1745	direct (N × H)
44. #97	C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#97×C₅₃	2
C₄³×C₅₃	2
C₅₃×#336	2
#1745	53

Subgroups — C53×#1745 #20270682

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#1745 #20270682