Subgroups — C₅₃×#17309 #27224821

Order 13568.

1466 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#1446	C₂	1	15	yes	maximal
2	C₂³×Q₈×C₅₃	C₂²	1	5	yes	—
3	C₂×#95×C₅₃	C₂²	1	30	yes	—
4	C₂²×Q₈×C₅₃	—	4	30	no	—
5	C₂²×Q₈×C₅₃	Q₈	1	20	yes	complemented
6	C₂³×C₄×C₅₃	C₂³	1	15	yes	—
7	#95×C₅₃	—	4	30	no	—
8	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
9	C₂×Q₈×C₅₃	—	4	60	no	—
10	C₂×Q₈×C₅₃	—	4	80	no	—
11	C₂²×C₄×C₅₃	—	2	90	no	—
12	C₂²×C₄×C₅₃	C₂×Q₈	1	30	yes	—
13	C₂³×C₅₃	C₂²×Q₈	1	5	yes	—
14	C₂³×C₅₃	H(5,𝔽₂)	1	10	yes	—
15	Q₈×C₅₃	—	4	80	no	—
16	C₂×C₄×C₅₃	—	2	90	no	—
17	C₂×C₄×C₅₃	—	4	60	no	—
18	#17309	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂²×C₅₃	#543	1	15	yes	—
20	C₂²×C₅₃	Q₈²	1	10	yes	—
21	C₂²×C₅₃	H(3,𝔽₄)	1	10	yes	—
22	C₄×C₅₃	—	4	30	no	—
23	#1446	C₂×C₅₃	1	15	yes	—
24	C₂×C₅₃	#2811	1	10	yes	—
25	C₂×C₅₃	#2840	1	5	yes	—
26	C₂³×Q₈	C₂²×C₅₃	1	5	yes	—
27	C₂×#95	C₂²×C₅₃	1	30	yes	—
28	C₅₃	#17309	1	1	yes	characteristic, direct product factor, Sylow
29	C₂²×Q₈	—	4	30	no	—
30	C₂²×Q₈	Q₈×C₅₃	1	20	yes	complemented
31	C₂³×C₄	C₂³×C₅₃	1	15	yes	—
32	#95	—	4	30	no	—
33	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
34	C₂×Q₈	—	4	60	no	—
35	C₂×Q₈	—	4	80	no	—
36	C₂²×C₄	—	2	90	no	—
37	C₂²×C₄	C₂×Q₈×C₅₃	1	30	yes	—
38	C₂³	C₂²×Q₈×C₅₃	1	5	yes	—
39	C₂³	H(5,𝔽₂)×C₅₃	1	10	yes	—
40	Q₈	—	4	80	no	—
41	C₂×C₄	—	2	90	no	—
42	C₂×C₄	—	4	60	no	—
43	C₂²	C₅₃×#543	1	15	yes	—
44	C₂²	Q₈²×C₅₃	1	10	yes	—
45	C₂²	C₅₃×H(3,𝔽₄)	1	10	yes	—
46	C₄	—	4	30	no	—
47	C₂	C₅₃×#2811	1	10	yes	—
48	C₂	C₅₃×#2840	1	5	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₅₃
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₂²×Q₈×C₅₃	Q₈	semidirect
18. #17309	C₅₃	direct (N × H)
28. C₅₃	#17309	direct (N × H)
30. C₂²×Q₈	Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#1446	2
#17309	53

Subgroups — C53×#17309 #27224821

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17309 #27224821