Subgroups — #27315294

Order 13568.

1278 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20272942	C₂	1	14	yes	maximal
2	C₂×C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133124	C₂²	1	28	yes	—
4	C₂²×C₄²×C₅₃	C₂²	1	7	yes	—
5	C₄³×C₅₃	C₄	1	8	yes	complemented
6	C₂³×Dic₅₃	C₂³	1	8	yes	—
7	#13133124	—	2	56	no	—
8	C₂³×C₄×C₅₃	C₂³	1	7	yes	—
9	C₂×C₄²×C₅₃	—	2	28	no	—
10	C₂×C₄²×C₅₃	C₂×C₄	1	28	yes	—
11	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
12	C₂²×C₄×C₅₃	—	2	28	no	—
13	C₂²×C₄×C₅₃	C₂²×C₄	1	14	yes	—
14	C₂²×C₄×C₅₃	#41	1	28	yes	—
15	C₂²×Dic₅₃	—	2	56	no	—
16	C₄²×C₅₃	—	2	56	no	—
17	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
18	C₂³×C₅₃	C₂×#41	1	14	yes	—
19	C₂×C₄×C₅₃	—	2	28	no	—
20	C₂×C₄×C₅₃	—	2	56	no	—
21	C₂×C₄×C₅₃	#117	1	28	yes	—
22	C₂×Dic₅₃	—	4	56	no	—
23	#14690	—	53	1	no	maximal, Sylow
24	C₂²×C₅₃	C₂×#117	1	7	yes	—
25	C₂²×C₅₃	#529	1	28	yes	—
26	C₄×C₅₃	—	2	28	no	—
27	Dic₅₃	—	8	8	no	—
28	C₂×C₄³	D₅₃	1	1	yes	characteristic
29	#1434	—	53	14	no	—
30	C₂×C₅₃	#2283	1	7	yes	—
31	C₂×C₅₃	#2865	1	8	yes	—
32	C₂²×C₄²	C₂×D₅₃	1	7	yes	—
33	C₂×#95	—	53	28	no	—
34	C₄³	Dic₅₃	1	8	yes	complemented
35	C₅₃	#14690	1	1	yes	characteristic, complemented, Sylow
36	C₂³×C₄	—	53	8	no	—
37	C₂³×C₄	C₂²×D₅₃	1	7	yes	—
38	C₂×C₄²	—	2	28	no	—
39	C₂×C₄²	C₂×Dic₅₃	1	28	yes	—
40	#95	—	106	56	no	—
41	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
42	C₂²×C₄	—	2	28	no	—
43	C₂²×C₄	—	106	56	no	—
44	C₂²×C₄	#11719945	1	28	yes	—
45	C₂²×C₄	C₂²×Dic₅₃	1	14	yes	—
46	C₄²	—	2	56	no	—
47	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
48	C₂³	C₂×#11719945	1	14	yes	—
49	C₂×C₄	—	2	28	no	—
50	C₂×C₄	—	2	56	no	—
51	C₂×C₄	—	212	56	no	—
52	C₂×C₄	#13133176	1	28	yes	—
53	C₂²	C₂×#13133176	1	7	yes	—
54	C₂²	#16811157	1	28	yes	—
55	C₄	—	2	28	no	—
56	C₄	—	424	8	no	—
57	C₂	#20277113	1	7	yes	—
58	C₂	#20283104	1	8	yes	—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Fitting	C₂×C₄³×C₅₃	C₂
socle, prime radical, cosocle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂×C₄³	D₅₃
hypocenter	C₅₃	#14690
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
5. C₄³×C₅₃	C₄	semidirect
34. C₄³	Dic₅₃	semidirect
35. C₅₃	#14690	semidirect

Maximal subgroups

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

subgroup	index
#20272942	2
C₂×C₄³×C₅₃	2
#14690	53