Subgroups — #28299153

Order 13568.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20289748	C₂	1	2	yes	maximal
2	#20272848	C₂	1	1	yes	maximal, characteristic
3	#16811486	—	2	2	no	—
4	C₂×#13133123	C₄	1	1	yes	characteristic
5	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
6	C₂×C₃₂×C₅₃	C₄	1	1	yes	characteristic
7	C₄⋊#88324	—	4	2	no	—
8	C₂×#11719908	Q₈	1	1	yes	characteristic
9	C₈×Dic₅₃	D₄	1	1	yes	characteristic
10	#13133123	—	2	1	no	—
11	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₃₂×C₅₃	C₈	1	2	yes	complemented
13	C₂×#88324	—	8	2	no	—
14	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
15	C₄×Dic₅₃	D₈	1	1	yes	characteristic
16	C₂×#88322	Q₁₆	1	1	yes	characteristic
17	C₁₆×C₅₃	C₂×C₈	1	1	yes	characteristic
18	C₁₆×C₅₃	M₁₆	1	1	yes	characteristic
19	#11719908	—	2	1	no	—
20	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
21	C₂×Dic₅₃	SD₃₂	1	2	yes	—
22	#88324	—	16	2	no	—
23	C₈×C₅₃	C₄⋊C₈	1	1	yes	characteristic
24	C₈×C₅₃	#108	1	1	yes	characteristic
25	#88322	—	2	1	no	—
26	#7463	—	53	1	no	maximal, Sylow
27	C₂²×C₅₃	#367	1	1	yes	characteristic
28	C₄×C₅₃	#335	1	1	yes	characteristic
29	C₄×C₅₃	#368	1	1	yes	characteristic
30	Dic₅₃	—	2	2	no	—
31	#1397	—	53	1	no	—
32	#1372	—	53	2	no	—
33	C₂×C₅₃	#1372	1	1	yes	characteristic
34	C₂×C₅₃	#1426	1	2	yes	—
35	C₂×C₃₂	—	53	1	no	—
36	C₂×C₃₂	#6157	1	1	yes	characteristic
37	C₄×C₁₆	—	53	1	no	—
38	#335	—	106	2	no	—
39	C₅₃	#7463	1	1	yes	characteristic, complemented, Sylow
40	C₂×C₁₆	—	53	1	no	—
41	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
42	C₄⋊C₈	—	212	2	no	—
43	C₄×C₈	—	53	1	no	—
44	C₃₂	—	106	1	no	—
45	C₃₂	#88324	1	2	yes	complemented
46	C₂×C₈	—	53	1	no	—
47	C₂×C₈	—	424	2	no	—
48	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
49	C₄²	—	53	1	no	—
50	C₁₆	—	106	1	no	—
51	C₁₆	#11719936	1	1	yes	characteristic
52	C₁₆	C₂×#88324	1	1	yes	characteristic
53	C₂×C₄	—	53	2	no	—
54	C₂×C₄	#13133286	1	1	yes	characteristic
55	C₈	—	848	2	no	—
56	C₈	—	106	1	no	—
57	C₈	C₄⋊#88324	1	1	yes	characteristic
58	C₈	#13133288	1	1	yes	characteristic
59	C₂²	#16811550	1	1	yes	characteristic
60	C₄	—	106	2	no	—
61	C₄	#16811486	1	1	yes	characteristic
62	C₄	#16811552	1	1	yes	characteristic
63	C₂	#20289748	1	1	yes	characteristic
64	C₂	#20289880	1	2	yes	—

	subgroup	G/H
center	C₂²	#16811550
commutator	C₁₆×C₅₃	C₂×C₈
Frattini	C₂×C₁₆	C₂×#6157
Fitting	C₂×C₃₂×C₅₃	C₄
socle, prime radical	C₂²×C₅₃	#367
cosocle	C₁₆×Dic₅₃	C₂²
hypercenter	C₂×C₃₂	#6157
hypocenter	C₅₃	#7463
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
12. C₃₂×C₅₃	C₈	semidirect
39. C₅₃	#7463	semidirect
45. C₃₂	#88324	semidirect

Maximal subgroups

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

subgroup	index
#20289748	2
#20272848	2
#7463	53