Subgroups — #27271973

Order 13568.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810342	C₂	1	1	yes	maximal, characteristic
2	#20272915	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×Q₆₄	C₂	1	1	yes	maximal, characteristic
4	#16810449	—	2	1	no	—
5	#16810342	—	2	1	no	—
6	C₂×Q₃₂×C₅₃	—	2	1	no	—
7	C₅₃×Q₆₄	—	2	1	no	—
8	C₅₃×Q₆₄	C₄	1	2	yes	complemented
9	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
10	#13133151	—	4	1	no	—
11	C₂×Q₁₆×C₅₃	—	4	1	no	—
12	Q₃₂×C₅₃	—	2	2	no	—
13	Q₃₂×C₅₃	—	4	1	no	—
14	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
15	C₃₂×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₅₃	D₄	1	1	yes	characteristic
17	C₂×Q₈×C₅₃	—	8	1	no	—
18	Q₁₆×C₅₃	—	4	2	no	—
19	Q₁₆×C₅₃	—	8	1	no	—
20	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
21	C₄⋊Dic₅₃	—	8	1	no	—
22	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₁₆×C₅₃	SD₁₆	1	1	yes	characteristic
24	Q₈×C₅₃	—	8	2	no	—
25	Q₈×C₅₃	—	16	1	no	—
26	C₂×C₄×C₅₃	—	16	1	no	—
27	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
28	C₂×Dic₅₃	—	16	1	no	—
29	C₈×C₅₃	#102	1	1	yes	characteristic
30	C₈×C₅₃	SD₃₂	1	1	yes	characteristic
31	#7538	—	53	1	no	maximal, Sylow
32	C₂²×C₅₃	D₃₂	1	1	yes	characteristic
33	C₄×C₅₃	—	16	2	no	—
34	C₄×C₅₃	#357	1	1	yes	characteristic
35	C₄×C₅₃	SD₆₄	1	1	yes	characteristic
36	Dic₅₃	—	32	1	no	—
37	C₂×Q₆₄	D₅₃	1	1	yes	characteristic
38	C₂×C₆₄	—	53	1	no	—
39	#1423	—	53	1	no	—
40	C₂×C₅₃	#1415	1	1	yes	characteristic
41	C₂×C₅₃	SD₁₂₈	1	1	yes	characteristic
42	C₂×C₅₃	Q₁₂₈	1	1	yes	characteristic
43	C₂×Q₃₂	—	2	1	no	—
44	Q₆₄	—	2	1	no	—
45	Q₆₄	Dic₅₃	1	2	yes	complemented
46	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
47	#366	—	106	1	no	—
48	C₆₄	—	106	1	no	—
49	C₅₃	#7538	1	1	yes	characteristic, complemented, Sylow
50	C₂×Q₁₆	—	4	1	no	—
51	Q₃₂	—	2	2	no	—
52	Q₃₂	—	4	1	no	—
53	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
54	#107	—	212	1	no	—
55	C₃₂	C₂×Dic₅₃	1	1	yes	characteristic
56	C₃₂	C₂²⋊D₅₃	1	1	yes	characteristic
57	C₂×Q₈	—	8	1	no	—
58	Q₁₆	—	4	2	no	—
59	Q₁₆	—	8	1	no	—
60	C₂×C₈	#11719922	1	1	yes	characteristic
61	C₄⋊C₄	—	424	1	no	—
62	C₁₆	#11719923	1	1	yes	characteristic
63	C₁₆	C₂²⋊Dic₅₃	1	1	yes	characteristic
64	Q₈	—	8	2	no	—
65	Q₈	—	16	1	no	—
66	C₂×C₄	—	16	1	no	—
67	C₂×C₄	—	848	1	no	—
68	C₂×C₄	#13133158	1	1	yes	characteristic
69	C₈	#13133138	1	1	yes	characteristic
70	C₈	#13133161	1	1	yes	characteristic
71	C₂²	#16810458	1	1	yes	characteristic
72	C₄	—	16	2	no	—
73	C₄	—	1696	1	no	—
74	C₄	#16810424	1	1	yes	characteristic
75	C₄	#16810461	1	1	yes	characteristic
76	C₂	#20272893	1	1	yes	characteristic
77	C₂	#20272929	1	1	yes	characteristic
78	C₂	#20272932	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16810458
commutator	C₃₂×C₅₃	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₅₃
Fitting	C₂×C₅₃×Q₆₄	C₂
socle, prime radical	C₂²×C₅₃	D₃₂
cosocle	C₂×C₃₂×C₅₃	C₂²
hypercenter	C₂×Q₆₄	D₅₃
hypocenter	C₅₃	#7538
LCS	G ▷ C₃₂×C₅₃ ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ C₂×Q₆₄

Complements

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

N	H ≅ G/N	type
8. C₅₃×Q₆₄	C₄	semidirect
45. Q₆₄	Dic₅₃	semidirect
49. C₅₃	#7538	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810342	2
#20272915	2
C₂×C₅₃×Q₆₄	2
#7538	53