Subgroups — #20272888

Order 6784.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810422	C₂	1	2	yes	maximal
2	C₅₃×#491	C₂	1	1	yes	maximal, characteristic
3	#13133136	—	2	2	no	—
4	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
5	#125×C₅₃	C₄	1	1	yes	characteristic
6	#103×C₅₃	—	2	2	no	—
7	#97×C₅₃	C₄	1	1	yes	characteristic
8	C₂×Q₈×C₅₃	D₄	1	2	yes	—
9	C₂×C₈×C₅₃	—	2	1	no	—
10	C₄⋊C₄×C₅₃	—	2	1	no	—
11	C₄⋊C₄×C₅₃	—	4	1	no	—
12	C₄⋊C₄×C₅₃	—	4	1	no	—
13	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
14	#11719917	—	4	2	no	—
15	Q₈×C₅₃	—	4	2	no	—
16	C₂×C₄×C₅₃	—	2	2	no	—
17	C₂×C₄×C₅₃	—	2	1	no	—
18	C₂×C₄×C₅₃	—	4	1	no	—
19	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
20	C₈×C₅₃	—	4	1	no	—
21	#88322	—	8	2	no	—
22	C₂²×C₅₃	#99	1	1	yes	characteristic
23	C₄×C₅₃	—	2	1	no	—
24	C₄×C₅₃	—	4	1	no	—
25	C₄×C₅₃	—	4	2	no	—
26	C₄×C₅₃	—	8	1	no	—
27	#1413	—	53	1	no	maximal, Sylow
28	C₂×C₅₃	—	2	1	no	—
29	C₂×C₅₃	#354	1	1	yes	characteristic
30	#491	D₅₃	1	1	yes	characteristic
31	#356	—	53	2	no	—
32	C₅₃	#1413	1	1	yes	characteristic, complemented, Sylow
33	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
34	#125	Dic₅₃	1	1	yes	characteristic
35	#103	—	2	2	no	—
36	#101	—	106	2	no	—
37	#97	Dic₅₃	1	1	yes	characteristic
38	C₂×Q₈	C₂²⋊D₅₃	1	2	yes	—
39	M₁₆	—	212	2	no	—
40	C₂×C₈	—	2	1	no	—
41	C₄⋊C₄	—	2	1	no	—
42	C₄⋊C₄	—	4	1	no	—
43	C₄⋊C₄	—	4	1	no	—
44	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
45	Q₈	—	4	2	no	—
46	C₂×C₄	—	2	1	no	—
47	C₂×C₄	—	2	2	no	—
48	C₂×C₄	—	4	1	no	—
49	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
50	C₈	—	4	1	no	—
51	C₈	—	424	2	no	—
52	C₂²	#13133132	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	4	1	no	—
55	C₄	—	4	2	no	—
56	C₄	—	8	1	no	—
57	C₂	—	2	1	no	—
58	C₂	#16810418	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16810418
commutator	C₄²×C₅₃	C₂×C₄
Frattini	C₄⋊Q₈	C₂×D₅₃
Fitting	C₅₃×#491	C₂
socle	C₂×C₅₃	#354
hypercenter	#491	D₅₃
hypocenter	C₅₃	#1413
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊Q₈ ◁ #491

Complements

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

N	H ≅ G/N	type
32. C₅₃	#1413	semidirect