Subgroups — #20274846

Order 6784.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810489	C₂	1	2	yes	maximal
2	#16810386	C₂	1	4	yes	maximal
3	C₂×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Dic₅₃	C₂²	1	4	yes	—
5	#13133124	—	2	4	no	—
6	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
7	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
8	C₄×C₈×C₅₃	C₄	1	4	yes	complemented
9	C₂²×C₄×C₅₃	D₄	1	2	yes	—
10	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
11	C₂²×Dic₅₃	—	2	4	no	—
12	C₂×C₈×C₅₃	—	2	4	no	—
13	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
14	C₄²×C₅₃	C₂×C₄	1	2	yes	—
15	C₄²×C₅₃	Q₈	1	2	yes	—
16	C₄⋊Dic₅₃	—	2	8	no	—
17	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
18	C₂×C₄×C₅₃	—	2	4	no	—
19	C₂×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
20	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
21	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
22	C₂×C₄×C₅₃	#41	1	4	yes	—
23	C₂×Dic₅₃	—	4	4	no	—
24	C₂×Dic₅₃	—	4	8	no	—
25	C₈×C₅₃	—	2	4	no	—
26	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₅₃	#124	1	2	yes	—
28	C₂²×C₅₃	#135	1	4	yes	—
29	C₄×C₅₃	—	2	4	no	—
30	C₄×C₅₃	#117	1	2	yes	—
31	C₄×C₅₃	#125	1	2	yes	—
32	Dic₅₃	—	8	4	no	—
33	#1835	—	53	1	no	maximal, Sylow
34	C₂×C₅₃	#382	1	1	yes	characteristic
35	C₂×C₅₃	#427	1	2	yes	—
36	C₂×C₅₃	#488	1	4	yes	—
37	C₂×C₄×C₈	D₅₃	1	1	yes	characteristic
38	#382	—	53	2	no	—
39	#340	—	53	4	no	—
40	C₅₃	#1835	1	1	yes	characteristic, complemented, Sylow
41	C₂²×C₈	C₂×D₅₃	1	2	yes	—
42	C₂×C₄⋊C₄	—	53	4	no	—
43	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
44	C₄×C₈	Dic₅₃	1	4	yes	complemented
45	#95	—	106	4	no	—
46	C₂²×C₄	—	106	4	no	—
47	C₂²×C₄	D₂₁₂	1	2	yes	—
48	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
49	C₂×C₈	—	2	4	no	—
50	C₂×C₈	C₂×Dic₅₃	1	4	yes	—
51	C₄⋊C₄	—	106	8	no	—
52	C₄²	Dic₁₀₆	1	2	yes	—
53	C₄²	C₂×Dic₅₃	1	2	yes	—
54	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
55	C₂×C₄	—	2	4	no	—
56	C₂×C₄	—	212	4	no	—
57	C₂×C₄	—	212	8	no	—
58	C₂×C₄	C₄⋊Dic₅₃	1	4	yes	—
59	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
60	C₂×C₄	#11719945	1	4	yes	—
61	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
62	C₈	—	2	4	no	—
63	C₂²	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
64	C₂²	#13133205	1	2	yes	—
65	C₂²	#13133239	1	4	yes	—
66	C₄	—	2	4	no	—
67	C₄	—	424	4	no	—
68	C₄	#13133176	1	2	yes	—
69	C₄	#13133209	1	2	yes	—
70	C₂	#16810489	1	1	yes	characteristic
71	C₂	#16810661	1	2	yes	—
72	C₂	#16810978	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×C₄×C₈	D₅₃
hypocenter	C₅₃	#1835
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
8. C₄×C₈×C₅₃	C₄	semidirect
40. C₅₃	#1835	semidirect
44. C₄×C₈	Dic₅₃	semidirect