Subgroups — #20282992

Order 6784.

3422 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133186	C₂	1	7	yes	maximal, complemented
2	#16810479	C₂	1	7	yes	maximal, complemented
3	C₂⁶×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₂²⋊D₅₃	C₂²	1	7	yes	complemented
5	#13133186	—	2	28	no	—
6	C₂×C₂²⋊Dic₅₃	C₂²	1	21	yes	complemented
7	C₂⁵×C₅₃	—	2	28	no	—
8	C₂⁵×C₅₃	C₂²	1	7	yes	—
9	C₂⁴×C₅₃	—	2	224	no	—
10	C₂⁴×C₅₃	—	2	84	no	—
11	C₂⁴×C₅₃	D₄	1	28	yes	complemented
12	C₂⁴×C₅₃	C₂³	1	7	yes	—
13	C₂³×D₅₃	C₂³	1	1	yes	characteristic, complemented
14	C₂×C₂²⋊D₅₃	—	2	42	no	—
15	C₂²×Dic₅₃	C₂³	1	7	yes	complemented
16	C₂²⋊Dic₅₃	—	2	42	no	—
17	C₂³×C₅₃	—	2	256	no	—
18	C₂³×C₅₃	—	2	224	no	—
19	C₂³×C₅₃	—	2	168	no	—
20	C₂³×C₅₃	—	2	28	no	—
21	C₂³×C₅₃	C₂×D₄	1	42	yes	—
22	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
23	C₂²×D₅₃	—	2	7	no	—
24	C₂²⋊D₅₃	—	4	28	no	—
25	C₂×Dic₅₃	—	2	21	no	—
26	C₂²×C₅₃	—	2	224	no	—
27	C₂²×C₅₃	—	2	84	no	—
28	C₂²×C₅₃	#120	1	28	yes	complemented
29	C₂²×C₅₃	C₂²×D₄	1	7	yes	—
30	C₂×D₅₃	—	4	7	no	—
31	Dic₅₃	—	4	7	no	—
32	#2846	—	53	1	no	maximal, Sylow
33	C₂×C₅₃	—	2	28	no	—
34	C₂×C₅₃	C₂×#120	1	7	yes	—
35	D₅₃	—	8	1	no	—
36	C₂⁶	D₅₃	1	1	yes	characteristic, complemented
37	C₂×#120	—	53	7	no	—
38	#379	—	53	7	no	—
39	C₅₃	#2846	1	1	yes	characteristic, complemented, Sylow
40	C₂⁵	—	2	28	no	—
41	C₂⁵	C₂×D₅₃	1	7	yes	—
42	C₂²×D₄	—	53	7	no	—
43	#120	—	106	28	no	—
44	C₂×C₂²⋊C₄	—	53	21	no	—
45	C₂⁴	—	2	224	no	—
46	C₂⁴	—	2	84	no	—
47	C₂⁴	—	53	1	no	—
48	C₂⁴	C₂²⋊D₅₃	1	28	yes	complemented
49	C₂⁴	C₂²×D₅₃	1	7	yes	—
50	C₂×D₄	—	106	42	no	—
51	C₂²×C₄	—	53	7	no	—
52	C₂²⋊C₄	—	106	42	no	—
53	C₂³	—	2	256	no	—
54	C₂³	—	2	224	no	—
55	C₂³	—	2	168	no	—
56	C₂³	—	2	28	no	—
57	C₂³	—	106	7	no	—
58	C₂³	C₂×C₂²⋊D₅₃	1	42	yes	—
59	C₂³	C₂³×D₅₃	1	1	yes	characteristic
60	D₄	—	212	28	no	—
61	C₂×C₄	—	106	21	no	—
62	C₂²	—	2	224	no	—
63	C₂²	—	2	84	no	—
64	C₂²	—	212	7	no	—
65	C₂²	#13133186	1	28	yes	complemented
66	C₂²	C₂²×C₂²⋊D₅₃	1	7	yes	—
67	C₄	—	212	7	no	—
68	C₂	—	2	28	no	—
69	C₂	—	424	1	no	—
70	C₂	C₂×#13133186	1	7	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂⁴
Fitting	C₂⁶×C₅₃	C₂
hypercenter	C₂⁶	D₅₃
hypocenter	C₅₃	#2846
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂⁶

Complements

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

N	H ≅ G/N	type
1. C₂×#13133186	C₂	semidirect
2. #16810479	C₂	semidirect
3. C₂⁶×C₅₃	C₂	semidirect
4. C₂²×C₂²⋊D₅₃	C₂²	semidirect
6. C₂×C₂²⋊Dic₅₃	C₂²	semidirect
11. C₂⁴×C₅₃	D₄	semidirect
13. C₂³×D₅₃	C₂³	semidirect
15. C₂²×Dic₅₃	C₂³	semidirect
28. C₂²×C₅₃	#120	semidirect
36. C₂⁶	D₅₃	semidirect
39. C₅₃	#2846	semidirect
48. C₂⁴	C₂²⋊D₅₃	semidirect
65. C₂²	#13133186	semidirect