Subgroups — #20283104

Order 6784.

418 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810514	C₂	1	7	yes	maximal
2	#16810507	C₂	1	7	yes	maximal, complemented
3	C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11719919	C₂²	1	7	yes	—
5	C₂×#11719921	C₂²	1	7	yes	—
6	#13133124	C₂²	1	14	yes	—
7	C₂×C₄²×C₅₃	C₂²	1	7	yes	—
8	C₂³×D₅₃	C₂³	1	1	yes	characteristic
9	C₂²×C₄×C₅₃	C₂³	1	7	yes	—
10	C₂²×Dic₅₃	C₂³	1	7	yes	—
11	C₄²×C₅₃	—	2	14	no	—
12	#11719921	—	2	14	no	—
13	#11719919	—	2	14	no	—
14	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
15	C₂²×D₅₃	—	2	7	no	—
16	C₂×C₄×C₅₃	—	2	14	no	—
17	C₂×C₄×C₅₃	#41	1	14	yes	—
18	C₂×Dic₅₃	—	2	14	no	—
19	C₂×Dic₅₃	—	2	7	no	—
20	C₂²×C₅₃	C₂×#41	1	7	yes	—
21	C₂×D₅₃	—	4	7	no	—
22	C₄×C₅₃	—	2	14	no	—
23	Dic₅₃	—	4	7	no	—
24	#2865	—	53	1	no	maximal, Sylow
25	C₂×C₅₃	#529	1	7	yes	—
26	D₅₃	—	8	1	no	—
27	#388	—	53	7	no	—
28	#387	—	53	7	no	—
29	C₄³	D₅₃	1	1	yes	characteristic, complemented
30	C₅₃	#2865	1	1	yes	characteristic, complemented, Sylow
31	C₂×C₄⋊C₄	—	53	7	no	—
32	C₂×C₂²⋊C₄	—	53	7	no	—
33	C₂×C₄²	C₂×D₅₃	1	7	yes	—
34	#95	—	53	14	no	—
35	C₂⁴	—	53	1	no	—
36	C₂²×C₄	—	53	7	no	—
37	C₂²×C₄	C₂²×D₅₃	1	7	yes	—
38	C₄⋊C₄	—	106	14	no	—
39	C₂²⋊C₄	—	106	14	no	—
40	C₄²	—	2	14	no	—
41	C₂³	—	106	7	no	—
42	C₂³	C₂³×D₅₃	1	1	yes	characteristic
43	C₂×C₄	—	2	14	no	—
44	C₂×C₄	—	106	7	no	—
45	C₂×C₄	—	106	14	no	—
46	C₂×C₄	#11719945	1	14	yes	—
47	C₂²	—	212	7	no	—
48	C₂²	C₂×#11719945	1	7	yes	—
49	C₄	—	2	14	no	—
50	C₄	—	212	7	no	—
51	C₂	—	424	1	no	—
52	C₂	#16811157	1	7	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂⁴
Fitting	C₄³×C₅₃	C₂
hypercenter	C₄³	D₅₃
hypocenter	C₅₃	#2865
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³

Complements

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

N	H ≅ G/N	type
2. #16810507	C₂	semidirect
3. C₄³×C₅₃	C₂	semidirect
29. C₄³	D₅₃	semidirect
30. C₅₃	#2865	semidirect