Subgroups — #20289722

Order 6784.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811515	C₂	1	2	yes	maximal
2	#16810993	C₂	1	1	yes	maximal, characteristic
3	#13133291	—	2	2	no	—
4	#13133217	C₄	1	1	yes	characteristic
5	#13133140	—	2	2	no	—
6	#13133128	C₂²	1	1	yes	characteristic
7	#125×C₅₃	C₄	1	1	yes	characteristic
8	C₂×Dic₁₀₆	—	2	1	no	—
9	C₄⋊C₄×C₅₃	—	2	1	no	—
10	C₄⋊C₄×C₅₃	—	4	1	no	—
11	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₄⋊Dic₅₃	—	4	1	no	—
13	C₂×#88322	D₄	1	2	yes	—
14	#11719910	—	2	4	no	—
15	C₂×C₄×C₅₃	—	2	1	no	—
16	C₂×C₄×C₅₃	—	4	1	no	—
17	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
18	C₂×Dic₅₃	—	4	1	no	—
19	Dic₁₀₆	—	4	2	no	—
20	#88322	—	2	2	no	—
21	C₂²×C₅₃	#100	1	1	yes	characteristic
22	C₄×C₅₃	—	4	1	no	—
23	C₄×C₅₃	—	8	1	no	—
24	C₄×C₅₃	#104	1	2	yes	—
25	Dic₅₃	—	8	1	no	—
26	#1356	—	53	1	no	maximal, Sylow
27	C₂×C₅₃	—	2	1	no	—
28	C₂×C₅₃	#329	1	1	yes	characteristic
29	#491	—	53	1	no	—
30	#347	—	53	2	no	—
31	C₅₃	#1356	1	1	yes	characteristic, complemented, Sylow
32	C₄⋊Q₈	—	53	1	no	—
33	#125	#6157	1	1	yes	characteristic
34	M₃₂	—	106	2	no	—
35	#103	—	106	2	no	—
36	#97	—	53	1	no	—
37	C₂×Q₈	—	106	1	no	—
38	C₂×C₈	—	53	2	no	—
39	C₄⋊C₄	—	2	1	no	—
40	C₄⋊C₄	—	4	1	no	—
41	C₄⋊C₄	—	212	1	no	—
42	C₄²	C₂×#6157	1	1	yes	characteristic
43	C₁₆	—	106	4	no	—
44	Q₈	—	212	2	no	—
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	4	1	no	—
47	C₂×C₄	—	212	1	no	—
48	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
49	C₈	—	106	2	no	—
50	C₂²	#13133277	1	1	yes	characteristic
51	C₄	—	4	1	no	—
52	C₄	—	8	1	no	—
53	C₄	—	424	1	no	—
54	C₄	#13133282	1	2	yes	—
55	C₂	—	2	1	no	—
56	C₂	#16811479	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16811479
commutator	C₄²×C₅₃	C₂×C₄
Frattini	C₄²	C₂×#6157
Fitting	#125×C₅₃	C₄
socle	C₂×C₅₃	#329
hypercenter	#125	#6157
hypocenter	C₅₃	#1356
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
31. C₅₃	#1356	semidirect