Subgroups — #20289724

Order 6784.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811546	C₂	1	2	yes	maximal
2	#16811002	C₂	1	1	yes	maximal, characteristic
3	#13133291	—	2	2	no	—
4	C₂×#11719925	—	2	2	no	—
5	#13133217	C₄	1	1	yes	characteristic
6	C₄×#88322	C₂²	1	1	yes	characteristic
7	C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
8	C₂×Q₈×C₅₃	—	2	1	no	—
9	C₂×Dic₁₀₆	—	2	1	no	—
10	C₄⋊C₄×C₅₃	—	4	1	no	—
11	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
12	#11719925	—	4	2	no	—
13	#11719925	—	4	2	no	—
14	C₄⋊Dic₅₃	—	4	1	no	—
15	C₂×#88322	D₄	1	2	yes	—
16	#11719910	—	4	2	no	—
17	Q₈×C₅₃	—	4	2	no	—
18	C₂×C₄×C₅₃	—	2	1	no	—
19	C₂×C₄×C₅₃	—	4	1	no	—
20	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
21	C₂×Dic₅₃	—	4	1	no	—
22	Dic₁₀₆	—	4	2	no	—
23	#88322	D₈	1	2	yes	—
24	#88322	SD₁₆	1	2	yes	—
25	C₂²×C₅₃	#100	1	1	yes	characteristic
26	C₄×C₅₃	—	2	2	no	—
27	C₄×C₅₃	—	8	1	no	—
28	C₄×C₅₃	#102	1	2	yes	—
29	Dic₅₃	—	8	1	no	—
30	#1358	—	53	1	no	maximal, Sylow
31	C₂×C₅₃	—	2	1	no	—
32	C₂×C₅₃	#331	1	1	yes	characteristic
33	#494	—	53	1	no	—
34	#364	—	53	2	no	—
35	C₅₃	#1358	1	1	yes	characteristic, complemented, Sylow
36	C₂×Q₁₆	—	106	2	no	—
37	C₄⋊Q₈	—	53	1	no	—
38	C₄⋊Q₈	#6157	1	1	yes	characteristic
39	M₃₂	—	106	2	no	—
40	C₄×C₈	—	53	1	no	—
41	C₂×Q₈	—	2	1	no	—
42	C₂×Q₈	—	106	1	no	—
43	Q₁₆	—	212	2	no	—
44	Q₁₆	—	212	2	no	—
45	C₂×C₈	—	53	2	no	—
46	C₄⋊C₄	—	4	1	no	—
47	C₄⋊C₄	—	212	1	no	—
48	C₄²	C₂×#6157	1	1	yes	characteristic
49	C₁₆	—	212	2	no	—
50	Q₈	—	4	2	no	—
51	Q₈	—	212	2	no	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	4	1	no	—
54	C₂×C₄	—	212	1	no	—
55	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
56	C₈	—	53	2	no	—
57	C₈	—	53	2	no	—
58	C₂²	#13133277	1	1	yes	characteristic
59	C₄	—	2	2	no	—
60	C₄	—	8	1	no	—
61	C₄	—	424	1	no	—
62	C₄	#13133279	1	2	yes	—
63	C₂	—	2	1	no	—
64	C₂	#16811481	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16811481
commutator	C₄²×C₅₃	C₂×C₄
Frattini	C₄²	C₂×#6157
Fitting	C₄⋊Q₈×C₅₃	C₄
socle	C₂×C₅₃	#331
hypercenter	C₄⋊Q₈	#6157
hypocenter	C₅₃	#1358
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
35. C₅₃	#1358	semidirect