Subgroups — #20289744

Order 6784.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#11719910	C₂	1	2	yes	maximal
2	C₈×#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×#11719910	—	2	2	no	—
4	C₄×#88322	C₄	1	1	yes	characteristic
5	C₄×#88322	C₂²	1	1	yes	characteristic
6	C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
7	C₂×C₈×C₅₃	C₈	1	1	yes	characteristic
8	C₂×C₈×C₅₃	C₈	1	1	yes	characteristic
9	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
10	C₂×#88322	—	2	1	no	—
11	C₂×#88322	D₄	1	1	yes	characteristic
12	C₂×#88322	Q₈	1	1	yes	characteristic
13	#11719910	—	4	2	no	—
14	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
16	C₂×C₄×C₅₃	M₁₆	1	1	yes	characteristic
17	C₈×C₅₃	—	2	1	no	—
18	C₈×C₅₃	C₁₆	1	2	yes	complemented
19	#88322	—	2	2	no	—
20	#88322	—	2	1	no	—
21	#88322	D₈	1	1	yes	characteristic
22	#88322	Q₁₆	1	1	yes	characteristic
23	C₂²×C₅₃	C₄⋊C₈	1	1	yes	characteristic
24	C₄×C₅₃	—	2	1	no	—
25	C₄×C₅₃	#107	1	1	yes	characteristic
26	C₄×C₅₃	#108	1	1	yes	characteristic
27	C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
28	C₄×C₅₃	M₃₂	1	1	yes	characteristic
29	#1370	—	53	1	no	maximal, Sylow
30	C₂×C₅₃	#335	1	1	yes	characteristic
31	C₂×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
32	C₂×C₅₃	#364	1	1	yes	characteristic
33	C₄⋊C₁₆	—	53	2	no	—
34	C₈²	—	53	1	no	—
35	C₅₃	#1370	1	1	yes	characteristic, complemented, Sylow
36	C₂×C₁₆	—	106	2	no	—
37	C₄×C₈	—	53	1	no	—
38	C₄×C₈	—	53	1	no	—
39	C₄×C₈	#6157	1	1	yes	characteristic
40	C₂×C₈	—	53	1	no	—
41	C₂×C₈	—	53	1	no	—
42	C₂×C₈	—	106	1	no	—
43	C₂×C₈	#88324	1	1	yes	characteristic
44	C₂×C₈	#88324	1	1	yes	characteristic
45	C₄²	C₂×#6157	1	1	yes	characteristic
46	C₁₆	—	212	2	no	—
47	C₂×C₄	#11719936	1	1	yes	characteristic
48	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
49	C₂×C₄	C₂×#88324	1	1	yes	characteristic
50	C₈	—	2	1	no	—
51	C₈	—	53	1	no	—
52	C₈	—	53	1	no	—
53	C₈	—	106	2	no	—
54	C₈	—	106	1	no	—
55	C₈	#11719910	1	2	yes	complemented
56	C₂²	C₄⋊#88324	1	1	yes	characteristic
57	C₄	—	2	1	no	—
58	C₄	#13133286	1	1	yes	characteristic
59	C₄	#13133288	1	1	yes	characteristic
60	C₄	C₂×#11719910	1	1	yes	characteristic
61	C₄	#13133291	1	1	yes	characteristic
62	C₂	#16811486	1	1	yes	characteristic
63	C₂	C₄⋊#11719910	1	1	yes	characteristic
64	C₂	#16811545	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄⋊#6157
commutator	C₄×C₅₃	C₂×C₁₆
Frattini	C₄²	C₂×#6157
Fitting	C₄×C₈×C₅₃	C₄
socle	C₂²×C₅₃	C₄⋊C₈
hypercenter	C₄×C₈	#6157
hypocenter	C₅₃	#1370
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
18. C₈×C₅₃	C₁₆	semidirect
35. C₅₃	#1370	semidirect
55. C₈	#11719910	semidirect