Subgroups — #14583704

Order 2752.

268 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13063371	C₂	1	2	yes	maximal, complemented
2	#13063353	C₂	1	6	yes	maximal, complemented
3	#13063347	C₂	1	3	yes	maximal, complemented
4	C₂×D₃₄₄	C₂	1	3	yes	maximal, complemented
5	#131×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#41×C₄₃	C₂²	1	1	yes	characteristic
7	#10621902	—	2	2	no	—
8	D₄×D₄₃	—	2	6	no	—
9	#10621898	C₂²	1	6	yes	complemented
10	C₂×D₁₇₂	C₂²	1	6	yes	complemented
11	M₁₆×C₄₃	C₂²	1	3	yes	complemented
12	C₂×C₈×C₄₃	C₂²	1	3	yes	complemented
13	Dic₁₇₂	C₂²	1	1	yes	characteristic, complemented
14	D₃₄₄	C₂²	1	3	yes	complemented
15	D₃₄₄	C₂²	1	6	yes	complemented
16	#10621873	C₂²	1	6	yes	complemented
17	C₂²×D₄₃	—	2	6	no	—
18	Q₈×C₄₃	D₄	1	1	yes	characteristic
19	D₄×C₄₃	D₄	1	3	yes	—
20	C₂×C₄×C₄₃	C₂³	1	3	yes	—
21	C₂²⋊D₄₃	—	2	6	no	—
22	D₁₇₂	—	2	6	no	—
23	D₁₇₂	C₂³	1	6	yes	—
24	C₄×D₄₃	—	2	6	no	—
25	Dic₈₆	C₂³	1	2	yes	—
26	C₈×C₄₃	C₂³	1	3	yes	—
27	C₈×C₄₃	C₂³	1	1	yes	characteristic
28	C₂²×C₄₃	C₂×D₄	1	3	yes	—
29	C₂×D₄₃	—	2	6	no	—
30	C₂×D₄₃	—	8	6	no	—
31	C₄×C₄₃	C₂×D₄	1	3	yes	—
32	C₄×C₄₃	C₂⁴	1	1	yes	characteristic
33	Dic₄₃	—	2	2	no	—
34	C₂×C₄₃	—	2	3	no	—
35	C₂×C₄₃	C₂²×D₄	1	1	yes	characteristic
36	D₄₃	—	4	6	no	—
37	#576	—	43	1	no	maximal, Sylow
38	C₄₃	#576	1	1	yes	characteristic, complemented, Sylow
39	H(5,𝔽₂)	—	43	2	no	—
40	AGL(1,ℤ/8)	—	43	6	no	—
41	#135	—	43	3	no	—
42	C₂×D₈	—	43	3	no	—
43	#131	D₄₃	1	1	yes	characteristic, complemented
44	#41	—	43	6	no	—
45	#41	—	86	2	no	—
46	#41	C₂×D₄₃	1	1	yes	characteristic
47	C₂×D₄	—	43	6	no	—
48	C₂×D₄	—	86	6	no	—
49	Q₁₆	—	43	1	no	—
50	SD₁₆	—	43	6	no	—
51	D₈	—	43	3	no	—
52	D₈	—	43	6	no	—
53	M₁₆	C₂×D₄₃	1	3	yes	complemented
54	C₂×C₈	C₂×D₄₃	1	3	yes	complemented
55	C₂³	—	86	6	no	—
56	Q₈	—	43	2	no	—
57	Q₈	D₁₇₂	1	1	yes	characteristic
58	D₄	—	43	6	no	—
59	D₄	—	86	6	no	—
60	D₄	—	86	6	no	—
61	D₄	D₁₇₂	1	3	yes	—
62	C₂×C₄	—	86	6	no	—
63	C₂×C₄	C₂²×D₄₃	1	3	yes	—
64	C₈	C₂²×D₄₃	1	1	yes	characteristic
65	C₈	C₂²×D₄₃	1	3	yes	—
66	C₂²	—	86	6	no	—
67	C₂²	—	344	6	no	—
68	C₂²	C₂×D₁₇₂	1	3	yes	—
69	C₄	—	86	2	no	—
70	C₄	C₂×D₁₇₂	1	3	yes	—
71	C₄	C₂³×D₄₃	1	1	yes	characteristic
72	C₂	—	2	3	no	—
73	C₂	—	172	6	no	—
74	C₂	C₂²×D₁₇₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₁₇₂
commutator, cosocle	C₄×C₄₃	C₂⁴
Frattini	C₄	C₂³×D₄₃
Fitting	#131×C₄₃	C₂
socle	C₂×C₄₃	C₂²×D₄
hypercenter	#131	D₄₃
hypocenter	C₄₃	#576
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

Complements

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

N	H ≅ G/N	type
1. #13063371	C₂	semidirect
2. #13063353	C₂	semidirect
3. #13063347	C₂	semidirect
4. C₂×D₃₄₄	C₂	semidirect
5. #131×C₄₃	C₂	semidirect
9. #10621898	C₂²	semidirect
10. C₂×D₁₇₂	C₂²	semidirect
11. M₁₆×C₄₃	C₂²	semidirect
12. C₂×C₈×C₄₃	C₂²	semidirect
13. Dic₁₇₂	C₂²	semidirect
14. D₃₄₄	C₂²	semidirect
15. D₃₄₄	C₂²	semidirect
16. #10621873	C₂²	semidirect
38. C₄₃	#576	semidirect
43. #131	D₄₃	semidirect
53. M₁₆	C₂×D₄₃	semidirect
54. C₂×C₈	C₂×D₄₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#13063371	2
#13063353	2
#13063347	2
C₂×D₃₄₄	2
#131×C₄₃	2
#576	43