Subgroups — #44417582

Order 29248.

268 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28410598	C₂	1	2	yes	maximal, complemented
2	#28410580	C₂	1	6	yes	maximal, complemented
3	#28410574	C₂	1	3	yes	maximal, complemented
4	C₂×D₃₆₅₆	C₂	1	3	yes	maximal, complemented
5	#131×C₄₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#20300742	—	2	2	no	—
7	#20300741	C₂²	1	6	yes	complemented
8	C₂×D₁₈₂₈	C₂²	1	6	yes	complemented
9	D₄×D₄₅₇	—	2	6	no	—
10	Dic₁₈₂₈	C₂²	1	1	yes	characteristic, complemented
11	#20300730	C₂²	1	6	yes	complemented
12	D₃₆₅₆	C₂²	1	3	yes	complemented
13	D₃₆₅₆	C₂²	1	6	yes	complemented
14	#41×C₄₅₇	C₂²	1	1	yes	characteristic
15	M₁₆×C₄₅₇	C₂²	1	3	yes	complemented
16	C₂×C₈×C₄₅₇	C₂²	1	3	yes	complemented
17	C₂²×D₄₅₇	—	2	6	no	—
18	Dic₉₁₄	C₂³	1	2	yes	—
19	D₁₈₂₈	—	2	6	no	—
20	D₁₈₂₈	C₂³	1	6	yes	—
21	C₂²⋊D₄₅₇	—	2	6	no	—
22	C₄×D₄₅₇	—	2	6	no	—
23	Q₈×C₄₅₇	D₄	1	1	yes	characteristic
24	D₄×C₄₅₇	D₄	1	3	yes	—
25	C₂×C₄×C₄₅₇	C₂³	1	3	yes	—
26	C₈×C₄₅₇	C₂³	1	1	yes	characteristic
27	C₈×C₄₅₇	C₂³	1	3	yes	—
28	C₂²×C₄₅₇	C₂×D₄	1	3	yes	—
29	C₂×D₄₅₇	—	2	6	no	—
30	C₂×D₄₅₇	—	8	6	no	—
31	C₄×C₄₅₇	C₂×D₄	1	3	yes	—
32	C₄×C₄₅₇	C₂⁴	1	1	yes	characteristic
33	Dic₄₅₇	—	2	2	no	—
34	C₂×C₄₅₇	—	2	3	no	—
35	C₂×C₄₅₇	C₂²×D₄	1	1	yes	characteristic
36	D₄₅₇	—	4	6	no	—
37	C₄₅₇	#576	1	1	yes	characteristic, complemented, Sylow
38	#576	—	457	1	no	maximal, Sylow
39	H(5,𝔽₂)	—	457	2	no	—
40	AGL(1,ℤ/8)	—	457	6	no	—
41	#135	—	457	3	no	—
42	C₂×D₈	—	457	3	no	—
43	#131	D₄₅₇	1	1	yes	characteristic, complemented
44	#41	—	914	2	no	—
45	#41	—	457	6	no	—
46	#41	C₂×D₄₅₇	1	1	yes	characteristic
47	C₂×D₄	—	914	6	no	—
48	C₂×D₄	—	457	6	no	—
49	Q₁₆	—	457	1	no	—
50	SD₁₆	—	457	6	no	—
51	D₈	—	457	3	no	—
52	D₈	—	457	6	no	—
53	M₁₆	C₂×D₄₅₇	1	3	yes	complemented
54	C₂×C₈	C₂×D₄₅₇	1	3	yes	complemented
55	C₂³	—	914	6	no	—
56	Q₈	—	457	2	no	—
57	Q₈	D₁₈₂₈	1	1	yes	characteristic
58	D₄	—	914	6	no	—
59	D₄	—	914	6	no	—
60	D₄	—	457	6	no	—
61	D₄	D₁₈₂₈	1	3	yes	—
62	C₂×C₄	—	914	6	no	—
63	C₂×C₄	C₂²×D₄₅₇	1	3	yes	—
64	C₈	C₂²×D₄₅₇	1	1	yes	characteristic
65	C₈	C₂²×D₄₅₇	1	3	yes	—
66	C₂²	—	3656	6	no	—
67	C₂²	—	914	6	no	—
68	C₂²	C₂×D₁₈₂₈	1	3	yes	—
69	C₄	—	914	2	no	—
70	C₄	C₂×D₁₈₂₈	1	3	yes	—
71	C₄	C₂³×D₄₅₇	1	1	yes	characteristic
72	C₂	—	2	3	no	—
73	C₂	—	1828	6	no	—
74	C₂	C₂²×D₁₈₂₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₁₈₂₈
commutator, cosocle	C₄×C₄₅₇	C₂⁴
Frattini	C₄	C₂³×D₄₅₇
Fitting	#131×C₄₅₇	C₂
socle	C₂×C₄₅₇	C₂²×D₄
hypercenter	#131	D₄₅₇
hypocenter	C₄₅₇	#576
LCS	G ▷ C₄×C₄₅₇ ▷ C₂×C₄₅₇ ▷ C₄₅₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

Complements

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

N	H ≅ G/N	type
1. #28410598	C₂	semidirect
2. #28410580	C₂	semidirect
3. #28410574	C₂	semidirect
4. C₂×D₃₆₅₆	C₂	semidirect
5. #131×C₄₅₇	C₂	semidirect
7. #20300741	C₂²	semidirect
8. C₂×D₁₈₂₈	C₂²	semidirect
10. Dic₁₈₂₈	C₂²	semidirect
11. #20300730	C₂²	semidirect
12. D₃₆₅₆	C₂²	semidirect
13. D₃₆₅₆	C₂²	semidirect
15. M₁₆×C₄₅₇	C₂²	semidirect
16. C₂×C₈×C₄₅₇	C₂²	semidirect
37. C₄₅₇	#576	semidirect
43. #131	D₄₅₇	semidirect
53. M₁₆	C₂×D₄₅₇	semidirect
54. C₂×C₈	C₂×D₄₅₇	semidirect

Maximal subgroups

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

subgroup	index
#28410598	2
#28410580	2
#28410574	2
C₂×D₃₆₅₆	2
#131×C₄₅₇	2
#576	457