Subgroups — #28410576

Order 14624.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20300740	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×D₄₅₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#20300730	C₂	1	1	yes	maximal, characteristic, complemented
4	#20300729	C₂	1	1	yes	maximal, characteristic, complemented
5	#20300726	C₂	1	1	yes	maximal, characteristic, complemented
6	#20300724	C₂	1	1	yes	maximal, characteristic, complemented
7	D₈×C₄₅₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₄₅₇	—	2	1	no	—
9	Dic₉₁₄	C₂²	1	1	yes	characteristic, complemented
10	D₁₈₂₈	C₂²	1	1	yes	characteristic
11	C₂²⋊D₄₅₇	—	2	1	no	—
12	C₂²⋊D₄₅₇	—	2	1	no	—
13	C₂×Dic₄₅₇	—	2	1	no	—
14	C₄×D₄₅₇	C₂²	1	1	yes	characteristic
15	#16814532	C₂²	1	1	yes	characteristic, complemented
16	D₄×C₄₅₇	C₂²	1	1	yes	characteristic
17	D₄×C₄₅₇	C₂²	1	1	yes	characteristic, complemented
18	C₈×C₄₅₇	C₂²	1	1	yes	characteristic, complemented
19	C₂²×C₄₅₇	—	2	1	no	—
20	C₂²×C₄₅₇	—	2	1	no	—
21	C₂×D₄₅₇	—	2	1	no	—
22	C₂×D₄₅₇	—	4	2	no	—
23	C₂×D₄₅₇	D₄	1	1	yes	characteristic
24	C₄×C₄₅₇	C₂³	1	1	yes	characteristic
25	Dic₄₅₇	—	2	1	no	—
26	Dic₄₅₇	D₄	1	1	yes	characteristic
27	C₂×C₄₅₇	—	4	1	no	—
28	C₂×C₄₅₇	—	4	1	no	—
29	C₂×C₄₅₇	C₂×D₄	1	1	yes	characteristic
30	D₄₅₇	—	2	1	no	—
31	D₄₅₇	—	4	1	no	—
32	C₄₅₇	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
33	AGL(1,ℤ/8)	—	457	1	no	maximal, Sylow
34	#41	—	457	1	no	—
35	C₂×D₄	—	457	1	no	—
36	SD₁₆	—	457	1	no	—
37	SD₁₆	—	457	1	no	—
38	D₈	—	457	1	no	—
39	D₈	D₄₅₇	1	1	yes	characteristic, complemented
40	M₁₆	—	457	1	no	—
41	C₂³	—	914	1	no	—
42	Q₈	—	457	1	no	—
43	D₄	—	914	1	no	—
44	D₄	—	914	1	no	—
45	D₄	—	457	1	no	—
46	D₄	C₂×D₄₅₇	1	1	yes	characteristic
47	D₄	C₂×D₄₅₇	1	1	yes	characteristic, complemented
48	C₂×C₄	—	914	1	no	—
49	C₂×C₄	—	457	1	no	—
50	C₈	—	457	1	no	—
51	C₈	C₂×D₄₅₇	1	1	yes	characteristic, complemented
52	C₂²	—	2	1	no	—
53	C₂²	—	2	1	no	—
54	C₂²	—	1828	2	no	—
55	C₂²	—	914	1	no	—
56	C₂²	—	457	1	no	—
57	C₄	—	914	1	no	—
58	C₄	—	457	1	no	—
59	C₄	C₂²×D₄₅₇	1	1	yes	characteristic
60	C₂	—	4	1	no	—
61	C₂	—	4	1	no	—
62	C₂	—	1828	1	no	—
63	C₂	—	914	1	no	—
64	C₂	D₄×D₄₅₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₄₅₇
commutator, cosocle	C₄×C₄₅₇	C₂³
Frattini	C₄	C₂²×D₄₅₇
Fitting	D₈×C₄₅₇	C₂
socle	C₂×C₄₅₇	C₂×D₄
hypercenter	D₈	D₄₅₇
hypocenter	C₄₅₇	AGL(1,ℤ/8)
LCS	G ▷ C₄×C₄₅₇ ▷ C₂×C₄₅₇ ▷ C₄₅₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. #20300740	C₂	semidirect
2. D₄×D₄₅₇	C₂	semidirect
3. #20300730	C₂	semidirect
4. #20300729	C₂	semidirect
5. #20300726	C₂	semidirect
6. #20300724	C₂	semidirect
7. D₈×C₄₅₇	C₂	semidirect
9. Dic₉₁₄	C₂²	semidirect
15. #16814532	C₂²	semidirect
17. D₄×C₄₅₇	C₂²	semidirect
18. C₈×C₄₅₇	C₂²	semidirect
32. C₄₅₇	AGL(1,ℤ/8)	semidirect
39. D₈	D₄₅₇	semidirect
47. D₄	C₂×D₄₅₇	semidirect
51. 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
#20300740	2
D₄×D₄₅₇	2
#20300730	2
#20300729	2
#20300726	2
#20300724	2
D₈×C₄₅₇	2
AGL(1,ℤ/8)	457