Subgroups — #28410580

Order 14624.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20300741	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₁₈₂₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#20300730	C₂	1	2	yes	maximal, complemented
4	D₃₆₅₆	C₂	1	2	yes	maximal, complemented
5	M₁₆×C₄₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄₅₇	—	2	1	no	—
7	Dic₉₁₄	C₂²	1	1	yes	characteristic, complemented
8	D₁₈₂₈	—	2	1	no	—
9	D₁₈₂₈	C₂²	1	2	yes	—
10	D₁₈₂₈	C₂²	1	1	yes	characteristic, complemented
11	C₂²⋊D₄₅₇	—	2	1	no	—
12	C₄×D₄₅₇	—	2	1	no	—
13	C₂×C₄×C₄₅₇	C₂²	1	1	yes	characteristic
14	C₈×C₄₅₇	C₂²	1	2	yes	complemented
15	C₂²×C₄₅₇	D₄	1	1	yes	characteristic
16	C₂×D₄₅₇	—	2	1	no	—
17	C₂×D₄₅₇	—	2	2	no	—
18	C₂×D₄₅₇	—	4	2	no	—
19	C₄×C₄₅₇	D₄	1	1	yes	characteristic
20	C₄×C₄₅₇	C₂³	1	1	yes	characteristic
21	Dic₄₅₇	—	2	1	no	—
22	C₂×C₄₅₇	—	2	1	no	—
23	C₂×C₄₅₇	C₂×D₄	1	1	yes	characteristic
24	D₄₅₇	—	4	1	no	—
25	D₄₅₇	—	4	2	no	—
26	C₄₅₇	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
27	AGL(1,ℤ/8)	—	457	1	no	maximal, Sylow
28	#41	—	457	1	no	—
29	C₂×D₄	—	457	1	no	—
30	SD₁₆	—	457	2	no	—
31	D₈	—	457	2	no	—
32	M₁₆	D₄₅₇	1	1	yes	characteristic, complemented
33	C₂³	—	914	1	no	—
34	Q₈	—	457	1	no	—
35	D₄	—	914	1	no	—
36	D₄	—	914	1	no	—
37	D₄	—	457	1	no	—
38	D₄	—	457	2	no	—
39	C₂×C₄	—	914	1	no	—
40	C₂×C₄	C₂×D₄₅₇	1	1	yes	characteristic
41	C₈	C₂×D₄₅₇	1	2	yes	complemented
42	C₂²	—	1828	2	no	—
43	C₂²	—	914	1	no	—
44	C₂²	—	914	2	no	—
45	C₂²	D₁₈₂₈	1	1	yes	characteristic
46	C₄	—	914	1	no	—
47	C₄	D₁₈₂₈	1	1	yes	characteristic
48	C₄	C₂²×D₄₅₇	1	1	yes	characteristic
49	C₂	—	2	1	no	—
50	C₂	—	1828	2	no	—
51	C₂	—	1828	1	no	—
52	C₂	C₂×D₁₈₂₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₁₈₂₈
commutator, cosocle	C₄×C₄₅₇	C₂³
Frattini	C₄	C₂²×D₄₅₇
Fitting	M₁₆×C₄₅₇	C₂
socle	C₂×C₄₅₇	C₂×D₄
hypercenter	M₁₆	D₄₅₇
hypocenter	C₄₅₇	AGL(1,ℤ/8)
LCS	G ▷ C₄×C₄₅₇ ▷ C₂×C₄₅₇ ▷ C₄₅₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
1. #20300741	C₂	semidirect
2. C₂×D₁₈₂₈	C₂	semidirect
3. #20300730	C₂	semidirect
4. D₃₆₅₆	C₂	semidirect
5. M₁₆×C₄₅₇	C₂	semidirect
7. Dic₉₁₄	C₂²	semidirect
10. D₁₈₂₈	C₂²	semidirect
14. C₈×C₄₅₇	C₂²	semidirect
26. C₄₅₇	AGL(1,ℤ/8)	semidirect
32. M₁₆	D₄₅₇	semidirect
41. 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
#20300741	2
C₂×D₁₈₂₈	2
#20300730	2
D₃₆₅₆	2
M₁₆×C₄₅₇	2
AGL(1,ℤ/8)	457