Subgroups — #37564415

Order 24224.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27131088	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₃₀₂₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#27131077	C₂	1	2	yes	maximal, complemented
4	D₆₀₅₆	C₂	1	2	yes	maximal, complemented
5	M₁₆×C₇₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₇₅₇	—	2	1	no	—
7	Dic₁₅₁₄	C₂²	1	1	yes	characteristic, complemented
8	D₃₀₂₈	—	2	1	no	—
9	D₃₀₂₈	C₂²	1	2	yes	—
10	D₃₀₂₈	C₂²	1	1	yes	characteristic, complemented
11	C₂²⋊D₇₅₇	—	2	1	no	—
12	C₄×D₇₅₇	—	2	1	no	—
13	C₂×C₄×C₇₅₇	C₂²	1	1	yes	characteristic
14	C₈×C₇₅₇	C₂²	1	2	yes	complemented
15	C₂²×C₇₅₇	D₄	1	1	yes	characteristic
16	C₂×D₇₅₇	—	2	1	no	—
17	C₂×D₇₅₇	—	2	2	no	—
18	C₂×D₇₅₇	—	4	2	no	—
19	C₄×C₇₅₇	D₄	1	1	yes	characteristic
20	C₄×C₇₅₇	C₂³	1	1	yes	characteristic
21	Dic₇₅₇	—	2	1	no	—
22	C₂×C₇₅₇	—	2	1	no	—
23	C₂×C₇₅₇	C₂×D₄	1	1	yes	characteristic
24	D₇₅₇	—	4	1	no	—
25	D₇₅₇	—	4	2	no	—
26	C₇₅₇	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
27	AGL(1,ℤ/8)	—	757	1	no	maximal, Sylow
28	#41	—	757	1	no	—
29	C₂×D₄	—	757	1	no	—
30	SD₁₆	—	757	2	no	—
31	D₈	—	757	2	no	—
32	M₁₆	D₇₅₇	1	1	yes	characteristic, complemented
33	C₂³	—	1514	1	no	—
34	Q₈	—	757	1	no	—
35	D₄	—	1514	1	no	—
36	D₄	—	1514	1	no	—
37	D₄	—	757	1	no	—
38	D₄	—	757	2	no	—
39	C₂×C₄	—	1514	1	no	—
40	C₂×C₄	C₂×D₇₅₇	1	1	yes	characteristic
41	C₈	C₂×D₇₅₇	1	2	yes	complemented
42	C₂²	—	3028	2	no	—
43	C₂²	—	1514	1	no	—
44	C₂²	—	1514	2	no	—
45	C₂²	D₃₀₂₈	1	1	yes	characteristic
46	C₄	—	1514	1	no	—
47	C₄	D₃₀₂₈	1	1	yes	characteristic
48	C₄	C₂²×D₇₅₇	1	1	yes	characteristic
49	C₂	—	2	1	no	—
50	C₂	—	3028	2	no	—
51	C₂	—	3028	1	no	—
52	C₂	C₂×D₃₀₂₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₃₀₂₈
commutator, cosocle	C₄×C₇₅₇	C₂³
Frattini	C₄	C₂²×D₇₅₇
Fitting	M₁₆×C₇₅₇	C₂
socle	C₂×C₇₅₇	C₂×D₄
hypercenter	M₁₆	D₇₅₇
hypocenter	C₇₅₇	AGL(1,ℤ/8)
LCS	G ▷ C₄×C₇₅₇ ▷ C₂×C₇₅₇ ▷ C₇₅₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
1. #27131088	C₂	semidirect
2. C₂×D₃₀₂₈	C₂	semidirect
3. #27131077	C₂	semidirect
4. D₆₀₅₆	C₂	semidirect
5. M₁₆×C₇₅₇	C₂	semidirect
7. Dic₁₅₁₄	C₂²	semidirect
10. D₃₀₂₈	C₂²	semidirect
14. C₈×C₇₅₇	C₂²	semidirect
26. C₇₅₇	AGL(1,ℤ/8)	semidirect
32. M₁₆	D₇₅₇	semidirect
41. C₈	C₂×D₇₅₇	semidirect

Maximal subgroups

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

subgroup	index
#27131088	2
C₂×D₃₀₂₈	2
#27131077	2
D₆₀₅₆	2
M₁₆×C₇₅₇	2
AGL(1,ℤ/8)	757