Subgroups — #35231347

Order 21088.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24902110	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₂₆₃₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#24902099	C₂	1	2	yes	maximal, complemented
4	D₅₂₇₂	C₂	1	2	yes	maximal, complemented
5	M₁₆×C₆₅₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₆₅₉	—	2	1	no	—
7	Dic₁₃₁₈	C₂²	1	1	yes	characteristic, complemented
8	D₂₆₃₆	—	2	1	no	—
9	D₂₆₃₆	C₂²	1	2	yes	—
10	D₂₆₃₆	C₂²	1	1	yes	characteristic, complemented
11	C₂²⋊D₆₅₉	—	2	1	no	—
12	C₄×D₆₅₉	—	2	1	no	—
13	C₂×C₄×C₆₅₉	C₂²	1	1	yes	characteristic
14	C₈×C₆₅₉	C₂²	1	2	yes	complemented
15	C₂²×C₆₅₉	D₄	1	1	yes	characteristic
16	C₂×D₆₅₉	—	2	1	no	—
17	C₂×D₆₅₉	—	2	2	no	—
18	C₂×D₆₅₉	—	4	2	no	—
19	C₄×C₆₅₉	D₄	1	1	yes	characteristic
20	C₄×C₆₅₉	C₂³	1	1	yes	characteristic
21	Dic₆₅₉	—	2	1	no	—
22	C₂×C₆₅₉	—	2	1	no	—
23	C₂×C₆₅₉	C₂×D₄	1	1	yes	characteristic
24	D₆₅₉	—	4	1	no	—
25	D₆₅₉	—	4	2	no	—
26	C₆₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
27	AGL(1,ℤ/8)	—	659	1	no	maximal, Sylow
28	#41	—	659	1	no	—
29	C₂×D₄	—	659	1	no	—
30	SD₁₆	—	659	2	no	—
31	D₈	—	659	2	no	—
32	M₁₆	D₆₅₉	1	1	yes	characteristic, complemented
33	C₂³	—	1318	1	no	—
34	Q₈	—	659	1	no	—
35	D₄	—	1318	1	no	—
36	D₄	—	1318	1	no	—
37	D₄	—	659	1	no	—
38	D₄	—	659	2	no	—
39	C₂×C₄	—	1318	1	no	—
40	C₂×C₄	C₂×D₆₅₉	1	1	yes	characteristic
41	C₈	C₂×D₆₅₉	1	2	yes	complemented
42	C₂²	—	1318	1	no	—
43	C₂²	—	1318	2	no	—
44	C₂²	—	2636	2	no	—
45	C₂²	D₂₆₃₆	1	1	yes	characteristic
46	C₄	—	1318	1	no	—
47	C₄	D₂₆₃₆	1	1	yes	characteristic
48	C₄	C₂²×D₆₅₉	1	1	yes	characteristic
49	C₂	—	2	1	no	—
50	C₂	—	2636	2	no	—
51	C₂	—	2636	1	no	—
52	C₂	C₂×D₂₆₃₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₂₆₃₆
commutator, cosocle	C₄×C₆₅₉	C₂³
Frattini	C₄	C₂²×D₆₅₉
Fitting	M₁₆×C₆₅₉	C₂
socle	C₂×C₆₅₉	C₂×D₄
hypercenter	M₁₆	D₆₅₉
hypocenter	C₆₅₉	AGL(1,ℤ/8)
LCS	G ▷ C₄×C₆₅₉ ▷ C₂×C₆₅₉ ▷ C₆₅₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
1. #24902110	C₂	semidirect
2. C₂×D₂₆₃₆	C₂	semidirect
3. #24902099	C₂	semidirect
4. D₅₂₇₂	C₂	semidirect
5. M₁₆×C₆₅₉	C₂	semidirect
7. Dic₁₃₁₈	C₂²	semidirect
10. D₂₆₃₆	C₂²	semidirect
14. C₈×C₆₅₉	C₂²	semidirect
26. C₆₅₉	AGL(1,ℤ/8)	semidirect
32. M₁₆	D₆₅₉	semidirect
41. C₈	C₂×D₆₅₉	semidirect

Maximal subgroups

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

subgroup	index
#24902110	2
C₂×D₂₆₃₆	2
#24902099	2
D₅₂₇₂	2
M₁₆×C₆₅₉	2
AGL(1,ℤ/8)	659