Subgroups — #35231352

Order 21088.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24902110	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₃₁₈	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₂₆₃₆	C₂	1	2	yes	maximal, complemented
4	#24902099	C₂	1	2	yes	maximal, complemented
5	M₁₆×C₆₅₉	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₁₃₁₈	—	2	1	no	—
7	Dic₁₃₁₈	C₂²	1	2	yes	—
8	Dic₁₃₁₈	C₂²	1	1	yes	characteristic
9	D₂₆₃₆	C₂²	1	1	yes	characteristic
10	C₂²⋊D₆₅₉	—	2	1	no	—
11	C₂×Dic₆₅₉	—	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	—
15	C₂²×C₆₅₉	D₄	1	1	yes	characteristic
16	C₂×D₆₅₉	—	2	1	no	—
17	C₄×C₆₅₉	D₄	1	1	yes	characteristic
18	C₄×C₆₅₉	C₂³	1	1	yes	characteristic
19	Dic₆₅₉	—	2	1	no	—
20	Dic₆₅₉	—	2	2	no	—
21	C₂×C₆₅₉	—	2	1	no	—
22	C₂×C₆₅₉	C₂×D₄	1	1	yes	characteristic
23	D₆₅₉	—	4	1	no	—
24	C₆₅₉	#137	1	1	yes	characteristic, complemented, Sylow
25	#137	—	659	1	no	maximal, Sylow
26	#41	—	659	1	no	—
27	C₂×Q₈	—	659	1	no	—
28	Q₁₆	—	659	2	no	—
29	SD₁₆	—	659	2	no	—
30	M₁₆	D₆₅₉	1	1	yes	characteristic, complemented
31	Q₈	—	1318	1	no	—
32	Q₈	—	659	2	no	—
33	Q₈	—	659	1	no	—
34	D₄	—	1318	1	no	—
35	D₄	—	659	1	no	—
36	C₂×C₄	—	1318	1	no	—
37	C₂×C₄	—	1318	1	no	—
38	C₂×C₄	C₂×D₆₅₉	1	1	yes	characteristic
39	C₈	C₂×D₆₅₉	1	2	yes	—
40	C₂²	—	1318	1	no	—
41	C₂²	D₂₆₃₆	1	1	yes	characteristic
42	C₄	—	1318	1	no	—
43	C₄	—	1318	2	no	—
44	C₄	D₂₆₃₆	1	1	yes	characteristic
45	C₄	C₂²×D₆₅₉	1	1	yes	characteristic
46	C₂	—	2	1	no	—
47	C₂	—	2636	1	no	—
48	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
prime radical	#24902110	C₂
hypercenter	M₁₆	D₆₅₉
hypocenter	C₆₅₉	#137
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
2. C₂×Dic₁₃₁₈	C₂	semidirect
3. Dic₂₆₃₆	C₂	semidirect
4. #24902099	C₂	semidirect
5. M₁₆×C₆₅₉	C₂	semidirect
24. C₆₅₉	#137	semidirect
30. M₁₆	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×Dic₁₃₁₈	2
Dic₂₆₃₆	2
#24902099	2
M₁₆×C₆₅₉	2
#137	659