Subgroups — #17949621

Order 4352.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14543945	C₂	1	2	yes	maximal
2	D₁₇×C₆₄	C₂	1	1	yes	maximal, characteristic
3	#11763735	—	2	2	no	—
4	C₃₂×D₁₇	C₂²	1	1	yes	characteristic
5	#11762528	C₄	1	1	yes	characteristic, complemented
6	C₁₇×C₆₄	C₄	1	1	yes	characteristic, complemented
7	#10587534	—	4	2	no	—
8	C₁₆×D₁₇	D₄	1	1	yes	characteristic
9	C₁₇×C₃₂	C₂×C₄	1	1	yes	characteristic
10	#10587465	Q₈	1	1	yes	characteristic
11	C₄⋊#595	—	8	2	no	—
12	C₈×D₁₇	D₈	1	1	yes	characteristic
13	C₁₆×C₁₇	C₄⋊C₄	1	1	yes	characteristic
14	#63214	Q₁₆	1	1	yes	characteristic
15	#7546	—	17	1	no	maximal, Sylow
16	C₂×#595	—	16	2	no	—
17	C₄×D₁₇	D₁₆	1	1	yes	characteristic
18	C₈×C₁₇	#107	1	1	yes	characteristic
19	#3622	Q₃₂	1	1	yes	characteristic
20	C₂×C₆₄	—	17	1	no	—
21	#1423	—	17	2	no	—
22	C₂×D₁₇	D₃₂	1	1	yes	characteristic
23	#595	—	32	2	no	—
24	C₄×C₁₇	#366	1	1	yes	characteristic
25	Dic₁₇	Q₆₄	1	1	yes	characteristic
26	C₂×C₃₂	—	17	1	no	—
27	#366	—	34	2	no	—
28	C₆₄	—	17	1	no	—
29	C₆₄	#595	1	1	yes	characteristic, complemented
30	C₂×C₁₇	#1423	1	1	yes	characteristic
31	D₁₇	SD₁₂₈	1	2	yes	—
32	C₂×C₁₆	—	17	1	no	—
33	#107	—	68	2	no	—
34	C₃₂	—	17	1	no	—
35	C₃₂	C₂×#595	1	1	yes	characteristic
36	C₁₇	#7546	1	1	yes	characteristic, complemented, Sylow
37	C₂×C₈	—	17	1	no	—
38	C₄⋊C₄	—	136	2	no	—
39	C₁₆	—	17	1	no	—
40	C₁₆	C₄⋊#595	1	1	yes	characteristic
41	C₂×C₄	—	272	2	no	—
42	C₂×C₄	—	17	1	no	—
43	C₈	—	17	1	no	—
44	C₈	#10587534	1	1	yes	characteristic
45	C₂²	—	17	1	no	—
46	C₄	—	17	1	no	—
47	C₄	—	544	2	no	—
48	C₄	#11763735	1	1	yes	characteristic
49	C₂	—	17	2	no	—
50	C₂	#14543945	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#14543945
commutator	C₁₇×C₃₂	C₂×C₄
Frattini	C₃₂	C₂×#595
Fitting	C₁₇×C₆₄	C₄
socle	C₂×C₁₇	#1423
prime radical	C₂×D₁₇	D₃₂
cosocle	C₃₂×D₁₇	C₂²
hypercenter	C₆₄	#595
hypocenter	C₁₇	#7546
LCS	G ▷ C₁₇×C₃₂ ▷ C₁₆×C₁₇ ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ C₆₄

Complements

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

N	H ≅ G/N	type
5. #11762528	C₄	semidirect
6. C₁₇×C₆₄	C₄	semidirect
29. C₆₄	#595	semidirect
36. C₁₇	#7546	semidirect

Maximal subgroups

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

subgroup	index
#14543945	2
D₁₇×C₆₄	2
#7546	17