Subgroups — #11717850

Order 816.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×D₁₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃⋊#595	C₂	1	2	yes	maximal, complemented
3	C₂²⋊#595	—	3	1	no	maximal, Hall
4	C₂²×C₃×C₁₇	C₄	1	1	yes	characteristic, complemented
5	C₂×C₃×D₁₇	—	2	2	no	—
6	C₂×C₃×D₁₇	C₄	1	1	yes	characteristic, complemented
7	C₂×C₃×D₁₇	C₂²	1	1	yes	characteristic
8	C₃⋊#595	—	2	2	no	—
9	C₂²×D₁₇	D₃	1	1	yes	characteristic
10	C₂×#595	—	3	2	no	—
11	C₂×C₃×C₁₇	—	2	1	no	—
12	C₂×C₃×C₁₇	C₂×C₄	1	1	yes	characteristic
13	C₃×D₁₇	—	2	1	no	—
14	C₃×D₁₇	D₄	1	2	yes	—
15	C₂²×C₁₇	Dic₃	1	1	yes	characteristic, complemented
16	C₂×D₁₇	—	2	2	no	—
17	C₂×D₁₇	Dic₃	1	1	yes	characteristic, complemented
18	C₂×D₁₇	C₂×D₃	1	1	yes	characteristic
19	#595	—	6	2	no	—
20	C₃×C₁₇	C₂²⋊C₄	1	1	yes	characteristic, complemented, Hall
21	#217	—	17	1	no	maximal, Hall
22	C₂×C₁₇	—	2	1	no	—
23	C₂×C₁₇	C₂×Dic₃	1	1	yes	characteristic
24	D₁₇	—	2	1	no	—
25	D₁₇	#67	1	2	yes	—
26	C₂³×C₃	—	17	1	no	—
27	C₂×Dic₃	—	17	2	no	—
28	C₁₇	#217	1	1	yes	characteristic, complemented, Sylow
29	C₂²⋊C₄	—	51	1	no	Sylow
30	C₂²×C₃	—	17	1	no	—
31	C₂²×C₃	—	17	1	no	—
32	C₂²×C₃	—	34	2	no	—
33	C₂²×C₃	#595	1	1	yes	characteristic, complemented
34	Dic₃	—	34	2	no	—
35	C₂³	—	17	1	no	—
36	C₂×C₄	—	51	2	no	—
37	C₂×C₃	—	2	1	no	—
38	C₂×C₃	—	17	2	no	—
39	C₂×C₃	—	34	1	no	—
40	C₂×C₃	C₂×#595	1	1	yes	characteristic
41	C₂²	—	17	1	no	—
42	C₂²	—	17	1	no	—
43	C₂²	—	34	2	no	—
44	C₂²	C₃⋊#595	1	1	yes	characteristic, complemented
45	C₄	—	102	2	no	—
46	C₃	C₂²⋊#595	1	1	yes	characteristic, complemented, Sylow
47	C₂	—	2	1	no	—
48	C₂	—	17	2	no	—
49	C₂	—	34	1	no	—
50	C₂	C₂×C₃⋊#595	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×C₃⋊#595
commutator, socle	C₂×C₃×C₁₇	C₂×C₄
Fitting	C₂²×C₃×C₁₇	C₄
prime radical	C₂²×C₃×D₁₇	C₂
cosocle	C₂×C₃×D₁₇	C₂²
hypercenter	C₂²	C₃⋊#595
hypocenter	C₃×C₁₇	C₂²⋊C₄
LCS	G ▷ C₂×C₃×C₁₇ ▷ C₃×C₁₇
UCS	C₁ ◁ C₂ ◁ C₂²

Complements

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

N	H ≅ G/N	type
2. C₂×C₃⋊#595	C₂	semidirect
4. C₂²×C₃×C₁₇	C₄	semidirect
6. C₂×C₃×D₁₇	C₄	semidirect
15. C₂²×C₁₇	Dic₃	semidirect
17. C₂×D₁₇	Dic₃	semidirect
20. C₃×C₁₇	C₂²⋊C₄	semidirect
28. C₁₇	#217	semidirect
33. C₂²×C₃	#595	semidirect
44. C₂²	C₃⋊#595	semidirect
46. C₃	C₂²⋊#595	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×D₁₇	2
C₂×C₃⋊#595	2
C₂²⋊#595	3
#217	17