Subgroups — #17953828

Order 4352.

486 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14544719	C₂	1	6	yes	maximal
2	C₈²×D₁₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#10587533	—	2	12	no	—
4	#11763755	C₂²	1	4	yes	—
5	C₄×C₈×D₁₇	C₂²	1	3	yes	—
6	C₈×#3622	C₄	1	3	yes	complemented
7	C₈²×C₁₇	C₄	1	1	yes	characteristic, complemented
8	C₂×C₄⋊#595	—	2	12	no	—
9	C₂×C₈×D₁₇	D₄	1	6	yes	—
10	C₄²×D₁₇	C₂³	1	1	yes	characteristic
11	#10587533	—	4	24	no	—
12	C₄×C₈×C₁₇	C₂×C₄	1	3	yes	—
13	C₈×Dic₁₇	Q₈	1	6	yes	—
14	C₄×#3622	C₂×C₄	1	3	yes	—
15	C₂²×#595	—	4	4	no	—
16	C₂×C₄×D₁₇	C₂×D₄	1	3	yes	—
17	C₄⋊#595	—	4	24	no	—
18	C₂×C₈×C₁₇	C₄⋊C₄	1	6	yes	—
19	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
20	C₄×Dic₁₇	C₂×Q₈	1	3	yes	—
21	C₂×#3622	C₄⋊C₄	1	6	yes	—
22	C₈×D₁₇	SD₁₆	1	24	yes	—
23	#8813	—	17	1	no	maximal, Sylow
24	C₂²×D₁₇	#127	1	1	yes	characteristic
25	C₂×#595	—	8	12	no	—
26	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	3	yes	—
27	C₂×Dic₁₇	C₄⋊Q₈	1	3	yes	—
28	C₄×D₁₇	C₂×SD₁₆	1	12	yes	—
29	C₈×C₁₇	#106	1	12	yes	complemented
30	#3622	#106	1	12	yes	complemented
31	#1844	—	17	6	no	—
32	C₂×C₈²	—	17	1	no	—
33	C₂²×C₁₇	#384	1	1	yes	characteristic
34	C₂×D₁₇	#492	1	6	yes	—
35	#595	—	16	4	no	—
36	C₄×C₁₇	C₂×#106	1	6	yes	—
37	Dic₁₇	#498	1	6	yes	—
38	C₂×#106	—	34	12	no	—
39	C₂×C₄×C₈	—	17	3	no	—
40	#384	—	17	4	no	—
41	C₈²	—	17	3	no	—
42	C₈²	#595	1	1	yes	characteristic, complemented
43	C₂×C₁₇	#1844	1	3	yes	—
44	D₁₇	#1705	1	4	yes	—
45	C₂²×C₈	—	17	6	no	—
46	C₂×C₄⋊C₄	—	34	12	no	—
47	C₂×C₄²	—	17	1	no	—
48	#106	—	68	24	no	—
49	C₄×C₈	—	17	6	no	—
50	C₄×C₈	—	17	3	no	—
51	C₄×C₈	C₂×#595	1	3	yes	—
52	C₁₇	#8813	1	1	yes	characteristic, complemented, Sylow
53	C₂²×C₄	—	17	3	no	—
54	C₂²×C₄	—	68	4	no	—
55	C₂×C₈	—	17	6	no	—
56	C₂×C₈	—	17	24	no	—
57	C₂×C₈	C₄⋊#595	1	6	yes	—
58	C₄⋊C₄	—	68	24	no	—
59	C₄²	—	17	3	no	—
60	C₄²	C₂²×#595	1	1	yes	characteristic
61	C₂³	—	17	1	no	—
62	C₂×C₄	—	17	3	no	—
63	C₂×C₄	—	17	12	no	—
64	C₂×C₄	—	136	12	no	—
65	C₂×C₄	C₂×C₄⋊#595	1	3	yes	—
66	C₈	—	17	12	no	—
67	C₈	#10587533	1	12	yes	complemented
68	C₂²	—	17	6	no	—
69	C₂²	#11763755	1	1	yes	characteristic
70	C₄	—	272	4	no	—
71	C₄	—	17	6	no	—
72	C₄	C₂×#10587533	1	6	yes	—
73	C₂	—	17	4	no	—
74	C₂	#14544719	1	3	yes	—

	subgroup	G/H
center	C₂²	#11763755
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₄²	C₂²×#595
Fitting	C₈²×C₁₇	C₄
socle	C₂²×C₁₇	#384
prime radical	C₂²×D₁₇	#127
cosocle	C₄²×D₁₇	C₂³
hypercenter	C₈²	#595
hypocenter	C₁₇	#8813
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
6. C₈×#3622	C₄	semidirect
7. C₈²×C₁₇	C₄	semidirect
29. C₈×C₁₇	#106	semidirect
30. #3622	#106	semidirect
42. C₈²	#595	semidirect
52. C₁₇	#8813	semidirect
67. C₈	#10587533	semidirect

Maximal subgroups

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

subgroup	index
#14544719	2
C₈²×D₁₇	2
#8813	17